Market Equilibrium Price: Existence, Properties and Consequences

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1 Market Equilibrium Price: Existence, Properties and Consequences Ram Singh Lecture 5 Ram Singh: (DSE) General Equilibrium Analysis 1 / 14

2 Questions Today, we will discuss the following issues: How does the Adam Smith s Invisible Hand work? Is increase in Prices bad? Do some people want increase in prices? If yes, who would want an increase in prices and of what type? Does increase in prices have distributive consequences? Ram Singh: (DSE) General Equilibrium Analysis 2 / 14

3 Individual UMP: Some Features I Notations: p = (p 1,..., p M ) is a M-component vector in R M. If p = (p 1,..., p M ) R M ++, then p j > 0 for all j = 1,..., M, i.e., (p 1,..., p M ) > (0,..., 0). If p = (p 1,..., p M ) R M +, then p j 0 for all j {1,..., M} and p j > 0 for some j {1,..., M}, i.e., (p 1,..., p M ) (0,..., 0) and (p 1,..., p M ) (0,..., 0). Let x = (x 1,..., x M ) and x = (x 1,..., x M ). If x x, then x j x j for all j {1,..., M} and x j > x j for some j {1,..., M}. Let x = (x 1,..., x M ) and x = (x 1,..., x M ). If x > x, then x j > x j for all j {1,..., M}. Ram Singh: (DSE) General Equilibrium Analysis 3 / 14

4 Individual UMP: Some Features II Take a price vector p = (p 1,..., p M ) R M ++. That is, (p 1,..., p M ) > (0,..., 0). The consumer i s OP (UMP) is to solves: max u i (x) s.t. p.x p.e i x R J + Definition u i is strongly increasing if for any two bundles x and x x x u i (x ) > u i (x). Assumption For all i I, u i is continuous, strongly increasing, and strictly quasi-concave on R M + Ram Singh: (DSE) General Equilibrium Analysis 4 / 14

5 Individual UMP: Some Features III In view of monotonicity, for given p = (p 1,..., p M ) >> (0,..., 0), consumer i solves: max u i (x) s.t. p.x = p.e i (1) x R M + Theorem Under the above assumptions on u i (.), for every (p 1,..., p M ) > (0,..., 0), (1) has a unique solution, say x i (p, p.e i ). Note: Existence follows from Monotonicity and Boundedness of the Budget set Uniqueness follows from strictly quasi-concavity Ram Singh: (DSE) General Equilibrium Analysis 5 / 14

6 Individual UMP: Some Features IV Note: x i (p, p.e i ) is the (Marshallian) Demand Function for individual i. For each i = 1,..., N, x i (p, p.e i ) : R M ++ R M +; x i (p, p.e i ) = (x i 1(p, p.e i ),..., x i j (p, p.e i ),..., x i M(p, p.e i )). In general, demand for jth good depends on price of kth good, k = 1,..., M Demand for jth good depends on price of kth good relative to the other prices Ram Singh: (DSE) General Equilibrium Analysis 6 / 14

7 Individual UMP: Some Features V Theorem Under the above assumptions on u i (.), for every (p 1,..., p M ) > (0,..., 0), x i (p, p.e i ) is continuous in p over R M ++. For all i = 1, 2,..., N, we have: x i (tp) = x i (p), for all t > 0. That is, demand of each good j by individual i satisfies the following property: x i j (tp) = x i j (p) for all t > 0. Question Given that u i (.) is strongly increasing, is x i (p) continuous over R M +? is the demand function x i j (p) defined at p j = 0? Is a Cobb-Douglas utility function strongly increasing over R M +? Ram Singh: (DSE) General Equilibrium Analysis 7 / 14

8 Excess Demand Function I Definition The excess demand for jth good by the ith individual is give by: z i j (p) = x i j (p, p.e i ) e i j. The aggregate excess demand for jth good is give by: z j (p) = N i=1 x i j (p, p.e i ) N ej i. i=1 So, Aggregate Excess Demand Function is a vector-valued function: z(p) = (z 1 (p),..., z j (p),..., z M (p)), Ram Singh: (DSE) General Equilibrium Analysis 8 / 14

9 Excess Demand Function II Theorem Under the above assumptions on u i (.), for any p >> 0, z(.) is continuous in p z(tp) = z(p), for all t > 0 p.z(p) = 0. (the Walras Law) For any given price vector p, the individual UMP gives us p.x i (p, p.e i ) p.e i = 0, i.e., M M p j xj i (p, p.e i ) p j ej i = 0, i.e., j=1 M j=1 j=1 p j [x i j (p, p.e i ) e i j ] = 0. Ram Singh: (DSE) General Equilibrium Analysis 9 / 14

10 Excess Demand Function III This gives: That is, j=1 N M i=1 j=1 M N j=1 i=1 [ M N p j xj i (p, p.e i ) i=1 p j [x i j (p, p.e i ) e i j ] = 0, i.e., p j [x i j (p, p.e i ) e i j ] = 0, i.e., N i=1 e i j ] M p j z j (p) = 0, i.e., j=1 p.z(p) = 0 = 0 Ram Singh: (DSE) General Equilibrium Analysis 10 / 14

11 Excess Demand Function IV So, p 1 z 1 (p) + p 2 z 2 (p) p j 1 z j 1 (p) + p j+1 z j+1 (p) + +p M z M (p) = p j z j (p) For a price vector p >> 0, if z k (p) = 0 for all k j, then z j (p) = 0 For two goods case p 1 z 1 (p) + p 2 z 2 (p) = 0., i.e., p 1 z 1 (p) = p 2 z 2 (p). Therefore, z 1 (p) = 0 z 2 (p) = 0 z 1 (p) > 0 z 2 (p) < 0. Ram Singh: (DSE) General Equilibrium Analysis 11 / 14

12 Walrasian Equilibrium Definition Walrasian Equilibrium Price: A price vector p is equilibrium price vector, if for all j = 1,..., J, z j (p ) = N i=1 x i j (p, p.e i ) N ej i = 0, i.e., if i=1 z(p ) = 0 = (0,..., 0). Proposition If p is equilibrium price vector, then p = tp, t > 0, is also an equilibrium price vector If p is equilibrium price vector, then p tp, t > 0, may or may not be an equilibrium price vector Ram Singh: (DSE) General Equilibrium Analysis 12 / 14

13 WE: Proof I Two goods: food and cloth Let (p f, p c ) be the price vector. We can work with p = ( p f p c, 1) = (p, 1). Why? Let p = (p, 1) and tp = (p f, p c ) We know that for all t > 0: z(tp) = z(p) that is (z f (tp), z c (tp)) = (z f (p), z c (p)) Therefore, tp.z(tp) = 0, i.e., tp f z f (tp) + tp c z c (tp) = 0 implies Assume: p.z(p) = 0, i.e., pz f (p) + z c (p) = 0. z i (p) is continuous for all p >> 0, i.e., for all p > 0. Ram Singh: (DSE) General Equilibrium Analysis 13 / 14

14 WE: Proof II Note Since utility function is monotonic, x f (p) will explode as p f = p 0. Therefore, there exists small p = ɛ > 0 s.t. z f (p, 1) >> 0 and z c (p, 1) < 0 (Why?). there exists another p > 1 ɛ s.t. z f (p, 1) < 0 and z c (p, 1) > 0. (Why?). Therefore, for a two goods case we have: There is a value of p such that z f (p, 1) = 0 and z c (p, 1) = 0 That is, there exists a WE price vector. In general, Equlibrium price is determined by Tatonnement process Ram Singh: (DSE) General Equilibrium Analysis 14 / 14

Market Equilibrium and the Core

Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange