Learning Objectives
- What happens in a general equilibrium when there are more than two people buying more than two goods?
- Does the Cobb-Douglas case provide insight?
We will illustrate general equilibrium for the case when all consumers have Cobb-Douglas utility in an exchange economy. An exchange economy is an economy where the supply of each good is just the total endowment of that good, and there is no production. Suppose that there are N people, indexed by \(n = 1, 2, … , N\) There are G goods, indexed by \(g=1,2, \ldots, G\). Person n has Cobb-Douglas utility, which we can represent using exponents α(n, g), so that the utility of person n can be represented as \(\Pi g=1 G \times(n, g) a(n, g)\), where x(n, g) is person n’s consumption of good g. Assume that \(a(n, g) \geq 0\) for all n and g, which amounts to assuming that the products are, in fact, goods. Without any loss of generality, we can require \(\sum g=1 G a(n, g)=1\) for each n. (To see this, note that maximizing the function U is equivalent to maximizing the function Uβ for any positive β.)
Let y(n, g) be person n’s endowment of good g. The goal of general equilibrium is to find prices p1, p2, … , pG for the goods in such a way that demand for each good exactly equals supply of the good. The supply of good g is just the sum of the endowments of that good. The prices yield a wealth for person n equal to W n = ∑ g=1 G p g y(n,g) .
We will assume that \(\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{y}(\mathrm{n}, \mathrm{i})>0\) for every pair of goods g and i. This assumption states that for any pair of goods, there is at least one agent that values good g and has an endowment of good i. The assumption ensures that there is always someone who is willing and able to trade if the price is sufficiently attractive. The assumption is much stronger than necessary but useful for exposition. The assumption also ensures that the endowment of each good is positive.
The Cobb-Douglas utility simplifies the analysis because of a feature that we already encountered in the case of two goods, which holds, in general, that the share of wealth for a consumer n on good g equals the exponent α(n, g). Thus, the total demand for good g is \(\mathrm{X} \mathrm{g}=\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{W} \mathrm{n} \mathrm{p} \mathrm{g}\).
The equilibrium conditions, then, can be expressed by saying that supply (sum of the endowments) equals demand; or, for each good g, \(\mathrm{X} \mathrm{g}=\Sigma \mathrm{n}=1 \mathrm{N} \mathrm{a}(\mathrm{n}, \mathrm{g}) \mathrm{W} \mathrm{n} \mathrm{p} \mathrm{g}\)
We can rewrite this expression, provided that pg > 0 (and it must be, for otherwise demand is infinite), to be
\[p g-\Sigma i=1 G p i \Sigma n=1 N y(n, i) a(n, g) \Sigma n=1 N y(n, g)=0 \nonumber \]
Let B be the G × G matrix whose (g, i) term is b \(g i=\sum n=1 N y(n, i) a(n, g) \sum n=1 N y(n, g)\)
Let p be the vector of prices. Then we can write the equilibrium conditions as (I – B) p = 0, where 0 is the zero vector. Thus, for an equilibrium (other than p = 0) to exist, B must have an eigenvalue equal to 1 and a corresponding eigenvector p that is positive in each component. Moreover, if such an eigenvector–eigenvalue pair exists, it is an equilibrium, because demand is equal to supply for each good.
The actual price vector is not completely identified because if p is an equilibrium price vector, then so is any positive scalar times p. Scaling prices doesn’t change the equilibrium because both prices and wealth (which is based on endowments) rise by the scalar factor. Usually economists assign one good to be a numeraire, which means that all other goods are indexed in terms of that good; and the numeraire’s price is artificially set to be 1. We will treat any scaling of a price vector as the same vector.
The relevant theorem is the Perron-Frobenius theorem.Oskar Perron (1880–1975) and Georg Frobenius (1849–1917). It states that if B is a positive matrix (each component positive), then there is an eigenvalue λ > 0 and an associated positive eigenvector p; and, moreover, λ is the largest (in absolute value) eigenvector of B.The Perron-Frobenius theorem, as usually stated, only assumes that B is nonnegative and that B is irreducible. It turns out that a strictly positive matrix is irreducible, so this condition is sufficient to invoke the theorem. In addition, we can still apply the theorem even when B has some zeros in it, provided that it is irreducible. Irreducibility means that the economy can’t be divided into two economies, where the people in one economy can’t buy from the people in the second economy because they aren’t endowed with anything that the people in the first economy value. If B is not irreducible, then some people may wind up consuming zero of things they value. This conclusion does most of the work of demonstrating the existence of an equilibrium. The only remaining condition to check is that the eigenvalue is in fact 1, so that (I – B) p = 0.
