Why Are Plumbers Paid More in the US?

The economist Anthony Lee Zhang recently asked what I think is a very good question on Twitter. How come a plumber in the US is paid several times more than a plumber in the third world even though both do exactly the same thing? As he notes, it follows that, according to the usual definition of productivity, the plumber in the US is much more productive than a plumber in the third world not because he fixes more toilets in the same amount of time or does a better job at it but simply because he plumbs for researchers at OpenAI and the other doesn’t, which just seems wrong. Most people who replied to him seemed to think that it was either not interesting or completely obvious, but I actually think it’s very interesting and not so obvious. In fact, I think that understanding why many workers make so much more in rich countries than in poor countries despite seemingly being no better at it is very important to understand growth, but that the usual explanation is only telling part of the story and probably not even the most important part of the story. I had actually been thinking about writing a post on the topic for a while, because a few years ago I asked myself exactly the same question as Anthony and it forced to think about it to figure it out, so I figured that it was the opportunity to go ahead and finally do it.

The Canonical Explanation and Why It’s Inadequate

The usual story is a supply-side explanation that rests on the mobility of labor across sectors and the idea that employers in different sectors compete for workers. For instance, Baumol proposed a simple model that distinguishes between a “progressive” sector where productivity increases rapidly and a “non-progressive” sector where it doesn’t, which leads to a rise in the relative price of the goods produced in the non-progressive sector. Indeed, since wages in the progressive sector will grow with productivity, firms in the non-progressive sector will also have to increase wages or nobody will agree to work for them. But unlike in the progressive sector, in the non-progressive sector, this increase in wages will not be matched by a rise in output since by definition productivity remains constant in the non-progressive sector. Hence the prices of goods and services produced in the non-progressive sector will increase relative to the prices of goods and services in the progressive sector. In that model, wages in the non-progressive sector rise despite the fact that productivity is stagnant because employers in that sector compete with employers in the progressive sector, where productivity growth pushes wages up, for workers.

The Balassa-Samuelson effect is about differences in the prices of non-tradables between different countries at a given point in time, rather than differences in the prices of products of the non-progressive sector relative to products of the progressive sector in the same country at different times as the Baumol effect, but the usual explanation for it rests on essentially the same supply-side channel and similarly assumes labor mobility across sectors. In that case, since they are subject to international competition, the price of tradables is assumed to be pinned down internationally. The price of non-tradables such as plumbing services and haircuts on the other hand, since they can’t be imported and can only be obtained from local providers, can vary between countries. If productivity in the tradable sector is higher in rich countries than in poor countries, workers in that sector have to be paid more in rich countries than in poor countries. But even if productivity in the non-tradable sector is the same in rich countries as in poor countries, because plumbers and hairdressers do just as good a job in poor countries as in rich countries, they still have to be paid more in rich countries because otherwise nobody would agree to work in the non-tradable sector.

The reason why I don’t like this mechanism, however, is that it assumes a lot of labor mobility across sectors and I think it’s implausible that labor mobility is high enough to explain all or even most of the effect. It’s not as plumbers could become researchers at OpenAI if they’re not happy about how much people are willing to pay them for fixing their toilets. Now, you don’t need to assume a perfectly homogeneous labor market for the supply-side mechanism typically used to explain the Baumol/Balassa-Samuelson effect to have some force, it can operate to some extent even with imperfect labor mobility. For instance, although plumbers can’t become researchers at OpenAI, many of them can probably become electricians. Many electricians in turn can probably become network technicians. It’s plausible that, through such chains of imperfect substitution on the labor market and competition for workers between employers in different sectors at each link on such chains, the high wages of researchers at OpenAI can indirectly raise the wages of plumbers to some extent.

But it doesn’t seem plausible to me that it can explain why plumbers make so much more in rich countries than in poor countries or why even in the same country plumbers make so much more today than they did a few decades ago. To be clear, I’m sure that labor mobility and competition for workers on the labor market is part of the story, but I don’t think it’s the whole story or even most of it. Other explanations seem even less plausible. For instance, I don’t think it’s even remotely plausible that, if plumbers or hairdressers make so much more in the US than in India, that’s because plumbers and hairdressers are so much better at their job in the US than in India. It’s possible that, because they are better trained, have better equipment or work in firms that are better managed, plumbers in the US are able to fix a somewhat greater number of toilets in the same amount of time and do it more reliably than plumbers in India, but the idea that it can explain the enormous wage premium that plumbers in the US enjoy over plumbers in India doesn’t pass a basic smell test. Moreover, if that were true, it wouldn’t be the case that Indian plumbers immediately make a lot more upon moving to the US. At least the supply-side mechanism based on labor mobility I just criticized can in theory explain that.

A Demand-Side Mechanism for the Transmission of Productivity Growth Across Sectors

In the rest of this post, I will show that, even if there were no labor mobility across sectors and the supply-side channel usually posited to explain the Baumol/Balassa-Samuelson effect therefore didn’t operate at all, this effect could still arise through a demand-side channel. Although so far I have been talking about why some workers like plumbers are paid more in rich countries than in poor countries despite not being more productive in a physical sense, it’s often easier to talk about why such workers are paid more today than they were in the past in the same country despite not having become more productive in that sense, but as we shall see that’s essentially the same question. In order to keep things simple, let’s assume there are only 2 sectors in the economy, sector A where productivity is rising over time and sector B where it remains constant. That’s basically the distinction between what Baumol called the progressive sector and the non-progressive sector, but whereas he assumed that both sectors employed the same kind of workers and therefore paid the same hourly wage, we’re going to assume that each sector employs a specific kind of workers that can only work in that sector and therefore allow the hourly wage to differ between sectors.¹

If the economy is competitive and the supply of the kind of labor used in sector A isn’t elastic enough, as productivity increases in sector A, wages in that sector must increase.² If some companies in sector A didn’t increase the hourly wage they pay, other companies in that sector could increase the hourly wage they pay just enough to attract the workers of the companies that kept it the same but not so much that it fully compensates for the increase in productivity, allowing them to make a profit in the process. As workers in sector A become richer, provided that the goods produced in sector B have an income elasticity of demand greater than zero, the demand for the goods produced in sector B will increase. Therefore, as long as the elasticity of supply of the goods produced in sector B is less than one, the supply of the goods produced in sector B will not increase enough to satisfy that additional demand and their price will have to rise in order to equalize demand with supply. In turn, for the same reason that a rise in productivity in sector A raised wages in that sector, this will lead wages in sector B to rise. Thus, although production only increased in sector A, the gains resulting from that expansion didn’t just accrue to workers in that sector but were shared with those in sector B.

Crucially, the rise of productivity in sector A led to a wage increase in sector B without any labor mobility across sectors, since we assumed it away at the outset. Of course, the point is not that labor mobility across sectors is literally zero, but that labor mobility is not necessary for the benefits of productivity growth in one sector to propagate to the rest of the economy. This means that, in addition to the supply-side channel for the transmission of productivity growth across sectors based on the competition for workers between employers in different sectors that people typically focus on, there is a demand-side channel based on the effect that a rise of productivity in one sector has on the demand for the products of other sectors. As I explained above, I personally suspect that this demand-side mechanism is actually more important, but that’s an empirical question and the only point I’m making is that labor mobility across sectors is not necessary for the transmission of productivity growth across sectors.

Switching back to a between-country comparison, this mechanism can also explain why plumbers and workers in many other occupations make a lot more in rich countries than in poor countries, despite not being much more productive. They benefit from the fact that, compared to poor countries, the rest of the economy is much more productive in rich countries, which results in higher wages in the other sectors of the economy and, because that induces a rise in the relative prices of the goods and services they produce, ends up raising their wages as well. Curiously, although I’m sure that I’m not the first person to think about it and I may simply not be familiar with the relevant literature, this demand-side mechanism seems to be almost completely absent from discussions of the Baumol/Balassa-Samuelson effect, which only talk about the supply-side channel. This makes it worthwhile to study the demand-side channel a bit more, so in what follows I will use a simple model to take a closer look at it and learn more about how it works.

A simple model

Don’t worry if you’re not familiar with the standard concepts of economics, for I’m going to slowly introduce everything you need before deriving the main results, so the presentation will be entirely self-contained. I’ll start by presenting the basic setup of the model, which is a two-sector economy with heterogeneous households, fixed labor endowments, no labor mobility across sectors and no capital. Then I’ll introduce the general concept of elasticity and explain how we’ll use it to study how the variables of the model respond to a change in the productivity of one sector when that of the other remain fixed. Next, I will introduce the concept of income elasticity of demand and that of elasticity of substitution, which are necessary to understand the supply-side mechanism I want to study with the model. Then I will give a quick and dirty introduction to index number theory, to explain how to deflate monetary quantities to ensure their purchasing power remains fixed as prices change. Finally, in the last subsection, I will tie everything together and derive various results about how variables in the model respond to change in the productivity of one sector when the productivity of the other stays constant. Unfortunately, I need to spend a lot of time introducing the necessary conceptual background before I can actually derive the results we’re really interested in, but again I will go very slowly to make sure that even people without any knowledge of economics can follow with some effort. If you’re already familiar with the relevant concepts though, you can probably just skim the first subsections and go directly to the last.

The basic setup

We’ll assume that the economy produces only two goods, A and B, that are produced with no capital and two kinds of labor. A-workers can only work in sector A and B-workers can only work in sector B, so there is no labor mobility across sectors. The quantities of both kinds of labor,

\(L_A > 0\text{ and } L_B > 0\)

are fixed. Production is described by

\(Y_A = aL_A \text{ and } Y_B = bL_B,\)

where Yᵢ is the quantity of good i produced. It follows that the marginal productivity of labor is equal to the average productivity of labor:

\(\frac{dY_A}{dL_A} = \frac{Y_A}{L_A} = a \text{ and } \frac{dY_B}{dL_B} = \frac{Y_B}{L_B} = b.\)

I will therefore talk about productivity when I really mean marginal productivity, because in this model they are the same thing. This is what I call the physical productivity of labor, to be distinguished from the value productivity of labor, which I introduce below.

Moreover,

\(p_A > 0 \text{ and } p_B > 0\)

are the prices of A and B respectively, while

\(q \equiv \frac{p_B}{p_A}\)

is the price of B relative to A. Nominal GDP is

\(N \equiv p_AY_A + p_BY_B,\)

while the share of sector A in GDP is

\(\theta \equiv \frac{p_AY_A}{N},\)

hence he share of sector B is

\(1 - \theta = \frac{p_BY_B}{N}.\)

I’ll introduce the notion of real GDP later, after we have seen how to define a price index to deflate N.

Above I defined the physical productivity of labor, i. e. the quantity of goods that a worker makes in one unit of time, but now we can introduce the value productivity of labor, which is just the physical productivity of labor in a sector multiplied by the price of goods produced in that sector:

\(\frac{dp_AY_A}{dL_A} = \frac{p_AY_A}{L_A} = p_Aa \text{ and } \frac{dp_BY_B}{dL_B} = \frac{p_BY_B}{L_B} = p_Bb.\)

I will come back to this distinction between the physical productivity of labor and the value productivity of labor later. We assume that the economy is competitive, so workers have to be paid their marginal value productivity, so the hourly wages paid in sector A and sector B are respectively

\(w_A \equiv p_Aa \text{ and } w_B \equiv p_Bb.\)

The total wage bill is

\(w_AL_A + w_BL_B = p_AaL_A + p_BbL_B = p_AY_A + p_BY_B,\)

which is equal to N, something to be expected since only labor is involved in production and therefore the value of each sector’s output is paid entirely to its workers.

There are H households in the economy, which are indexed by h. Each household h owns

\(L_{Ah} \geq 0 \text{ and } L_{Bh} \geq 0\)

of the two kinds of labor and we further assume that

\(L_{Ah} + L_{Bh} > 0\)

to ensure that each household has positive income. The labor endowments of households satisfy

\(\sum_{h}^{H} L_{Ah} = L_A \text{ and } \sum_{h}^{H} L_{Bh} = L_B.\)

I make the standard assumptions on the preferences of households, which are represented by ordinal utility functions. The preferences of household h are represented by the utility function Uₕ, which is such that Uₕ(C) ≥ Uₕ(C’) if and only h prefers C to C’ or is indifferent between them, where

\(\mathbf{C} = (C_A, C_B) \text{ and }\mathbf{C}^{'} = (C^{'}_A, C^{'}_B)\)

are consumption baskets.

Let

\(m_h = w_AL_{Ah} + w_BL_{Bh}\)

be the budget of household h, which it derives from the wages it receives for its labor. Since the economy is competitive, there is no unemployment, hence each household receives wages in proportion to its endowments in both types of labor. I’m making the usual assumption that, subject to their budget constraint, households choose the consumption basket that maximizes their utility. So the demand for A and B of household h is defined by

\(\bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr) = \underset{\mathbf{C} = (C_A,C_B)}{\arg\max}\ U_h(\mathbf{C}) \text{ such that }p_AC_A + p_BC_B \leq m_h\)

This is also called the Marshallian demand of h, by contrast with the Hicksian demand of h, which I will introduce later. The standard assumptions about preferences guarantee that this is well-defined. It will also be useful to define

\(\omega_{Ah} \equiv \frac{p_Ac_Ah}{m_h} \text{ and }\omega_{Bh} \equiv \frac{p_Bc_Bh}{m_h},\)

which are household h’s expenditure shares on A and B.

Elasticities and the equilibrium path of the economy

We’re going to allow the physical productivity of A to change while keeping that of B fixed and look at what the model predicts about how various quantities, starting with the wages in each sector, change in response to that. Unfortunately, I need to introduce a few concepts first, so just hang in there for a while and it will soon pay off. The sort of quantity I was just talking about, which measures how sensitive a variable u is to a change in another variable v, is known as the elasticity of u with respect to v and can be defined intuitively as the percentage change in u resulting from a one-percent change in v. Formally, it’s defined as

\(\frac{d\log u}{d\log v}.\)

This quantity is the local rate of change in log u with respect to log v at the point where it’s evaluated, so we have

\(\Delta \log u \approx \frac{d\log u}{d\log v}\Delta \log v\)

for a small Δlog v.³ Now, since

\(\Delta\log x = \log(x_1) - \log(x_0) = \log(\frac{x_1}{x_0}) = \log(1 + \delta) \approx \delta \text{ where }\delta = \frac{x_1 - x_0}{x_0} = \frac{\Delta x}{x_0},\)

for any x it follows that, for a small change Δv around the point at which elasticity is evaluated,

\(\frac{\Delta u}{u} \approx \frac{d \log u}{d \log v}\frac{\Delta v}{v}.\)

But that’s precisely the relationship that should hold between Δu / u, Δv / v and elasticity as I defined it above in intuitive terms.

This can also be seen in another way. It can also be shown that

\(\frac{d\log u}{d\log v} = \frac{v}{u}\frac{du}{dv}.\)

If you think for a moment about this formula, you will see that it corresponds precisely to the concept of elasticity I defined informally above. Indeed, du / dv says how u changes in response to a change in v, but it’s expressed in units of u per units of v. However, since elasticity is a ratio of percentage changes, it’s supposed to be a dimensionless quantity. This is why du / dv is multiplied by v / u, which makes it dimensionless, resulting in precisely the quantity we wanted. For instance, if u is the weekly labor income of some individual and v is the number of hours worked by that individual in a week, du / dv is the additional weekly labor income he will earn by working one extra hour at the point where elasticity is evaluated, so it’s measured in units of income per unit of time. Thus, depending on whether income is measured in dollars or cents and time is measured in hours or minutes, it will have a different value. By multiplying that value by v / u, we turn that into a dimensionless quantity, which is equal to the percentage change in u produced by a one-percent increase in v. For example, if du / dv is $150 per hour and the individual of interest currently makes $7,000/week by working 35 hours/week, the elasticity is 35 hours / $7,000 * $150/hour = 0.75. If we measure income in cents and time in minutes, du / dv is approximately ¢250 per minute, but the elasticity is still 2,100 minutes / ¢700,000 * ¢250/minute = 0.75. What it means is that, at the point where that individual currently is on his labor supply curve, the income that he earns from his labor would increase by approximately 0.75% if he worked 1% longer.

