General Equilibrium Partial equilibrium: a single-market story Determine the market price of the output How are the input prices determined? Not answered. General equilibrium: All prices are determined simultaneously. All markets clear (aggregate demand = aggregate supply in all markets).
GE theory takes account of the functioning of all individual markets the interactions between markets Our plan of study GE in pure exchange economy: consumers with endowments; no producers GE in production economy: consumers + producers
GE in Exchange Economy Model environment: n markets for n goods. I agents with utility u i ( x i), where i = 1,, I. Each receives endowments of n goods. e i = ( e i 1, ei 2,, ei n) : vector of endowments for agent i, where e i j 0, for all j = 1,, n. x i = ( x i 1, xi 2,, xi n) : consumption bundle of agent i, where x i j 0, for all j = 1,, n.
Feasibility: For each type of goods, the total amount consumed is no more than the total endowments: I i=1 x i k I i=1 e i k, k = 1,, n. We assume that the utility function u i ( ) is continuous and strictly increasing; there exists a unique solution to the utility-maximization problem.
Walrasian equilibrium An agent i takes prices p = (p 1, p 2,, p n ) as given and chooses the consumption bundle x i = ( x i 1, xi 2,, xi n) to maximize utility: max x i u i ( x i ) s.t. p x i p e i. Optimal solution: x i ( p, p e i), Marshallian demand function. p e i : consumer i s wealth, market value of the initial endowment
Definition The aggregate excess demand function for good k is z k (p) I i=1 x i k ( p, p e i ) I i=1 e i k. The aggregate excess demand is z (p) = (z 1 (p), z 2 (p),, z n (p)).
Theorem 16 (Properties of Aggregate Excess Demand) The aggregate excess demand z (p) is a continuous function, and satisfies 1. homogeneity: z (λp) = z (p), λ > 0. 2. Walras law: p z (p) = 0.
Proof of Part 1 of Theorem 16 (Homogeneity) Given prices λp, an agent s utility-maximization problem is: max x i u i ( x i ) s.t. λp x i λp e i max x i u i ( x i ) s.t. p x i p e i. Therefore, the agent s problem is the same under prices λp as under p. Thus, x i ( λp, λp e i) = x i ( p, p e i).
It follows that ( λp, λp e i ) z k (λp) = i x i k ( p, p e i ) i e i k = i x i k i e i k = z k (p), k. Thus, z (λp) = z (p), λ > 0. QED
Proof of Part 2 of Theorem 16 (Walras law) For each agent, non-satiation implies that p x i = p e i at the optimum. Thus, That is, p (x i e i) = 0, i n { pk [ x i k ( p, p e i ) e i k ]} = 0, i k=1 Sum up the above equation over all agents i = 1,, I: I i=1 ( n k=1 { pk [ x i k ( p, p e i ) e i k ]}) = 0.
Reversing the order of summations, we get n k=1 ( I i=1 ( n I p k=1 k i=1 { pk [ x i k ( p, p e i ) e i k [( x i k ( p, p e i ) e i k ]} ) = 0 )] ) = 0 n k=1 p k z k (p) = 0 p z (p) = 0. QED
Corollary If vector p = ( p 1, p 2,, p n 1, p n) is such that prices p 1, p 2,, p n 1 clear n 1 markets, then p n must clear the nth market. Definition A vector p is called a Walrasian equilibrium if z (p ) = 0. In light of the Corollary, it is convenient to use the price of a certain good as the numeraire when solving for a Walrasian equilibrium. That is, set one of the good prices as one, and then solve for the rest n 1 equilibrium prices.
Theorem 17 (First Welfare Theorem) If u i ( ) is strictly increasing, then every Walrasian equilibrium is Pareto optimal. That is, given p, u i ( x i ( p, p e i)) u i ( x i), for all feasible x i x i ( p, p e i), where ">" holds for at least one i.
