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Unformatted text preview: THE GEORGE WASHINGTON UNIVERSITY Department of Economics B.L. Boulier Supply Side-Short Run Production
I. Production Handout 6: Short-Run Costs with More than One Input Suppose that the firm's production function is: (1) Q = f(K,L,R), where L is labor, K is capital services, and R is other inputs which are assumed to be fixed, so that only K and L are variable inputs. As shown in the text, we can display the different combinations of L and K which can be used to produce a given level of output in the following isoquant diagram: An isoquant is a line showing all possible combinations of inputs which can be used to produce a given level of output. 2 The marginal rate of technical substitution of labor for capital is defined as the amount of capital which would substitute for the loss of a unit of labor and still keep output constant or, alternatively, the required reduction in K to keep output constant if L is increased by one unit. The marginal rate of technical substitution of labor for capital is the negative of the slope of the isoquant. Thus, in the above example, K1 - K2 = 2 would just substitute for the loss of L2 - L1 = 1 units of labor. The marginal rate of technical substitution of L for K is MPL/MPK. (Proof). Isoquants usually exhibit decreasing marginal rates of substitution of L for K. That is, the slope of the isoquant is usually flatter as L increases. (Reason) The following is a real example. The amount of oil that can be pumped through a pipeline depends upon the diameter of the pipe and the horsepower of the engine used to pump the oil. Other things being, the larger the pipeline the greater the oil that can be pumped in a give time period. On the other hand, for a given diameter of pipe, the larger the engine the greater the amount of oil that can be pumped. What happens if the price of an input changes? (Say the price of labor falls.) Application: Allocation of Production among Multiple Plants
3 Yet another approach. Maximize output for each level expenditures. Rearrange Equation (1) as:
Horsepower of Engine (in 1000's) 5 10 20 Diameter of Pipeline in Inches Quantity of Oil Pumped per Day 75,000 Barrels 150,000 Barrels 300,000 Barrels (3) 12.3 10.6 9.1 18.3 15.8 13.6 27.4 23.6 20.3 or, maximum output, given expenditures, is obtained when the marginal product of an input 30 8.4 12.5 18.6 per dollar spent on the input is the same.
40 50 60 70 80 90 100 7.9 7.5 7.2 7.0 6.8 6.6 6.5 11.7 11.2 10.8 10.4 10.1 9.9 9.7 17.5 16.7 16.1 15.6 15.1 14.7 14.4 Special case: No substitutability among inputs. (Salt) © Bryan L. Boulier, 2011. All rights reserved. ...
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