The yak-wool industry

1. The yakwool industry has ten identical firms. Each firm has a total cost function of �(�, �, �) = 100 + (��),..�/ where y indicates the number of bags of yakwool and r and w are input prices. This implies that the marginal cost of production is 2(��),..�. There are a hundred identical consumers who each have income of $1000, and who consume yakwool and xenon. Their preferences are summarized in the utility function � = �,..�,... This implies that ��� = 0.5�7,..�,.. and ��� = 0.5�,..�7,... Because xenon is produced through a process with constant returns to scale, its price is constant, and we will assume it is one. We will denote the price of y by p. (33 marks) (a) Find firm (5 marks) and industry (2 marks) supply functions of y in the short run. (b) Find the individual (5 marks) and market (2 marks) demand functions for y. (c) Find the equilibrium price. (5 marks) (d) Find the SR profits (or losses) of each producer. Assume that r=w=1 to make life easy. (5 marks) (e) Now assume that we are in the long run, and firms can enter and exit. Find the long run competitive price (5 marks) and total quantity supplied (2 marks). Continue to assume that r=w=1. How many firms will operate in equilibrium? (2 marks) 2. Suppose that competitive firms produce yogourt according to the supply function � = � + �. The prices of K and L are r and w, respectively. There are a hundred identical consumers who each have income of $1000, and who consume yogourt and xenon. Their preferences are summarized in the utility function � = �,..�,... This implies that ��� = 0.5�7,..�,.. and ��� = 0.5�,..�7,... Because xenon is produced through a process with constant returns to scale, its price is constant, and we will assume it is one. We will denote the price of y by p. (21 marks) (a) Assume r=10 and w=5. Find conditional input demand functions for L and K (2 marks) the long-run price and total quantity consumed of yogourt. (Tip:to make life easier for the net questions, just solve with w and then swap out w for 5 at the end.) (5 marks) (b) Assume that the government imposes a tax on L, so that for every unit of L used, firms must pay a tax of 3, as well as the wage of 5. Find the long-run price and total quantity consumed of yogourt. (5 marks) How much money does the government earn? (2 marks) (c) Assume that the government was so happy with the result in (b) that it increases the tax to 7. Find the long-run price and total quantity consumed of yogourt. (5 marks) How much money does the government earn? (2 marks) SEE NEXT PAGE 3 3. We claimed in class that firms minimize costs. (15 marks) a. If that is so, does that mean that they must invariably deliver low-quality products? Briefly explain. (5 marks) b. The oilsands cost data I showed indicated that not every firm seems to have the same cost. Does that mean that some must be not cost-minimizing? Briefly explain. (5 marks) c. What should we expect must happen to firms that do not cost-minimize? Briefly explain. (5 marks)                                                                                                                                        

Sample Solution