Production Theory

Topics • The Production Function • The Short-Run Production Function • Optimal Use of the Variable Input • Long-Run Production Functions • Optimal Combination of Inputs • Returns to Scale © 2006 by Nelson, a division of Thomson Canada Limited 2006 Thomson Nelson

Overview • 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. © 2006 by Nelson, a division of Thomson Canada Limited

The Production Function • A Production Function is the maximum quantity from any amounts of inputs or • Q = f (L,K) • If L is labour and K is capital, one popular functional form is known as the Cobb-Douglas Production Function • Q = a • L b1• K 2is a Cobb-Douglas Production Function • The number of inputs is often large. But economists simplify by suggesting some, like materials or labour, are variable, whereas plant and equipment is fairly fixed in the short run. © 2006 by Nelson, a division of Thomson Canada Limited

The Short-Run Production Function • Short Run Production Functions: • MAX output, from anyset 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 has only one variable input, labour, is easily analyzed. The one variable input is labour, L. © 2006 by Nelson, a division of Thomson Canada Limited

Total Product = Q * L • Average Product = Q / L • output per labour • Marginal Product = ΔQ/ΔL or Q/L • output attributable to last unit of labour applied • Similar to profit functions, the Peak of MP occurs before the Peak of average product • When MP = AP, we are at the peak of the AP curve © 2006 by Nelson, a division of Thomson Canada Limited

Elasticities of Production • The production elasticity of labour, • 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 labour elasticity, EL > 1. • A 1 percent increase in labour will increase output by more than 1percent. • When MPL < APL, then the labour elasticity, EL < 1. • A 1 percent increase in labour will increase output by less than 1 percent. • When MPL = APL , then the labour elasticity, EL = 1 • A 1 percent increase in labour will increase output by 1 percent. © 2006 by Nelson, a division of Thomson Canada Limited

Short-Run Production Function Numerical Example Marginal Product Average Product L Labour Elasticity is greater then one, for labour use up through L = 3 units 1 2 3 4 5 © 2006 by Nelson, a division of Thomson Canada Limited

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 TORONTO BLUE JAYS, YOUR ADDITION TO THE TEAM WOULD LOWER THE JAY’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 © 2006 by Nelson, a division of Thomson Canada Limited

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 © 2006 by Nelson, a division of Thomson Canada Limited

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. Stages of Production – TP, AP and MP Stage 2 0<Ep<1 Stage 1 Ep>1 Stage 3 Ep<0 TP Q Pt of Marginal Returns Ep=0 TP Ep=1 Increasing Returns Negative Returns Decreasing Returns L1 L2 L3 L AP,MP AP © 2006 by Nelson, a division of Thomson Canada Limited L1 L2 L3 MP L

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, MRPL = W  MFC MRPL MPL • MRQ= W Optimal Use of the Variable Input wage • W  MFC W MRPL L optimal labour © 2006 by Nelson, a division of Thomson Canada Limited

If Labour is MORE productive, demand for labour increases If Labour is LESS productive, demand for labour decreases Suppose an EARTHQUAKEdestroys capital  MPL declines with less capital, wages and labour are HURT MRPL is the Demand for Labour S L W D L D’ L L’ L © 2006 by Nelson, a division of Thomson Canada Limited

Long-Run Production Functions • All inputs are variable • greatest output from any set of inputs • Q = f (L, K) is two input example • MP of capital and MP of labour are the derivatives of the production function • MPL = DQ /DL or Q /L • MP of labour declines as more labour is applied. Also the MP of capital declines as more capital is applied. © 2006 by Nelson, a division of Thomson Canada Limited

In the LONG RUN, ALL factors are variable Q = f (L, K) ISOQUANTS -- locus of input combinations which produces the same output (A & B or on the same isoquant) MAGNITUDE of SLOPEof ISOQUANT is ratio of Marginal Products, called the MRTS, the marginal rate of technical substitution MRTS = (K1-K2)/(L1-L2) = <>K/<>L MRTS = MPL/MPK ISOQUANT MAP Isoquants & LR Production Functions K Q3 C B Q2 A Q1 L © 2006 by Nelson, a division of Thomson Canada Limited

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 4.9 Optimal Combination of Inputs at D, slope of isocost = slope of isoquant D K Q(1) C(1) © 2006 by Nelson, a division of Thomson Canada Limited L

Example – Isocost / Isoquant • Deep Creek Mining Co. • Cost per worker is $50 per period CL • Mining Equipment can be leased at $0.2 per brake per horse power. • Recall C0 = CL·L + CK·K • C = 50L + 0.2K, rearranging K = C/.2 – (250/.2)L • K = C/0.2 – 250L K Q= f(L,K) K=C/0.2 – CL © 2006 by Nelson, a division of Thomson Canada Limited L

Q: Is the following firm EFFICIENT? Suppose that: MP L = 30 MPK = 50 CL = 10 (labour cost) CK = 25 (capital cost) Labour: 30/10 = 3 Capital: 50/25 = 2 A: No! A dollar spent on labour produces 3, and a dollar spent on capital produces 2. USE RELATIVELY MORE LABOUR! If spend $1 less in capital, output falls 2 units, but rises 3 units when spent on labour Shift to more labour until the equimarginal condition holds. That is peak efficiency. Use of the Equimarginal Criterion © 2006 by Nelson, a division of Thomson Canada Limited

Allocative & Technical Efficiency • Allocative Efficiency – asks if the firm is using the least cost combination of inputs • 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) © 2006 by Nelson, a division of Thomson Canada Limited

Scale Efficiency • Scale Efficiency – asks if the firm is using the lowest possible minimum average cost for all production processes – defined as the ratio of this lowest cost to the potential average cost of production process chosen © 2006 by Nelson, a division of Thomson Canada Limited

Overall Production Efficiency • Overall Production Efficiency – the product of allocative, technical, and scale efficiencies • (Overall Production Efficiency) = (Allocative Efficiency) x (Technical Efficiency) x (Scale Efficiency) © 2006 by Nelson, a division of Thomson Canada Limited

Returns to Scale • A function is homogeneous of degree n • if multiplying all inputs by (lambda) increases the dependent variable byn • Q = f (L, K) • So, f ( L, K) = 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1 + 2 © 2006 by Nelson, a division of Thomson Canada Limited

Cobb-Douglas Production Functions • Q = α • L 1 • K2is a Cobb-Douglas Production Function • IMPLIES: • Can be CRS, DRS, or IRS if 1+ 2 1, then constant returns to scale if 1+ 2< 1, then decreasing returns to scale if 1+ 2> 1, then increasing returns to scale • Coefficients are elasticities 1 is the labour elasticity of output, often about 0.33 2 is the capital elasticity of output, often about 0.67 which are E L and EK Most firms have some slight increasing returns to scale © 2006 by Nelson, a division of Thomson Canada Limited

Problem Suppose: Q = 1.4 L 0.70 K 0.35 • Is this function homogeneous? • Is the production function constant returns to scale? • What is the production elasticity of labour? • What is the production elasticity of capital? • What happens to Q, if L increases 3% and capital is cut 10%? © 2006 by Nelson, a division of Thomson Canada Limited

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. • 0.70 is the production elasticity of labour • 0.35 is the production elasticity of capital • %Q = EL• %L+ EK • %K = 0.70(+3%) + 0.35(-10%) = 2.1% -3.5% = -1.4% © 2006 by Nelson, a division of Thomson Canada Limited

Production Theory