Production EconomicsChapter 6 • 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. 2005 South-Western Publishing
The Production Function • A Production Function is the maximum 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 • Q = a • K b1• L 2is a Cobb-Douglas Production Function • The number of inputs is often large. 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 yset 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 as Fixed • A Production Function is has only one variable input, labor, is easily analyzed. The one variable input is labor, L.
Average Product = Q / L • output per labor • Marginal Product =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, we’re at the peak of the AP curve
Elasticities of Production • The production elasticity of labor, • EL = MPL / APL = (DQ/DL) / (Q/L) = (DQ/DL)·(L/Q) • The production elasticity of capital has the identical in form, except K appears in place of L. • When MPL > APL, then the labor elasticity, EL > 1. • A 1 percent increase in labor will increase output by more than 1 percent. • When MPL < APL, then the labor elasticity, EL < 1. • A 1 percent increase in labor will increase output by less than 1 percent.
Short Run Production Function Numerical Example Marginal Product Average Product L Labor Elasticity is greater then one, for labor use up through L = 3 units 1 2 3 4 5
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
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
Three stages of production Stage 2 Total Output • Stage 1: average product rising. • Stage 2: average product declining (but marginal product positive). • Stage 3: marginal product is negative, or total product is declining. Stage 1 Stage 3 L Figure 7.4 on Page 306
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 Optimal Use of the Variable Input wage • W MFC W MRPL MPL L optimal labor
If Labor is MORE productive, demand for labor increases If Labor is LESS productive, demand for labor decreases Suppose an EARTHQUAKEdestroys capital MP L declines with less capital, wages and labor are HURT MRP L is the Demand for Labor S L W D L D’ L L’ L
Long Run Production Functions • All 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.
In the LONG RUN, ALL factors are variable Q = f ( K, L ) ISOQUANTS -- locus of input combinations which produces the same output (A & B or on the same isoquant) SLOPEof ISOQUANT is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution ISOQUANT MAP Isoquants& LR Production Functions K Q3 C B Q2 A Q1 L
The objective is to minimize cost for a given output ISOCOSTlines 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 Equimarginal Criterion:Produce where MPL/CL = MPK/CKwhere marginal products per dollar are equal Figure 7.9 on page 316 Optimal Combination of Inputs at D, slope of isocost = slope of isoquant D K Q(1) C(1) L
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! 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. Use of the Equimarginal Criterion
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, they firm is not producing as much as is technically possible. D T Q(1) Q(0)
Returns to Scale • A function is homogeneous of degree-n • if multiplying all inputs by (lambda) increases the dependent variable byn • Q = f ( K, L) • So, f(K, L) = n • Q • Constant Returns to Scale is homogeneous of degree 1. • 10% more all inputs leads to 10% more output. • Cobb-Douglas Production Functions are homogeneous of degree +
Cobb-Douglas Production Functions • Q = A • K • Lis 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 • Coefficients are elasticities is the capital elasticity of output, often about .67 is the labor elasticity of output, often about .33 which are EK and E L Most firms have some slight increasing returns to scale
Problem Suppose: Q = 1.4 L .70 K .35 • Is this function homogeneous? • Is the production function constant returns to scale? • What is the production elasticity of labor? • What is the production elasticity of capital? • What happens to Q, if L increases 3% and capital is cut 10%?
Answers • Yes. Increasing all inputs by , increases output by 1.05. It is homogeneous of degree 1.05. • No, it is not constant returns to scale. It is increasing Returns to Scale, since 1.05 > 1. • .70 is the production elasticity of labor • .35 is the production elasticity of capital • %Q = EL• %L+ EK • %K = .7(+3%) + .35(-10%) = 2.1% -3.5% = -1.4%