EconS 301 Review Session #6 Chapter 8: Cost Curves

Transcription

1 EconS 01 Revie Session #6 Chapter 8: Cost Curves 8.1. Consider a production function ith to inputs, labor and capital, given by (. The marginal products associated ith this production function are as follos: et and r 1 a) Suppose the firm is required to produce units of output. Sho ho the costminimizing quantity of labor depends on the quantity. Sho ho the costminimizing quantity of capital depends on quantity. Starting ith the tangency condition e have MP MPK r 1/ 1/ 1/ + K 1/ 1/ 1/ + K K 1 K 4 K 4 Plugging this into the total cost function yields + (4 ) 1/ 1/ 1/ Inserting this back into the solution for K above gives 4 K b) Find the equation of the firm s long-run total cost curve. 4 TC + TC 1

2 c) Find the equation of the firm s long-run average cost curve. TC AC AC d) Find the solution to the firm s short-run cost-minimization problem hen capital is fixed at a quantity of units (i.e. ). When the firm needs no labor. If > the firm must hire labor. Setting K and plugging in for capital in the production function yields 1/ 1/ + 1/ 1/ + 1/ 1/ Thus, 1/ 1 if > 0 if e) Find the short-run total cost curve, and graph it along ith the long-run total cost curve. ( 1/ ) + hen > TC hen Graphically, short-run and long-run total cost are shon in the folloing figure. Total Cost SRTC RTC

3 f) Find the associated short-run average cost curve. ( 1/ ) + if > TC AC if A hat manufacturing firm has the folloing production function ith capital and labor being the inputs: min(4, 7K) that is it has a fixed-proportions production function. If is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm s long-run total cost curve and average cost curve in terms of the input prices and. The fixed proportions production function implies that for the firm to be at a cost minimizing optimum, 4 7K and both of these equal. Therefore, /4 and r K /7. So the firm s total cost is + rk / 4 + r / 7 [ + ]. 4 7 r The average cost curve is RAC TC / +. Note that this average cost 4 7 curve is independent of and is simply a straight line Consider a production function of three inputs, labor, capital, and materials, given by KM. The marginal products associated ith this production function are as follos MP KM, MP K M, and MP M K. et, r 1, and m, here m is the price per unit of materials. a) Suppose that the firm is required to produce units of output. Sho ho the costminimizing quantity of labor depends on the quantity. Sho ho the cost-minimizing quantity of capital depends on the quantity. Sho ho the cost-minimizing quantity of materials depends on the quantity. Equating the bang for the buck beteen labor and capital implies MP MPK r KM M 1 K Equating the bang for the buck beteen labor and materials implies

4 MP MP M m KM K M Plugging these into the production function yields ( ) Substituting into the tangency condition results above implies K and M b) Find the equation of the firm s long-run total cost curve. TC + + TC 1 c) Find the equation of the firm s long-run average cost curve. TC 1 AC d) Suppose that the firm is required to produce units of output, but that its capital is fixed at a quantity of 0 units (i.e. 0). Sho ho the cost-minimizing quantity of labor depends on the quantity. Sho ho the cost-minimizing quantity of materials depends on the quantity. Beginning ith the tangency condition 4

5 MP MP M m KM K M Setting K 0 and substituting into the production function yields (0) 1 1 Substituting this result into the tangency condition result above implies M 1 M 0 e) Find the equation of the short-run total cost curve hen capital is fixed at a quantity of 0 units (i.e. 0) and graph it along ith the long-run total cost curve. In the short run, TC TC + 0 Graphically, short-run and long-run total cost curves are shon in the folloing figure.

6 Total Cost SRTC RTC f) Find the equation of the associated short-run average cost curve. Short run average cost is given by + 0 TC AC 6