The other condition is the slope condition also holds

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The other condition is the “slope” condition also holds, i.e. MRS A = p x /p y = 3 or, equivalently, MU A X p X = MU A Y p Y , i.e. Y 1 / 3 A 3 = 1 3 X A Y 2 / 3 A 1 , i.e. X A = Y A . Combining the two equations gives X A = Y A = 2 . (d) STEP 1: < A optimizes > From (c2) we know that p X = 3 and p Y = 1 implies ( X A , Y A ) = (2 , 2) . STEP 2: < B optimizes > We can show through the same procedure for Brigit (i.e. combine B’s budget line 3 X B + Y B = M B = X E B 3 + Y E B 1 = 1 3 3 + 5 1 = 6 and MRS B = p x /p y = 3) to show that ( X B , Y B ) = (1 , 3) STEP 3 < market equilibrium > : Check whether for both goods we get that demand = supply. Good X : demand = X A + X B = 2 + 1 ? = X E A + X E B = 8 3 + 1 3 = 3 = supply < check! > . Good Y : demand = Y A + Y B = 2 + 3 ? = Y E A + Y E B = 0 + 5 = 5 = supply < check! >. SUMMARY: We have fi rst shown (Steps 1+2) that optimization of A and B with prices p X = 3 and p Y = 1 implies the allocation will be ( X A , Y A ) = (2 , 2) and ( X B , Y B ) = (1 , 3). We have next (Step 3) seen that this allocation is a feasible allocation. Thus, the prices p X = 3 and p Y = 1 are com- petitive equilibrium prices, and the competitive equilibrium allocation is ( X A , Y A ) = (2 , 2) and ( X B , Y B ) = (1 , 3) . Question 2-2 (a) Isocost line 2 K + 2 L = C. With C = 100 , 000 we have K = L + 50 , 000 . Picture is trivial. (b) min K,L 2 K + 2 L subject to KL 1 / 2 = q. (c) Two conditions should hold in the optimum: (1) ”slope condition” and (2) isoquant. The slope condition can be written down as MP K r = MP L w L 1 / 2 2 = 1 2 KL 1 / 2 2 L = 1 2 K Substituting this into the isoquant gives K ( 1 2 K ) 1 / 2 = q ³ 1 2 ´ 1 / 2 K 3 / 2 = q 2 1 / 2 K 3 / 2 = q K 3 / 2 = 2 1 / 2 q K = (2 1 / 2 q ) 2 / 3 = 2 1 / 3 q 2 / 3 . So L = 1 2 2 1 / 3 q 2 / 3 . If you like, change 2 1 / 3 into 3 2. (d) C ( q ) = 2 K + 2 L = 3 3 2 q 2 / 3 (e) The fi rm’s average cost AC ( q ) = C ( q ) q = 3 3 q 2 q decreases because the production tech- nology exhibits increasing returns to scale. 2
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1 Economics 203, Fall 2006 Intermediate microeconomics Instructor: Paul Schure QUIZ 3 Monday 20 November 2006 (50 minutes) Quiz 1 is 3 pages long and consists of two parts. Part 1 contains two short-answer questions that are worth 5 marks each. Part 2 has two long-answer questions that are worth 20 marks each. In total you can earn 50 marks, i.e. a minute per mark. You must fill out your name and student number before you start. Give your answers only on this exam sheet. Good luck! Name: ...................................... Student #: ................................ PART 1: TWO SHORT-ANSWER QUESTIONS [5 marks each] Question 1-1. Consider a monopolistic firm facing identical consumers. Consider the following three statements and tell me whether they are TRUE, FALSE or UNCERTAIN. Please, briefly clarify (only) if you answer UNCERTAIN. (a) If the monopolist charged a single price to each consumer, the price would be higher than the welfare-maximizing price (b) In case arbitrage is not possible, the monopolist could well choose a two-part tariff to capture all consumer surplus (c) For the two-part tariff in (b) welfare is maximized Question 1-2.
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