Markets and efficiency We analyse the e ffi ciency properties of a - - PDF document

March 12, 2014 c Massimo D’Antoni — http://www.econ-pol.unisi.it/~dantoni Lecture notes for the course of Public Economics

Markets and efficiency

We analyse the efficiency properties of a market economy, as described in the neo- classical general economic equilibrium model. This mode assumes a complete set of competitive markets, where individuals make their consumption and production plans under the constraint given by available resources (on which property rights are clearly defined and enforced) and technology. The model is not meant to be realistic. It is an ideal reference point, useful to provide a rigorous framework for the analysis of the relation between market and effi- ciency, and understand the reasons for public intervention.

• 1. The first fundamental theorem (pure exchange)

We want to show that every competitive equilibrium is a pareto optimum. We refer to a pure exchange economy (i.e. an economy in which we have no production) with n commodities and H individuals. In this simplified case, we define allocation a description of the quantities con- sumed by all individuals. Let zh = (zh

1,. . . , zh n) be the n-dimensional vector represent-

ing h’s consumption of each good; an allocation is the n × H-dimensional vector: z = (z1,. . . , zH). Let Z = (Z1,. . . , Zn) be the n-dimensional vector of aggregate quantities of commodi- ties available in the economy. We will say that z is feasible if aggregate consumption

• f all goods does not exceed the available quantity Z, or

H

• h=1

zh Z. Definition 1 (Pareto optimal allocation). A feasible allocation z represents Pareto

• ptimal if there is no other feasible allocation ¯

z which is strictly preferred to z by all individuals. In formal terms: (a) H

h=1 zh Z;

(b) there is no ¯ z such that H

h=1 ¯

zh Z and such that uh(¯ zh) > uh(zh) for all h.

• Definition 2 (Competitive equilibrium). An allocation z and a price vector p = (p1,. . . ,pn)

are a competitive equilibrium if: (A) H

h=1 zh = Z, the allocation is feasible (we may also describe this condition as

saying that demand equals supply for each commodity); (B) for each individual h, there is no vector ¯ zh such that uh(¯ zh) > uh(zh) and at the same time n

i=1 pi ¯

zh

i n i=1 pizh i (i.e. we assume that zh is the preferred vector

for h among those of equal or lower cost at given prices).

• Hence, the competitive equilibrium is defined by the compatibility of optimizing

choices of price-taking individuals. 1

Theorem 1 (First fundamental theorem of welfare economics). If (z,p) is a com- petitive equilibrium, the allocation z is Pareto optimal.

• Proof. We deny the theorem by assuming there is ¯

z feasible and Pareto dominating z: in other words, we assume that uh(¯ zh) > uh(zh) for every individual in the economy. If z is a competitive equilibrium, then from property (B) follows that

n

• i=1

pi ¯ zh

i > n

• i=1

pizh

i

for all h. Summing the inequalities for all hs and using property (A), we have

n

• i=1

pi

H

• h=1

¯ zh

i > n

• i=1

pi

H

• h=1

zh

i = n

• i=1

piZi but this is not compatible with the assumption that ¯ z is feasible, because in this case, with prices pi all positive, we should have n

i=1 pi

H

h=1 ¯

zh

i n i=1 piZi.

• REMARK: the theorem has been proved using the weak notion of Pareto optimality (z is a weak

Pareto optimum if there is no allocation ¯ z strictly preferred by all individuals). Could you extend the proof to the case of a strong Pareto optimum (z is a strong Pareto optimum if there is no alloca- tion ¯ z which is not worse than z for all individuals and is strictly preferred by some individuals)? Hint: if ¯ z is not worse than z for h—so that uh(¯ zh) uh(zh)—and if (z,p) is a competitive market equilibrium, then we must have n

i=1 pi ¯

zh

i n i=1 pizh i . Indeed, if we had n

• i=1

pi ¯ zh

i < n

• i=1

pizh

i ,

it would be possibile to find another bundle ˆ zh satisfying the budget contraint of individual h— hence, from property (B), such that uh(zh) uh(ˆ zh)—and including more of some commodities and no less of all commodities with respect to ¯ zh—hence such that uh(ˆ zh) > uh(¯ zh). In other words, we would have uh(zh) uh(ˆ zh) > uh(¯ zh). With this in mind, try to proof the 1st theorem with strong Pareto optimality. REMARK: Note that the theorem does not say that a competitive equilibrium exists, only that if it exist, it must be Pareto optimal. In order to make sure the equilibrium exists we need some additional conditions on preferences and technology.

It is important to emphasize the assumption implicit in the theorem. When one of such assumption is not verified, efficiency of a market equilibrium is not warranted:

• 1. Every individual is interested only in the quantities of good consumed by him/herself;

individual h is indifferent between two allocations differing only for quantities in zk, k h.

