Production Economics Chapter 6

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## Production Economics Chapter 6

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**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%