Suppose that the eigenvalue is λ. Then λp = Bp. Thus for each g,
\[λ p g = ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) ∑ m=1 N y(m,g) p i \nonumber \]
or
\[λ p g ∑ m=1 N y(m,g) = ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) p i . \nonumber \]
Summing both sides over g,
\[λ ∑ g=1 G p g ∑ m=1 N y(m,g) = ∑ g=1 G ∑ i=1 G ∑ n=1 N α(n,g)y(n,i) p i= ∑ i=1 G ∑ n=1 N ∑ g=1 G α(n,g) y(n,i) p i = ∑ i=1 G ∑ n=1 N y(n,i) p i . \nonumber \]
Thus, λ = 1 as desired.
The Perron-Frobenius theorem actually provides two more useful conclusions. First, the equilibrium is unique. This is a feature of the Cobb-Douglas utility and does not necessarily occur for other utility functions. Moreover, the equilibrium is readily approximated. Denote by Bt the product of B with itself t times. Then for any positive vector v, \(\lim t \rightarrow \infty \text { B } t v=p\). While approximations are very useful for large systems (large numbers of goods), the system can readily be computed exactly with small numbers of goods, even with a large number of individuals. Moreover, the approximation can be interpreted in a potentially useful manner. Let v be a candidate for an equilibrium price vector. Use v to permit people to calculate their wealth, which for person n is \(\mathrm{W} \mathrm{n}=\Sigma \mathrm{i}=1 \mathrm{G} \mathrm{v} \text { i } \mathrm{y}(\mathrm{n}, \mathrm{i})\). Given the wealth levels, what prices clear the market? Demand for good g is \(x g (v)= ∑ n=1 N α(n,g) W n = ∑ i=1 G v i ∑ n=1 N α(n,g)y(n,i) \), and the market clears, given the wealth levels, if \(p g = ∑ i=1 G v i ∑ n=1 N α(n,g)y(n,i) ∑ n=1 N y(n,g)\), which is equivalent to p = Bv. This defines an iterative process. Start with an arbitrary price vector, compute wealth levels, and then compute the price vector that clears the market for the given wealth levels. Use this price to recalculate the wealth levels, and then compute a new market-clearing price vector for the new wealth levels. This process can be iterated and, in fact, converges to the equilibrium price vector from any starting point.
We finish this section by considering three special cases. If there are two goods, we can let αn = αα(n, 1), and then conclude that α(n, 2) = 1 – an. Then let Y g = ∑ n=1 N y(n,g) be the endowment of good g. Then the matrix B is
\[B=( 1 Y 1 ∑ n=1 N y(n,1) a n 1 Y 1 ∑ n=1 N y(n,2) a n 1 Y 2 ∑ n=1 N y(n,1)(1− a n ) 1 Y 2 ∑ n=1 N y(n,2)(1− a n ) )=( 1 Y 1 ∑ n=1 N y(n,1) a n 1 Y 1 ∑ n=1 N y(n,2) a n 1 Y 2 ( Y 1 − ∑ n=1 N y(n,1) a n ) 1− 1 Y 2 ∑ n=1 N y(n,2) a n ). \nonumber \]
The relevant eigenvector of B is \(p=( ∑ n=1 N y(n,2) a n ∑ n=1 N y(n,1)(1− a n ) ) .\)
The overall level of prices is not pinned down—any scalar multiple of p is also an equilibrium price—so the relevant term is the price ratio, which is the price of Good 1 in terms of Good 2, or
\[p 1 p 2 = ∑ n=1 N y(n,2) a n ∑ n=1 N y(n,1)(1− a n ) . \nonumber \]
We can readily see that an increase in the supply of Good 1, or a decrease in the supply of Good 2, decreases the price ratio. An increase in the preference for Good 1 increases the price of Good 1. When people who value Good 1 relatively highly are endowed with a lot of Good 2, the correlation between preference for Good 1, an, and endowment of Good 2 is higher. The higher the correlation, the higher is the price ratio. Intuitively, if the people who have a lot of Good 2 want a lot of Good 1, the price of Good 1 is going to be higher. Similarly, if the people who have a lot of Good 1 want a lot of Good 2, the price of Good 1 is going to be lower. Thus, the correlation between endowments and preferences also matters to the price ratio.
