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handout - CRS LR costs

# handout - CRS LR costs - ECO 100B Handout on CRS Long-Run...

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ECO 100B Handout on CRS Long-Run Costs Suppose f is a CRS technology: i.e., f ( &L; ) = ( L;K ) . Let w be the wage rate and r the rental rate of capital. 1 unit of output: that this would involve L units of labor, K units of capital, and cost C = wL + rK (1) So this would imply that C = C (1) where C ( q ) is the cost function that tells you the cheapest way to produce q units of output. Claim : C ( q ) = qC for all q . Proof : We can argue this by contradiction. Pick some arbitrary q , and for convenience, let q > 1 . Suppose C q ) 6 qC : then Case 1: C q ) > qC or Case 2: C q ) < qC . We will argue that each of these cases will lead to a logical contradiction. Case 1 : Suppose C q ) > qC . Now, f ( L ;K ) = 1 , so f qL ; qK qf ( L ;K q , because f is CRS. So one way to produce q units of output is to scale up production and use qL units of labor and

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handout - CRS LR costs - ECO 100B Handout on CRS Long-Run...

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