The first two features of general equilibrium models that we presented in this chapter were technical. They are of some help in computing equilibria, but taken for themselves they do not provide any deep new insights that could be applied to the real world. The situation is different with the last feature that we are going to address, the efficiency of outcomes in general equilibrium economies. This result has important implications for the welfare properties of economic models, and it plays a key role in the theory of comparative eco-nomic systems.
Before we can show that equilibria in our model are efficient, we have to make precise what exactly is meant by efficiency. In economics, we usually use the concept of Pareto efficiency. Another term for Pareto efficiency is Pareto optimality, and we will use both versions interchangeably. An allocation is Pareto efficient if it satisfies the market-clearing conditions and if there is no other allocation that: (1) also satisfies the market-clearing con-ditions; and (2) makes everyone better off. In our model, an allocationfci1;ci2;:::;ciN
g I
i=1is therefore Pareto efficient if the market-clearing constraint in equation (5.2) holds for each of theNgoods and if there is no other allocationfc¯i1;c¯i2;:::;c¯iN
g I
i=1that also satisfies market-clearing and such that:
u( ¯ci1;c¯i2;:::;c¯iN)>u(ci1;ci2;:::;ciN)
for every consumeri.3 Notice that the concept of Pareto optimality does not require us to take any stand on the issue of distribution. For example, if utility functions are strictly in-creasing, one Pareto-optimal allocation is to have one consumer consume all the resources in the economy. Such an allocation is clearly feasible, and every alternative allocation makes this one consumer worse off. A Pareto-efficient allocation is therefore not neces-sarily one that many people would consider “fair” or even “optimal”. On the other hand, many people would agree that it is better to make everyone better off as long as it is pos-sible to do so. Therefore we can interpret Pareto efficiency as a minimal standard for a
“good” allocation, rather than as a criterion for the “best” one.
We now want to show that any equilibrium allocation in our economy is necessarily Pareto optimal. The equilibrium consists of an allocationfci1;ci2;:::;ciN
g I
i=1 and a price system
fp1;p2;:::;pNg. Since market-clearing conditions hold for any equilibrium allocation, the first requirement for Pareto optimality is automatically satisfied. The second part takes a little more work. We want to show that there is no other allocation that also satisfies market-clearing and that makes everyone better off. We are going to prove this by contra-diction. That is, we will assume that such a better allocation actually exists, and then we will show that this leads us to a contradiction. Let us therefore assume that there is another allocationfc¯i1;c¯i2;:::;c¯iN
g I
i=1that satisfies market-clearing and such that:
u( ¯ci1;c¯i2;:::;c¯iN)>u(ci1;ci2;:::;ciN)
3A weaker notion of Pareto efficiency replaces the strict inequality with weak inequalities plus the requirement that at least one person is strictly better off. The proof of the First Welfare Theorem still goes through with the weaker version, but for simplicity we use strict inequalities.
5.4 The First Welfare Theorem 45
for every consumeri. We know that consumerimaximizes utility subject to the budget constraint. Since the consumer choosesfci1;ci2;:::;ciN
geven thoughfc¯i1;c¯i2;:::;c¯iN
gyields higher utility, it has to be the case thatfc¯i1;c¯i2;:::;c¯iN
gviolates the consumer’s budget con-straint:
Otherwise, the optimizing consumers would not have chosen the consumptions in the al-locationfci1;ci2;:::;ciN
g I
i=1 in the first place. Summing equation (5.4) over all consumers and rearranging yields:
i=1satisfied market-clearing. Therefore the terms inside the brackets are all zero. This implies 0>0, which is a contradiction. There-fore, no such allocationfc¯i1;c¯i2;:::;c¯iN
g I
i=1can exist, and the original equilibrium allocation
fc
i=1is Pareto optimal.
Since any competitive equilibrium is Pareto optimal, there is no possibility of a redistribu-tion of goods that makes everyone better off than before. Individual optimizaredistribu-tion together with the existence of markets imply that all gains from trade are exploited.
There is also a partial converse to the result that we just proved, the “Second Welfare Theo-rem”. While the First Welfare Theorem says that every competitive equilibrium is Pareto ef-ficient, the Second Welfare Theorem states that every Pareto optimum can be implemented as a competitive equilibrium, as long as wealth can be redistributed in advance. The Sec-ond Welfare Theorem rests on some extra assumptions and is harder to prove, so we omit it here. In economies with a single consumer there are no distributional issues, and the two theorems are equivalent.
Variable Definition
N Number of goods
p
n Price of goodn
I Number of consumers
c i
n Consumption of goodnby consumeri
u
i() Utility function of consumeri
e i
n Endowment with goodnof consumeri
Arbitrary proportionality factor Table 5.1: Notation for Chapter 5
Exercises
Exercise 5.1(Easy)
Show that Walras’ law holds for the credit-market economy that we discussed in Chapter 3.2. That is, use the consumer’s budget constraints and the market-clearing conditions for goods to derive the market-clearing condition for bonds in equation (3.9).
