Production functions production Function



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PRODUCTION FUNCTIONS


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 (MP)

  • To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant









Diminishing Marginal Productivity

  • The marginal physical product of an input depends on how much of that input is used

  • In general, we assume diminishing marginal productivity








  • Because of diminishing marginal productivity, 19th century economist Thomas Malthus worried about the effect of population growth on labor productivity

  • But changes in the marginal productivity of labor over time also depend on changes in other inputs such as capital

  • we need to consider flk which is often > 0

  • In fact, increases in capital input have offset the impact of diminishing marginal productivity

Average Physical Product (AP)



  • 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





  • Each isoquant represents a different level of output

      • output rises as we move northeast


Marginal Rate of Technical Substitution (RTS)

  • The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. This is, in fact, the slope of the isoquant.




  • Take the total differential of the production function:







  • Along an isoquant dq = 0, so






Diminishing RTS

  • It is generally not possible to derive a diminishing RTS (convex isoquants) from the assumption of diminishing marginal productivity alone

  • To show that isoquants are convex, we would like to show that d(RTS)/dl < 0

  • Since RTS = fl/fk













  • Intuitively, it seems reasonable that fkl = flk should be positive

  • if workers have more capital, they will be more productive (have higher marginal productivity)

  • But some production functions have fkl < 0 over some input ranges

  • when we assume diminishing RTS we will assume that MPl and MPk diminish quickly enough to compensate for any possible negative cross-productivity effects

Returns to Scale

  • How does output respond to increases in all inputs together?

  • suppose that all inputs are doubled, would output double?

  • Returns to scale have been of interest to economists since the days of Adam Smith




  • Smith identified two forces that come into operation as inputs are doubled

  • greater division of labor and specialization of function

  • loss in efficiency because management may become more difficult given the larger scale of the firm

  • If the production function is given by

q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then







  • It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels

Generalization

q = f(x1,x2,…,xn)

  • If all inputs are multiplied by a positive constant t, we have

f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq

  • If k = 1, we have constant returns to scale

  • If k < 1, we have decreasing returns to scale

  • If k > 1, we have increasing returns to scale

Constant Returns to Scale

  • Constant returns-to-scale production functions are homogeneous of degree one in inputs

f(tk,tl) = t1f(k,l) = tq




  • This implies that the marginal productivity functions are homogeneous of degree zero

  • if a function is homogeneous of degree k, its derivatives are homogeneous of degree k-1

  • The marginal productivity of any input depends on the ratio of capital and labor (not on the absolute levels of these inputs)

  • The RTS between k and l depends only on the ratio of k to l, not the scale of operation

  • The production function will be homothetic.

  • Geometrically, all of the isoquants are radial expansions of one another. Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants



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



Both RTS and k/l will change as we move from point A to point B






  • If is low, the RTS will change by a substantial amount as k/l changes

  • the isoquant will be sharply curved




  • It is possible for to change along an isoquant or as the scale of production changes

The Linear Production Function

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

  • =

Capital and labor are perfect substitutes



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


k per period

k/l is fixed at b/a


= 0
q3/a



q3


q2
q1
l per period

q3/b
Substitution between capital and labor is not possible.

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

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


k per period

q3

q1

q2

l per period

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

Technical Progress

same level of output can be produced with fewer inputs

  • the isoquant shifts in



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