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© John Riley 28 July 2011 ANSWERS TO EVEN NUMBERED EXERCISES IN APPENDIX C SECTION C.1: TWO VARIABLES Exercise C.1-2: Firm with interdependent demands A firm sells two products. Demand prices for these products are as follows. 11 2 2 1 2 22 180 , 180 p qq p q q =− =−− The cost of production is 2 2 () Cq q qq q α =+ + . (a) If 1 = , show that the profit function is concave and solve for the outputs that satisfy the first order conditions. (b) Show that the first order conditions for profit maximization are satisfied at (20,20) if 4 q == . (c) Are the second order (necessary) conditions satisfied at (20,20) q = ? (d) Show that there are two corner solutions to the FOC. Explain why profit must be maximized at these outputs. (e) Show that, fixing 2 q , profit is a concave function of 1 q and hence solve for * 12 , the profit maximizing output of commodity 1. (f) Totally differentiate * 12 2 (() ,) Π and explain why ** d dq q Π∂ Π = Hence show that *2 1 2 4 ( , ) 180 45(1 ) (4 (1 ) ) ) d q dq q αα Π + + (g) Hence characterize the function * Π for different values of .

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© John Riley 28 July 2011 ANSWER (a) 22 12 1 1 ( ) 180( ) R qq q q q q q =+ , 11 2 2 () Cq q qq q α + Hence 1 1 2 2 ( ) 180( ) 2 (1 ) 2 q q q q q Π= +− + . The matrix of second partial derivatives is 4( 1 ) ) 4 −− + −+ . If 1 = the determinant of the nth principal minor has sign ( 1) n so the function is concave. The FOC for an interior maximum are as follows. 1 180 4 ) 0 q ∂Π =−− + = 2 180 (1 ) 4 0 q ∂Π =− + −= . If 1 = the solution is * (30,30) q = . Since the function is concave the FOC are sufficient. (b) If 4 = , 0 (20,20) q = satisfies the FOC. However the maximand is not concave so we cannot be sure that this is the solution. (c) The second order necessary conditions are that the determinants of the principal minors should alternate in sign. This is NOT the case with 4. = Thus 0 q is not a local maximum. (d) Since there is no interior maximum there must be a corner solution. Suppose that only 1 0 q > . The FOC are 1 1 180 4 0 q q ∂Π =−= 1 2 180 (1 ) 0 q q ∂Π + .
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