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...UNIVERSITY YORK FACULTY OF ARTS DEPARTMENT OF ECONOMICS Economics 2350.03A Assignment 2: Answer Professor S. Chiang 1. P = 100 Q T C = Q2 + 16 Q max = (100 Q)Q Q2 16 = 100 2Q 2Q = 0 Q2 2 2 . .. Q = 25 P = 100 25 = 75. Thus, = (100 25)25 252 16 = 1234 2. = (10 Q1 Q2 )(Q1 + Q2 ) (Q2 + 2Q1 ) ( 1 + 4Q2 ) = 100 Q1 Q2 (Q1 + Q2 ) 2Q1 2 = 0 Q1 = Q2 ...
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UNIVERSITY YORK FACULTY OF ARTS DEPARTMENT OF ECONOMICS Economics 2350.03A Assignment 2: Answer Professor S. Chiang 1. P = 100 Q T C = Q2 + 16 Q max = (100 Q)Q Q2 16 = 100 2Q 2Q = 0 Q2 2 2 . .. Q = 25 P = 100 25 = 75. Thus, = (100 25)25 252 16 = 1234 2. = (10 Q1 Q2 )(Q1 + Q2 ) (Q2 + 2Q1 ) ( 1 + 4Q2 ) = 100 Q1 Q2 (Q1 + Q2 ) 2Q1 2 = 0 Q1 = Q2 10 Q1 Q2 (Q1 + Q2 ) Q2 4 = 0 Q = 1.5 1 Solving : Q = 1 2 Hence, P = 10 Q1 Q2 = 7.5. 3. Shared Monopoly: max = (15 Q)Q 3Q Q=6 ie. Q1 = Q2 = 3 and Pro ts are therefore Q = 15 2Q 3 = 0 P = 15 Q = 9 1 = (15 6)3 3 3 = 18 2 = 18 1 = 15 2q1 q2 3 = 0 q1 2 = 15 q1 2q2 3 = 0 q2 Cournot: 1 = (15 q1 q2 )q1 3q1 2 = (15 q1 q2 )q2 3q2 ... q = 4 = q . and P = 15 4 4 = 7 1 2 1 = 2 = 7 4 3 4 = 16 Bertrand: P = M C ie. 15 q1 q2 = 3 In addition, q1 = q2 . Thus q1 = q2 = 6 and 1 = 2 = 0 15 q1 3 2 Stackelberg: Firm 1 max 1 = (15 q1 q2 )q1 3q1 subject to q2 = 1 q1 = 15 q1 15 q1 3 2 q1 + 1 2 q1 3 = 0 = 3. . .. q1 = 6. Given this, Hence q2 = 15 q1 3 2 P = 15 6 3 = 6 1 = 6 6 3 6 = 18 2 = 6 3 3 3 = 9 4. (a) W = 12 2L W = 4L Wmin = 7 W = 12 2L W = 2L 12 2L = 4L L = 2 and W = 2L = 4 (b) 7 = 12 2L . .. L = (12 7)/2 = 2.5 and W = 7. L =3 W = 12 2 3 = 6 (c) 12 2L = 2L 5. (a) The contract curve is Oa Ob (the 450 line; see the diagram) (b) The core of the economy is Oa E. (c) Goods will be traded at 1:1 ratio. Person A utility improves from U A (x, y) = min[0, 10] = 0 to U A (x, y) = min[5, 5] = 5. Person B s utility remains unchanged: U B (x, y) = x+y = 5+5 (note that his initial utility is U B (x, y) = x+y = 10+0 = 10 before trade). 6. (a) E cient division them requires to stay on the contract curve. That is, M RSJohn = M RSJoan . (b) U John = P and U Joan = C where P = pork and C = chicken. 7. Let R = pq(p). Thus, R/ p = q (p)p + q(p) = q[ qqp + 1] = q[1 + ]. It is clear that as price increases, R increases (decreases) as is smaller (greater) than 1. Its e ect depends not only on but also on q. Therefore, there are two factors that are important: elasticity of demand and the current level of output before the change is made. 8. (a) Firm sets Q to max = [12/Q0.5 ]Q 5 Q0.5 Q. This yields the following rst order condition: 6Q 0.5 1.5Q0.5 = 0. Thus, Q = 4. Substituting this into the price equaition and the pro t function, we have P = 12/Q0.5 12/3 = 6 and = 11 > 0. (b) Given that P cannot exceed 4 dollars, the rst order condition becomes 4 1.5Q0.5 = 0. Solving: Q = 64/9. Pro t is therefore = 4 (64/9) 5 (64/9) (8/3) = 4.5 > 0. This compars with the xed cost of 5 dollars. The rm will thus stay on. (c) As long as the price is set by the government, the rm would determine its output according to p = M C. Note that M C = 1.5Q0.5 . Hence, p = 1.5Q0.5 is represents the rm s suppy curve. This along with the demand function (p = 12/Q0.5 ) determines the price ceiling that induces the monopolist to produce the largest possible output. That is, Q is determined according to 12/Q0.5 = 1.5Q0.5 . This gives Q = 8 and p = 12/ 8. 9. (a) Firm s optimal behavior is characterized by M R1 = M R2 = M C. Note that M R1 = 55 2Q1 , M R2 = 35 Q2 , and M C = 5. It is easy to verify that Q = 25 1 and Q = 30. By substitution, we have p = 30 and p = 20. Firm makes 625 2 1 2 dollars from market 1 and 450 dollars from market 2. 2 (b) If resale is possible, people will move goods from the low-price market to the high-price market (i.e., market 1 in the present case). This process will go on until their prices di er by 5 dollars. If the transportation cost is zero, then it is impossible for the rm to practice price discrimination. The rm will pool these two markets together and charge a uniform price. 3
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