• 2. Individuals make their choices taking prices p as given.
• 3. There are markets for all possible commodities; i.e. all interactions among indi-

viduals take place in competitive markets. These assumptions rule out externalities, market power and asymmetric informa- tion. 2

• 2. Marginal conditions in a simplified model with production

When utility and production functions are differentiable (and convex), efficient alloca- tions and market equilibria can be characterized in terms of marginal conditions. We develop the analysis in the case of an economy with production. Efficient allocation. Consider an economy with two commodities x and y and H 2

• individuals. Let xh and yh be the quantities of the two commodities consumed by h;

let aggregate quantities be X =

h xh and Y = h yh. Commodities are produced

using an input L with a technology described by the relation X + F(Y) = L, where X and F(Y) represent the cost of producing respectively X and Y measured in units

• f L: the marginal cost of x is constant and equal to one, while F′(Y) is the marginal

cost of y (we assume that the marginal cost is increasing in Y, or F′′(Y) > 0). The relation between the marginal costs of the two commodities is the marginal rate of transformation, which indicates how many units of one commodity we must give up to increase the production of the other commodity by one unit. In our case, the marginal rate of transformation (how many units of x for each unit of y) is given by F′(Y). An allocation (x1 . . . , xH, y1,. . . , yH) is efficient if it solves the following maxi- mization problem: max

x1,...,xH y1,...,yH

U1(x1, y1) s.t Uh(xh, yh) uh h = 2,. . . , H H

h=1 xh = X

H

h=1 yh = Y

X + F(Y) = L We assume for simplicity that the utility functions are quasi-linear, i.e. we can write them in the form Uh(xh, yh) = vh(yh) + xh. This assumption implies that the utility of one commodity (hence its demand) is independent of the quantity of the other commodity consumed by the individual. The assumption of quasi-linearity of utilities (and of the production function) allows us to substitute for xh from all constraints, so that the problem can be restated as follows: max

y1,...,yH H

• h=1

vh (yh) + L − F

h yh

−

H

• h=2

uh; under the hypothesis of quasi-linear utility, x1,. . . , xh have only redistributive effects and do not affect the optimal choice of y1,. . . , yh. We obtain the first order condition for maximization: v′

h(yh) = F′(Y)

h = 1,. . . , H this is the well known condition of equality among the marginal rates of substitution

• f all individuals and the marginal rate of transformation.

It is useful to decompose the problem above in two stages. In the first stage we consider the optimal allocation among consumers as a function of the total quantities produced X and Y (given aggregate quantities, the problem is one of pure exchange); in the second stage we determine the efficient level of X and Y. As a first step we define the function φ1(X,Y,u2,. . . ,uH) = max

x1,...,xH y1,...,yH

Uh(x1, y1) s.t. Uh(xh, yh) uh h = 2,. . . , H H

h=1 xh = X

H

h=1 yh = Y

3

This function gives the maximal utility for individual 1 that we can obtain given the aggregate quantities X and Y, and taking into account that from such quantities must be used to secure given levels of utilities (u2,. . . ,uH) to the other individuals. The equation u1 = φ1(X,Y,u2,. . . ,uH) identifies all combinations of quantities X and Y which allow to reach a vector of utilities (u1,. . . ,uH). In other words, X and Y satisfying such condition are indifferent from the point of view of all individuals in the society; in the (X,Y) space, they give rise to the community indifference contours.1 The contours identify a decreasing relation between X and Y (this is obvious: if the utility levels must remain unchanged as the quantity of X decreases, there must be an increase in Y) and they are convex;2 moreover, if we increase the level of uh for individual h while we keep all other levels unchanged, the curve moves outwards. Given a H-ple of utilities (u1,. . . ,uH), all combinations (X,Y) above the contour defined by u1 = φ1(X,Y,u2,. . . ,uH), when properly allocated, make individuals better

• ff with respect to the given H-ple.

With quasi-linear utilities, the function φ1 can be expressed as follows (we substi- tute constraints in the objective function): φ1(X,Y,u2,. . . ,uH) = max

y1,...,yH H

• h=1

vh(yh) + X −

H

• h=2

uh s.v.

• h

yh = Y. = ψ(Y) + X −

H

• h=2

uh where ψ(Y) = max

y1,...,yH H

• h=1

vh(yh) s.v.

H

• h=1

yh = Y is a function which can be interpreted as the (maximum) aggregate benefit individuals can obtain from consuming an aggregate quantity Y when such quantity is allocated

• ptimally among them.

The condition u1 = φ1 identifying the community indifference contours can now be expressed as X = H

h=1 uh − ψ(Y). If we consider the implicit function in the (X,Y)

space, we see that its slope is −ψ′(Y). The fact thatY is optimally allocated among the individuals requires that the marginal rates of substitution are equal for all, or v′

h(yh) = v′ k(yk) for each h,k. It follows

that also the marginal increase of ψ(Y) due to a marginal increase in Y is v′(yh), i.e. ψ′(Y) = v′

h(yh) for each h.3

Finally, the efficient level of X and Y can be identifies by opposing the map of community indifference contours to the constraint X + F(Y) = L, which reflects the availability of resources and the technology. In the (X,Y) space, such constraint is represented by the production possibility frontier X = L − F(Y), whose concavity reflects the assumption of decreasing returns in the use of factor L, i.e. F′′(Y) > 0. The optimum will be the point where such constraint is tangent to the highest commu- nity indifference contours among those corresponding to increasing levels of u1. The

1 This contours are somehow analogous to the individual indifference curves; while indifference curves give the combination of commodities consumed by one individuals which secure a given level of utilites, community indifference contours give the combinations of aggregate commodities which secure given levels of utilites to H individuals. 2 This can be proven as an exercise. 3 This is immediately clear if we consider the optimization problem which defines ψ(Y), as ψ′(Y) is equal to the Lagrange multiplier and the moltiplier is equal to v′

h(yh) for each h.

