Production Theory and Estimation Chapter 7

Managers must decide not only what to produce for the market, but also how to produce it in the most efficient or least cost manner. Economics offers widely accepted tools for judging whether the production choices are least cost. A production function relates the most that can be produced from a given set of inputs. Production functions allow measures of the marginal product of each input. 1

Green Power Initiatives California permitted and encouraged buying cheap power from other states. So, PG&E and Southern Cal Edison scaled back their expansion of production facilities. Off-Peak and Peak costs per MWh ranged from $25 to over $65, but regulators tried to keep prices low. Resulting in PG&E bankruptcy

Green Power Initiatives Carbon dioxide emission trading schemes in Europe encouraged construction of greener nuclear & wind generation. Q: If you were asked to pay 3 times more for electricity in the day than night, would you change your usage?

The Organization of Production Inputs Labor, Capital, Land Fixed Inputs Variable Inputs Short Run At least one input is fixed Long Run All inputs are variable

Production Function With Two Inputs Q = f(L, K)

Production Function With One Variable Input Total Product TP = Q = f(L) MPL = TP L Marginal Product APL = TP L Average Product EL = MPL APL Production or Output Elasticity

Production Function With One Variable Input Total, Marginal, and Average Product of Labor, and Output Elasticity

Production Function With One Variable Input

Production Function With One Variable Input

Optimal Use of the Variable Input Marginal Revenue Product of Labor MRPL = (MPL)(MR) Marginal Resource Cost of Labor TC L MRCL = Optimal Use of Labor MRPL = MRCL

Optimal Use of the Variable Input Use of Labor is Optimal When L = 3.50

Optimal Use of the Variable Input

Production With Two Variable Inputs Isoquants show combinations of two inputs that can produce the same level of output. Firms will only use combinations of two inputs that are in the economic region of production, which is defined by the portion of each isoquant that is negatively sloped.

Production With Two Variable Inputs Isoquants

Production With Two Variable Inputs Economic Region of Production

Production With Two Variable Inputs Marginal Rate of Technical Substitution MRTS = -K/L = MPL/MPK

Production With Two Variable Inputs MRTS = -(-2.5/1) = 2.5

Production With Two Variable Inputs Perfect Substitutes Perfect Complements

Optimal Combination of Inputs Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

Optimal Combination of Inputs Isocost Lines AB C = $100, w = r = $10 A’B’ C = $140, w = r = $10 A’’B’’ C = $80, w = r = $10 AB* C = $100, w = $5, r = $10 Prepared by Robert F. Brooker, Ph.D. Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved. Slide 21

Optimal Combination of Inputs MRTS = w/r Prepared by Robert F. Brooker, Ph.D. Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved. Slide 22

Optimal Combination of Inputs Effect of a Change in Input Prices

Production Function Q = f(L, K) Returns to Scale Production Function Q = f(L, K) Q = f(hL, hK) If  = h, then f has constant returns to scale. If  > h, then f has increasing returns to scale. If  < h, the f has decreasing returns to scale.

Returns to Scale Constant Returns to Scale Increasing Returns to Scale Decreasing Returns to Scale Prepared by Robert F. Brooker, Ph.D. Copyright ©2004 by South-Western, a division of Thomson Learning. All rights reserved. Slide 25

Empirical Production Functions Cobb-Douglas Production Function Q = AKaLb Estimated using Natural Logarithms ln Q = ln A + a ln K + b ln L

The Production Function A Production Function is the maximum feasible quantity from any amounts of inputs If L is labor and K is capital, one popular functional form is known as the Cobb-Douglas Production Function 2

The Production Function (con’t) Q = a • K b1• L 2 is a Cobb-Douglas Production Function The number of inputs is typically greater than just K & L. But economists simplify by suggesting some, like materials or labor, is variable, whereas plant and equipment is fairly fixed in the short run.

The Short Run Production Function Short Run Production Functions: Max output, from a n y set of inputs Q = f ( X1, X2, X3, X4, X5 ... ) FIXED IN SR VARIABLE IN SR _ _ Q = f ( K, L) for two input case, where K is Fixed © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2

The Short Run Production Function (con’t) A Production Function with only one variable input is easily analyzed. The one variable input is labor, L. Q = f( L )

Marginal Product = DQ/DL =Q/L = dQ/dL Average Product = Q / L output per labor Marginal Product = DQ/DL =Q/L = dQ/dL output attributable to last unit of labor applied Similar to profit functions, the Peak of MP occurs before the Peak of average product When MP = AP, this is the peak of the AP curve 3

Law of Diminishing Returns INCREASES IN ONE FACTOR OF PRODUCTION, HOLDING ONE OR OTHER FACTORS FIXED, AFTER SOME POINT, MARGINAL PRODUCT DIMINISHES. MP A SHORT RUN LAW point of diminishing returns Variable input 7

Bottlenecks in Production Plants Boeing found diminishing returns in ramping up production. It sought ways to adopt lean production techniques, cut order sizes, and outsourced work at bottlenecked plants.