A number of rules about elasticities will be useful later, so I will state them here. First, if w = uv, then

\(\frac{d\log w}{d\log x} = \frac{d(\log u + \log v)}{d\log x} = \frac{d\log u}{d\log x} + \frac{d\log v}{d\log x}.\)

Similarly, if w = u/v, then

\(\frac{d\log w}{d\log x} = \frac{d(\log u - \log v)}{d\log x} = \frac{d\log u}{d\log x} - \frac{d\log v}{d\log x}.\)

Finally, if w = u + v, then

\(\frac{d\log w}{d\log x} = \frac{x}{w}\frac{dw}{dx} = \frac{u}{w}\frac{x}{u}\frac{du}{dx} + \frac{v}{w}\frac{x}{v}\frac{dv}{dx} = \frac{u}{u + v}\frac{d\log u}{d\log x} + \frac{v}{u + v}\frac{d\log v}{d\log x}.\)

As you can see, those rules follow straightforwardly from ordinary calculus and the properties of logarithm.

Since we are interested in how, keeping the productivity of sector B and the labor endowments of households fixed, various quantities change in response to a change in the productivity of sector A, we need to refer to a path for the productivity of sector A. We describe that path as

\(a(s) = a_0e^s.\)

At every point s along that path, the economy is assumed to have adjusted to the general equilibrium associated with a(s), which means that demand and supply for both A and B are equal. Note that

\(s = \log\bigl((\frac{a(s)}{a(0)}\bigr) = \log(a(s)) - \log(a(0)),\)

so s is measured in log-productivity units, which is convenient for reasons to be explained shortly. Thus, s is not a temporal index and a₀ can be any reference point, not necessarily a point in time. We can interpret a(s) as a trajectory in time, but it can also be seen as describing a continuum of economies with different values of a but the same number of households with the same preferences and the same labor endowments. How you interpret it depends on whether you’re interested in how changes in the productivity of sector A over time affects the other variables of the model or in what the model predicts about how those variables will have different values in economies with different values of a.

Thus, every variable in the model is therefore a function of s, which indexes points along the path that a is taking. However, since we are assuming that the productivity of sector B and the labor endowments of households are fixed, I will usually not write those variables as functions of s. In other words, we have

\(b(s) = b\)

and

\(L_{Ah}(s) = L_{Ah}, L_{Bh}(s) = L_{Bh},\)

hence

\(L_A(s) = \sum_{h}^{H} L_{Ah}(s) = \sum_{h}^{H} L_{Ah} = L_A, L_B(s) = \sum_{h}^{H} L_{Bh}(s) = \sum_{h}^{H} L_{Bh} = L_B.\)

I will therefore only write the other variables of the model as a function of s.

Since we’re interested in how the variables of the model that aren’t fixed, such as prices and the wage rate in each sector, change in response to a change in the productivity of sector A, we want to know the elasticities of those variables with respect to a. It will be useful to introduce a specific notation

\(\epsilon_x \equiv \frac{d\log x}{d\log a}.\)

for the elasticity of a variable x with respect to a.⁴ Using that notation, the rules about elasticity derived previously are:

\(\epsilon_{xy} = \epsilon_x + \epsilon_y, \epsilon_{x/y} = \epsilon_x - \epsilon_y, \epsilon_{x + y} = \frac{x}{x + y}\epsilon_x + \frac{y}{x + y}\epsilon_y.\)

As we have seen above in the discussion of elasticity, εₓ(s) is the local rate of change in log x in response to a change in log a (which is approximately equal to the percentage change in x produced by a one-percent change in a at s), but for a finite change we need to integrate εₓ over the relevant path because it will not in general remain constant along that path. For instance, if we’re interested in how the variable x changes as a rises from a(0) = a₀ to a(s) = a₀eˢ (so that a(s) = a₀λ with λ = eˢ and s = log λ), we must use

\(\frac{x(s) - x(0)}{x(0)} = e^{\int_{0}^{\log \lambda} \epsilon_x(s) \,ds} - 1\)

to recover the exact proportional change.⁵ If εₓ is constant over the interval, i. e. εₓ(s) = ε̅ₓ for s ∈ [0, log λ], this reduces to

\(\frac{x(s) - x(0)}{x(0)} = \lambda^{\bar{\epsilon}_x} - 1,\)

but that is not generally the case.

The income elasticity of demand and the elasticity of substitution

Now let’s introduce the concept of income elasticity of demand for good i, which is defined as

\(\eta_{ih} \equiv \left. \frac{\partial \log c_{ih}(p_A,p_B,m_h)}{\partial \log m_h}\right|_{p_A,p_B}\text{ for }i\in\{A,B\},\)

where cᵢₕ is the Marshallian demand for i of h defined above. This is just a generalization of the concept of elasticity we have introduced previously to the case of a function of several variables, where the elasticity is defined with respect to a single variable while the others are held constant. The elasticity of cᵢₕ with respect to mₕ is the partial derivative of cᵢₕ with respect to mₕ, which gives the local change in log cᵢₕ in response to a change in log mₕ while holding the prices constant. By the same argument as in the case of a function of a single variable, this quantity is approximately equal to the percentage change in the demand of h for i produced by a one-percent change in h’s budget at mₕ when the prices of both A and B are held constant. When ηᵢₕ > 0, which means that h wants more of i as it becomes richer, i is called a “normal good”. Normal goods are further subdivided between necessities and luxuries, which are respectively defined by 0 < ηᵢₕ ≤ 1 and ηᵢₕ > 1, respectively. Examples of necessities are groceries and basic clothing, while examples of luxuries are vacations and fancy restaurants. When ηᵢₕ < 0, which means that h wants less of i as its income rises, i is called an inferior good. For a typical household in a high-income country, examples of inferior goods include public transportation and fast food. Since the income elasticity of demand is evaluated at a particular income, what counts as a normal goods, necessities, luxuries and inferior goods depend on the household.

Because utility is strictly increasing and we’re assuming that households maximize utility, every household h consumes its entire budget, so Marshallian demands must satisfy

\(p_Ac_{Ah} + p_Bc_{Bh} = m_h\)

for every h. If that weren’t the case and h had some income left over, it could reach a higher utility by consuming more of at least one good, which would contradict the assumption that Marshallian demand maximizes utility. Since there are only two goods, it follows from budget exhaustion that, once the income elasticity of demand for one good is fixed, the income elasticity of demand for the other is also determined. More precisely, it can be shown that

\(\omega_{Ah}\eta_{Ah} = 1 - \omega_{Bh}\eta_{Bh}\text{ and }\eta_{Ah} = \frac{1 - \omega_{Bh}\eta_{Bh}}{\omega_{Ah}},\)

where

\(\omega_{ih} \equiv \frac{p_ic_{ih}}{m_h}\)

is h’s expenditure share of good i.⁶ This makes sense intuitively because, if h’s income elasticity of demand for one good tells you how much h’s demand for that good increases in response to a small increase of h’s budget, h must spend that whole extra income and there is only one other good on which it can spend it, then it also tells you how much h’s demand for that other good will rise in response to that small increase of its budget.

Another type of elasticity, the price-elasticity of demand, will also be useful below. In order to define it, however, we must first introduce the concept of Hicksian demand. Whereas Marshallian demand gives the consumption basket that will maximize a household’s utility among those it can afford given the prices and its budget, Hicksian demand gives the consumption basket that will ensure a certain level of utility to h given the prices. It’s therefore defined as

\(\bigl(h_{Ah}(p_A,p_B,\bar{u}),h_{Bh}(p_A,p_B,u)\bigr) = \underset{\mathbf{C} = (C_A,C_B)}{\arg\min}\ p_AC_A + p_BC_B \text{ such that }U_h(\mathbf{C}) \geq u.\)

When households choose their consumption baskets so as to maximize utility, Hicksian demand is closely related to Marshallian demand. More precisely, if household h maximizes utility, then

\(\bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr) = \bigl(h_{Ah}(p_A,p_B,u),h_{Bh}(p_A,p_B,u)\bigr)\)

with

\(u = U_h(\mathbf{C})\text{ and }\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr).\)

This is easy to prove by reductio. Suppose that household h attains utility u by choosing the consumption basket that maximizes its utility among those it can afford given the prices and its budget. This consumption basket must also be the cheapest that h could afford at those prices that allowed it to attain u, because if there were a cheaper consumption basket that also allowed it to reach utility level u, h could have reached a level of utility above u for the same budget by spending the remaining income on additional consumption, contradicting the assumption that h initially chose the consumption basket that maximized its utility.

Note that Hicksian demand is homogeneous of degree zero. This means that, for any λ > 0,

\(\begin{aligned} \bigl(h_{Ah}(\lambda p_A,\lambda p_B,\bar{u}),h_{Bh}(\lambda p_A,\lambda p_B,u)\bigr) &= \underset{\mathbf{C} = (C_A,C_B)}{\arg\min}\ \lambda(p_AC_A + p_BC_B) \text{ such that }U_h(\mathbf{C}) \geq u\\ &= \underset{\mathbf{C} = (C_A,C_B)}{\arg\min}\ p_AC_A + p_BC_B \text{ such that }U_h(\mathbf{C}) \geq u\\ &= \bigl(h_{Ah}(p_B,\bar{u}),h_{Bh}(\lambda p_A,\lambda p_B,u)\bigr) \end{aligned},\)

so

\(h_{ih}(\lambda p_A, \lambda p_B, u) = h_{ih}(p_A, p_B, u).\)

In particular, setting

\(\lambda = \frac{1}{p_A}\)

and using

\(q = \frac{p_B}{p_A}\)

we have

\(h_{ih}(p_A, p_B, u) = h_{ih}(1, q, u).\)

In other words, when u is fixed, Hicksian demand only depends on prices through their ratio q.

When prices change, it affects demand in two different ways. First, if their nominal income remains constant, it changes what consumption baskets they can afford. Second, if the price change affects relative prices (i. e. it’s not the case that the price of every good is multiplied by the same number), it makes households substitute away from goods that have become relatively more expensive toward goods that have become relatively cheaper. The substitution effect is captured by the elasticity of Hicksian demand with respect to q, the price of B relative to A,

\(\sigma_h \equiv \left. - \frac{\partial\log\bigl(h_{Bh}(p_A,p_B,u)/h_{Ah}(p_A,p_B,u)\bigr)}{\partial\log q}\right|_u.\)

This is called the elasticity of substitution between A and B of household h. It measures how readily h substitutes between A and B as their relative price changes under the assumption that it maximizes utility. It’s approximately equal to the percentage change in the quantity of B relative to A that h consumes produced by a one-percent change in q when h’s utility is held constant at u. So the elasticity of substitution measures how easily households replace one good with another as their relative price changes.

In practice, the level of utility at which σₕ is evaluated will be

\(u = U_h(\mathbf{C})\)

with

\(\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr),\)

because we’re interested in how h substitutes between A and B as their relative price changes when h’s budget is held constant. There is a minus sign in the definition to enforce the convention that the elasticity of substitution between A and B is positive when h substitutes away from B toward A when the price of the former increases relative to that of the former, since in that case the partial derivative in the definition is negative.⁷ The elasticity of substitution ranges between 0 and ∞. When σₕ = 0, the goods are perfect complement, which means that no substitution takes place as the relative price changes. When 0 < σₕ < 1, the goods are weak substitutes for each other, meaning that h substitutes one for the other less than proportionally as the price of the former increases relative to that of the latter. When σₕ = 1, h spends a constant share of its budget on each good as that budget rises, which changes proportionally with the relative price. When σₕ > 1, the goods are strong substitutes, which means that a small rise in the price of one good relative to that of the other leads to a large shift in the quantities demanded by h in favor of the latter. Finally, when σₕ = ∞, the goods are perfect substitute and h will only buy the cheapest one.

Price indices and deflators

So far, we have only been using nominal variables obtained by multiplying quantities by current prices, but price changes affect the purchasing power of money and we’re ultimately interested in how a change of productivity in sector A affects those variables in real terms. For instance, if a rise in the productivity of sector A doubles a household’s nominal income, but it also halves the purchasing power of money, this increase of the productivity in sector A actually leaves that household’s income unchanged. So we need to define a price index such that, when it’s used to deflate a nominal budget, the purchasing power of that nominal budget is preserved. If we define equality of purchasing power as utility-preservation, this is called a Konüs index. In order to define it, we must first introduce the expenditure function, which is defined as

\(E_h(\mathbf{p},u) \equiv \min_{\mathbf{C} = (C_A,C_B)} \bigl\{p_A C_A+p_B C_B:U_h(\mathbf{C}) \ge u\bigr\}.\)

The expenditure function of a household h is the cost at given prices of the cheapest consumption basket that allows h to attain a given utility level. If h is utility-maximizing and u is the utility level it attains with budget mₕ at prices p, then it’s equal to the cost of the consumption basket picked by Marshallian demand:

\(E_h(\mathbf{p},u) = \mathbf{p}\cdot\mathbf{C} = m_h\)

with

\(u = U_h(\mathbf{C})\text{ and }\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr).\)

In order to see that, suppose there were a consumption basket C’ such that Eₕ(𝐩,u) = 𝐩⋅𝐂’ < 𝐩⋅𝐂. Since u = Uₕ(C) and the cost of C is mₕ, h could buy C’ and attain the same utility level it can get with C while still having some income left over, so it could use that income to consume more of at least one good and reach a higher level of utility, contradicting the hypothesis that C maximizes h’s utility. If on the other hand there were a consumption basket C’ such that Eₕ(𝐩,u) = 𝐩⋅𝐂’ > 𝐩⋅𝐂, then h could reach the same utility level u by purchasing the cheaper C, contradicting the hypothesis that C’ is the cheapest consumption basket with which h can attain utility level u.

Let’s temporarily forget about how the economy adjusts as the productivity of sector A moves along the trajectory a(s) and ask the more general question of how a nominal budget must be adjusted to preserve its purchasing power as prices change. Suppose that prices change from p* at the reference point to p and let h be a household and u be a utility level. Intuitively, it makes sense to say that a nominal budget m has the same purchasing power for h when prices are p than another nominal budget m* did when prices were p if and only if h can reach the same utility level with m when prices are p than it could with m* when prices were p*, so we’re looking for a factor such that m will be equal to m* when that factor is used to deflate it it has the same purchasing power in that sense. The exact compensation factor of h at u is

\(K_h(\mathbf{p}, \mathbf{p^*}, u) \equiv \frac{E_h(\mathbf{p}, u)}{E_h(\mathbf{p^*}, u)}\)

and by fixing a reference price vector p* it defines what is called a Konüs index. If a budget of m* allows h to attain utility level u when prices are p* but prices shift to p, what h can buy with m* will also change and in general so will the utility level it can attain. However, if after the price change h now has the budget

\(m = m^*K_h(\mathbf{p}, \mathbf{p^*}, u),\)

then h can attain the same utility level u as before. Equivalently, if we use Kₕ(p, p*, u) to deflate h’s income, we’ll have

\(\frac{m}{K_h(\mathbf{p}, \mathbf{p^*}, u)} = m^*.\)

This shows that a Konüs index has precisely the property that a price index should have according to our pre-theoretic notion.

