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# prelims_Micro Prelim ANSWERS June 2006 - 5 2.1(d If the...

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5 2.1(d). If the production function is strictly quasiconvex, then the cost-minimization problem will have only corner solutions, i.e., the firm will either buy only Type 1 labor ( z j 1 > 0 and zj 1 = 0) or only Type 2 labor ( z j 1 = 0 and zj 2 > 0). Because the production function is homogeneous, whether the firm chooses to buy only Type one or only Type 2 does not depend on the amount of output to be produced. This, we can restrict the cost-minimization analysis to the unit isoquant. Suppose that the firm considers producing one unit of output using only Type 1. The required quantity of Type 1 is then determined by solving 1 ) ( ) ( 1 1 1 = θ β A h z h j , i.e., β β θ + β = 1 1 1 ) ( A h z j , requiring the firm to spend β β θ + β 1 1 1 A h w . Similarly, using only Type 2 entails an expenditure of β β θ + β 1 2 2 A h w . Therefore: the firm uses only Type 1 if β β θ + β 1 1 1 A h w < β β θ + β 1 2 2 A h w , only Type 2 if β β θ + β 1 1 1 A h w > β β θ + β 1 2 2 A h w , and is indifferent between using only Type 1 or only Type 2 if β β θ + β 1 1 1 A h w = β β θ + β 1 2 2 A h w . It is clear that, given h 1 , h 2 and β , the firm will choose only Type 1 if w 1 / w 2 is low enough, and only Type 2 if w 1 / w 2 is high enough.

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prelims_Micro Prelim ANSWERS June 2006 - 5 2.1(d If the...

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