Cost Minimization An alternative approach to the decision of the firm Long run and short run costs Returns to scale and the cost function Different types of costs Alternative Approach Thus far firm chooses inputs in order to maximize profits Alternative approach 1 Firm chooses inputs in order to minimize the cost of producing given level of output 2 Firm chooses level of output that maximizes profits Cost Minimization Given factors of production K and L with rental prices wK and wL find cheapest way to produce a given level of output y min wL L wK K L K such that y F L K Cost Function Solution to minimization problem is cost function c wL wK y wL L wL wK y wK K wL wK y Finding the Cost Function Cost of using K and L C wL L wK K Isocost lines C wL K L wK wK C wL L K Isocost Lines wK wK K C1 wK C2 wK C3 wK C1 C 2 C 3 wL wK L Cost Minimization K isoquant K y F L K L L Cost Minimization Optimal choice wL TRS L K wK where MPL L K TRS L K MPK L K Cost Minimization To find solution use optimality condition plus production function 2 equations in 2 unknowns wL TRS L K wK y F L K Short Run and Long Run Cost Functions In the short run some factors of production are fixed short run cost function gives the minimum cost to produce a given level of output only adjusting the variable factors of production In the long run all factors are variable long run cost function gives the minimum cost to produce a given level of output adjusting all factors of production Example Short Run Find the short run cost function in the example of consulting firm wL 70 y 3000 0 2 xl 0 6 Quantity of labor used to produce y y xl 0 2 3000 1 0 6 Example Short Run Quantity of labor used to produce y y xl 0 2 3000 1 0 6 Short run cost function y c y 70 0 2 3000 1 0 6 Example Inputs are Perfect Substitutes y L 2 K K isoquant wL 1 wk 2 0 L y L Perfect Substitutes wL 1 If wk 2 labor only input L y Cost function is c wL wK y wL y Perfect Substitutes wL 1 If wk 2 capital only input y K 2 Cost function is wK c wL wK y y 2 Perfect Substitutes Summarizing cost function is wK c wL wK y min wL 2 y Fixed Proportions Production Function y min L K K isoquant K L L Fixed Proportions No matter what input prices are K L y min K L L K Cost function c wL wK y wL L wK K wL wK y Cost Function and Returns to Scale Constant returns to double output need to double all inputs double cost Decreasing returns to double output need to more than double inputs more than double cost Increasing returns to double output need to less than double inputs less than double cost Cost Function and Returns to Scale Constant returns c wL wK 2 2c wL wK 1 Decreasing returns c wL wK 2 2c wL wK 1 Increasing returns c wL wK 2 2c wL wK 1