Exercise 5.2 (Hard)
Assume that the equilibrium price of one of theN goods is zero. What is the economic interpretation of this situation? Which of our assumptions ruled out that a price equals zero? Why? Does Walras’ Law continue to hold? What about the First Welfare Theorem?
Chapter 6
The Labor Market
This chapter works out the details of two separate models. Section 6.1 contains a one-period model in which households are both demanders and suppliers of labor. Market clearing in the labor market determines the equilibrium wage rate. Section 6.2 further develops the two-period model from Chapter 3. In this case, the households are permitted to choose their labor supply in each period.
6.1Equilibrium in the Labor Market
This economy consists of a large number of identical households. Each owns a farm on which it employs labor to make consumption goods, and each has labor that can be sup-plied to other farmers. For each unit of labor supsup-plied to others, a household receives a wagew, which is paid in consumption goods. Households take this wage as given. In order to make the exposition clear, we prohibit a household from providing labor for its own farm. (This has no bearing on the results of the model.)
The first task of the representative household is to maximize the profit of its farm. The output of the farm is given by a production functionf(ld), whereldis the labor demanded (i.e., employed) by that farm. The only expense of the farm is its labor costs, so the profit of the farm is:=f(ld) wld. The household that owns the farm chooses how much labor
l
dto hire. The first-order condition with respect toldis:
@
@l
d
=f0(l?d) w= 0; so:
w=f0(l?d): (6.1)
This implies that the household will continue to hire laborers until the marginal product
of additional labor has fallen to the market wage. Equation (6.1) pins down the optimal labor inputl?d. Plugging this into the profit equation yields the maximized profit of the household:?=f(ld?) wld?.
After the profit of the farm is maximized, the household must decide how much to work on the farms of others and how much to consume. Its preferences are given byu(c;ls), wherec is the household’s consumption, andlsis the amount of labor that the household supplies to the farms of other households. The household gets income? from running its own farm and labor income from working on the farms of others. Accordingly, the household’s budget is:
c=?+wls; so Lagrangean for the household’s problem is:
L=u(c;ls) +[?+wls c]: The first-order condition with respect tocis:
u1(c?;l?s) +?[ 1] = 0; (FOCc)
and that with respect tolsis:
u2(c?;l?s) +?[w] = 0: (FOCls)
Solving each of these forand setting them equal yields:
u2(c?;l?s)
u1(c?;l?s) =w;
(6.2)
so the household continues to supply labor until its marginal rate of substitution of labor for consumption falls to the wage the household receives.
Given particular functional forms foru() andf(), we can solve for the optimal choicesld?
andl?sand compute the equilibrium wage. For example, assume:
u(c;l) = ln(c) + ln(1 l); and:
f(l) =Al : Under these functional forms, equation (6.1) becomes:
w=A (l?d) 1; so:
l
?
d =
A
w
1 1
(6.3) :
This implies that the profit?of each household is:
? =A
A
1
w
A
1 1
:
6.1 Equilibrium in the Labor Market 49
After some factoring and algebraic manipulation, this becomes:
budget equation impliesc=+wls. Plugging these into equation (6.2) gives us:
Plugging in?from equation (6.4) yields:
l
Now we have determined the household’s optimal supply of laborls?as a function of the market wagew, and we have calculated the household’s optimal choice of labor to hireld?
for a given wage. Since all household’s are identical, equilibrium occurs where the house-hold’s supply equals the househouse-hold’s demand. Accordingly, we setl?s = l?d and call the resulting wagew?:
We gather like terms to get:
1
Finally, we plug this equilibrium wage back into our expressions forls?andl?d, which were in terms ofw. For example, plugging the formula forw?into equation (6.3) gives us:
l
Of course, we get the same answer forl?s, since supply must equal demand in equilibrium.
Given these answers forl?s,ld?, andw?, we can perform comparative statics to determine how the equilibrium values are influenced by changes in the underlying parameters. For example, suppose the economy experiences a positive shock to its productivity. This could be represented by an increase in theAparameter to the production function. We might be interested in how that affects the equilibrium wage:
@w
?
@A
=
+ 1
1
>0;
so the equilibrium wage will increase. Just by inspecting the formula forl?
d
and l?s, we know that labor supply and labor demand will be unchanged, sinceAdoes not appear. The intuition of this result is straightforward. With the new, higher productivity, households will be more inclined to hire labor, but this is exactly offset by the fact that the new wage is higher. On the other side, households are enticed to work more because of the higher wage, but at the same time they are wealthier, so they want to enjoy more leisure, which is a normal good. Under these preferences, the two effects cancel.
Variable Definition
w Wage in consumption goods per unit of labor
l
d Labor demanded by owner of farm
f(ld) Output of farm
Profit of farm
c Consumption of household
l
s Labor supplied by household
u(c;ls) Utility of household
Lagrange multiplier
L Lagrangean
A Parameter of the production function
Parameter of the production function Table 6.1: Notation for Section 6.1