Production Function The firms 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 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 q marginal physical product of capital MPK fK K q marginal physical product of labor MPL fL L 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 Average Physical Product Labor productivity is often measured by average productivity output q f (K , L ) APL
labor input L L Note that APL also depends on the amount of capital employed A Two-Input Production Function Suppose the production function for flyswatters can be represented by q = f(K,L) = 600K 2L2 - K 3L3
To construct MPL and APL, we must assume a value for K Let K = 10 The production function becomes q = 60,000L2 - 1000L3 A Two-Input Production Function The
marginal productivity function is MPL = q/L = 120,000L - 3000L2 which diminishes as L increases This implies that q has a maximum value: 120,000L - 3000L2 = 0 40L = L2 L = 40 Labor input beyond L=40 reduces output A Two-Input Production Function To
find average productivity, we hold K=10 and solve APL = q/L = 60,000L - 1000L2 APL reaches its maximum where APL/L = 60,000 - 2000L = 0 L = 30 A Two-Input Production Function In fact, when L=30, both APL and MPL are equal to 900,000
Thus, when APL is at its maximum, APL and MPL are equal 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 K per period q = 30 q = 20 L per period
Marginal Rate of Technical Substitution (MRTS) The slope of an isoquant shows the rate at which L can be substituted for K K per period KA - slope = marginal rate of technical substitution (MRTS) A B
KB MRTS > 0 and is diminishing for increasing inputs of labor q = 20 LA LB L per period Marginal Rate of Technical Substitution (MRTS) The
marginal rate of technical substitution (MRTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant dK MRTS ( L for K ) dL q q0 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 Returns to Scale Smith identified two forces that come into operation as inputs are doubled greater division of labor and specialization of labor loss in efficiency because management may
become more difficult given the larger scale of the firm Returns to Scale 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 economists refer to the degree of returns to scale with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered
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 No substitution between labor and capital is possible K/L is fixed at b/a K per period q3 q3/a q2 q1 q3/b 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(mK,mL) = A(mK)a(mL) b = Ama+b KaLb = ma+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 Suppose that hamburgers are produced according to the Cobb-Douglas function q = 10K 0.5 L0.5 Since a+b=1 constant returns to scale The isoquant map can be derived q = 50 = 10K 0.5 L0.5 KL = 25
q = 100 = 10K 0.5 L0.5 KL = 100 The isoquants are rectangular hyperbolas Cobb-Douglas Production Function The MRTS can easily be calculated f L 5 L 0.5 K 0.5 K MRTS ( L for K ) 0.5 0.5 f K 5L K L The MRTS declines as L rises and K falls
The MRTS depends only on the ratio of K and L 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
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