Chapter 9 Production Functions Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.

Production Function The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l) q = f(k,l)

Marginal Physical Product Marginal physical product is the additional output that can be produced by employing one more unit of that input holding other inputs constant

Diminishing Marginal Productivity Marginal physical product depends on how much of that input is used In general, we assume diminishing marginal productivity

Average Physical Product Labor productivity is often measured by average productivity Note that APl also depends on the amount of capital employed

Isoquant Maps To illustrate the possible substitution of one input for another, we use an isoquant map An isoquant shows those combinations of k and l that can produce a given level of output (q0) f(k,l) = q0

Isoquant Map Each isoquant represents a different level of output output rises as we move northeast q = 30 q = 20 k per period l per period

Marginal Rate of Technical Substitution (RTS) The slope of an isoquant shows the rate at which l can be substituted for k lA kA kB lB A B k per period - slope = marginal rate of technical substitution (RTS) RTS > 0 and is diminishing for increasing inputs of labor q = 20 l per period

Marginal Rate of Technical Substitution (RTS) The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital holding output constant along an isoquant

RTS and Marginal Productivities Take the total differential of the production function: Along an isoquant dq = 0, so

RTS and Marginal Productivities Because MPl and MPk will both be nonnegative, RTS will be positive (or zero) However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone

RTS and Marginal Productivities To show that isoquants are convex, we would like to show that d(RTS)/dl < 0 Since RTS = fl/fk

RTS and Marginal Productivities Using the fact that dk/dl = -fl/fk along an isoquant and Young’s theorem (fkl = flk) Because we have assumed fk > 0, the denominator is positive Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is positive

Returns to Scale If the production function is given by q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then

Constant Returns to Scale Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants k per period q = 3 q = 2 q = 1 The isoquants are equally spaced as output expands l per period

Elasticity of Substitution The elasticity of substitution () measures the proportionate change in k/l relative to the proportionate change in the RTS along an isoquant The value of  will always be positive because k/l and RTS move in the same direction

The Linear Production Function Suppose that the production function is q = f(k,l) = ak + bl This production function exhibits constant returns to scale f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l) All isoquants are straight lines RTS is constant  = 

The Linear Production Function Capital and labor are perfect substitutes k per period RTS is constant as k/l changes slope = -b/a q1 q2 q3  =  l per period

Fixed Proportions Suppose that the production function is q = min (ak,bl) a,b > 0 Capital and labor must always be used in a fixed ratio the firm will always operate along a ray where k/l is constant Because k/l is constant,  = 0

Fixed Proportions No substitution between labor and capital is possible k/l is fixed at b/a q3/b q3/a k per period q1 q2 q3  = 0 l per period

Cobb-Douglas Production Function Suppose that the production function is q = f(k,l) = Akalb A,a,b > 0 This production function can exhibit any returns to scale f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l) if a + b = 1  constant returns to scale if a + b > 1  increasing returns to scale if a + b < 1  decreasing returns to scale

Cobb-Douglas Production Function The Cobb-Douglas production function is linear in logarithms ln q = ln A + a ln k + b ln l a is the elasticity of output with respect to k b is the elasticity of output with respect to l

CES Production Function Suppose that the production function is q = f(k,l) = [k + l] /   1,   0,  > 0  > 1  increasing returns to scale  < 1  decreasing returns to scale For this production function  = 1/(1-)  = 1  linear production function  = -  fixed proportions production function  = 0  Cobb-Douglas production function

CES variants

A Generalized Leontief Production Function Suppose that the production function is q = f(k,l) = k + l + 2(kl)0.5 Marginal productivities are fk = 1 + (k/l)-0.5，fl = 1 + (k/l)0.5

A Generalized Leontief Production Function This function has a CES form with  = 0.5 and  = 1 The elasticity of substitution is

Technical Progress Methods of production change over time Following the development of superior production techniques, the same level of output can be produced with fewer inputs the isoquant shifts in

Technical Progress Suppose that the production function is q = A(t)f(k,l) where A(t) represents all influences that go into determining q other than k and l changes in A over time represent technical progress A is shown as a function of time (t) dA/dt > 0

Technical Progress Differentiating the production function with respect to time we get

Technical Progress Dividing by q gives us

Technical Progress For any variable x, [(dx/dt)/x] is the proportional growth rate in x denote this by Gx Then, we can write the equation in terms of growth rates

Technical Progress Since