Adv. Micro Theory, ECON 6202-090
Assignment 8 Answers, Fall 2010 Due: Not to be turned in
Directions: Some general equilibrium with production questions for you to answer.
1. Consider the following Robinson Crusoe economy. Robinson the consumer is endowed with zero units 2 of coconuts, x, and 24 hours of time, h, so that e = (0; 24) . His preferences are de…ned over R+ and represented by u(x; h) = xh. Robinson the producer uses the consumer’s labor services, l, to produce coconuts, y, according to the production function y = pl. The producer sells the coconuts to the consumer. All pro…ts from the production and sale of coconuts are distributed to the consumer. Find the Walrasian equilibrium prices and allocation of this economy. Answer: Let p denote the price of coconuts and w denote the price of Robinson’s time. The …rm wishes to maximize pro…t by choosing an amount of output y and labor l so that the …rm’sproblem is:
max py wl s.t. y = pl y 0;l 0 Substituting in for output (since the constraint will hold with equality) we can make the …rm’sproblem a function of only l so that we have: max ppl wl l 0 The …rst-order condition is:
d 1 1=2 = pl w dl 2 1 1=2 0 = pl w 2 1=2 2w = pl 2w = l 1=2 p 2w 2 = l 1 p p2 = l 4w2 The producer’ssupply is:
y = pl p2 y = 4w2 rp y = 2w
1 Firm pro…ts are: = py wl p p2 = p w 2w 4w2 p2 p2 = 2w 4w 2p2 p2 = 4w 4w p2 = 4w Notice that if p > 0 and w > 0 then > 0 so it is okay that we imposed that the producer’s…rst-order condition is equal to zero. The consumer’sproblem is: 1 max xh s.t. px + wh = 24w + x 0;24 h 0 4w Since the consumer’s budget constraint holds with equality we can substitute in for either x or h (I will choose x), so that: 24w 1 wh x = + p 4wp p The consumer now solves: 24w 1 wh max + h 24 h 0 p 4wp p 24wh h wh2 max + 24 h 0 p 4wp p The consumer’s…rst-order condition is: du 24w 1 2wh = + dh p 4wp p 24w 1 2wh 0 = + p 4wp p 0 = 96w2 + 1 8w2h 8w2h = 96w2 + 1 96w2 + 1 h = 8w2 1 h = 12 + 8w2 So that now we have that x is: 24w 1 wh x = + p 4wp p 1 24w 1 w 12 + 2 x = + 8w p 4wp p 24w 1 12w w x = + p 4wp p 8w2p 12w 1 1 x = + p 4wp 8wp 12w 2 1 x = + p 8wp 12w 1 x = + p 8wp
2 Now, setting the equilibrium condition for the hours market we have:
h + l = 24 1 p2 12 + + = 24 8w2 4w2 1 2p2 + = 12 8w2 8w2 1 + 2p2 = 96w2
Normalize p to 1 to …nd:
1 + 2 12 = 96w2 3 = 96w2 1 = w2 32 1 = w 4p2
So that the equilibrium price vector (p ; w ) = 1; 1 . Now …nding l, h, y, x, and we have: 4p2 p2 1 1 1 l = 2 = 2 = 1 = 1 = 8 4w 1 4 4 32 8 4p2 1 1 1 1 h = 12 + 2 = 12 + 2 = 12 + 1 = 12 + 1 = 16 8w 1 8 8 32 4 4p2 p 1 1 y = = = = 2p2 2w 2 1 1 4p2 2p2 12 1 12w 1 4p2 1 3 1 3 p2 3p2 + p2 4p2 x = + = + = + = + = = = 2p2 p 8wp 1 8 1 1 p2 2 p2 2 2 2 4p2 p2 p2 1 1 = = = = p2 4w 4 1 1 4p2 p2 2. In the Robinson Crusoe economy described in Exercise 1, suppose that Robinson does not think about a market, but simply chooses to enjoy h hours of leisure and spend 24 h hours collecting coconuts. What is his optimal choice of h? How many coconuts does he get? Compare your answer to the answer to Exercise 1. Answer: Robinson’sproblem is
max x h s.t. x = p24 h x 0;h 0 max hp24 h h 0
3 Taking the derivative with respect to h we have:
du 1 1=2 = p24 h + h (24 h) ( 1) dh 2 1 1=2 0 = p24 h + h (24 h) ( 1) 2 1 1=2 1=2 h (24 h) = (24 h) 2 h = 2 (24 h) h = 48 2h 3h = 48 h = 16 Since h = 16 we have x = p24 16 = p8 = 2p2. So the answer is the same as in question 1. 3. Consider the following economy. There are two …rms, …rm 1 and …rm 2. Firm 1 produces commodity 1 out of labor, l, according to the production function y1 = pl. Firm 2 produces commodity 2 out of l according to the production function y2 = l. There are two agents, A and B, with identical utility function u(x1; x2; h) = x1x2, where, xk; k = 1; 2, denotes commodity k, and h denotes the leisure time. Each consumer is endowed with 6 units of time. There is no initial endowment of any of the two commodities. Finally, both consumers own half of each …rm. Compute the Walrasian equilibrium prices and allocation. Answer:
Let p1 be the price of good 1, p2 be the price of good 2, and w be the price of labor.