Ideally, we’d like a price index to be a function of prices alone, but as you can see the Konüs index is also a function of a utility level u even for a single household h. It will only reduce to a function of prices alone when h’s preferences are homothetic. The preferences of h are homothetic if and only if, for any λ > 0 and any consumption baskets C and D, Uₕ(C) ≥ Uₕ(D) ⇔ Uₕ(λC) ≥ Uₕ(λD). In other words, if the quantity of goods in consumption baskets is multiplied by the same positive factor, it doesn’t change which basket h prefers. It can be shown that, when h’s preferences are homothetic, the expenditure function can be written as

\(E_h(\mathbf{p},u) = G_h(u)E(\mathbf{p},1).\)

In other words, the expenditure function can be decomposed into the product of a function depending only on utility and a function depending only on prices, which turns out to be very convenient.⁸ Indeed, it means that h’s Konüs index

\(K_h(\mathbf{p}, \mathbf{p^*}, u) = \frac{G_h(u)E_h(\mathbf{p}, 1)}{G_h(u)E_h(\mathbf{p^*}, 1)} = \frac{E_h(\mathbf{p}, 1)}{E_h(\mathbf{p^*}, 1)}\)

doesn’t depend on u, but is a function of prices alone. When h’s preferences aren’t homothetic, on the other hand, its expenditure function can’t be decomposed into a utility-only factor and a price-only factor, so its Konüs index doesn’t reduce to a function of prices alone but also depends on the level of utility. The condition that h’s preferences must be homothetic, however, is highly restrictive and completely implausible. Indeed, if preferences are homothetic, it means that ηᵢₕ = 1 for every good i.⁹ In other words, as h’s income rises, h’s demand for any good increases proportionally to it. For instance, it implies that if a household’s budget is multiplied tenfold, the amount of bread it consumes is also multiplied tenfold, which is absurd.¹⁰ So in practice what this means is that, even for a single household, there is no price index that is both a function of prices alone and utility-preserving.

Things are even worse once you consider the fact that the economy contains several heterogeneous households. Indeed, even if the preferences of every household in the economy are homothetic, there isn’t in general a price index that is a function of prices alone and preserves utility for every household. In other words, it can be shown that a function K(p, p*) such that

\(K(\mathbf{p}, \mathbf{p^{*}}) = K_h(\mathbf{p}, \mathbf{p^*}, u)\)

for every household h, every pair of price vectors p and p* and every utility level u exists if and only if not only the preferences of every household are homothetic but those preferences are the same as those of every other household.¹¹ Obviously, this condition is even more implausible than what is required for the existence of a utility-preserving price index that is a function of prices alone for a single household,¹² so what this means in practice is that no function of prices alone is a universal cost-of-living deflator.¹³ Thus, although our pre-theoretic concept of price index arguably entails that a price index should be a function of prices alone and the same for every household no matter their budget, it can be proven that no function satisfying those desiderata actually exists.

In practice, economists simply ignore the problem and posit what they call a “representative household” to model the economy, which is a fictional household assumed to hold all the endowments and income. This representative household’s utility function can then be used to generate aggregate demand and define a Konüs index to deflate nominal aggregates. As we have seen however, even in that case, preferences must still be assumed to be homothetic for the price index to be a function of prices alone, but at least the aggregation problem is avoided. This analytical sleight of hand is merely a way to sidestep the problem and not really a solution to it though. Indeed, for such a representation to be justified, there needs to be a utility function Uᴬ with the usual properties such that

\(\sum_{h=1}^{H}\mathbf{c_h}(\mathbf{p},m_h) = \mathbf{c^A}\!\left(\mathbf{p},\sum_{h=1}^{H}m_h\right)\text{ for every }\mathbf{p}\text{ and every }(m_1, \ldots, m_H),\)

where cₕ is the vector of Marshallian demands generated by Uₕ and cᴬ is the vector of Marshallian demands generated by Uᴬ. In other words, for aggregation to be possible, it must be the case that how aggregate income is distributed across households doesn’t affect aggregate demand as long as it remains the same.

Now, this can be shown to be the case if and only if, for every household h,

\(\mathbf{c_h}(\mathbf{p}, m_h) = \mathbf{\alpha_h}(\mathbf{p}) + \mathbf{\beta}(\mathbf{p})m_h\)

where cₕ is the vector of h’s Marshallian demands.¹⁴ This condition for the possibility of aggregation is known as the Gorman condition. What it says is that every household has the same marginal propensity to consume each good. It makes sense intuitively that it’s the condition for aggregation to be possible, because if different households had different marginal propensities to consume some goods, then transferring income from one household to another would not in general leave aggregate demand unchanged. But this condition is extremely restrictive and highly implausible. For instance, it requires that whether you give one extra dollar to Elon Musk or a homeless person, they will spend it on exactly the same things. But that is ridiculous, so the Gorman condition doesn’t hold and there isn’t a utility function such that the Marshallian demands generated by that utility function are equal to the sum of the Marshallian demands generated by the utility functions of individual households, which also means that we can’t justify the representative household sleight of hand in that way. Moreover, even if the Gorman condition were satisfied, it wouldn’t generally be enough for the Konüs index defined in terms of Uᴬ to be utility-preserving for every household. Indeed, as we have seen above, there is such a universal cost-of-living index if and only if every household also had the same preferences and those preferences were homothetic. But the Gorman condition doesn’t guarantee that, since although β is the same for every household αₕ needn’t be.¹⁵

To sum things up, when the Gorman condition fails (as it invariably does in any realistic economy), the sum of individual demands can’t be represented as the Marshallian demand generated by a single utility function Uᴬ. The sum of individual demands

\(\sum_{h=1}^{H}\mathbf{c_h}(\mathbf{p},m_h)\)

still exists, but in general it depends on how income is distributed across households. Thus, when household heterogeneity is taken seriously rather than swept under the rug by adopting the fiction of a representative household, the actual economy has neither a utility function Uᴬ representing the preferences of society as a whole nor a cost-of-living index defined in terms of that utility function that is a function of prices alone and preserves utility for every household. As a result, for any function K(p, p*) of prices alone that we use to deflate monetary quantities, there will be some household h, some price vectors p and p* and some utility level u attainable by h such that

\(K(\mathbf{p}, \mathbf{p^{*}}) \neq K_h(\mathbf{p}, \mathbf{p^*}, u).\)

Thus, no matter how we define a price index, as long as it’s a function of prices alone (hence the same for every household), if we use it to deflate household budgets as the economy moves along the path a(s) then in general some households will be overcompensated and others will be undercompensated.

However, each household’s expenditure function and the Konüs index derived from it remain well-defined, so we can still construct a price index that combines them in a somewhat principled way even if the result will not be utility-preserving for every household or even for any of them.¹⁶ Let u be a vector of utility levels for each household in the economy and let’s define

\(m^*_h \equiv E_h(\mathbf{p^*}, u_h), M^* \equiv \sum^H_h m^*_h \text{ and } \mu^*_h \equiv \frac{m^*_h}{M^*}.\)

The plutocratic cost-of-living index is

\(\begin{aligned} K^{\mathrm{pl}}(\mathbf{p},\mathbf{p^*}, \mathbf{u}) &\equiv \sum^H_h \frac{E_h(\mathbf{p},u_h)}{\sum^H_h E_h(\mathbf{p^*},u_h)}\\ &= \sum^H_h \frac{m^*_hK_h(\mathbf{p}, \mathbf{p^*}, u_h)}{M^*}\\ &= \sum^H_h\mu^*_hK_h(\mathbf{p}, \mathbf{p^*}, u_h) \end{aligned}.\)

So the plutocratic cost-of-living index is a weighted average of the exact compensation factors of the different households in the economy, where the weight of each household is equal to its share of aggregate income/consumption that allows each of them to attain the utility level uₕ at the reference prices p*. A household whose income is ten times larger than that of another therefore receives ten times as much weight in the index, which is why it’s called plutocratic. This is similar to what national statistical institutes do when they create price indices such as the CPI, except that it’s impossible for them to observe the consumption of every household so the weights they use only approximate expenditure shares and they also don’t observe utility so they use fixed-basket indices.¹⁷

Note that

\(M^*K^{\mathrm{pl}}(\mathbf{p},\mathbf{p^*}, \mathbf{u}) = \sum^H_h E_h(\mathbf{p}, u_h),\)

so if the aggregate income after prices shifted to p was

\(M = M^*K^{\mathrm{pl}}(\mathbf{p},\mathbf{p^*}, \mathbf{u}),\)

it follows that

\(\frac{M}{K^{\mathrm{pl}}(\mathbf{p},\mathbf{p^*}, \mathbf{u})} = M^*.\)

So if the income of each household h was sufficient to attain utility uₕ at prices p* and aggregate income was multiplied by the plutocratic cost-of-living index after prices shifted to p, it’s possible to distribute aggregate income after the price so that each household will have enough to recover its original utility level at the new prices. Of course, aggregate income may not actually be distributed in that way, so this doesn’t mean that no household is worse off, but as we have seen no price index has this property and it’s reasonable to see the plutocratic cost-of-living index as a second-best.

Now let’s go back to the model and think about how we should deflate variables expressed in monetary units as the economy travels along the equilibrium path induced by the trajectory of productivity in sector A. As we have seen, this trajectory is indexed by s, which is equal to log(a(s) / a(0) = log(a(s)) - log(a(0)) and therefore measured in log-productivity units. At every point on that path, the economy is assumed to have adjusted to be in equilibrium, so prices and other variables of the model such as the utility attained by each household change endogenously along the path so that at every point they have the values implied by the assumption that every market clears.¹⁸ As we have seen above, except in the special case where a household’s preferences are homothetic (which is not a realistic case), its cost-of-living price index is relative to a particular utility level. In other words, even for a single household, the question of how to combine prices to create a utility-preserving index is not well-posed, because the answer will generally be different depending on that household’s utility level and therefore on its purchasing power. Even for the same household, a price index that preserves utility as prices change when it’s poor will not preserve it when it’s rich, and vice versa. Thus, if we use a Konüs index, we’ll have to choose a utility target at which to evaluate the expenditure function, but the utility level reached by each household also changes as the economy travels along the equilibrium path and there doesn’t seem to be a non-arbitrary way to choose that utility target.

We could pick Uₕ(0), the utility that household h attains at the reference point, but that utility changes as the economy moves from that point to s. If we’re interested in deflating h’s nominal budget along that path, so that it reflects h’s purchasing power at each point, there doesn’t seem to be any principled reason to prefer Uₕ(0) to Uₕ(s) or any other utility level attained by h at some point on that path for that matter. A chained Divisia index therefore adopts a different approach, consisting in integrating h’s local proportional expenditure response to the induced price movement over the equilibrium path, holding utility fixed during each local comparison at the level h attains at that point. First, let’s define

\(F_{h,\tau}(r) \equiv E_h\bigl(p_A(r),p_B(r),U_h(\tau)\bigr),\)

which is a family of functions indexed by h and τ. Each function in the family is the expenditure function of h at r when the utility level is fixed at Uₕ(τ). Next we define

\(\begin{aligned} g^K_h(s) &\equiv \epsilon_{F_{h,s}}(s)\\ &= \frac{d \log F_{h,s}(r)}{d \log a(r)}(s)\\ &= \frac{d \log F_{h,s}(r)}{d r}(s) \end{aligned}.\)

As we have seen, as the productivity of sector A changes, the model determines the trajectory of the other variables, such as prices and the utility of every household in the economy. At each point s on the equilibrium path, gᴷₕ(s) therefore gives the elasticity of h’s expenditure function with respect to the productivity of sector A when the utility target is fixed at s.

If we apply the chain rule to gᴷₕ(s), we have

\(\begin{aligned} g^K_h(s) &= \frac{1}{F_{h,s}(r)}\frac{d \log F_{h,s}(r)}{d r}\\ &= \left. \frac{1}{F_{h,s}(r)}\bigl(\frac{\partial E_h}{\partial p_A}\frac{d p_A(r)}{d r} + \frac{\partial E_h}{\partial p_B}\frac{d p_B(r)}{d r}\bigr)\right|_{r = s} \end{aligned}.\)

Now, Shephard’s lemma states that

\(\frac{\partial E_h(p_A, p_B, u)}{\partial p_A} = h_{Ah}(p_A, p_B, u)\text{ and }\frac{\partial E_h(p_A, p_B, u)}{\partial p_B} = h_{Bh}(p_A, p_B, u)\)

for any utility level u.¹⁹ Moreover, as we have seen above, if

\(u = U_h(\mathbf{C})\text{ and }\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr)\)

then

\(\bigl(h_{Ah}(p_A,p_B,u),h_{Bh}(p_A,p_B,u)\bigr) = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr)\)

and

\(E_h(\mathbf{p},u) = m_h.\)

It follows that

\(\begin{aligned} g^K_h(s) &= \frac{1}{m_h(s)}\bigl(c_{Ah}\bigl(p_A(s), p_B(s), m_h(s)\bigr)\frac{d p_A(r)}{d r} + c_{Bh}\bigl(p_A(s), p_B(s), m_h(s)\bigr)\frac{d p_B(r)}{d r}\bigr)\\ &= \frac{1}{m_h(s)}\bigl(p_A(s)c_{Ah}\bigl(p_A(s), p_B(s), m_h(s)\bigr)\frac{1}{p_A(s)}\frac{d p_A(r)}{d r} + p_B(s)c_{Bh}\bigl(p_A(s), p_B(s), m_h(s)\bigr)\frac{1}{p_B(s)}\frac{d p_B(r)}{d r}\bigr)\\ &= \omega_{Ah}(s)\left.\frac{d \log p_A(r)}{d r}\right|_{r=s} + \omega_{Bh}(s)\left.\frac{d \log p_B(r)}{d r}\right|_{r=s}\\ &= \omega_{Ah}(s)\epsilon_{p_A}(s) + \omega_{Bh}(s)\epsilon_{p_B}(s) \end{aligned}.\)

Thus, h’s local proportional expenditure response to the price movements induced by the change in the productivity of sector A at s is a weighted average of the elasticities of the prices with respect to the change in the productivity of sector A at this point, using h’s expenditure shares at s as weights.

Finally, we define the chained Divisia index as

\(\log P^D_h(s) = \int_{0}^{s} g^K_h(r) \, dr,\)

so

\(P^D_h(s) = e^{\int_{0}^{s} g^K_h(r) \, dr}, \frac{\log P^D_h(s)}{ds} = g^K_h(s)\text{ and }P^D_h(0) = 1.\)

By integrating gᴷₕ(r) over the path from 0 to s, we obtain the cumulative expenditure response as the productivity of sector A moves from a(0) to a(s) in logarithm, fixing the utility target in the expenditure function at Uₕ(s) at each point along the path. Thus, by exponentiating that integral, we get the cumulative factor by which a budget expressed in prices at 0 must be multiplied at s to preserve utility at each point along the path between 0 to s when the utility target in the expenditure function is fixed at the level of utility attained by h at each point. The utility target is therefore updated as the productivity of sector A moves from a(0) to a(s), being fixed at each point at the equilibrium level attained by h at this point. As the expression for gᴷₕ(s) derived above shows, by making the utility target follow the equilibrium utility of h along the path, we are continuously updating the weights of the price movements so they are equal to h’s expenditure shares at equilibrium as we integrate over the path.