Proof of Theorem 17 Suppose there exist some allocations { x i} I i=1 such that u i ( x i) u i ( x i ( p, p e i)), where ">" holds for at least one agent i. Since u i ( ) is strictly increasing, it must be true that p x i p x i ( p, p e i), > holds for at least one i. Summing over all i, p ( I i=1 xi ) > p ( I i=1 xi ( p, p e i)) = p ( I i=1 ei ),
which is equivalent to ( n I k=1 p k i=1 xi k ) I i=1 ei k > 0. This contradicts the feasibility constraint: I i=1 xi k I i=1 ei k, k, given positive prices. Therefore, the supposition cannot be true. QED
Theorem 18 (Second Welfare Theorem) Any effi cient allocation can be sustainable by a competitive equilibrium, under certain assumptions. [Proof of the Second Welfare Theorem is not required for this course.] Despite the apparent symmetry, the first welfare theorem is much more general than the second one, requiring far weaker assumptions.
Example 2 goods; 20 agents with the same indirect utility function: v (p 1, p 2, y) = y3 p 1 p 2. 2 Endowments: 10 agents with e = (3, 1) and 10 with e = (1, 3). Normalize p 1 = 1. Calculate the Walrasian equilibrium price p 2.
Solution Calculate the individual Marshallian demand of good 1 using Roy s identity: x 1 (p 1, p 2, y) = v p 1 v y = y 3 p 2 1 p2 2 3y 2 p 1 p 2 2 = y 3p 1. Calculate the aggregate demand of good 1, using p 1 = 1: 10 3p 1 + p 2 + 10 p 1 + 3p 2 = 40 + 40p 2. 3p 1 3p 1 3
In the equilibrium, market of good 1 clears so that the aggregate demand is equal to aggregate supply. This implies 40 + 40p 2 3 = 40 p 2 = 2. [Note: One can also calculate the above price from the market-clearing condition for good 2.] [End of solution.]
Example Consider the same environment as in the previous Example. Now endowments: 10 agents with e = (3, 1) and 10 with e = (1, 2). Calculate the Walrasian equilibrium prices. (Normalize p 1 = 1.)
Solution From the previous example, the Marshallian demands are given by x 1 (p 1, p 2, y) = y 3p 1, x 2 (p 1, p 2, y) = 2y 3p 2. Given p 1 = 1, the market-clearing condition for good 1 implies: 10 3p 1 + p 2 3p 1 + 10 p 1 + 2p 2 3p 1 = 40 p 2 = 8 3. [Again, one can also calculate the above price from the market-clearing condition for good 2.] [End of solution.]
Example Consider a two-consumer, two-good exchange economy. Consumers have the following utility function: u (x 1, x 2 ) = min [x 1, x 2 ]. Endowments are e 1 = (5, 0) and 10 with e 2 = (0, 20). Does the Walrasian equilibrium exist in this economy? (Normalize p 1 = 1.)
Solution Let (x 1, x 2 ) be consumer 1 s Marshallian demand and (y 1, y 2 ) consumer 2 s Marshallian demand. Then let m i be a consumer s income, where i = 1, 2. Consider consumer 1 s utility-maximization problem: max (x 1,x 2 ) {min [x 1, x 2 ]}, s.t. p 1 x 1 + p 2 x 2 m 1. First suppose x 1 x 2 at the optimum. Then the problem is simplified to max (x 1,x 2 ) x 1, s.t. p 1 x 1 + p 2 x 2 = m 1,
where the budget constraint must hold with equality because the objective function is strictly increasing. Then it is obvious that it is optimal to have x 1 = x 2, which solves for Now suppose x 1 x 2 simplified to x 1 = x 2 = m 1 p 1 + p 2 = m 1 1 + p 2. (1) at the optimum. Then the consumer s problem is max (x 1,x 2 ) x 2, s.t. p 1 x 1 + p 2 x 2 = m 1. By similar argument as in the first case, the optimal solutions are still given by (1). Therefore, consumer 1 Marshallian demands are indeed given by x 1 = x 2 = 5p 1 p 1 + p 2 = 5 1 + p 2.
Similarly, consumer 2 s Marshallian demands are y 1 = y 2 = 20p 2 p 1 + p 2 = 20p 2 1 + p 2, where consumer 1 s income is m 2 = 20p 2. The market-clearing condition for good 1 implies: 5 1 + p 2 + 20p 2 1 + p 2 = 5, p 2 = 0. Therefore, there does not exist any strictly positive price of good 2 that supports a Walrasian equilibrium. Hence no Walrasian equilibrium for this economy. [End of solution.]