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 f_{l}_{k} 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 AP_{l} 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 (q_{0}) f(k,l) = q_{0}

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 = f_{l}/f_{k}
Using the fact that dk/dl = f_{l}/f_{k} along an isoquant and Young’s theorem (f_{k}_{l} = f_{l}_{k})
Intuitively, it seems reasonable that f_{k}_{l}_{ }= f_{l}_{k} should be positive if workers have more capital, they will be more productive (have higher marginal productivity) But some production functions have f_{k}_{l} < 0 over some input ranges when we assume diminishing RTS we will assume that MP_{l} and MP_{k} diminish quickly enough to compensate for any possible negative crossproductivity effects Returns to Scale How does output respond to increases in all inputs together? suppose that all inputs are doubled, would output double? 
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(x_{1},x_{2},…,x_{n}) If all inputs are multiplied by a positive constant t, we have f(tx_{1},tx_{2},…,tx_{n}) = t^{k}f(x_{1},x_{2},…,x_{n})=t^{k}q If k = 1, we have constant returns to scale If k < 1, we have decreasing returns to scale 
Constant Returns to Scale Constant returnstoscale production functions are homogeneous of degree one in inputs f(tk,tl) = t^{1}f(k,l) = tq

if a function is homogeneous of degree k, its derivatives are homogeneous of degree k1 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 high, the RTS will not change much relative to k/l 
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 
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
q_{3}/a
q_{3}
q_{2}
q_{1}
l per period
q_{3}/b
Substitution between capital and labor is not possible.
CobbDouglas Production Function Suppose that the production function is q = f(k,l) = Ak^{a}l^{b} A,a,b > 0 This production function can exhibit any returns to scale f(tk,tl) = A(tk)^{a}(tl)^{b} = At^{a}^{+}^{b }k^{a}l^{b} = t^{a}^{+}^{b}f(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 CobbDouglas 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
q_{3}
q_{1}
q_{2}
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 CobbDouglas production function same level of output can be produced with fewer inputs 