Increasing Returns and Network Effects There are exceptions to the law of diminishing returns. When the installed base of a network product makes efforts to acquire new customers increasing more productive, we have network effects Outlook and Microsoft Office

Table 7.2: Total, Marginal & Average Products

Total, Marginal & Average Products Marginal Product Average Product 3 4 5 6 7 8 The maximum MP occurs before the maximum AP

When MP > AP, then AP is RISING IF YOUR MARGINAL GRADE IN THIS CLASS IS HIGHER THAN YOUR GRADE POINT AVERAGE, THEN YOUR G.P.A. IS RISING When MP < AP, then AP is FALLING IF YOUR MARGINAL BATTING AVERAGE IS LESS THAN THAT OF THE NEW YORK YANKEES, YOUR ADDITION TO THE TEAM WOULD LOWER THE YANKEE’S TEAM BATTING AVERAGE When MP = AP, then AP is at its MAX IF THE NEW HIRE IS JUST AS EFFICIENT AS THE AVERAGE EMPLOYEE, THEN AVERAGE PRODUCTIVITY DOESN’T CHANGE 6

Three stages of production

Three stages of production Stage 1: average product rising. Increasing returns Stage 2: average product declining (but marginal product positive). Decreasing returns Stage 3: marginal product is negative, or total product is declining. Negative returns 8

Determining the Optimal Use of the Variable Input HIRE, IF GET MORE REVENUE THAN COST HIRE if TR/L > TC/L HIRE if the marginal revenue product > marginal factor cost: MRP L > MFC L AT OPTIMUM, MRP L = W  MFC MRP L  MP L • P Q = W wage W • W  MFC MRPL MPL optimal labor L 9

Optimal Input Use at L = 6 Table 7.3 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Production Functions with multiple variable inputs Suppose several inputs are variable greatest output from any set of inputs Q = f( K, L ) is two input example MP of capital and MP of labor are the derivatives of the production function MPL = Q/L = DQ/DL MP of labor declines as more labor is applied. Also the MP of capital declines as more capital is applied. 11

Isoquants & LR Production Functions ISOQUANT MAP In the LONG RUN, ALL factors are variable Q = f ( K, L ) ISOQUANTS -- locus of input combinations which produces the same output Points A & B are on the same isoquant SLOPE of ISOQUANT from A to B is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution = -DK /DL K Q3 B C Q2 A Q1 L 16

Optimal Combination of Inputs The objective is to minimize cost for a given output ISOCOST lines are the combination of inputs for a given cost, C0 C0 = CL·L + CK·K K = C0/CK - (CL/CK)·L Optimal where: MPL/MPK = CL/CK· Rearranged, this becomes the equimarginal criterion

Optimal Combination of Inputs Equimarginal Criterion: Produce where MPL/CL = MPK/CK where marginal products per dollar are equal at D, slope of isocost = slope of isoquant 17

Use of the Equimarginal Criterion Q: Is the following firm EFFICIENT? Suppose that: MP L = 30 MPK = 50 W = 10 (cost of labor) R = 25 (cost of capital) Labor: 30/10 = 3 Capital: 50/25 = 2 A: No!

Use of the Equimarginal Criterion A dollar spent on labor produces 3, and a dollar spent on capital produces 2. USE RELATIVELY MORE LABOR! If spend $1 less in capital, output falls 2 units, but rises 3 units when spent on labor Shift to more labor until the equimarginal condition holds. That is peak efficiency. 19

Production Processes and Process Rays under Fixed Proportions If a firm has five computers and just one person, typically only one computer is used at a time. You really need five people to work on the five computers. The isoquants for processes with fixed proportions are L-shaped. Small changes in the prices of input may lead to no change in the process. M is the process ray of one worker and one machine people 5 4 3 2 1 M 1 2 3 4 5 6 7 8 9 computers

Allocative & Technical Efficiency Allocative Efficiency – asks if the firm using the least cost combination of input It satisfies: MPL/CL = MPK/CK Technical Efficiency – asks if the firm is maximizing potential output from a given set of inputs When a firm produces at point T rather than point D on a lower isoquant, that firm is not producing as much as is technically possible. D Q(1) T Q(0)

Overall Production Efficiency Suppose a plant produces 93% of what the technical efficient plant (the benchmark) would produce. Suppose a plant produces 85.7% of what an allocatively efficient plant would produce, due to a misaligning the input mix. Overall Production Efficiency = (technical efficiency)*(allocative efficiency) In this case: overall production efficiency = (.93)(.857) = 0.79701 or about 79.7%.

Returns to Scale If multiplying all inputs by  (lambda) increases the dependent variable by, the firm has constant returns to scale (CRS). Q = f ( K, L) So, f(K,  L) =  • Q is Constant Returns to Scale So if 10% more all inputs leads to 10% more output the firm is constant returns to scale. Cobb-Douglas Production Functions are constant returns if  +  =1 12

Cobb-Douglas Production Functions Q = A • K • L is a Cobb-Douglas Production Function IMPLIES: Can be CRS, DRS, or IRS if  +  1, then constant returns to scale if  + < 1, then decreasing returns to scale if  + > 1, then increasing returns to scale Suppose: Q = 1.4 K .35 L .70 Is this production function constant returns to scale? No, it is Increasing Returns to Scale, because 1.05 > 1. 13

Reasons for Increasing & Decreasing Returns to Scale Some Reasons for IRS Some Reasons for DRS The advantage of specialization in capital and labor – become more adept at a task Engineering size and volume effects – doubling the size of motor more than doubles its power Network effects Pecuniary advantages of buying in bulk Problems with coordination and control – as a organization gets larger, harder to get everyone to work together Shirking increases Bottlenecks appear – a form of the law of diminishing returns appears CEO can’t oversee a gigantically complex operation

Interpreting the Exponents of the Cobb-Douglas Production Functions The exponents a and b are elasticities a is the capital elasticity of output The a is [% change in Q / % change in K] b is the labor elasticity of output The b is a [% change in Q / % change in L] These elasticities can be written as EK and E L Most firms have some slight increasing returns to scale. 13

Empirical Production Elasticities Table 7 Empirical Production Elasticities Table 7.4: Most are statistically close to CRS or have IRS ˟ such as management or other staff personnel.