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# Estimation Of Production And Cost Function

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### Estimation Of Production And Cost Function

1. 1. ESTIMATION OF PRODUCTION AND COST FUNCTION <ul><li>For practical decision-making purposes it is necessary to obtain estimates of production and cost functions. </li></ul><ul><li>In economics, it is usually hard to perform controlled laboratory experiments. Instead, actual operating data are used with some statistical procedures to derive these estimates. </li></ul>
2. 2. Estimation of Production and Costs involves: <ul><li>1. Data collection (time series, cross-sectional data). </li></ul><ul><li>2. Have to assume some mathematical form for the function. </li></ul><ul><li>3. Have to determine the estimation method for finding the parameter values (regression analysis for instance). </li></ul>
3. 3. However, data collection can be difficult (even more so than in demand estimation) : <ul><ul><li>it may be relatively easy to estimate the use of labor in production, but to estimate capital usage can be very difficult </li></ul></ul><ul><ul><li>most cost data are obtained from accounting records, so they do not necessarily conform to costs as defined by economists </li></ul></ul><ul><ul><li>how to treat the assumption of constant technology for a given production or cost function </li></ul></ul>
4. 4. Estimation of Production Functions <ul><li>A short review of theory </li></ul><ul><li>SHORT-RUN: Linear Production Function </li></ul><ul><li>Q = a + bL </li></ul><ul><li>- very simple, but does not take into account the law of diminishing returns </li></ul>L Q = a + bL MP L Q
5. 5. Estimation of Production Functions continued <ul><li>Quadratic Production Function </li></ul><ul><li>Q = a + bL - cL 2 </li></ul><ul><li>MP L = b - 2cL </li></ul><ul><li>AP L = a/L + b -cL </li></ul><ul><li>- implies diminishing returns, but not rising marginal product at the beginning </li></ul>
6. 6. Quadratic Production Function continued Q L L AP MP MP AP a) b)
7. 7. <ul><li>Cubic Production Function </li></ul><ul><li>Q = a + bL + cL 2 - dL 3 </li></ul><ul><li>MP L = b + 2cL - 3dL 2 </li></ul><ul><li>AP L = a/L +b + cL - dL 2 </li></ul><ul><li>- implies first increasing marginal returns and then diminishing returns </li></ul>
8. 8. Cubic Production Function continued Q L L AP MP MP AP a) b)
9. 9. Production Function as Power Function <ul><li>Q = aL b </li></ul><ul><ul><li>if </li></ul></ul><ul><ul><li>B > 1, Q increasing at increasing rate: MP L increasing </li></ul></ul><ul><ul><li>B = 1, Q increasing at constant rate: MP L constant </li></ul></ul><ul><ul><li>B < 1, Q increasing at decreasing rate: MP L decreasing </li></ul></ul><ul><ul><li>Major advantage of the power function is the fact that it can be transformed in a log-linear function </li></ul></ul><ul><ul><li>log Q = log a + b log L </li></ul></ul>
10. 10. Power Function continued Q L b<1 b=1 b>1
11. 11. Power Function continued <ul><li>Power function is the most frequently used type of production function in empirical work, even though it cannot exhibit two directions for marginal product on the same function. </li></ul><ul><li>One reason for its popularity is that it can be readily transformed into a function with two or more independent variables: </li></ul>
12. 12. The Cobb-Douglas Production Function <ul><li>A special case of power functions: </li></ul><ul><li>Q = aL b K 1-b , </li></ul><ul><li>Original version with constant returns to scale ( b + 1 - b = 1) introduced by Cobb in 1928 </li></ul><ul><li>He estimated the production function of U.S. manufacturing output for years 1899-1922 </li></ul>
13. 13. Reformulation by Cobb and Douglas: <ul><li>Q = aL b K c </li></ul><ul><li>b + c = 1, constant returns </li></ul><ul><li>b + c > 1, increasing returns </li></ul><ul><li>b + c < 1, decreasing returns </li></ul><ul><li>Can only use one of these at a time… so which one to choose? </li></ul>
14. 14. Properties of the Cobb-Douglas function that have kept it so popular for 90 years <ul><li>1. Both inputs have to be used simultaneously to get an output </li></ul><ul><li>2. Can exhibit different returns to scale (even though can not show a unit or an industry to move through all three stages) </li></ul>
15. 15. <ul><li>3. Allows to investigate MP for any factor while holding all others constant. So it is useful both in short-run and long-run analysis. </li></ul><ul><li>4. Elasticities are equal to the exponents b and c. (constant in this formulation) </li></ul>
16. 16. Elasticity of Production <ul><li>Measures the sensitivity of total product to a change in an input in percentage terms: </li></ul>
17. 17. Estimation of Production Functions <ul><li>Regression analysis often used </li></ul><ul><ul><li>time-series or cross-sectional analysis? </li></ul></ul><ul><li>Both methods have their advantages and disadvantages </li></ul>
18. 18. Estimation of Production Functions continued <ul><li>Time-series </li></ul><ul><li>1. If data in monetary terms, an inflation adjustment is necessary. </li></ul><ul><li>2. Technology may change over time. </li></ul><ul><li>3. Production function assumes that production takes place where input combination is most efficient. </li></ul>
19. 19. Estimation of Production Functions continued <ul><li>Cross-sectional </li></ul><ul><li>1. No technological change over time, but all plants in the investigation are assumed to have same technology. </li></ul><ul><li>2. Adjustments across different geographical areas must be made. </li></ul><ul><ul><li>wages and price level </li></ul></ul><ul><li>3. No guarantee that each plant operates at the most efficient input combination for the period examined. </li></ul>
