hw06a_2008 (1)

# hw06a_2008 (1) - UNC-Wilmington Department of Economics and...

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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Homework 6 Solutions 1) Because Q can be produced with either L or K alone , we know that the production technology for Q is that of perfect substitutes . Given that each L produces 4 units of Q and each K produces (1/2) unit of Q, the firm’s production technology can be described as: Q = (4)L + (1/2)K. The firm’s profit-maximizing problem is therefore: subject to: Q = (4)L + (1/2)K There are now three possible cases: Recall that we can produce Q using L or K alone. If it is not profitable to produce Q with either L or K, then we should not produce at all: CASE 1: If ( 29 0 P (4) P L Q < - and ( 29 0 P ) 2 / 1 ( P K Q < - , then L*=0, K*=0, Q*=0 On the other hand, if it is profitable to produce Q with either L or K, then we want to produce using the input that is most profitable: CASE 2: If ( 29 0 P ) 4 ( P L Q - and ( 29 ( 29 K Q L Q P ) 2 / 1 ( P P ) 4 ( P - - , then = L* , K*=0, = Q* CASE 3: If ( 29 0 P ) 2 / 1 ( P K Q - and ( 29 ( 29 L Q K Q P ) 4 ( P P ) 2 / 1 ( P - - , then 0 L* = , = K* , = Q* In this problem, since ( 29 \$30 P ) 4 ( P L Q = - and ( 29 \$1 P ) 2 / 1 ( P K Q = - , we have CASE 2; so, the answer is = L* , K * =0, = Q* . 1 ( 29 K P L P Q P ofit Pr K L Q , , max + - = Q K L ( 29 ( 29 K P L P - K ) 2 / 1 ( L (4) P ofit Pr K L Q , max + + = K L ( 29 ( 29 K P - ) 2 / 1 ( P L P (4) P ofit Pr K Q L Q , max + - = K L

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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas 2) Because L and K must be used in fixed-proportions in the production of Q, we know that the production technology for Q is that of perfect complements , or the “Leontief ” technology . Given that the firm can produce 1 unit of Q for every 8 units of L used in production, and 4 units of K must be used with each unit of L, the firm’s production technology can be described as: = = L 4 K L 8 1 Q or = K 32 1 L, 8 1 min Q The firm’s profit-maximization problem is therefore:
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hw06a_2008 (1) - UNC-Wilmington Department of Economics and...

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