1 © 2006 by Nelson, a division of Thomson Canada Limited Production Theory LECTURE 4 ECON 340 MANAGERIAL ECONOMICS Christopher Michael Trent University Department of Economics

2 © 2006 by Nelson, a division of Thomson Canada Limited  2006 Thomson Nelson 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 Topics

3 © 2006 by Nelson, a division of Thomson Canada Limited 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.

4 © 2006 by Nelson, a division of Thomson Canada Limited 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 =  L   K   is 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. The Production Function

5 © 2006 by Nelson, a division of Thomson Canada Limited Short Run Production Functions: »MAX output, from any 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 as Fixed A Production Function has only one variable input, labour, is easily analyzed. The one variable input is labour, L. The Short-Run Production Function

6 © 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

7 © 2006 by Nelson, a division of Thomson Canada Limited Elasticities of Production The production elasticity of labour, »E L = MP L / AP L = (  Q/  L) / (Q/L) = (  Q/  L)·(L/Q) »The production elasticity of capital has the identical in form, except K appears in place of L. When MP L > AP L, then the labour elasticity, E L > 1. »A 1 percent increase in labour will increase output by more than 1percent. When MP L < AP L, then the labour elasticity, E L < 1. »A 1 percent increase in labour will increase output by less than 1 percent. When MP L = AP L, then the labour elasticity, E L = 1 »A 1 percent increase in labour will increase output by 1 percent.

8 © 2006 by Nelson, a division of Thomson Canada Limited Short-Run Production Function Numerical Example Marginal Product L Average Product Labour Elasticity is greater then one, for labour use up through L = 3 units

9 © 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

10 © 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. A Short-Run Law point of diminishing returns Variable input MP

11 © 2006 by Nelson, a division of Thomson Canada Limited Stages of Production – TP, AP and MP Increasing Returns Decreasing Returns Negative Returns TP Q L AP, MP L AP MP TP L3L2L1 L2 L3 Stage 1 Ep>1 Stage 2 0<Ep<1 Stage 3 Ep<0 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. Pt of Marginal Returns Ep=0 Ep=1

12 © 2006 by Nelson, a division of Thomson Canada Limited 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 MR Q = W optimal labour MRP L W W  MFC L wage

13 © 2006 by Nelson, a division of Thomson Canada Limited MRP L is the Demand for Labour If Labour is MORE productive, demand for labour increases If Labour is LESS productive, demand for labour decreases EARTHQUAKESuppose an EARTHQUAKE destroys capital  MP L declines with less capital, wages and labour are HURT D L D’ L S L WW L’ L

14 © 2006 by Nelson, a division of Thomson Canada Limited 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 »MP L =  Q /  L or  Q /  L MP of labour declines as more labour is applied. Also the MP of capital declines as more capital is applied. Long-Run Production Functions

15 © 2006 by Nelson, a division of Thomson Canada Limited Isoquants & LR Production Functions 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 SLOPE of 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 B A C Q1 Q2 Q3 K L

16 © 2006 by Nelson, a division of Thomson Canada Limited The objective is to minimize cost for a given output ISOCOST lines are the combination of inputs for a given cost, C 0 C 0 = C L ·L + C K ·K K = C 0 /C K - (C L /C K )·L Optimal where: » MP L /MP K = C L /C K · »Rearranged, this becomes the equimarginal criterion Equimarginal Criterion: Produce where MP L /C L = MP K /C K where marginal products per dollar are equal Figure 4.9 Q (1) D L K at D, slope of isocost = slope of isoquant C (1) Optimal Combination of Inputs

17 © 2006 by Nelson, a division of Thomson Canada Limited 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 C 0 = C L ·L + C K ·K »C = 50L + 0.2K, rearranging K = C/.2 – (250/.2)L »K = C/0.2 – 250L K L K=C/0.2 – CL Q= f(L,K)

18 © 2006 by Nelson, a division of Thomson Canada Limited Use of the Equimarginal Criterion Q: Is the following firm EFFICIENT? Suppose that: »MP L = 30 »MP K = 50 »C L = 10 (labour cost) »C K = 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.

19 © 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: MP L /C L = MP K /C K 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. Q (1) D Q (0) T

20 © 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

21 © 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)

22 © 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,  ) = 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   +  

23 © 2006 by Nelson, a division of Thomson Canada Limited Cobb-Douglas Production Functions Q = α L   K   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 Coefficients are elasticities    is the labour elasticity of output, often about 0.33   is the capital elasticity of output, often about 0.67 which are E L and E K Most firms have some slight increasing returns to scale

24 © 2006 by Nelson, a division of Thomson Canada Limited Problem Suppose: Q = 1.4 L 0.70 K Is this function homogeneous? 2.Is the production function constant returns to scale? 3.What is the production elasticity of labour? 4.What is the production elasticity of capital? 5.What happens to Q, if L increases 3% and capital is cut 10%?

25 © 2006 by Nelson, a division of Thomson Canada Limited Answers 1.Yes. Increasing all inputs by, increases output by It is homogeneous of degree No, it is not constant returns to scale. It is increasing Returns to Scale, since 1.05 > is the production elasticity of labour is the production elasticity of capital 5.%  Q = E L %  L+ E K %  K = 0.70(+3%) (-10%) = 2.1% -3.5% = - 1.4%