20. 20. Cost Estimation <ul><li>Short-run cost functions are estimated to help managers to determine optimal pricing policy for the company </li></ul><ul><ul><li>used to determine marginal cost of producing additional units of output </li></ul></ul>
21. 21. <ul><li>Long-run cost functions are used in planning firm’s investment decisions </li></ul><ul><ul><li>To determine the extent of economies and diseconomies of scale in order to select the optimal plant size </li></ul></ul>
22. 22. Estimation of Short-Run Cost Functions <ul><li>Techniques used: </li></ul><ul><li>mostly regression analysis with time series data </li></ul><ul><li>problems and adjustments </li></ul><ul><ul><li>economic vs. accounting costs </li></ul></ul><ul><ul><li>rate changes; such as tax rates, social security contributions etc. </li></ul></ul><ul><ul><li>output homogeneity </li></ul></ul><ul><ul><li>timing of costs </li></ul></ul><ul><ul><li>accounting changes (have deprecation methods changed…) </li></ul></ul>
23. 23. Shapes of Short-Run Cost Functions <ul><li>Cubic cost function: </li></ul><ul><li>A cubic cost function represents the normal theoretical cost function, which exhibits both decreasing marginal and average costs and increasing marginal and average costs </li></ul>
24. 24. Cubic cost function continued <ul><li>TC = a + bQ - cQ 2 + dQ 3 </li></ul><ul><li>AC = a/Q + b - cQ + dQ 2 </li></ul><ul><li>MC = b - 2cQ + 3dQ 2 </li></ul>
25. 25. Cubic cost function \$ TC Q
26. 26. First marginal and average costs decrease and then increase: \$ MC Q AC
27. 27. Quadratic cost function <ul><li>If data does not fit to a cubic cost function, we can try to fit it to a quadratic one. </li></ul><ul><li>Quadratic cost function: </li></ul><ul><li>TC = a + bQ + cQ 2 </li></ul><ul><li>AC = a/Q + b + cQ </li></ul><ul><li>MC = b + 2cQ </li></ul>
28. 28. Costs increasing at increasing rate \$ TC Q
29. 29. No decreasing marginal cost! \$ MC Q AC
30. 30. Linear cost function <ul><li>Also a linear total cost function can be fitted. Then the three functions get the following form </li></ul><ul><li>TC = a + bQ </li></ul><ul><li>AC = a/Q + b </li></ul><ul><li>MC = b </li></ul>
31. 31. No Law of Diminishing Marginal Returns \$ TC Q \$ MC Q AC
32. 32. Note! <ul><li>Many empirical short-run cost studies have found a linear relationship between total cost and output, indicating a constant marginal cost. </li></ul>
33. 33. So… <ul><li>Should economist revise their view of U-shaped average and marginal cost curves? </li></ul><ul><ul><li>data employed concentrate on output levels of limited range </li></ul></ul><ul><ul><li>capital inputs may not be fixed even in short-run </li></ul></ul><ul><ul><li>regression is not a perfect tool </li></ul></ul>
34. 34. Estimation of Long-Run Cost Functions <ul><li>Techniques used: </li></ul><ul><li>Regression analysis </li></ul><ul><li>Engineering cost method </li></ul><ul><li>Survivor technique </li></ul>
35. 35. Regression analysis in L-R Cost Estimation <ul><li>Mostly with cross-sectional data </li></ul><ul><li>Pluses: </li></ul><ul><ul><li>since data comes from different firms, quantity of output can vary over relatively wide ranges </li></ul></ul><ul><ul><li>all data from same point of time, so technology will not change </li></ul></ul><ul><ul><li>do not have to regard price changes </li></ul></ul>
36. 36. Regression analysis in L-R Cost Estimation continued <ul><li>Minuses: </li></ul><ul><ul><li>interregional cost differences </li></ul></ul><ul><ul><li>all firms not necessarily operating at optimal level of technology </li></ul></ul><ul><ul><li>costs may be recorded differently in different firms </li></ul></ul><ul><ul><li>different companies may pay their cost factors differently </li></ul></ul>
37. 37. Engineering cost method <ul><li>Based on understanding of inputs and outputs and their relationships </li></ul><ul><li>In this approach, the analysis begins with an ”engineering production function”: optimal production input combinations for producing any given level of production is identified. </li></ul><ul><li>Cost can be obtained by multiplying each level of input usage by current price of the input and summing over the inputs. </li></ul>
38. 38. Engineering cost method continued <ul><li>Pluses: </li></ul><ul><ul><li>technology held constant </li></ul></ul><ul><ul><li>no problem with inflation (current input prices) </li></ul></ul><ul><ul><li>less error from measurement </li></ul></ul><ul><li>Minuses: </li></ul><ul><ul><li>cost estimates are normative </li></ul></ul><ul><ul><li>only direct output costs are estimated </li></ul></ul><ul><ul><li>often made based on pilot plant operations, not actual production </li></ul></ul>
39. 39. Survivor technique <ul><li>This method, suggested by G. Stigler, bases its findings on the change in the proportion of total industry output produced by firms of different size categories </li></ul><ul><ul><li>look at company size that is successful in an industry! </li></ul></ul><ul><li>Used for deciding optimal plant size </li></ul>
40. 40. Survivor technique continued <ul><li>Pluses: </li></ul><ul><ul><li>simple </li></ul></ul><ul><ul><li>avoids unreliable data </li></ul></ul><ul><li>Minuses: </li></ul><ul><ul><li>no help in measuring costs for planning purposes </li></ul></ul><ul><ul><ul><li>just tells which company size appears to be more efficient </li></ul></ul></ul><ul><ul><li>implicitly assumes that the industry highly competitive, so survival and prosperity are solely a function of efficient use of resources, not the market power or erection of barriers of entry </li></ul></ul>