By contrast, if we use a Konüs index of the form

\(\begin{aligned} P^K_{h,u}(s) &= K_h\bigl((\mathbf{p}(s), \mathbf{p}(0), u\bigr)\\ &= \frac{E_h\bigl((\mathbf{p}(s), u\bigr)}{E_h\bigl((\mathbf{p}(0), u\bigr)} \end{aligned},\)

the utility target would have been held constant at some level u as prices move along the equilibrium path of the economy in response to the rise of the productivity in sector A. For instance, we could choose u = Uₕ(0), in which case the utility target would have been kept at the level attained by h at s = 0 along the entire path. If h’s preferences are homothetic, then no matter how u is chosen, the Konüs index and the chained Divisia index are identical. Indeed, since in that case Eₕ(p, u) = Gₕ(u)Π(p), we have

\(\begin{aligned} g^K_h(s) &= \frac{d \log F_{h,s}(r)}{d r}\\ &= \frac{d \log E_h\bigl(\mathbf{p}(r), U_h(s)\bigr)}{d r}\\ &= \frac{d \log \bigl(G_h\bigl(U_h(s)\bigr)\Pi_h\bigl(\mathbf{p}(r)\bigr)\bigr)}{d r}\\ &= \frac{d \log G_h\bigl(U_h(s)\bigr) + \log \Pi_h\bigl(\mathbf{p}(r)\bigr)}{d r}\\ &= \frac{d \log \Pi_h\bigl(\mathbf{p}(r)\bigr)}{d r} \end{aligned}.\)

It follows that

\(\begin{aligned} \log P^D_h(s) &= \int_{0}^{s} g^K_h(r) \, dr\\ &= \int_{0}^{s} \frac{d \log \Pi_h\bigl(\mathbf{p}(r)\bigr)}{d r}\\ &= \log \Pi_h\bigl(\mathbf{p}(s)\bigr) - \log \Pi_h\bigl(\mathbf{p}(0)\bigr)\\ &= \log \frac{\Pi_h\bigl(\mathbf{p}(s)\bigr)}{\Pi_h\bigl(\mathbf{p}(0)\bigr)}\\ P^D_h(s) &= \frac{\Pi_h\bigl(\mathbf{p}(s)\bigr)}{\Pi_h\bigl(\mathbf{p}(0)\bigr)}\\ &= \frac{G_h(u)\Pi_h\bigl(\mathbf{p}(s)\bigr)}{G_h(u)\Pi_h\bigl(\mathbf{p}(0)\bigr)}\\ &= \frac{E_h(\mathbf{p}(s), u)}{E_h(\mathbf{p}(0), u)}\\ &= P^K_{h,u}(s) \end{aligned}\)

for any u, so the chained Divisia index is equal to the Konüs index regardless of the utility target chosen. This makes sense given that, as we have seen, the Konüs index for a household h doesn’t depend on the utility target when h’s preferences are homothetic. However, when h’s preferences aren’t homothetic, this result doesn’t hold in general and the chained Divisia index differs from a Konüs index.

Each chained Divisia index is specific to a particular household, but they can be combined to define a chained aggregate Divisia index in the same way I defined the plutocratic cost-of-living index above. Let

\(M(s) \equiv \sum^H_h m_h(s), \mu_h(s) \equiv \frac{m_h(s)}{M(s)}\)

and

\(S_i(s) \equiv \frac{p_i \sum^H_h c_{ih}\bigl(\mathbf{p}(s), m_h(s)\bigr)}{M(s)}.\)

Since

\(p_i c_{ih}\bigl(\mathbf{p}(s), m_h(s)\bigr) = \sum^H_h m_h(s)\omega_{ih},\)

it follows that

\(S_i(s) = \mu_h(s)\omega_{ih}(s).\)

The aggregate elasticity of expenditure with respect to the productivity of sector A is defined as

\(\begin{aligned} g_P(s) &\equiv \sum_{I \in \{A, B\}} S_i(s)\frac{d \log p_i(s)}{ds}\\ &= \sum^H_h \mu_h(s) \sum_{I \in \{A, B\}} \omega_{ih}(s)\frac{d \log p_i(s)}{ds}\\ &= \sum^H_h \mu_h(s)g^K_h(s) \end{aligned},\)

so it’s just the weighted average of the elasticities of expenditure for the different households where the weights are their share of aggregate income.

The chained aggregate Divisia index is then defined as

\(\log P(s) = \int_{0}^{s} g_P(r) \, dr,\)

so

\(P(s) = e^{\int_{0}^{s} g_P(r) \, dr}, \frac{d \log P(s)}{ds} = g_P(s)\text{ and }P(0) = 1.\)

Let

\(V_h(s) \equiv U_h\bigl(c_{Ah}(\mathbf{p}(s), m_h(s), c_{Ah}(\mathbf{p}(s), m_h(s)\bigr)\)

and

\(N_s(r) \equiv \sum^H_h E_h\bigl(\mathbf{p}(r), V_h(s)\bigr).\)

By definition of the plutocratic cost-of-living index, we have

\(\begin{aligned} \left.\frac{d \log K^{\mathrm{pl}}\bigl(\mathbf{p}(r),\mathbf{p}(s), V_1(s), \ldots, V_H(s)\bigr)}{d r}\right|_{r = s} &= \left.\frac{d \log \frac{N_s(r)}{N_s(s)}}{d r}\right|_{r = s}\\ &= \left.\frac{d \log N_s(r) - \log N_s(s)}{d r}\right|_{r = s}\\ &= \frac{1}{N_s(s)} \left.\frac{d N_s(r)}{d r}\right|_{r = s}\\ &= \frac{1}{\sum^H_h E_h\bigl(\mathbf{p}(s), V_h(s)\bigr)} \left.\frac{d \sum^H_h E_h\bigl(\mathbf{p}(r), V_h(s)\bigr)}{d r}\right|_{r = s}\\ &= \frac{1}{\sum^H_h E_h\bigl(d \mathbf{p}(s), V_h(s)\bigr)} \sum^H_h E_h\bigl(\mathbf{p}(s), V_h(s)\bigr)g^K_h(s)\\ &= \frac{\sum^H_h m_h(s)g^K_h(s)}{M(s)}\\ &= \sum^H_h \mu_h(s)g^K_h(s)\\ &= g_P(s) \end{aligned},\)

so the chained aggregate Divisia index and the plutocratic cost-of-living index have the same local compensation rate with respect to the productivity of sector A. However, it’s still the case that

\(K^{\mathrm{pl}}(\mathbf{p}(s),\mathbf{p}(r), V_1(r), \ldots, V_H(r)) \neq \frac{P(s)}{P(r)}\)

over a finite change in log-productivity, unless every household have the same homothetic preferences.

In what follows, since there is no such thing as a universal cost-of-living index and the choice of a utility target for each household is arbitrary, I will be using P to deflate aggregate monetary variables. As we have seen, the equilibrium path can be seen as describing the same economy at different points in time where productivity in sector A is different, but it also has a spatial interpretation according to which it describes a continuum of economies with different productivity levels in sector A but the same number of households with the same preferences and labor endowments. The path a(s), where s is measured in log-productivity units, just describes a trajectory for the productivity of sector A. From a mathematical point of view, it’s indifferent whether each point on that path is interpreted as describing the same economy at a different time or a different country at the same time, what matters is just the level of productivity in sector A at that point. This means that P(s) can similarly be interpreted both as a utility-preserving price index, with the caveat as we have seen it doesn’t actually preserve utility for any particular household (because no such price index exists unless every household has the same homothetic preferences), used to deflate monetary quantities in the same economy at different times or in different economies at the same time. Regardless of whether one adopts a temporal or spatial interpretation of the equilibrium path, P(s) just says how monetary quantities must be deflated to keep purchasing power constant relative to the reference point when productivity in sector A differs from that point by s, where this difference is measured in log-productivity units.

The response to a change in the productivity of sector A

Now that we have introduced the necessary conceptual machinery, we can finally derive various results about how the economy responds to a change in the productivity of sector A. Concretely, we are going to use the model presented above to derive a bunch of elasticities with respect to the productivity of sector A, which will shed light on the demand-side mechanism I explained in intuitive terms at the beginning of this post.

The relative price elasticity

Since

\(m_h = p_AL_{Ah} + p_BL_{Bh},\)

and

\(q = \frac{p_B}{p_A},\)

by using the rules on the elasticity of a sum and the elasticity of a product, we have

\(\begin{aligned} \epsilon_{m_h} &= \alpha_{Ah}\epsilon_{p_AaL_A} + \alpha_{Bh}\epsilon_{p_BbL_B}\\ &= \alpha_{Ah}(\epsilon_{p_A} + 1) + \alpha_{Bh}\epsilon_{p_B}\\ &= \alpha_{Ah}(\epsilon_{p_A} + 1) + \alpha_{Bh}(\epsilon_{p_A} + \epsilon_q)\\ &= \epsilon_{p_A}(\alpha_{Ah} + \alpha_{Bh}) + \alpha_{Ah} + \alpha_{Bh}\epsilon_q\\ &= \epsilon_{p_A} + \alpha_{Ah} + \alpha_{Bh}\epsilon_q\\ \end{aligned}\)

where

\(\alpha_{Ah} = \frac{p_AaL_{Ah}}{m_h} \text{ and } \alpha_{Bh} = \frac{p_BbL_{Bh}}{m_h}\)

are the shares of household h’s income from A-labor and B-labor respectively.

Moreover, since

\(\epsilon_{P^D_h} = g^K_h,\)

it follows that

\(\begin{aligned} \epsilon_{P^D_h} &= \omega_{Ah}\epsilon_{p_A} + \omega_{Bh}\epsilon_{p_B}\\ &= \omega_{Ah}\epsilon_{p_A} + \omega_{Bh}(\epsilon_{p_A} + \epsilon_q)\\ &= \epsilon_{p_A}(\omega_{Ah} + \omega_{Bh}) + \omega_{Bh}\epsilon_q\\ &= \epsilon_{p_A} + \omega_{Bh}\epsilon_q\\ \end{aligned}.\)

Therefore, by applying the elasticity rule for a quotient, we have

\(\begin{aligned} \epsilon_{m_H/P^D_h} &= \epsilon_{m_h} - \epsilon_{P^D_h}\\ &= \epsilon_{p_A} + \alpha_{Ah} + \alpha_{Bh}\epsilon_q - (\epsilon_{p_A} + \omega_{Bh}\epsilon_q)\\ &= \alpha_{Ah} + \epsilon_q(\alpha_{Bh} - \omega_{Bh}) \end{aligned}.\)

This formula shows how h’s purchasing power increases in response to a rise in the productivity of sector A. The first term implies that it increases with the share of h’s income that it derives from A-labor, which makes sense because the rise of productivity in sector A increases wages in that sector. The second term shows that the rise in the price of B relative to A pulls in two opposite direction.²⁰ On the one hand, it increases h’s purchasing power to the extent that it derives a large share of its income from B-labor, which makes sense because the rise of the price of B increases wages in that sector. But on the other hand, it reduces h’s purchasing power to the extent that it spends a large share of its income on B, which makes sense because this makes h feel the increase in the price of B more intensely.

We have seen that, since Hicksian demand was homogeneous of degree zero, it was only affected by prices through their ratio q. Therefore, we can define

\(\zeta^H_{ih} \equiv \left.\frac{\partial \log h_{ih}}{\partial \log q}\right|_{u},\)

the elasticity of Hicksian demand of h for i with respect to q when the utility target is held fixed. In what follows, utility is fixed at u while

\(E_h\text{ and }h_{ih}\)

are evaluated at

\((p_A(s), p_B(s), u)\)

so we suppress their arguments. We have

\(E_h = p_Ah_{Ah} + p_Bh_{Bh},\)

so differentiating with respect to s the product rule gives

\(\begin{aligned} \frac{dE_h}{d s} &= \frac{d(p_Ah_{Ah})}{d s} + \frac{d(p_Bh_{Bh})}{d s}\\ &= p_A \frac{d h_{Ah}}{d s} + h_{Ah} \frac{d p_A}{d s} + p_B \frac{d h_{Bh}}{d s} + h_{Bh} \frac{d p_B}{d s}\\ \end{aligned}.\)

If we apply the multivariable chain rule and Shephard’s lemma instead, we get

\(\begin{aligned} \frac{d E_h}{d s} &= \frac{d E_h}{d p_A}\frac{d p_A}{d s} + \frac{d E_h}{d p_B}\frac{d p_B}{d s}\\ &= h_{Ah}\frac{d p_A}{d s} + h_{Bh}\frac{d p_B}{d r}\\ \end{aligned}.\)

Equating those expressions and canceling their common terms yields

\(p_A \frac{d h_{Ah}}{d s} + p_B \frac{d h_{Bh}}{d s} = 0.\)

Since we are keeping utility fixed as the productivity of sector A increases, whatever relative price change is induced by the rise in the productivity of sector A, the cost of substituting toward one good must be exactly offset by the income freed by substituting away from the other.

By the chain rule,

\(\frac{d\log h_{ih}}{d s} = \zeta^H_{ih}\frac{d \log q}{d s}\)

and

\(\frac{d h_{ih}}{d s} = h_{ih}\frac{d \log h_{ih}}{d s},\)

so

\(\frac{d h_{ih}}{d s} = h_{ih}\zeta^H_{ih}\frac{d \log q}{d s}.\)

Substituting into the equality derived above, we get

\(\frac{d \log q}{d s}(p_A h_{Ah}\zeta^H_{Ah} + p_B h_{Bh}\zeta^H_{Bh}) = 0,\)

which divided by d log q / ds and Eₕ yields

\(\frac{p_A h_{Ah}}{E_h}\zeta^H_{Ah} + \frac{p_B h_{Bh}}{E_h}\zeta^H_{Bh} = 0.\)

As we have seen, if households maximize utility, Hicksian demand is equal to Marshallian demand and each household’s expenditure function is equal to its budget at the equilibrium. In other words,

\(h_{ih} = c_{ih}\text{ and }E_h = m_h,\)

so by substituting in the previous equality we get

\(\omega_{Ah}\zeta^H_{Ah} + \omega_{Bh}\zeta^H_{Bh} = 0.\)

We can now use this result to relate the elasticities of Hicksian demands and the expenditure shares to the elasticity of substitution.

By definition of σₕ,

\(\zeta^H_{Bh} - \zeta^H_{Ah} = -\sigma_h,\)

so

\(\zeta^H_{Bh} = \zeta^H_{Ah} - \sigma_h\)

and

\(\zeta^H_{Ah} = \omega_{Bh}\sigma_h\)

by substituting into the equality derived above. In turn, by substituting this equality into

\(\zeta^H_{Bh} - \zeta^H_{Ah} = -\sigma_h,\)

we get

\(\zeta^H_{Bh} = -\omega_{Ah}\sigma_h.\)

As we have seen, when a household responds to a change in relative prices by substituting away from one good toward the other, the additional expenditure caused by purchasing more of one must be exactly offset by the income freed by purchasing less of the other. Thus, to the extent that h currently spends a larger share of its budget on one good, the same proportional reduction in the quantity of that good h buys will free more income to buy more of the other good and conversely the same proportional increase in the quantity of that good h buys must be offset by a greater reduction in the quantity of the other good it buys.

We now turn to the elasticity of Marshallian demand with respect to productivity in sector A. The Slutsky identity, which states that

\(\frac{\partial c_{ih}}{\partial p_j} = \left.\frac{\partial h_{ih}}{\partial p_j}\right|_u - c_{ih}\frac{\partial c_{ih}}{\partial m_h},\)

for i, j ∈ {A, B} decomposes the effect of a price change on Marshallian demand into the substitution effect caused by the change in relative prices and the income effect caused by the change in purchasing power that change of price induces.²¹ Thus, since

\(\frac{d c_{ih}}{d s} = \sum_{j \in \{A, B \}} \frac{\partial c_{ih}}{\partial p_j}\frac{d p_j}{d s} + \frac{\partial c_{ih}}{\partial m_h}\frac{d m_h}{d s}\)

by the multivariable chain rule, we have

\(\frac{d c_{ih}}{d s} = \sum_{j \in \{A, B \}} \left.\frac{\partial h_{ih}}{\partial p_j}\right|_u\frac{d p_j}{d s} + \frac{\partial c_{ih}}{\partial m_h}\bigl(\frac{d m_h}{d s} - \sum_{j \in \{A, B \}} c_{ih}\frac{d p_j}{d s}\bigr).\)

Since

\(\frac{1}{c_{ih}}\frac{d c_{ih}}{d s} = \frac{d \log c_{ih}}{d s},\)

We can recover the elasticities of Marshallian demands by dividing this equation by cᵢₕ.