Begin with what needs to be found. We need to …nd …rm 1’samount of labor units (l1) and its output (y1) and its pro…t (1). We need to …nd …rm 2’samount of labor units (l2) and its output (y2) and its 1 1 1 pro…t (2). We need to …nd consumer 1’sdemand for x1, x2, and h as well as consumer 2’sdemand 2 2 2 for x1, x2, and h . Begin with …rm 1’spro…t maximization problem:
max p1 l1 wl1 l1 0 The …rst-order condition is: p
d 1 1=2 = p1l1 w dl1 2 1 1=2 0 = p1l w 2 1 2w 1=2 = l1 p1 p2 1 = l 4w2 1 The …rm’soutput is then:
y1 = l1 p p2 y = 1 1 4w2 rp y = 1 1 2w and …rm 1’spro…ts are: 2 p1 p1 1 = p1 w 2w 4w2 p2 = 1 1 4w
4 This is just the same as in question 1. For …rm 2 we have: max p2l2 wl2 l2 0 When we …nd the …rst-order condition for this …rm (if we set the derivative equal to zero) we have p2 = w. This may or may not be true. If p2 > w then the demand for labor is and so we will not 1 have an equilibrium. If p2 < w then y2 = 0. If p2 = w then we will have the supply of y2 as anything along the constraint. Since we cannot have p2 > w we will have p2 w which means that 2 = 0. Consumer 1’sproblem is: 1 1 max x1x2 s.t. p1x1 + p2x2 = 6w + 1 + 2 x1 0;x2 0 2 2 2 p1 max x1x2 s.t. p1x1 + p2x2 = 6w + x1 0;x2 0 8w Since this utility function is of the Cobb-Douglas form we should know that for any Cobb-Douglas utility function we have: 1 1 1y x1 = ( 1 + 2) p1 1 where y is consumer 1’sincome and i is the exponent on the respective goods in the utility function (in this case 1 = 2 = 1 since u (x1; x2) = x1x2). Thus we have:
2 p1 1 6w + 8w x1 = 2p1 2 p1 1 6w + 8w x2 = 2p2 For consumer 2 we have a similar problem and a similar demand function:
2 p1 2 6w + 8w x1 = 2p1 2 p1 2 6w + 8w x2 = 2p2
Since we have positive demand for good 2, it must be the case that p2 = w so that production of good 2 occurs which means that we are in the case of w = p2. This also means that l2 = y2. Since we already know that p2 = w, we can normalize the price of w to 1 and we already know that p2 = 1. The only thing price left to …nd is p1 and we can …nd this from the market clearing condition (or equilibrium condition) for the good 1 market:
1 2 x1 + x1 = y1 p2 p2 6w + 1 6w + 1 p 8w + 8w = 1 2p1 2p1 2w p2 6w + 1 p 2 8w = 1 2p1 ! 2w p2 6w + 1 p 8w = 1 p1 2w 1 12w2 + p2 = p2 4 1 1 3 12w2 = p2 4 1
5 Since we have w = 1 we know: 3 12 = p2 4 1 2 16 = p1
4 = p1
Now the equilibrium price vector is (p1; p2; w) = (4; 1; 1) and the equilibrium allocations are: p2 16 l = 1 = = 4 1 4w2 4 1 p 4 y = 1 = = 2 1 2w 2 1 p2 16 = 1 = = 4 1 4w 4 1 p2 6w + 1 6 + 16 1 2 8w 8 8 x1 = x1 = = = = 1 2p1 2 4 8 p2 6w + 1 6 + 16 1 2 8w 8 8 x2 = x2 = = = = 4 2p2 2 1 2 y2 = 8
l2 = 8
2 = 0
The key is to see that the consumers demand 4 units of good 2 each, and so total demand is 8. This means that y2 = 8, which means that l2 = 8 as well. Note that there are 12 total hours of time that are endowed to the consumers, and that the amount of labor hours is equal to 12. This makes sense because the consumers have no utility for these hours (their utility is only a function of x1 and x2) and so they should save no hours for leisure.
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