4

Y X X = P

h

u

h

` ( Y )

X = L ` F(Y )

0(Y ) = F 0(Y )

Figure 1 Community indifference contours and production possibility frontier

Y p 0(Y ) F 0(Y )

Figure 2 Market demand and supply for commodity Y

tangency condition is express by the equality F′(Y) = ψ′(Y), which implies the usual condition F′(Y) = v′

h(yh). Such situation is represented in figure 1.

Market equilibrium with perfect competition. Consider a system of competitive mar-

• kets. When in competitive markets, all individuals make their choices based on the

same set of prices (which they take as given) so that these choices are mutually com- patible (demand equals supply). With only two commodities, is is sufficient to secure that we have equilibrium in the market for one commodity, and we will have equilibrium also in the other market (Walras’ Law). Hence, an equilibrium is given by a price p for commodity y (the price of x is fixed an equal to one), on whose basis each consumer h chooses xh and yh so that max

xh,yh xh + vh(yh)

s.t. xh + pyh = Rh where Rh is the income of the individual, measured in units of x. Rh dipends on individual h’s initial endowment of property rights: in a private property system, each individual is endowed a given amount ℓh of the production factor, such that

h ℓh = L,

and a share of the firms (from which they receive profits). By substituting the budget constraint in the objective function, we see that the optimal choice of yh must solve max

yh

vh(yh) + Rh − pyh considering that the condition xh 0 implies Rh pyh. Each individual maximizes utility by choosing yh Rh/p which satisfies the con- dition v′

h(yh) = p. The corresponding optimal value of xh is xh = Rh − pyh 0.4 The

values xh and yh we have determined represent individual h’s market demands. Aggregate (market) demand for good y at price p is defined as: ˆ Y (p) =

h

yh v′

h(yh) = p

4 We are assuming that at the desired value yh and price p income Rh is large enough to secure that yh Rh/p. If at price p we had instead pyh > Rh, our consumer would spend all of his income in the commodity y, or yh = Rh/p, and xh = 0. In this case, we would have v′

h(yh) > p and we should

modify in a corresponding manner the analysis which follows.

5

It should be clear that ˆ Y (p) is the inverse of ψ′(Y) (remember the definition of ψ(Y) above and its properties). Therefore, ψ′(Y) represents the (inverse) market demand function for commodity, which we can represent grafically in the usual quantity/price space (figure 2). As to the production side, the hypothesis of competitive markets requires that we assume many price-taking firms. However, in order to characterize our equilibrium conditions, we can proceed as if we had only one firm producing the whole quantity Y by buying a quantity F(Y) of factor L (given that L is taken as the numeraire, F(Y) rappresents the cost of producing Y) and selling Y at a price the firm takes as given. We can assume that the remaining quantity of L is directly transformed into X by the individuals (think of L as total time available, F(Y) as time spent working, and X as consumption of leisure time). Hence, the maximization problem of the firm is: max

Y

pY − F(Y) so that the supplied quantity will satisfy p = F′(Y); supply will be increasing in p if F′′ > 0 (decreasing returns) as depicted in figure 2). The market equilibrium, i.e. the condition of equality between demand and supply

• f commodity y, requires that the following condition is satisfied:

ψ′(Y) = p = F′(Y). Finally, consider that in the aggregate individuals’ incomes satisfy

• h

Rh = [pY − F(Y)] + L (this must be the case, as individuals own factor L and firms). From the individual budget cxh + pyh = Rh follows that

h xh = L − F(Y) = X, which implies the

equilibrium between demand and supply for commodity x.5 Therefore, in an equilibrium the following conditions are satisfied:

• h yh = Y

v′

h(yh) = p

for all h F′(Y) = p and this implies that the efficiency conditions above are satisfied as well. Figure 2 illustrates in a very simple way the correspondence between the compet- itive market equilibrium and efficiency; in both cases we have a level of production Y such that F′(Y) = ψ′(Y), and this corresponds to the level where market demand is equal to market supply. Note that in our characterization of the market equilibrium the levels of income Rh play no role. The irrelevance of income distribution is a consequence of the assumption

• f quasi-linear utilities and of the assumption that pyh < Rh for al h, which imply that

the choice of yh is independent of Rh for all individuals. Without either assumption, the income levels (and hence distribution) would affect the demand and the efficient level of X e Y; marginal conditions should be qualified to take into account this cir-

• cumstance. However, the general conclusion that in a competitive market equilibrium

the allocation of resources is efficient remains valid.

5 The property that, in a market system with n commodities, the equilibrium in n − 1 markets implies equilibrium in the nth market is known as Walras’ Law.

6