Taking the first term and dividing it by the Marshallian demand for i, using the multivariable rule again, we have

\(\begin{aligned} \frac{1}{c_{ih}}\sum_{j \in \{A, B \}} \left.\frac{\partial h_{ih}}{\partial p_j}\right|_u\frac{d p_j}{d s} &= \frac{1}{c_{ih}}\left.\frac{d h_{ih}}{d s}\right|_u\\ &= \left.\frac{d \log h_{ih}}{d s}\right|_u\\ &= \left.\frac{d \log h_{ih}}{d \log q}\right|_u \frac{d \log q}{d s}\\ &= \zeta^H_{ih}\epsilon_q \end{aligned}.\)

Taking the second term, since

\(\eta_{ih} = \frac{m_h}{c_{ih}}\frac{\partial c_{ih}}{\partial m_h}\)

and

\(\epsilon_{P^D_h} = \omega_{Ah}\epsilon_{p_A} + \omega_{Bh}\epsilon_{p_B},\)

we also have

\(\begin{aligned} \frac{1}{c_{ih}}\frac{\partial c_{ih}}{\partial m_h}\bigl(\frac{d m_h}{d s} - \sum_{j \in \{A, B \}} c_{ih}\frac{d p_j}{d s}\bigr) &= \eta_{ih}\bigl(\frac{d \log m_h}{d s} - \omega_{Ah}\frac{d \log p_A}{d s} - \omega_{Bh}\frac{d \log p_B}{d s}\bigr)\\ &= \eta_{ih}\bigl(\epsilon_{m_h} - \omega_{Ah}\epsilon_{p_A} - \omega_{Bh}\epsilon_{p_B}\bigr)\\ &= \eta_{ih}\bigl(\epsilon_{m_h} - \epsilon_{P^D_h}\bigr) \end{aligned}.\)

Substituting into the equality derived above, we therefore have

\(\begin{aligned} \epsilon_{c_{ih}} &= \frac{1}{c_{ih}}\frac{d c_{ih}}{d s}\\ &= \zeta^H_{ih}\epsilon_q + \eta_{ih}\bigl(\epsilon_{m_h} - \epsilon_{P^D_h}\bigr)\\ &= \zeta^H_{ih}\epsilon_q + \eta_{ih}\epsilon_{m_h/P^D_h} \end{aligned},\)

which after substituting the identities previously established yields

\(\begin{aligned} \epsilon_{c_{Ah}} &= \omega_{Bh}\sigma_h\epsilon_q + \eta_{Ah}\bigl(\alpha_{Ah} + \epsilon_q(\alpha_{Bh} - \omega_{Bh})\bigr)\\ &= \eta_{Ah}\alpha_{Ah} + \bigl(\eta_{Ah}(\alpha_{Bh} - \omega_{Bh}) + \omega_{Bh}\sigma_h\bigr)\epsilon_q \end{aligned}\)

and

\(\begin{aligned} \epsilon_{c_{Bh}} &= -\omega_{Ah}\sigma_h\epsilon_q + \eta_{Bh}\bigl(\alpha_{Ah} + \epsilon_q(\alpha_{Bh} - \omega_{Bh})\bigr)\\ &= \eta_{Bh}\alpha_{Ah} + \bigl(\eta_{Bh}(\alpha_{Bh} - \omega_{Bh}) - \omega_{Ah}\sigma_h\bigr)\epsilon_q \end{aligned}.\)

Those results show how the demand for A and B responds locally to the rise in the productivity of sector A.

As those formulas show, the elasticities of Marshallian demand with respect to the productivity of sector A combine two effects, each of them corresponding to a term in the formulas. There is a direct income effect, corresponding to the change in demand for A and B resulting from a rise in income derived by the household from A-labor, as wages in sector A increase with productivity in that sector. There is also a relative price effect, which can be subdivided into a relative price income effect and a substitution effect. The relative price income effect results from the fact that, as the productivity of sector A increases, it induces a change in the price of B relative to A. If this change is positive, which is what happens in the interpretation of the model that motivated it, h’s income will rise to the extent that it derives more income from B-labor than it spends on B and it will fall otherwise, while the opposite will be true if the price of B relative to A decreases in response to a rise of productivity in sector A. This in turn will affect the demand for each good in a way that is determined by the income elasticity of demand for them. Finally, the change in the price of B relative to A also results in a substitution effect, as h substitutes away from the good that became relatively more expensive toward the one that became relatively cheaper.²² If this substitution effect is sufficiently larger than the effect that the rise in productivity of sector A has on the demand for B through the increase of income it induces, both directly by increasing wages in that sector and indirectly by increasing the relative price of B, it can make the elasticity of the demand for B negative.²³

We can now derive the elasticity of q, the price of B relative to A. Since we are assuming that markets clear at each point on the equilibrium path, we have

\(Y_B = \sum^H_{h} c_{Bh}.\)

Moreover, since we are assuming that the production of B is fixed, the elasticity rule for sums gives

\(\begin{aligned} \epsilon_{Y_B} &= \sum^H_h \frac{c_Bh}{Y_B}\epsilon_{c_{Bh}}\\ &= \sum^H_h \nu_{Bh}\epsilon_{c_{Bh}}\\ &= 0 \end{aligned},\)

where

\(\nu_{ih} \equiv \frac{c_{ih}}{Y_i}\)

is h’s consumption of good i as a share of total production in that sector. Substituting the identity about the elasticity of demand for B derived above into this equation, we get

\(\begin{aligned} 0 &= \sum^H_h \nu_{Bh}\bigl(\eta_{Bh}\alpha_{Ah} + \bigl(\eta_{Bh}(\alpha_{Bh} - \omega_{Bh}) - \omega_{Ah}\sigma_h\bigr)\epsilon_q\bigr)\\ &= \sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah} + \epsilon_q\sum^H_h \nu_{Bh}\bigl(\eta_{Bh}(\alpha_{Bh} - \omega_{Bh}) - \omega_{Ah}\sigma_h\bigr)\\ &= \sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah} - \epsilon_q\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr) \end{aligned}.\)

Now, by rearranging this equation, we obtain

\(\sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah} = \epsilon_q\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr).\)

This result makes sense intuitively. Since the supply of B is fixed, for supply to equal demand, the direct income effect of the rise of productivity in sector A on the demand for B must be exactly offset by the effect of a change in relative prices, which combine the relative price income effect and the substitution effect.

Now, provided that

\(\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr) \neq 0,\)

we can rearrange the previous equation to get

\(\epsilon_q = \frac{\sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah}}{\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr)}.\)

This formula shows how the price of B relative to A responds to a rise in the productivity of sector A and it’s worth studying in a little bit detail. Both the numerator and the denominator are weighted averages, using the share of each household’s consumption of B in the total production of B as weights.

It can be shown that

\(\sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah} = \left.\frac{\partial \log D_B}{\partial \log a}\right|_{p_A, p_B}\)

and

\(\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr) = -\left.\frac{\partial \log D_B}{\partial \log q}\right|_a,\)

so

\(\epsilon_q = - \frac{\left.\frac{\partial \log D_B}{\partial \log a}\right|_{p_A, p_B}}{\left.\frac{\partial \log D_B}{\partial \log q}\right|_a},\)

where

\(D_B = \sum^H_h c_{Bh} = Y_B\)

is the aggregate demand for B, which is equal to the total production of B since we’re assuming that the economy is at equilibrium.²⁴

This result makes it easier to interpret the formula we have derived for the elasticity of q intuitively. Since the model assumes that employment and productivity in sector B are fixed, any pressure on the demand for B caused by higher income in sector A as a result of the rise of productivity in that sector must be exactly offset by pressure in the opposite direction caused by a rise in the price of B relative to A. The numerator in the formula, which corresponds to the direct income effect I described above, measures the pressure on the demand for B caused by the rise of income in sector A induced by the increase of productivity in that sector. The denominator measures the countervailing pressure caused by the change in the price of B relative to A induced by the increase of productivity in sector A. As we have seen, this can be decomposed into a relative price income effect that increases that countervailing pressure and a substitution effect that attenuates it. Intuitively, holding the direct income effect on the demand for B constant, q must rise less as the countervailing pressure on the demand for B resulting from the change in the relative price of B induced by the increase of productivity in sector A is larger.

In the interpretation of the model that motivated it, the rise in the productivity of sector A results in upward pressure on the demand for B through higher income in sector A, but since the supply of B is fixed this results in a rise of the relative price of B, which in turn creates downward pressure on the demand for B that exactly offers the upward pressure resulting from higher income in sector A. In that case, the elasticity of q is positive, but this is not a consequence of the model which is consistent with a negative elasticity of q. In order to see how the sign of that elasticity is determined, it’s useful to write the formula we derived above as

\(\epsilon_q = \frac{I}{S - D},\)

where

\(I = \sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah}, S = \sum_h\nu_{Bh}\omega_{Ah}\sigma_h\text{ and }D = \sum^H_h\nu_{Bh}\eta_{Bh}(\alpha_{Bh} - \omega_{Bh}).\)

I is the direct income effect of a rise in the productivity of sector A, S is the substitution effect caused by the change in the relative price of B it induces and D is the relative price income effect.

This way of writing the formula makes it clear how household heterogeneity affects the elasticity of q. The direct income effect is larger as households that account for a large share of the consumption of B have a high income elasticity of demand for B and derive a large share of their income from A-labor. If enough households that account for a large share of the consumption of B don’t treat it as a normal good, this effect could in theory be negative, in which case the elasticity of q may be negative if the denominator is positive. The substitution effect is larger as the households that account for a large share of the consumption of B spend a large share of their income on A and have a high elasticity of substitution between A and B. The relative price income effect is larger as the households that account for a large share of the consumption of B receive more income from that sector than they spent on it and have a high income elasticity of demand for B. When the relative price of B increases, the purchasing power of households that receive more income from sector B than they spend on it also rises, so as long as they treat B as a normal good this will create upward pressure on the demand for that good that will mitigate the substitution effect and the higher their income elasticity of demand for B is the larger that countervailing pressure will be. In theory, this effect could even be strong enough to overwhelm the substitution effect, reversing the sign of the denominator and making the elasticity of q negative. However, if the households that account for a large share of the consumption of B tend to spend more on it than they receive from sector B, this effect will be negative and it will instead reinforce the substitution effect.

Although the representative household case is not realistic, it’s useful to look at what happens to that formula when there is only one household, because the contrast with the more general formula will shed light on how household heterogeneity affects the elasticity of q. For a representative household, there is no need to sum over households in the formula and we can drop the subscript we used to index by household. Moreover, since the representative household has all the labor endowments and we’re assuming that markets clear,

\(\alpha_A = \omega_A = \theta\text{ and }\alpha_B = \omega_B = 1 - \theta.\)

Therefore, substituting into

\(\epsilon_q = \frac{\sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah}}{\sum^H_h \nu_{Bh}\bigl( \omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr)},\)

we get

\(\begin{aligned} \epsilon_q &= \frac{\eta_B\theta}{\theta\sigma - \eta_B\bigl((1 - \theta) - (1 - \theta)\bigr)}\\ &= \frac{\eta_B\theta}{\theta\sigma}\\ &= \frac{\eta_B}{\sigma} \end{aligned}.\)

Thus, in the absence of household heterogeneity, the elasticity of q reduces to the ratio of the income elasticity of demand for B to the elasticity of substitution between A and B.

We could derive the same result without resorting to the representative household fiction by using the Gorman condition, although it would be more complicated. This would show that, although the terms in I and S are different for different households, their sums are still equal to the aggregate income elasticity of demand for B times the share of sector A in GDP and the aggregate elasticity of substitution times the share of sector A in GDP respectively, while the terms in D still sum to zero even though they don’t in general vanish for each household. In the special case where not only the Gorman condition holds, but preferences are homothetic and therefore the income elasticity of demand for B is equal to 1, the previous equation reduces to

\(\epsilon_q = \frac{1}{\sigma}.\)

If moreover the elasticity of substitution is also equal to 1, which means that preferences are Cobb-Douglas and households spend the same constant share of their budget on each good regardless of their income, it further reduces to

\(\epsilon_q = 1.\)

In that case, the relative price of B changes one-to-one with productivity in sector A, so that a rise of productivity in that sector results in a rise of the relative price of B in the same proportion.

The elasticity of q tells us how the relative price of B changes locally, for an infinitesimal change in the productivity of sector A, but how does it change for a finite change in the productivity of sector A? Let the productivity of sector A rise from a(0) = a₀ to a(z) = a₀λ, where λ = eᶻ and therefore z = log λ.²⁵ In general, we have

\(\log\frac{q(z)}{q(0)} = \int_0^z\epsilon_q(s)\,ds,\)

so

\(\frac{q(z)}{q(0)} = \exp\left(\int_0^z\epsilon_q(s)\,ds\right).\)

Thus, the finite response to a change in productivity of sector A depends on the value taken by the elasticity of q along the path, which in general changes with income sources, expenditure patterns and behavioral parameters.

If we define the average value of the elasticity of q over the interval as

\(\bar{\epsilon}_q \equiv \frac{1}{z}\int_0^z\epsilon_q(s)\,ds,\)

then

\(\begin{aligned} \frac{q(z)}{q(0)} &= \exp\left(\int_0^z\epsilon_q(s)\,ds\right)\\ &= \exp(\bar{\epsilon}_q z)\\ &= \lambda^{\bar{\epsilon}_q} \end{aligned}.\)

In the special case where every household has the same preferences and they are Cobb-Douglas, so that

\(\eta_B = \sigma = 1\)

and

\(\epsilon_q = 1,\)

we have

\(\frac{q(z)}{q(0)} = \lambda.\)

In other words, in that case, the relative price of B rises in proportion to productivity in sector A.

The sectoral share elasticities

Let

\(O \equiv \frac{\theta}{1 - \theta}\)

be the odds of θ, the share of sector A in GDP. By definition of θ, the share of sector A in GDP, we have

\(\begin{aligned} O &= \frac{\frac{p_AY_A}{N}}{\frac{p_BY_B}{N}}\\ &= \frac{p_AY_A}{p_BY_B}\\ &= \frac{Y_A}{qY_B}\\ &= \frac{aL_A}{qbL_B} \end{aligned}.\)

Thus, by applying the elasticity rules for quotients and products, we get

\(\begin{aligned} \epsilon_O &= \epsilon_{aL_A} - \epsilon_{qbL_B}\\ &= (\epsilon_a + \epsilon_{L_A}) - (\epsilon_{q} + \epsilon_b + \epsilon_{L_B})\\ &= 1 - \epsilon_{q} \end{aligned}.\)

This result is what one would expect intuitively. The share of sector A relative to sector B increases one-for-one with productivity in sector A holding relative prices constant, but relative prices also respond to the rise of productivity in sector A. If q responds by increasing, the production of sector B becomes relatively more valuable, despite remaining quantitatively fixed. If q responds by falling, on the other hand, the value of the production of sector B changes in a way that reinforces the physical expansion of the production of sector A.