In our second special case, we consider people with the same preferences but who start with different endowments. Hypothesizing identical preferences sets aside the correlation between endowments and preferences found in the two-good case. Since people are the same, α(n, g) = Ag for all n. In this case, \( b gi = ∑ n=1 N y(n,i)α(n,g) ∑ n=1 N y(n,g) = A g Y i Y g\), whereas before \(Y g = ∑ n=1 N y(n,g)\) is the total endowment of good g. The matrix B has a special structure, and in this case, \(p g=A g Y g\) is the equilibrium price vector. Prices are proportional to the preference for the good divided by the total endowment for that good.
Now consider a third special case, where no common structure is imposed on preferences, but endowments are proportional to each other; that is, the endowment of person n is a fraction wn of the total endowment. This implies that we can write \(\mathrm{y}(\mathrm{n}, \mathrm{g})=\mathrm{w}_{\mathrm{n}} \mathrm{Y}_{\mathrm{g}}\), an equation assumed to hold for all people n and goods g. Note that by construction, \(∑ n=1 N w n =1\), since the value wn represents n’s share of the total endowment. In this case, we have
\[b gi = ∑ n=1 N y(n,i)α(n,g) ∑ n=1 N y(n,g) = Y i Y g ∑ n=1 N w n α(n,g) \nonumber \]
These matrices also have a special structure, and it is readily verified that the equilibrium price vector satisfies \(p g = 1 Y g ∑ n=1 N w n α(n,g) .\)
This formula receives a similar interpretation—the price of good g is the strength of preference for good g, where strength of preference is a wealth-weighted average of the individual preference, divided by the endowment of the good. Such an interpretation is guaranteed by the assumption of Cobb-Douglas preferences, since these imply that individuals spend a constant proportion of their wealth on each good. It also generalizes the conclusion found in the two-good case to more goods, but with the restriction that the correlation is now between wealth and preferences. The special case has the virtue that individual wealth, which is endogenous because it depends on prices, can be readily determined.
Key Takeaways
- General equilibrium puts together consumer choice and producer theory to find sets of prices that clear many markets.
- For the case of an arbitrary number of goods and an arbitrary number of consumers—each with Cobb-Douglas utility—there is a closed form for the demand curves, and the price vector can be found by locating an eigenvector of a particular matrix. The equilibrium is unique (true for Cobb-Douglas but not true more generally).
- The actual price vector is not completely identified because if p is an equilibrium price vector, then so is any positive scalar times p. Scaling prices doesn’t change the equilibrium because both prices and wealth (which is based on endowments) rise by the scalar factor.
- The intuition arising from one-good models may fail because of interactions with other markets—increasing preferences for a good (shifting out demand) changes the values of endowments in ways that then reverberate through the system.
EXERCISE
- Consider a consumer with Cobb-Douglas utility \( U= ∏ i=1 G x i a i\),where \(\Sigma i=1 \mathrm{G} \mathrm{a} \mathrm{i}=1\) and facing the budget constraint \(W= ∑ i=1 G p i x i\).Show that the consumer maximizes utility by choosing \(x i = a i W p i\) for each good i. (Hint: Express the budget constraint as \(x G = 1 p G ( W− ∑ i=1 G−1 p i x i )\),and thus utility as \(U=( ∏ i=1 G−1 x i a i ) ( 1 p G ( W− ∑ i=1 G−1 p i x i ) ) a G.)\) This function can now be maximized in an unconstrained fashion. Verify that the result of the maximization can be expressed as \(p i x i = a i a G p G x G\), and thus \(W= ∑ i=1 G p i x i = ∑ i=1 G a i a G p G x G = p G x G a G \),which yields \(p i x i = a i W \)
