McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 10: Production and Cost Estimation

Managerial Economics 10-2 MP & AP Curves for the Short-Run Cubic Production Function (Figure 10.1) Q = AL 3 + BL 2 Dr. Chen’s notes: In order to obtain SR production function with one variable input (L), a cubic function without constant is preferred for math convenience. In Excel regression operation, please create two columns (L 3 and L 2 ) as indep. Variables then run “Regression” by choosing “Constant is Zero”. Please apply Table 10.1 on the textbook to verify the signs of A and B. A cubic production function ensures a quadratic marginal product (MP) curve which implies the law of diminishing returns.

Managerial Economics 10-3 Properties of a Short-Run Cubic Cost Function Average variable cost & marginal cost functions are, respectively: Dr. Chen’s notes: Please be aware that the fixed cost (TFC) is not considered here. In SR, the total cost, TC = TFC+ TVC. Variable cost and marginal cost can be derived from TVC. In reality, the data for cost function construction may contain either (TC, Q) or (TVC, Q). For (TC, Q), you may run “Regression” to obtain the whole total cost function then eliminate the intercept (TFC) to get TVC. For example, if you run the cubic regression function for TC = Q  5Q Q 3, then 300=TFC and TVC= 20Q  5Q Q 3. If the data contain (TVC, Q) directly, just run regression for cubic function by choosing “Constant is Zero”.

Managerial Economics 10-4 Estimation of a Short-Run Cost Function Estimate using data for which the level of usage of one or more inputs is fixed Usually time series data are used Data collection may be complicated by the fact that accounting data do not include firm’s opportunity costs Capital costs should reflect not only acquisition cost but any foregone rental income, depreciation, & capital gains/losses Must eliminate effects of inflation Divide by appropriate price index

Managerial Economics 10-5 Derivative for “Marginal” Dr. Chen’s notes: Well, let’s talk about “Calculus” here. In math, the derivative, dy/dx, denotes the change of y from a tiny change of x. In economics, the marginal cost (product) is also the derivative of total cost (product) to show the impact. You can learn the calculation of derivative immediately here. SMC (the derivative) comes from each term’s exponent times the variable which has been reduced by one degree. Two more examples: Ex 1 : Derive SMC from TVC = 20Q  5Q Q 3. Sol : SMC =1*20Q 1-1  2*5Q *0.2Q 3-1 = 20  10Q + 0.6Q 2 Ex 2 : Derive MP from Q =  0.5L 3 + 3L 2. Sol : MP =3*(  0.5L 3-1 ) + 2*3L 2-1 =  1.5L 2 + 6L Got it? You just need to know how to derive the marginal term for decision making.

Managerial Economics 10-6 Properties of a Short-Run Cubic Cost Function Average variable cost reaches its minimum value at: To conform to theoretical properties, parameters must satisfy the following restrictions:

Managerial Economics 10-7 Properties of a Short-Run Cost Function Cubic specification produces S-shaped TVC curve & U-shaped AVC & SMC curves All three cost curves employ the same parameters Only necessary to estimate one of these functions to obtain estimates of all three In the short-run cubic specification, input prices are assumed constant Not explicitly included in cost equation

Managerial Economics 10-8 Summary of Short-Run Empirical Cost Functions Short-run cubic cost equations Total variable cost Average variable cost Marginal cost Average variable cost reaches minimum at Restrictions on parameters