Now, by applying the elasticity rules on sums and products, we have

\(\begin{aligned} \epsilon_O &= \epsilon_{\frac{\theta}{1 - \theta}}\\ &= \epsilon_{\theta} - \epsilon_{1 - \theta}\\ &= \epsilon_{\theta} - [\frac{1}{1 - \theta}\epsilon_{1} - \frac{\theta}{1 - \theta}\epsilon_{-\theta}]\\ &= \epsilon_{\theta} - [\frac{1}{1 - \theta}\epsilon_{1} - \frac{\theta}{1 - \theta}(\epsilon_{-1} + \epsilon_{\theta})]\\ &= \epsilon_{\theta} + \frac{\theta}{1 - \theta}\epsilon_{\theta}\\ &= \frac{1}{1 - \theta}\epsilon_{\theta}\\ \end{aligned}.\)

Therefore, by equating this result with the expression for the elasticity of O we derived above, we get

\(\epsilon_{\theta} = (1 - \theta)(1 - \epsilon_{q}).\)

The factor 1 - θ appears because, since θ is the share of sector A in GDP and everything that’s not produced by sector A must be produced by sector B, any change in the share of sector A must be exactly offset by a change in the share of sector B. When the share of sector B is small, even a large increase in the odds of A’s share of GDP can’t result in a large increase in the share of sector A, because it doesn’t have much room to improve. When the share of sector B is large, on the other hand, even a small increase in the odds of A’s share of GDP can result in a substantial increase in the share of sector A. If the elasticity of q is 1, then the physical expansion of the production of sector A is exactly offset by the rise of the relative price of B, hence the share of sector A remains constant. If the elasticity of q is positive but less than 1, then the share of sector A increases, but that rise is mitigated by the increase of the relative price of B. If the elasticity of q is greater than 1, then the share of sector A decreases in spite of its physical expansion, because that expansion is more than offset by the rise in the relative price of B. Finally, if the elasticity of q is less than 0, the fall of the relative price of B reinforces the physical expansion of production in sector A instead of attenuating it.

Now let’s examine the response to a finite change in the productivity of sector A. Again we consider what happens as the productivity of sector A rises from a(0) = a₀ to a(z) = a₀λ, where λ = eᶻ and therefore z = log λ. Since the production of B is fixed and

\(O = \frac{Y_A}{qY_B},\)

we have

\(\begin{aligned} \frac{O(z)}{O(0)} &= \frac{\frac{Y_A(z)}{q(z)Y_B}}{\frac{Y_A(0)}{q(0)Y_B}}\\ &= \frac{Y_A(z)}{Y_A(0)}\frac{q(0)}{q(z)}\\ &= \frac{a(z)L_A}{a(0)L_A}\frac{q(0)}{q(z)}\\ &= \frac{a(z)}{a(0)}\frac{q(0)}{q(z)}\\ &= \frac{\lambda}{\frac{q(z)}{q(0)}} \end{aligned}.\)

Now,

\(\lambda = \exp\left(\int_0^z 1\,ds\right)\)

and

\(\frac{q(z)}{q(0)} = \exp\left(\int_0^z\epsilon_q(s)\,ds\right),\)

so

\(\begin{aligned} \frac{O(z)}{O(0)} &= \frac{\exp\left(\int_0^z 1\,ds\right)}{\exp\left(\int_0^z\epsilon_q(s)\,ds\right)}\\ &= \exp\left(\int_0^z [1 - \epsilon_q(s)]\,ds\right) \end{aligned}\)

So the odds of the share of sector A depends on how the value taken by the elasticity of q along the path.

Using the average elasticity along the path, since

\(\frac{O(z)}{O(0)} = \lambda^{1 - \bar{\epsilon}_q}.\)

and

\(\frac{q(z)}{q(0)} = \lambda^{\bar{\epsilon}_q},\)

it follows that

\(\frac{O(z)}{O(0)} = \lambda^{1 - \bar{\epsilon}_q}.\)

Finally, since

\(\theta = \frac{O}{1 + O},\)

we have

\(\begin{aligned} \theta(z) &= \frac{O(z)}{1 + O(z)}\\ &= \frac{O(0)\lambda^{1 - \bar{\epsilon}_q}}{1 + O(0)\lambda^{1 - \bar{\epsilon}_q}}\\ &= \frac{\frac{\theta(0)}{1 - \theta(0)}\lambda^{1 - \bar{\epsilon}_q}}{1 + \frac{\theta(0)}{1 - \theta(0)}\lambda^{1 - \bar{\epsilon}_q}}\\ &= \frac{\theta(0)\lambda^{1 - \bar{\epsilon}_q}}{1 - \theta(0) + \theta(0)\lambda^{1 - \bar{\epsilon}_q}} \end{aligned}.\)

However, to see what happens to θ as the productivity of sector A increases, it’s easier to look at the formula for 0. This shows that θ rises with the productivity of sector A if the average elasticity of q over the interval is less than 1, it stays constant if the average elasticity of q is equal to 1 and it falls if the average elasticity of q is greater than 1. This makes sense because, although the rise in productivity of sector A results in the physical expansion of that sector, the elasticity of q determines how its output is valued relative to the fixed output of sector B. If the rise of productivity in sector A increases the relative price of B even more proportionally than it physically expands the production of sector A, the share of sector A in GDP falls. If it increases the relative price of B exactly in proportion to the physical expansion of production in sector A, which is what happens in the special case where every household has the same preferences and they are Cobb-Douglas, the share of sector A remains constant. If the rise of productivity in sector A increases the relative price of B proportionally less than it physically expands the production of sector A, the share of sector A in GDP rises.

The real GDP elasticity

As we have seen above, the chained aggregate Divisia index is defined in such a way that

\(\epsilon_P = S_A\epsilon_{p_A} + S_B\epsilon_{p_B},\)

where

\(S_i \equiv \frac{p_i \sum^H_h c_{ih}}{M}\text{ and }M \equiv \sum^H_h m_h.\)

Since we are assuming that, at each point s, the economy is at equilibrium, it follows that

\(p_i\sum^H_h c_{ih} = p_iY_i\text{ and }\sum^H_h m_h = N\)

with

\(N \equiv p_AY_A + p_BY_B.\)

Therefore,

\(S_i = \frac{p_iY_i}{N}\text{ and }\epsilon_P = \theta\epsilon_{p_A} + (1 - \theta)\epsilon_{p_B}\)

where

\(\theta \equiv \frac{p_AY_A}{N}\text{ and }1 - \theta = \frac{p_BY_B}{N}\)

are the shares of each sector in nominal GDP.

As we have seen,

\(\epsilon_{p_B} = \epsilon_{p_A} + \epsilon_q,\)

hence

\(\begin{aligned} \epsilon_P &= \theta\epsilon_{p_A} + (1 - \theta)\epsilon_{p_B}\\ &= \theta\epsilon_{p_A} + (1 - \theta)(\epsilon_{p_A} + \epsilon_q)\\ &= \epsilon_{p_A} + (1 - \theta)\epsilon_q \end{aligned}.\)

Moreover,

\(\epsilon_{Y_A} = 1\)

and

\(\epsilon_{Y_B} = 0.\)

so by applying the elasticity rules about sums and products to derive the elasticity of nominal GDP, we get

\(\begin{aligned} \epsilon_N &= \theta\epsilon_{p_AY_A} + (1 - \theta)\epsilon_{p_BY_B}\\ &= \theta(\epsilon_{p_A} + \epsilon_{Y_A}) + (1 - \theta)(\epsilon_{p_B} + \epsilon_{Y_B})\\ &= \theta(\epsilon_{p_A} + 1) + (1 - \theta)(\epsilon_{p_A} + \epsilon_q)\\ &= \epsilon_{p_A} + \theta + (1 - \theta)\epsilon_q \end{aligned}.\)

We have thus derived the elasticity with respect to the productivity of sector A of both nominal GDP and the price index used to deflate it.

Finally, since real GDP is defined as

\(R \equiv \frac{N}{P},\)

the elasticity rule about quotients gives

\(\begin{aligned} \epsilon_R &= \epsilon_N - \epsilon_P\\ &= [\epsilon_{p_A} + \theta + (1 - \theta)\epsilon_q] - [\epsilon_{p_A} + (1 - \theta)\epsilon_q]\\ &= \theta \end{aligned}.\)

In the model, physical productivity is only assumed to rise in sector A, while the productivity of sector B is assumed to be constant. Since labor endowments are also assumed to be fixed, real GDP growth is purely the result of productivity growth in sector A, hence its elasticity with respect to the productivity of sector A is equal to the share of that sector in nominal GDP.

If we turn to the effect of a finite change in the productivity of sector A and consider what happens as the productivity of sector A rises from a(0) = a₀ to a(z) = a₀λ with λ = eᶻ and z = log λ, we have

\(\log\frac{R(z)}{R(0)} = \int_0^z\theta(s)\,ds,\)

hence

\(\frac{R(z)}{R(0)} = \exp\left(\int_0^z\theta(s)\,ds\right).\)

Thus, the whole trajectory of θ over the interval matters and not just the initial and final shares, because the response of real GDP to a rise of productivity in sector A at each point is equal to the share of sector A in GDP at that point, the value of the integral can be different for different trajectories even if they have the same initial and final shares. For instance, if θ rises quickly with the productivity of sector A and stays close to its final value for most of the interval, the real GDP compounds at a higher rate over a longer portion of the interval than if θ stays low for most of the interval and only rises to reach its final value toward the end. As we have seen, in the special case where all the households have identical Cobb-Douglas preferences, the share of sector A in GDP is constant, so we have

\(\begin{aligned} \frac{R(z)}{R(0)} &= \exp\left(\int_0^z\theta(s)\,ds\right)\\ &= \exp\left(z \theta\right)\\ &= \lambda^{\theta}\\ \end{aligned}.\)

In the general case, however, the share of sector A changes as the productivity of sector A rises and how much productivity growth in that sector increases real GDP depends on the whole trajectory.

Even if sector A initially accounts for a small share of GDP, provided that the relative price of B doesn’t respond so strongly that it overwhelms the physical expansion of sector A, its share of GDP can increase enough for productivity growth in sector A to eventually have a very strong impact on real GDP. In that case, for a large value of z, R(z) will be much larger than R(0). If we adopt the spatial interpretation of the equilibrium path, it means that, thanks to the fact that productivity is much higher in sector A, the economy at z on the continuum of economies described by a(s) is much larger in real terms than the economy at the reference point even though sector B is no more productive. However, if the relative price of B responds so strongly to the rise of productivity in sector A that it keeps the share of that sector in GDP low, productivity growth in sector A never have a large effect on real GDP even once the productivity of sector A has risen enormously. For instance, if sector A only accounts for 1% of GDP at the reference point and that share remains constant as productivity in sector A rises, then even a thousandfold productivity rise in sector A will only increase real GDP by a factor of 1000⁰ᐧ⁰¹ ≈ 1.071. This shows that, although productivity growth in one sector is transmitted to the rest of the economy, it will only have a large impact on the overall size of the economy if that sector accounts for a large share of GDP or physical expansion in that sector eventually increases its share of GDP enough. Often this will not happen because, past a certain point, households will become satiated with the good produced by that sector, so their income elasticity of demand for it will fall and they will no longer respond to a rise of the relative price of the other goods by substituting away from them toward the good produced by the sector whose productivity is rising a lot.

The real wage elasticities

As we have seen, since the economy is assumed to be competitive and workers are therefore paid at their marginal productivity (which in the model is the same as their average productivity), nominal wages are

\(w_A = p_Aa\text{ and }w_B = p_Ba.\)

Real wages are obtained by deflating the nominal wages by the chained aggregate Divisia index, so we have

\(w^*_A = \frac{p_Aa}{P}\text{ and }w^*_B = \frac{p_Bb}{P}.\)

Therefore, by using the quotient and product elasticity rules, we get

\(\begin{aligned} \epsilon_{w^*_A} &= \epsilon_{p_Aa} - \epsilon_{P}\\ &= [\epsilon_{p_A} + \epsilon_{a}] - [\epsilon_{p_A} + (1 - \theta)\epsilon_q]\\ &= [\epsilon_{p_A} + 1] - [\epsilon_{p_A} + (1 - \theta)\epsilon_q]\\ &= 1 - (1 - \theta)\epsilon_q \end{aligned}.\)

Thus, if q rises as the productivity of sector A increases, the nominal wages of workers in that sector increase with productivity, but they lose some purchasing power because good B becomes relatively more expensive. On the contrary, if q falls as productivity in sector A rises, the declining relative price of B reinforces the increase of their nominal wages due to productivity growth in sector A. Meanwhile, since b is fixed and

\(\epsilon_{p_B} = \epsilon_{p_A} + \epsilon_q,\)

the quotient and product elasticity rules yields

\(\begin{aligned} \epsilon_{w^*_B} &= \epsilon_{p_Bb} - \epsilon_{P}\\ &= [\epsilon_{p_B} + \epsilon_{b}] - [\epsilon_{p_A} + (1 - \theta)\epsilon_q]\\ &= [\epsilon_{p_A} + \epsilon_q] - [\epsilon_{p_A} + (1 - \theta)\epsilon_q]\\ &= \theta\epsilon_q \end{aligned}.\)

As q changes in response to the rise of productivity in sector A, the nominal wages of workers in sector B change one-for-one with the relative price of B after removing the common nominal price movement, but the cost of living decreases with q in proportion to the share of B in total consumption after removing the common nominal price movement. Hence, their real income changes with the relative price of B in proportion to the remaining expenditure share, which is equal to the share of A. So what is happening is that, although real GDP only grows because of productivity growth in sector A, the gains resulting from that productivity growth don’t just increase wages in that sector, but are partly transmitted to sector B through the change in relative prices that must happen to keep supply and demand equal. This is precisely the demand-side channel that I set out to study in this article.

Finally, let’s consider the response to a finite change in the productivity of sector A, by looking at what happens as it rises from a(0) = a₀ to a(z) = a₀λ with λ = eᶻ and z = log λ. In that case, we have

\(\frac{w^*_A(z)}{w^*_A(0)} = \exp\left(\int_0^z\left[1 - [1 - \theta(s)]\epsilon_q(s)\right]ds\right)\)

and

\(\frac{w^*_A(z)}{w^*_A(0)} = \exp\left(\int_0^z \theta(s)\epsilon_q(s)\,ds\right).\)

Now, since

\(w^*_AL_A = \frac{p_AY_A}{P}, R = \frac{p_AY_A + p_BY_B}{P}\text{ and }\theta = \frac{p_AY_A}{p_AY_A + p_BY_B},\)

we have

\(w^*_AL_A = \theta R\text{ and }w^*_A = \frac{\theta R}{L_A}.\)

Therefore, since the quantity of A-labor is fixed, it follows that

\(\begin{aligned} \frac{w^*_A(z)}{w^*_A(0)} &= \frac{\frac{\theta(z) R(z)}{L_A}}{\frac{\theta(0) R(0)}{L_A}}\\ &= \frac{R(z)}{R(0)}\frac{\theta(z)}{\theta(0)} \end{aligned}.\)

By the same argument, since

\(w^*_BL_B = \frac{p_BY_B}{P}\)

and the quantity of B-labor is fixed, we have

\(\frac{w^*_B(z)}{w^*_B(0)} = \frac{R(z)}{R(0)}\frac{1 - \theta(z)}{1 - \theta(0)}.\)

So the response of the real wage of each sector to a finite change in the productivity of sector A is equal to the product of the response of the real GDP and the response of the share of that sector. We have seen above how those quantities were determined by the other parameters of the model.

Conclusion

Plumbers and workers in similar occupations aren’t much better at fixing toilets if at all in rich countries than in low-income ones and they aren’t better at it than plumbers in rich countries were a few decades ago. In the physical sense of productivity, plumbers may be slightly more productive in rich countries because they have better equipment and they’re better managed, but that’s unlikely to explain the enormous differences in how much they make relative to plumbers in poor countries. While competition for labor, i. e. the supply-side channel that most people invoke to explain phenomena such as the Balassa-Samuelson effect or Baumol’s cost disease, is arguably part of the story, it’s also dubious that it’s the whole story. But even if that supply-side channel didn’t operate at all, plumbers would still be paid more in rich countries because higher productivity in the rest of the economy is transmitted to the plumbing sector through higher demand for plumbing services, which raises their relative price. This demand-side mechanism can result in higher wages for plumbers in rich countries even if productivity in the physical sense, i. e. how many toilets a plumber can fix in one hour and how well he can do it, is exactly the same as in poor ones and even if plumbers had no way to move to another, better-paid occupation.

