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 e ffi ciency 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 e ffi - 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 z h = ( z h 1 ,. . . , z h n ) be the n -dimensional vector represent- ing h ’s consumption of each good; an allocation is the n × H -dimensional vector: z = ( z 1 ,. . . , z H ) . Let Z = ( Z 1 ,. . . , Z n ) 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 of all goods does not exceed the available quantity Z , or H � z h � Z . h = 1 Definition 1 (Pareto optimal allocation). A feasible allocation z represents Pareto optimal if there is no other feasible allocation ¯ z which is strictly preferred to z by all individuals. In formal terms: ( a ) � H h = 1 z h � Z ; z h � Z and such that u h (¯ z such that � H z h ) > u h ( z h ) for all h . ( b ) there is no ¯ h = 1 ¯ � Definition 2 (Competitive equilibrium). An allocation z and a price vector p = ( p 1 ,. . . , p n ) are a competitive equilibrium if: h = 1 z h = Z , the allocation is feasible (we may also describe this condition as (A) � H saying that demand equals supply for each commodity); z h such that u h (¯ z h ) > u h ( z h ) and at the (B) for each individual h , there is no vector ¯ i (i.e. we assume that z h is the preferred vector same time � n z h i � � n i = 1 p i z h i = 1 p i ¯ 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. � P roof . We deny the theorem by assuming there is ¯ z feasible and Pareto dominating z : in other words, we assume that u h (¯ z h ) > u h ( z h ) for every individual in the economy. If z is a competitive equilibrium, then from property (B) follows that n n � � z h p i z h p i ¯ i > for all h . i i = 1 i = 1 Summing the inequalities for all h s and using property (A), we have n H n H n � � � � � z h z h p i ¯ i > p i i = p i Z i i = 1 h = 1 i = 1 h = 1 i = 1 but this is not compatible with the assumption that ¯ z is feasible, because in this case, with prices p i all positive, we should have � n � H z h i � � n i = 1 p i h = 1 ¯ i = 1 p i Z i . � 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)? z is not worse than z for h —so that u h (¯ z h ) � u h ( z h )—and if ( z , p ) is a competitive market Hint: if ¯ equilibrium, then we must have � n i � � n z h i = 1 p i z h i = 1 p i ¯ i . Indeed, if we had n n � � z h p i z h p i ¯ i < i , i = 1 i = 1 z h satisfying the budget contraint of individual h — it would be possibile to find another bundle ˆ hence, from property (B), such that u h ( z h ) � u h (ˆ z h )—and including more of some commodities z h —hence such that u h (ˆ z h ) > u h (¯ z h ). In other and no less of all commodities with respect to ¯ words, we would have u h ( z h ) � u h (ˆ z h ) > u h (¯ z h ). 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, e ffi ciency 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 indi ff erent between two allocations di ff ering only for quantities in z k , 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 di ff erentiable (and convex), e ffi cient 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. E ffi cient allocation. Consider an economy with two commodities x and y and H � 2 individuals. Let x h and y h be the quantities of the two commodities consumed by h ; let aggregate quantities be X = � h x h and Y = � h y h . 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 of 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 ( x 1 . . . , x H , y 1 ,. . . , y H ) is e ffi cient if it solves the following maxi- mization problem: U 1 ( x 1 , y 1 ) U h ( x h , y h ) � u h max s.t h = 2 ,. . . , H x 1 ,..., x H y 1 ,..., y H � H h = 1 x h = X � H h = 1 y h = 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 U h ( x h , y h ) = v h ( y h ) + x h . 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 x h from all constraints, so that the problem can be restated as follows: H H � − � � max v h ( y h ) + L − F �� u h ; h y h y 1 ,..., y H h = 1 h = 2 under the hypothesis of quasi-linear utility, x 1 ,. . . , x h have only redistributive e ff ects and do not a ff ect the optimal choice of y 1 ,. . . , y h . We obtain the first order condition for maximization: v ′ h ( y h ) = F ′ ( Y ) h = 1 ,. . . , H this is the well known condition of equality among the marginal rates of substitution of 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 e ffi cient level of X and Y . As a first step we define the function U h ( x 1 , y 1 ) U h ( x h , y h ) � u h φ 1 ( X , Y , u 2 ,. . . , u H ) = max s.t. h = 2 ,. . . , H x 1 ,..., x H y 1 ,..., y H � H h = 1 x h = X � H h = 1 y h = Y 3
Recommend
More recommend
