Last week we looked at a consumer i I s Utility Maximization Problem: max. s.t. k p kxk i k p kek

Transcription

1 Last week we looked at a consumer i I s Utility Maximization Problem: max { u i (x1 i,..., x n) i s.t. k p kxk i k p kek i (1) xk i 0 for k = 1,..., n

2 Last week we looked at a consumer i I s Utility Maximization Problem: The central assumption was: max { u i (x1 i,..., x n) i s.t. k p kxk i k p kek i (1) xk i 0 for k = 1,..., n

3 Last week we looked at a consumer i I s Utility Maximization Problem: The central assumption was: max { u i (x1 i,..., x n) i s.t. k p kxk i k p kek i (1) xk i 0 for k = 1,..., n Assumption 5.1. (JR p.188) The utility function u i : R n + R is continuous, strongly increasing, and strictly quasi-concave.

4 Under this assumption, we get theorem 5.1. (JR p.189). This says that when the price vector is positive in all coordinates (written p 0), the consumer s decision problem ((1) above or (5.2) in JR) has a unique solution: x i (p, pe i )

5 Under this assumption, we get theorem 5.1. (JR p.189). This says that when the price vector is positive in all coordinates (written p 0), the consumer s decision problem ((1) above or (5.2) in JR) has a unique solution: x i (p, pe i ) Furthermore, the function x i is continuous in p (for p R n ++).

6 Under this assumption, we get theorem 5.1. (JR p.189). This says that when the price vector is positive in all coordinates (written p 0), the consumer s decision problem ((1) above or (5.2) in JR) has a unique solution: x i (p, pe i ) Furthermore, the function x i is continuous in p (for p R n ++). x i is (of course) the demand function.

7 Note that there are two statements here:

8 Note that there are two statements here: Existence: The consumer s decision problem actually has a solution. There certainly are cases where this would not apply. One such case is when u i is strictly increasing and one of the prices p k equals zero (Optional exercise: Explain why!). Good news: A theory which leads to non-existence is empty.

9 Note that there are two statements here: Existence: The consumer s decision problem actually has a solution. There certainly are cases where this would not apply. One such case is when u i is strictly increasing and one of the prices p k equals zero (Optional exercise: Explain why!). Good news: A theory which leads to non-existence is empty. Uniqueness: The consumer s decision problem has at most one solution. So if we ask the consumer what she is going to buy, she ll say 7 apples and 4 oranges, she won t say well, I might buy 7 apples and 4 oranges, or 2 apples and 6 oranges, or.... We can t even write the demand function x i (p, pe i ) without uniqueness.

10 The k th coordinate of x i (p, pe i ) (the demand for the k th good) is written x i k (p, pei ).

11 The k th coordinate of x i (p, pe i ) (the demand for the k th good) is written x i k (p, pei ). Definition 5.4. (JR p.189) The aggregate excess demand function for good k {1,..., n} is the real-valued function: z k (p) = i I x i k (p, pei ) i I e i k The aggregate excess demand function is the vector-valued function: z(p) = (z 1 (p),..., z n (p))

12 A Walrasian Equilibrium (often written simply WE), is a price vector p R n ++ such that aggregate demand i xi (p, p e i ) equals aggregate supply i ei. That is to say: x i (p, p e i ) = i i e i

13 A Walrasian Equilibrium (often written simply WE), is a price vector p R n ++ such that aggregate demand i xi (p, p e i ) equals aggregate supply i ei. That is to say: x i (p, p e i ) = i i Or written in terms of the n goods: xk i (p, p e i ) = i i e i k e i, for k = 1,..., n

14 A Walrasian Equilibrium (often written simply WE), is a price vector p R n ++ such that aggregate demand i xi (p, p e i ) equals aggregate supply i ei. That is to say: x i (p, p e i ) = i i Or written in terms of the n goods: xk i (p, p e i ) = i i e i k e i, for k = 1,..., n Note that p 0 is a WE if and only if z(p ) = 0.

15 Definition 5.6. Let p be a WE for an economy with initial endowments e. Then x(p ) defined as follows is called a Walrasian Equilibrium Allocation (WEA): x(p ) = (x 1 (p, p e i ),..., x I (p, p e i )

16 Theorem 5.2. (Properties of Excess ) If for each consumer i, u i satisfies assumption 5.1., then for all p 0. 1 Continuity: z(p) will be continuous in p. 2 Homogeneity: z(λp) = z(p) for all λ > 0 3 Walras law: pz(p) = 0.

17 Theorem 5.2. (Properties of Excess ) If for each consumer i, u i satisfies assumption 5.1., then for all p 0. 1 Continuity: z(p) will be continuous in p. 2 Homogeneity: z(λp) = z(p) for all λ > 0 3 Walras law: pz(p) = 0. This theorem has a little star attached to it in the notes... which means that we are going to prove it, and you may be asked to prove it at the exam :-)

18 Theorem 5.5. (Existence of WE) If each consumer s utility function satisfies assumption 5.1. and i I ei 0, then there exists at least one WE, i.e., a price vector p 0 such that z(p ) = 0.

19 Theorem 5.5. (Existence of WE) If each consumer s utility function satisfies assumption 5.1. and i I ei 0, then there exists at least one WE, i.e., a price vector p 0 such that z(p ) = 0. This theorem is important, also from a socio-economic perspective.

20 Theorem 5.5. (Existence of WE) If each consumer s utility function satisfies assumption 5.1. and i I ei 0, then there exists at least one WE, i.e., a price vector p 0 such that z(p ) = 0. This theorem is important, also from a socio-economic perspective. Note we get existence here - not uniqueness!

21 Consider an exchange economy with I consumers, n goods, and endowment vector e = (e 1,..., e I ). The set of feasible allocations in this economy is given by: F (e) = {x = (x 1,..., x I ) R In + : I x i = i=1 I e i } i=1

22 Consider an exchange economy with I consumers, n goods, and endowment vector e = (e 1,..., e I ). The set of feasible allocations in this economy is given by: F (e) = {x = (x 1,..., x I ) R In + : I x i = i=1 I e i } Note that here it is implicit that each agent s consumption set is R n +. i=1

23 Consider an exchange economy with I consumers, n goods, and endowment vector e = (e 1,..., e I ). The set of feasible allocations in this economy is given by: F (e) = {x = (x 1,..., x I ) R In + : I x i = i=1 I e i } Note that here it is implicit that each agent s consumption set is R n +. A feasible allocation x F (e) is Pareto optimal (or Pareto efficient) if there is no other feasible allocation y F (e), such that: u i (y i ) u i (x i ) for all i = 1,..., I with at least one strict inequality. (2) i=1

24 Theorem () Consider an exchange economy (u i, e i ) i I and assume that each u i is strongly/strictly increasing on R n + (the consumption set). Then every Walrasian equilibrium allocation is Pareto optimal.

25 Very strong result because it only requires strong monotonicity.

26 Very strong result because it only requires strong monotonicity. Says something about efficiency - not about fairness.

27 Very strong result because it only requires strong monotonicity. Says something about efficiency - not about fairness. Result breaks down if there are externalities (if my utility depends on your consumption ).

28 Very strong result because it only requires strong monotonicity. Says something about efficiency - not about fairness. Result breaks down if there are externalities (if my utility depends on your consumption ). The first welfare theorem is also not valid, of course, if there is not perfect competition.