Although the physical productivity of plumbers may not be higher in rich countries than in low-income economies, their value productivity is, because value productivity doesn’t just depend on physical productivity but also on prices and the demand-side mechanism I described in this article raises the price of plumbing services. They don’t produce more in the same amount of time, but the value of what they produce is higher, because their services command a higher price on the market due to the fact that people in other sectors do produce more in the same amount of time that people in the same occupations in poor countries or they produce goods and services that people in other countries want but can’t make at all. This in turn is because, in a competitive economy, higher productivity in one sector is transmitted to the rest of the economy through both the demand-side mechanism I have examined in this article and the supply-side mechanism that economists usually emphasize when they talk about the Balassa-Samuelson effect or Baumol’s cost disease. This is important because it explains why productivity growth, even though it happens unequally in the economy, benefits everyone including people in sectors where productivity is stagnant. In other words, although people often oppose the market to redistribution, there is actually a redistribution mechanism built into the market.

This redistribution can occur through the supply-side channel that is usually invoked to explain the Balassa-Samuelson/Baumol effect, employers in sectors where productivity is stagnant compete with sectors where it increases for workers, but as we have seen it can also happen through a demand-side channel and that mechanism is arguably more important despite the fact that for some reason economists almost never talk about it. It’s also worth pointing out that, at no point in this article, the distinction between tradables and non-tradables or services and manufacturing played any role. The story doesn’t have anything to do with trade or the opposition between services and manufacturing fundamentally. It’s about how uneven productivity growth is transmitted across sectors. In fact, I think invoking the distinction between tradables and non-tradables or services and manufacturing is often more confusing than helpful, because it encourages the idea that countries only become rich by increasing the productivity of their tradables or manufacturing sector, when in fact rich countries also tend to have a much more productive service sector and improving the productivity of that sector is very important for development because it employs a large share of the labor force and accounts for a large share of GDP.

1. Once again, as Baumol himself made clear, he doesn’t need to assume that wages are exactly the same in both sectors. The effect he identified can still be shown to exist even if labor mobility across sectors is imperfect, allowing for differences in wages, but the mechanism Baumol highlighted nevertheless required some labor mobility, resulting in a degree of wage equalization.

2. Of course, the assumption that the economy is competitive in the relevant sense is doing a lot of work here and, as many people often point out, is a very strong assumption that no real world economy satisfies. However, this doesn’t really matter to the point I make in this post, because the mechanism I discuss here could also be shown to operate if we relaxed that assumption and making it just simplifies the exposition.

3. The equality is only approximate because the elasticity is a local rate of change at the point where it’s evaluated and, since Δlog v is finite, this rate of change will not in general remain exactly constant over that interval even if it’s small.

4. The fact that we’re interested in the elasticities of variables in the model with respect to a, by the way, shows why it’s convenient to parametrize the productivity of sector A as a₀eˢ. Indeed, this implies that

\(s(a) = \log a - \log a_0,\)

so it follows that

\(\frac{d\log x}{d\log a} = \frac{a}{x}\frac{dx}{da} = \frac{a}{x}\frac{dx}{ds}\frac{ds}{da} = \frac{a}{x}\frac{dx}{ds}\frac{1}{a} = \frac{1}{x}\frac{dx}{ds} = \frac{d\log x}{ds}.\)

In other words, when s is measured in log-productivity units because a(s) = a₀eˢ, the ordinary derivative of log x with respect to s is already an elasticity with respect to a. If we had indexed productivity by time instead, writing a(s) = a₀eᵍᵗ (so that g would be the growth rate of a over time), the same argument would have shown that

\(\frac{d\log x}{d\log a} = \frac{1}{g} \frac{d\log x}{dt}.\)

Thus, in order to recover the elasticity of x with respect to a, it would have been necessary to divide the ordinary derivative of log x with respect to s by g.

5. In order to see why, first note that

\(\log\bigl(\frac{x(s)}{x(0)}\bigr) = \log x(s) - \log x(0) = \int_{0}^{\log \lambda} \epsilon_x(s) \,ds,\)

hence

\(\frac{x(s)}{x(0)} = e^{\int_{0}^{\log \lambda} \epsilon_x(s) \,ds}.\)

The result follows by subtracting 1 from both sides.

6. In order to derive that result, start by differentiating h’s budget identity with respect to mₕ, which gives

\(p_A\frac{\partial c_{Ah}}{\partial m_h} + p_B\frac{\partial c_{Bh}}{\partial m_h} = 1\)

Now, each partial derivative on the left-hand side can be rewritten as

\(p_i\frac{\partial c_{ih}}{\partial m_h} = \frac{p_ic_{ih}}{m_h}\frac{m_h}{c_{ih}}\frac{\partial c_{ih}}{\partial m_h} = \omega_{ih}\eta_{ih},\)

which by substituting back into the previous equation yields

\(\omega_{Ah}\eta_{Ah} + \omega_{Bh}\eta_{Bh} = 1.\)

7. Note that the elasticity of substitution between A and B is symmetric in the sense that

\(\frac{\partial\log\bigl(h_{Bh}(p_A,p_B,\bar{u})/h_{Ah}(p_A,p_B,\bar{u})\bigr)}{\partial\log q} = \frac{\partial\log\bigl(h_{Ah}(p_A,p_B,\bar{u})/h_{Bh}(p_A,p_B,\bar{u})\bigr)}{\partial\log q^{'}}\text{ with }q^{'} \equiv \frac{p_A}{p_B},\)

so it doesn’t matter if we define σₕ in terms of the price of B relative to A and the ratio of h’s Hicksian demand for B to its Hicksian demand for A or if we define σₕ in terms of the price of A relative to B and the ratio of h’s Hicksian demand for A to its Hicksian demand for B.

8. The proof of that result is a little bit involved, so I won’t give it here, but if you’re curious you can just ask a LLM.

9. As we have seen, if h’s preferences are homothetic, then

\(E_h(\mathbf{p},u) = G_h(u)E_h(\mathbf{p},1).\)

We have also seen that, when h is utility-maximizing,

\(\bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr) = \bigl(h_{Ah}(p_A,p_B,u),h_{Bh}(p_A,p_B,u)\bigr)\)

and

\(E_h(\mathbf{p},u) = m_h\)

with

\(u = U_h(\mathbf{C})\text{ and }\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr).\)

It follows that

\(G_h\bigl(U_h(\mathbf{C})\bigr) = \frac{m_h}{E(\mathbf{p},1)}.\)

Now, by Shephard’s lemma, the Hicksian demand for good i is

\(h_{ih}(\mathbf{p},u) = \frac{\partial E_h(\mathbf{p}, u)}{\partial p_i}.\)

Now, since Eₕ(𝐩,u) = Gₕ(u)Eₕ(𝐩,1), it follows that

\(h_{ih}(\mathbf{p},u) = G_h(u)\frac{\partial E_h(\mathbf{p}, 1)}{\partial p_i}.\)

But as we have already noted above, if

\(u = U_h(\mathbf{C})\text{ and }\mathbf{C} = \bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr),\)

then

\(\bigl(c_{Ah}(p_A,p_B,m_h),c_{Bh}(p_A,p_B,m_h)\bigr) = \bigl(h_{Ah}(p_A,p_B,u),h_{Bh}(p_A,p_B,u)\bigr),\)

hence

\(c_{ih}(\mathbf{p},m_h) = G_h(U_h\bigl(\mathbf{C})\bigr)\frac{\partial E_h(\mathbf{p}, 1)}{\partial p_i}.\)

Substituting into that equation the expression derived above for Gₕ(Uₕ(𝐂)), we have

\(c_{ih}(\mathbf{p},m_h) = \frac{m_h}{E_h(\mathbf{p},1)}\frac{\partial E_h(\mathbf{p}, 1)}{\partial p_i} = m_h\frac{\partial \log E_h(\mathbf{p}, 1)}{\partial p_i}.\)

It follows that, for any λ > 0,

\(c_{ih}(\mathbf{p},\lambda m_h) = \lambda m_h\frac{\partial \log E_h(\mathbf{p}, 1)}{\partial p_i} = \lambda c_{ih}(\mathbf{p},m_h).\)

In other words, if h’s budget is multiplied by any positive factor, the Marshallian demand of h for any good i will be multiplied by the same factor. Finally, since

\(\eta_{ih} = \frac{\partial \log c_{ih}(\mathbf{p},m_h)}{\partial \log m_h} = \frac{m_h}{c_{ih}(\mathbf{p},m_h)}\frac{\partial c_{ih}(\mathbf{p},m_h)}{\partial \log m_h}\)

and

\(\frac{\partial c_{ih}(\mathbf{p},m_h)}{\partial p_i} = \frac{\partial \log E_h(\mathbf{p}, 1)}{\partial p_i},\)

it follows that

\(\eta_{ih} = \frac{m_h}{m_h\frac{\partial \log E_h(\mathbf{p}, 1)}{\partial p_i}}\frac{\partial \log E_h(\mathbf{p}, 1)}{\partial p_i} = 1.\)

10. In order to appreciate how restrictive that condition is, one can also note that homothetic preferences imply that expenditure shares don’t vary with income. Indeed, as we have just seen, if a household h’s preferences are homothetic then ηᵢₕ = 1 for every good i. Since we have also seen that

\(\omega_{Ah}\eta_{Ah} = 1 - \omega_{Bh}\eta_{Bh},\)

this implies that

\(\omega_{Ah} = 1 - \omega_{Bh}.\)

Moreover, since

\(\omega_{ih}(\mathbf{p}, m_h) = \frac{p_ic_{ih}(\mathbf{p}, m_h)}{m_h},\)

and

\(c_{ih}(\mathbf{p}, m_h) = m_hc_{ih}(\mathbf{p}, 1)\)

when h’s preferences are homothetic, it follows that

\(\omega_{ih}(\mathbf{p}, m_h) = \frac{p_im_hc_{ih}(\mathbf{p}, 1)}{m_h} = p_ic_{ih}(\mathbf{p}, 1)\)

Since this is not a function of mₕ, it means that when a household’s preferences are homothetic, its expenditure shares on each good don’t change with mₕ as long as prices stay constant.

11. Again, I’m not giving the proof here because it’s a bit involved, but if you’re curious you can ask a LLM for it.

12. As we have seen above, when a household’s preferences are homothetic, its expenditure shares on each good don’t change with its budget as long as prices stay constant. This is already wildly implausible, but for household to have the same preferences on top of that, it would have to be the case that every household in the economy allocates exactly the same share of its budget to the same goods.

13. It should be noted that, even when such a universal cost-of-living price index exists, it doesn’t by itself provide a measure of social welfare. Let K(p, p*) be such a universal cost-of-living price index and

\(M(s) = \frac{\sum_{h=1}^{H} m_h(s)}{K\bigl(\mathbf{p}(s), \mathbf{p}(0)\bigr)}\)

be the aggregate income in the economy as it moves along the path a(s) deflated by that index. This aggregate can rise even though the deflated income of some households falls, provided that the deflated income of the other households rise even more. In that case, a universal price index can tell us which households are better off and which households are worse off, but it can’t tell us whether the gains outweigh the losses because utility is a purely ordinal scale.

In other words, the utility levels a household attains with different consumption baskets are just numbers that are ranked in the same way as the baskets the utility function assigns to them, which don’t have any meaning beyond that. As a result, the fact that when deflated by the universal index a household’s income has risen only means that it can afford a consumption basket it prefers to what it could afford before and vice versa when its deflated income has fallen, but it can’t say anything about how intensely h feels the gain or loss it made. Indeed, if Uₕ is a utility function that represents the preferences of a household h, then f ∘ Uₕ is also a utility function that represents its preferences for any strictly increasing function f, so the size of utility change induced by a rise or fall of deflated income will be different depending on what utility function is used to represent h’s preferences.

Similarly, while deflating household incomes with the universal price index whose existence we are positing allows us to express gains and losses in real income units, such as constant dollars, we can’t draw any conclusion from the fact that the gain of a household is greater or less than the loss of another in constant dollars, because this would implicitly assumes that each dollar is valued as much no matter who receives it and there is no basis for such a judgment. It’s only when the deflated income of every household increases, so that everyone is better off, that we can unambiguously conclude that social welfare has improved under the Pareto criterion, but that is not generally the case. This is why, even in the unrealistic case where every household has the same preferences and they are homothetic (so that a universal cost-of-living price index exists), social welfare judgments can’t be made based on the deflated aggregate income alone but additional ethical principles are needed to weight the gains and losses of different households.

14. I omit the proof of that result, but again if you’re curious you can ask a LLM to explain it to you. I should note however that, strictly speaking, this result is restricted to the region ℛₕ ⊆ ℝ³₊, where (p, mₕ) ∈ ℛₕ if and only if cᵢₕ(p, mₕ) > 0 for every good I and the restriction of cᵢₕ to ℛₕ is continuously differentiable with respect to every component of p and with respect to mₕ.

15. On the other hand, if every household has the same preferences and those preferences are homothetic, then the Gorman condition holds. This follows straightforwardly from the fact that, as I already demonstrated above, when h’s preferences are homothetic,

\(c_{ih}(\mathbf{p},\lambda m_h) = \lambda c_{ih}(\mathbf{p},m_h)\)

for every good i and every λ > 0.

16. Indeed, not only is the representative household fiction a sleight of hand that sweeps under the rug the aggregation problem, but it also doesn’t provide any practical guidance on how to construct a price index from real data on consumption to deflate monetary quantities, so we don’t really have a choice but to look elsewhere for such guidance.

17. A fixed-basket index such as the Laspeyres index considers how the price of a fixed basket as the price of the various goods in that basket change, so it doesn’t allow households to substitute away from goods that became relatively more expensive toward goods that became relatively cheaper to mitigate the impact on their utility, unlike a Konüs index. Increasingly, national statistical institutes also use chained indices to mitigate the issues arising from the use of the fixed-basket approach, but this isn’t the place to explain that. I have been thinking of writing a post specifically on price indices and purchasing power parity exchange rates, because I think it’s a fascinating topic in itself and it’s relevant to many debates, so maybe I’ll revisit the issue more at length eventually.

18. To be clear, the model doesn’t actually describe how this adjustment takes place and in that sense is a purely comparative static exercise, but just assumes that a general equilibrium has been reached at each point. There is a lot to be said about this method, which is called “solving for equilibrium” and is standard in economics despite the fact that it’s not so easy to justify epistemologically, but that’s a story for another day.

19. I omit the proof of Shephard’s lemma, but if you’re curious, you ask a LLM for it.

20. I’m assuming that q, the price of B relative to A, will increase in response to a rise in the productivity of sector A because that’s what the model predicts for the parameter regime that generates the demand-side mechanism this post is about, but as we shall see shortly the model doesn’t rule out that in some cases it may fall.

21. I don’t prove the Slutsky identity, which is derived by using Shephard’s lemma, but again you can ask a LLM for it.

22. Note that in the derivation of

\(\frac{d \log c_{ih}}{d s} = \zeta^H_{ih}\frac{d \log q}{d s} + \eta_{ih}\frac{d \log R_h}{d s},\)

where

\(R_h\equiv\frac{m_h}{P_h^D}, \zeta^H_{ih} \equiv \left.\frac{\partial\log h_{ih}}{\partial\log q}\right|_u \text{ and } \eta_{ih} \equiv \left.\frac{\partial\log c_{ih}}{\partial\log m_h}\right|_{p_A,p_B},\)

the variable s played no special mathematical role, so it could be any variable indexing a differentiable path of prices and income. Thus, we may as well write

\(\frac{d \log c_{ih}}{d x} = \zeta^H_{ih}\frac{d \log q}{d x} + \eta_{ih}\frac{d \log R_h}{d x},\)

which is exactly the same equation where s has been replaced by x to make clear that it’s not necessarily the log-change of the productivity in sector A but can be any variable indexing a differentiable path of prices and income.

It’s immediate from the definition of Marshallian demand that, for any λ > 0,

\(c_{ih}(\lambda p_A,\lambda p_B,\lambda m_h) = c_{ih}(p_A,p_B,m_h).\)

Thus, setting

\(\lambda = \frac{1}{p_A},\)

we have

\(c_{ih}(p_A,p_B,m_h) = c_{ih}\left(1,q,\frac{m_h}{p_A}\right).\)

Thus, if we define

\(r_h \equiv \frac{m_h}{p_A},\)

we can write the Marshallian demand of h for i as a function of q and rₕ.

As we have already seen, again I proved this result with respect to the variable s but the nature of the index didn’t play any role in the proof, we also have

\(\frac{d \log P^D_h}{d s} = \omega_{Ah}\frac{d\log p_A}{dx} + \omega_{Bh}\frac{d\log p_B}{dx}.\)

We therefore have

\(\begin{aligned} \frac{d\log R_h}{dx} &= \frac{d\log \frac{m_h}{P^D_h}}{dx}\\ &= \frac{d\log m_h}{dx} - \frac{d\log P^D_h}{dx}\\ &= \frac{d\log m_h}{dx} - \omega_{Ah}\frac{d\log p_A}{dx} - \omega_{Bh}\frac{d\log p_B}{dx} \end{aligned}.\)

Now, since

\(\log r_h = \log m_h - \log p_A\)

and

\(\log q=\log p_B - \log p_A,\)

we have

\(\frac{d\log m_h}{dx} = \frac{d\log r_h}{dx} + \frac{d\log p_A}{dx}\)

and

\(\frac{d\log p_B}{dx} = \frac{d\log q}{dx} + \frac{d\log p_A}{dx}.\)

Thus, by substitution, we have

\(\begin{aligned} \frac{d \log R_h}{dx} &= \frac{d\log r_h}{d x} + \frac{d \log p_A}{dx} - \omega_{Ah}\frac{d \log p_A}{d x} -\omega_{Bh} \bigl(\frac{d \log q}{d x} + \frac{d \log p_A}{d x}\bigr)\\ &= \frac{d \log r_h}{d x} -\omega_{Bh}\frac{d \log q}{d x} \end{aligned},\)

where I have used the fact that h’s expenditure shares must sum to 1.

If q is held fixed, we have

\(\left.\frac{d \log q}{d x}\right|_q = 0,\)

so the previous equation becomes

\(\left.\frac{d \log R_h}{d x}\right|_q = \left.\frac{d \log r_h}{d x}\right|_q.\)

Thus, at fixed q, the change in the logarithm of Rₕ is locally equal to the change in the logarithm of rₕ. Now, as we have seen above, the Marshallian demand of h for i can be written as a function of q and rₕ. It follows that, at any point along the equilibrium path of the economy, the Marshallian demand of h for i can be written as a function of q and Rₕ when the analysis concerns local elasticities.

Let (q̅, R̅ₕ) be some equilibrium, then define the path

\(q(x) = \bar{q} e^x, R_h(x)=\bar{R}_h,\)

along which Rₕ is held constant but q is allowed to vary. When q is parametrized in that way, we have

\(x = \log \frac{q(x)}{q(0)} = \log \bigl(q(x) - q(0)\bigr),\)

so x measures the change in the logarithm of q along the path. Moreover,

\(\log q(x)=\log\bar q+x,\)

hence

\(\left.\frac{d \log q}{d x} \right|_{x = 0} = 1.\)

Moreover, since Rₕ is constant,

\(\left.\frac{d \log R_h}{d x}\right|_{x = 0} = 0.\)

Therefore, by applying the chain rule, we get

\(\begin{aligned} \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0} &= \left.\frac{\partial \log c_{ih}}{\partial \log q}\right|_{R_h}\left.\frac{d \log q}{d x}\right|_{x = 0} + \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q\left. \frac{d \log R_h}{d x}\right|_{x = 0}\\ &= \left.\frac{\partial \log c_{ih}}{\partial\log q}\right|_{R_h}(1) + \left.\frac{\partial \log c_{ih}}{\partial \log R_h} \right|_q(0) \end{aligned},\)

so

\(\left.\frac{\partial \log c_{ih}}{\partial \log q} \right|_{R_h} = \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0}. \)

Using the identity we started with, we get

\(\begin{aligned} \left.\frac{\partial \log c_{ih}}{\partial \log q}\right|_{R_h} &= \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0}\\ &= \left.\left[\zeta^H_{ih}\frac{d \log q}{dx} + \eta_{ih}\frac{d \log R_h}{d x}\right]\right|_{x = 0}\\ &= \zeta^H_{ih} \left.\frac{d (\log\bar q + x)}{d x}\right|_{x = 0} + \eta_{ih}\left.\frac{d \log\bar R_h}{d x}\right|_{x = 0}\\ &= \zeta^H_{ih}(1) + \eta_{ih}(0)\\ &= \zeta^H_{ih} \end{aligned}.\)

Thus, at any point along the equilibrium path of the economy, the elasticity of the Marshallian demand of h for i with respect to q when Rₕ is held constant is equal to the elasticity of Hicksian demand of h for i with respect to q when utility is held constant at the value h attains at that equilibrium. This makes sense when you consider that, at equilibrium, the utility maximization problem solved by Marshallian demand coincides with the expenditure minimization problem solved by Hicksian demand and the utility attained by a household is determined by its purchasing power at that equilibrium. By holding the household’s purchasing power constant, we are in effect requiring that nominal income change at the same rate locally as the household’s Divisia price index, which is exactly the compensation required to preserve utility at that point.

Now define the path

\(q(x) = \bar{q}, R_h(x)=\bar{R}_he^x,\)

along which q is held constant but Rₕ is allowed to vary. When Rₕ is parametrized in that way, we have

\(x = \log \frac{R_h(x)}{R_h(0)} = \log \bigl(R_h(x) - R_h(0)\bigr),\)

so x measures the change in h’s purchasing power along the path. Along this path,

\(\left.\frac{d \log q}{d x}\right|_{x = 0} = 0,\)

while

\(\begin{aligned} \left.\frac{d \log R_h(x)}{d x}\right|_{x = 0} &= \left.\frac{d(\log \bar{R}_h + x)}{d x} \right|_{x = 0}\\ & = 1 \end{aligned}.\)

Thus, by the chain rule, we have

\(\begin{aligned} \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0} &= \left.\frac{\partial \log c_{ih}}{\partial \log q}\right|_{R_h}(0) + \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q(1)\\ &= \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q \end{aligned},\)

so

\(\left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q = \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0}.\)

Then, using the identity we started with, we get

\(\begin{aligned} \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q &= \left.\frac{d \log c_{ih}}{d x}\right|_{x = 0}\\ &= \left.\left[\zeta^H_{ih}\frac{d \log q}{d x} + \eta_{ih}\frac{d \log R_h}{d x}\right]\right|_{x = 0}\\ &= \zeta^H_{ih}\left.\frac{d \log\bar{q}}{d x}\right|_{x = 0} + \eta_{ih} \left.\frac{d(\log \bar{R}_h + x)}{d x}\right|_{x = 0}\\ &= \zeta^H_{ih}(0) + \eta_{ih}(1)\\ &= \eta_{ih} \end{aligned}.\)

Thus, at any point along the equilibrium path of the economy, the elasticity of the Marshallian demand of h for i with respect to h’s deflated income when the price of B relative to A is held constant is equal to the elasticity of h’s Marshallian demand for i with respect to nominal income at that equilibrium when prices are held constant. This makes sense given that, when prices are held constant, a change in the nominal income of h is equivalent to a change in h’s deflated income when the price of B relative to A is held constant.

Now, if we substitute

\(\zeta^H_{ih} = \left.\frac{\partial \log c_{ih}}{\partial \log q}\right|_{R_h}\)

and

\(\eta_{ih} = \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q\)

into

\(\frac{d \log c_{ih}}{d s} = \zeta^H_{ih}\frac{d \log q}{d s} + \eta_{ih}\frac{d \log R_h}{d s},\)

we obtain

\(\frac{d \log c_{ih}}{d s} = \left.\frac{\partial \log c_{ih}}{\partial \log q}\right|_{R_h}\frac{d \log q}{d s} + \left.\frac{\partial \log c_{ih}}{\partial \log R_h}\right|_q\frac{d \log R_h}{d s}.\)

What this shows is that the formula we derived in the text for the elasticity of Marshallian demand with respect to the productivity of sector A is simply the decomposition of demand growth into a relative price response and a purchasing power response one gets by applying the multivariable chain rule to the Marshallian demand when it’s treated as a function of q and Rₕ.

23. For instance, this is arguably what happened with domestic cooks, whose services became relatively so expensive that everyone but the richest people substituted away from them services toward kitchen appliances and the occasional restaurant meal, whereas in the past and even today in poorer countries it was very common even for middle class people to have a domestic cook.

24. As we have already seen,

\(\frac{d \log c_{ih}}{d x} = \zeta^H_{ih}\frac{d \log q}{d x} + \eta_{ih}\frac{d \log \frac{m_h}{P^D_h}}{d x}\)

where x is any variable indexing a differentiable path of prices and income. By the elasticity rule on sums, we have

\(\frac{d \log m_h}{d x} = \alpha_{Ah} \left( \frac{d \log p_A}{d x} + \frac{d \log a}{d x} \right) + \alpha_{Bh}\frac{d\log p_B}{d x}.\)

Since

\(p_B = p_Aq\)

by definition of q, we also have

\(\frac{d \log p_B}{d x} = \frac{d \log p_A}{d x} + \frac{d \log q}{d x},\)

hence

\(\frac{d \log m_h}{d x} = \frac{d \log p_A}{d x} + \alpha_{Ah}\frac{d \log a}{d x} + \alpha_{Bh}\frac{d \log q}{d x}.\)

Since as we have already seen

\(\frac{d \log P_h^D}{d x} = \omega_{Ah}\frac{d \log p_A}{d x} + \omega_{Bh}\frac{d \log p_B}{d x},\)

by the same token,

\(\begin{aligned} \frac{d \log P_h^D}{d x} &= \omega_{Ah}\frac{d \log p_A}{d x} + \omega_{Bh} \left( \frac{d \log p_A}{d x} + \frac{d \log q}{d x} \right)\\ &= \frac{d \log p_A}{d x} + \omega_{Bh}\frac{d \log q}{d x} \end{aligned}.\)

Now, since

\(\frac{d \log \frac{m_h}{P^D_h}}{d x} = \frac{d \log m_h}{d x} - \frac{d \log P^D_h}{d x},\)

it follows that

\(\frac{d \log \frac{m_h}{P^D_h}}{d x} = \alpha_{Ah}\frac{d \log a}{d x} + (\alpha_{Bh} - \omega_{Bh}) \frac{d \log q}{d x}.\)

Therefore, by substituting into the identity we started with, we get

\(\frac{d \log c_{Bh}}{d x} = \eta_{Bh}\alpha_{Ah}\frac{d \log a}{d x} + \bigl(\zeta^H_{Bh} + \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr)\frac{d \log q}{d x}.\)

We are now going to use that identity to derive the result in the text.

Let

\((\bar{a}, \bar{p}_A, \bar{p}_B)\text{ with }\bar{q} = \frac{\bar{p}_B}{\bar{p}_A}\)

be some equilibrium, then define the path

\(a(x) = \bar{a}e^x, p_A(x) = \bar{p}_A, p_B(x) = \bar{p}_B,\)

along which the productivity of sector A varies but the prices of A and B is held constant and therefore so is q. In that case, we have

\(\frac{d \log q}{d x} = \frac{d \log p_B}{d x} - \frac{d \log p_A}{d x} = 0\)

and

\(\frac{d a}{d x} = a,\)

so

\(\frac{d \log a}{d x} = \frac{1}{a}\frac{d a}{d x} = 1.\)

Thus, substituting into the identify derived previously, we get

\(\frac{d \log c_{Bh}}{d x} = \eta_{Bh}\alpha_{Ah}.\)

Moreover, since along the path defined above only a varies, we also have

\(\begin{aligned} \frac{d \log c_{Bh}}{d x} &= \frac{1}{c_{Bh}}\frac{d c_{Bh}}{d x}\\ &= \frac{1}{c_{Bh}} \left.\frac{\partial c_{Bh}}{\partial a}\right|_{p_A,p_B}\frac{d a}{d x}\\ &= \frac{a}{c_{Bh}} \left.\frac{\partial c_{Bh}}{\partial a} \right|_{p_A,p_B}\\ &= \left.\frac{\partial \log c_{Bh}}{\partial \log a}\right|_{p_A,p_B} \end{aligned}.\)

It follows that

\(\left.\frac{\partial \log c_{Bh}}{\partial \log a}\right|_{p_A,p_B} = \eta_{Bh}\alpha_{Ah}.\)

Using the elasticity rule on sums, we obtain

\(\begin{aligned} \left.\frac{\partial \log D_B}{\partial \log a} \right|_{p_A,p_B} &= \sum_h \frac{c_{Bh}}{D_B}\left.\frac{\partial \log c_{Bh}}{\partial \log a}\right|_{p_A,p_B}\\ &= \sum^H_h \nu_{Bh}\eta_{Bh}\alpha_{Ah} \end{aligned},\)

which completes the proof for the numerator.

Now let’s define the path

\(a(x) = \bar{a}, q(x) = \bar{q}e^x,\)

along which the productivity of sector A is fixed but the price of B relative to A is allowed to vary. In that case, we have

\(\frac{d q}{d x} = q,\)

so

\(\frac{d \log q}{d x} = \frac{1}{q}\frac{d q}{d x} = 1.\)

Moreover, since a is fixed,

\(\frac{d \log a}{d x} = 0.\)

Therefore, substituting into the identity derived previously, we have

\(\frac{d \log c_{Bh}}{d x} = \zeta^H_{Bh} + \eta_{Bh}(\alpha_{Bh} - \omega_{Bh}).\)

Moreover, since along the path defined above only q varies, we also have

\(\begin{aligned} \frac{d \log c_{Bh}}{d x} &= \frac{1}{c_{Bh}}\frac{d c_{Bh}}{d x}\\ &= \frac{1}{c_{Bh}}\left.\frac{\partial c_{Bh}}{\partial q}\right|_a\frac{d q}{d x}\\ &= \frac{q}{c_{Bh}}\left.\frac{\partial c_{Bh}}{\partial q}\right|_a\\ &= \left.\frac{\partial \log c_{Bh}}{\partial \log q}\right|_a \end{aligned}.\)

It follows that

\(\left.\frac{\partial \log c_{Bh}}{\partial \log q}\right|_a = \zeta^H_{Bh} + \eta_{Bh}(\alpha_{Bh} - \omega_{Bh}).\)

As we have seen,

\(\zeta^H_{Bh} = -\omega_{Ah}\sigma_h,\)

so

\(\left.\frac{\partial \log c_{Bh}}{\partial \log q}\right|_a = -\omega_{Ah}\sigma_h + \eta_{Bh}(\alpha_{Bh} - \omega_{Bh}).\)

Using the elasticity rule on sums, we obtain

\(\begin{aligned} \left.\frac{\partial \log D_B}{\partial \log q}\right|_a &= \sum_h\nu_{Bh}\left[-\omega_{Ah}\sigma_h + \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\right]\\ &= -\sum^H_h \nu_{Bh}\bigl(\omega_{Ah}\sigma_h - \eta_{Bh}(\alpha_{Bh} - \omega_{Bh})\bigr) \end{aligned},\)

which completes the proof for the denominator.

25. By saying that the productivity of sector A rises from a(0) to a(z), I’m suggesting a temporal interpretation of the equilibrium path, but again we can also interpret a(s) spatially, in which case a(0) and a(s) are just two economies with different levels of productivity in sector A that are otherwise identical.