Egwald Economics: Microeconomics
Production Functions
by
Elmer G. Wiens
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CobbDouglas
 CES
 Generalized CES
 Translog
 Diewert
 Translog vs Diewert
 Diewert vs Translog
 Estimate Translog
 Estimate Diewert
 References and Links
Cost Functions:
CobbDouglas Cost
 Normalized Quadratic Cost
 Translog Cost
 Diewert Cost
 Generalized CESTranslog Cost
 Generalized CESDiewert Cost
 References and Links
Duality: Production / Cost Functions:
CobbDouglas Duality
 CES Duality
 Theory of Duality
 Translog Duality  CES
 Translog Duality  Generalized CES
While CobbDouglas production functions are great, because they are easy to estimate, the elasticity of substitution between factors is always equal to 1. CES production functions permit you to vary the elasticity of substitution.
B. CES (Constant Elasticity of Substitution) Production Function
The three factor CES production function is:
q = A * [alpha * (L^^{rho}) + beta * (K^^{rho}) + gamma *(M^^{rho})]^^{(nu/rho)} = f(L,K,M).
where L = labour, K = capital, M = materials and supplies, and q = product. The parameter nu is a measure of the economies of scale, while the parameter rho yields the elasticity of substitution:
sigma = 1/(1 + rho).
Separating the elasticity of scale, nu, from the other parameters is facilitated when:
alpha + beta + gamma = 1.
I. Decreasing returns to scale: nu < 1
With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant.
Let's assume the CES production function's parameters are:
A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=.92.
Note that rho = .17647 > sigma = .85
The CES production function has the form:
q = 1.0 * [.3 * (L^^{.17647}) + .4 * (K^^{.17647}) + .3 *(M^^{.17647})]^^{(.92/.17647)}
Suppose the firm can buy its factors at the prices:
wL = 7, wK = 13, wM = 6. Its costs will be:
c(q) = wL * L + wK * K + wM * M = 7 * L + 13 * K + 6 * L
To produce 35 units of product at minimum cost, it should use: L = 51.45, K = 38.82, and M = 58.86 units of inputs.
Notes:
1. 35 = 1. * [.3 * (51.45^{^.17647}) + .4 * (38.82^{^.17647}) + .3 * (58.86 ^{^.17647})]^(^{.92/.17647})
2.
c(q) = wL * L + wK * K + wM * M >
1216.78 = 7 * 51.45 + 13 * 38.82 + 6 * 58.86
3. Average cost = c(q)/q > 1216.78 / 35 = 34.77
Graph of the Average Cost and Marginal Cost for a CES Production Function  


Average cost function 

Marginal cost function 

Decreasing returns to scale  
Since our CES production function has decreasing returns to scale, the average cost and marginal cost are increasing, and marginal cost is greater than average cost.
CES Production Function Data Decreasing Returns to Scale 
q  L  K  M  cost  ave.cost  marg.cost  s_{LK}  s_{LM}  s_{KM} 
5  6.21  4.68  7.07  146.77 
29.35  31.91  0.85  0.85  0.85 
10  13.18  9.95  15.03  311.77 
31.18  33.89  0.85  0.85  0.85 
15  20.48  15.46  23.35  484.44 
32.3  35.1  0.85  0.85  0.85 
20  28  21.13  31.93  662.28 
33.11  35.99  0.85  0.85  0.85 
25  35.69  26.93  40.69  844.07 
33.76  36.7  0.85  0.85  0.85 
30  43.51  32.83  49.61  1029.07 
34.3  37.29  0.85  0.85  0.85 
35  51.45  38.82  58.66  1216.78 
34.77  37.79  0.85  0.85  0.85 
40  59.49  44.89  67.82  1406.85 
35.17  38.23  0.85  0.85  0.85 
45  67.61  51.02  77.08  1599 
35.53  38.62  0.85  0.85  0.85 
50  75.82  57.21  86.43  1793.02 
35.86  38.98  0.85  0.85  0.85 
The terms s_{LK}, s_{LM}, and s_{KM} are the Allen partial elasticities of substitution. Here they all equal .85, because this is the constant elasticity of substitution between inputs given the value of rho = .17647. These measures are important for such production functions as the translog and Diewert, where they are not necessarily constant.
II. Constant returns to scale: nu = 1
With constant returns to scale, a proportional increase in all inputs will increase output by the proportional constant.
A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=1.0, sigma=.85.
factor prices: wL = 7, wK = 13, wM = 6.
The CES production function has the form:
q = 1. * [.3 * (L^{^.17647}) + .4 * (K^{^.17647}) + .3 * (M ^{^.17647})]^(^{1/.17647})
Graph of the Average Cost and Marginal Cost for a CES Production Function  


Average cost function 

Marginal cost function 

Constant returns to scale  
The average cost and marginal cost curves coincide, a consequence of constant returns to scale.
CES Production Function Data Constant Returns to Scale 
q  L  K  M  cost  ave.cost  marg.cost  s_{LK}  s_{LM}  s_{KM} 
5  5.4  4.07  6.15  127.6 
25.52  25.52  0.85  0.85  0.85 
10  10.79  8.14  12.3  255.2 
25.52  25.52  0.85  0.85  0.85 
15  16.19  12.21  18.45  382.8 
25.52  25.52  0.85  0.85  0.85 
20  21.58  16.28  24.6  510.4 
25.52  25.52  0.85  0.85  0.85 
25  26.98  20.36  30.75  638 
25.52  25.52  0.85  0.85  0.85 
30  32.37  24.43  36.91  765.59 
25.52  25.52  0.85  0.85  0.85 
35  37.77  28.5  43.06  893.19 
25.52  25.52  0.85  0.85  0.85 
40  43.16  32.57  49.21  1020.79 
25.52  25.52  0.85  0.85  0.85 
45  48.56  36.64  55.36  1148.39 
25.52  25.52  0.85  0.85  0.85 
50  53.96  40.71  61.51  1275.99 
25.52  25.52  0.85  0.85  0.85 
III. Increasing returns to scale: nu = 1.08 > 1
With increasing returns to scale, a proportional increase in all inputs will increase output by more than the proportional constant.
A = 1.0, alpha = .3, beta = .4, gamma = .3, rho = .17647, nu=1.08, sigma=.85.
factor prices: wL = 7, wK = 13, wM = 6.
The CES production function has the form:
q = 1. * [.3 * (L^{^.17647}) + .4 * (K^{^.17647}) + .3 * (M ^{^.17647})]^(^{1.08/.17647})
Graph of the Average Cost and Marginal Cost for a CES Production Function  


Average cost function 

Marginal cost function 

Increasing returns to scale  
Both average cost and marginal cost are decreasing, with marginal cost below average cost, a consequence of increasing returns to scale.
CES Production Function Data Increasing Returns to Scale 
q  L  K  M  cost  ave.cost  marg.cost  s_{LK}  s_{LM}  s_{KM} 
5  4.79  3.61  5.46  113.26 
22.65  20.97  0.85  0.85  0.85 
10  9.1  6.87  10.37  215.18 
21.52  19.92  0.85  0.85  0.85 
15  13.24  9.99  15.1  313.22 
20.88  19.33  0.85  0.85  0.85 
20  17.29  13.04  19.71  408.82 
20.44  18.93  0.85  0.85  0.85 
25  21.25  16.04  24.23  502.65 
20.11  18.62  0.85  0.85  0.85 
30  25.16  18.99  28.69  595.09 
19.84  18.37  0.85  0.85  0.85 
35  29.02  21.9  33.09  686.39 
19.61  18.16  0.85  0.85  0.85 
40  32.84  24.78  37.44  776.72 
19.42  17.98  0.85  0.85  0.85 
45  36.63  27.64  41.76  866.22 
19.25  17.82  0.85  0.85  0.85 
50  40.38  30.47  46.04  954.99 
19.1  17.68  0.85  0.85  0.85 
IV. Short Run:
Economists often assume that capital is fixed in the shortrun. While the quantities of labour, and materials and supplies can be adjusted, changing the amount of capital services, quickly, is costly. To model the shortrun production activities of a firm, capital will be set at the level that is associated with producing q = 30 units of product. Note that the values of the coefficients alpha, beta, and gamma have been changed below, to facilitate comparison with the translog and Diewert production functions.
1. Decreasing returns to scale:
A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = .92, sigma = .85.
factor prices: wL = 7, wK = 13, wM = 6.
Set capital
K = 32.82, the amount of capital associated with producing q = 30 units of product.
Graph of Average Cost and Marginal Cost CES Production Function  Capital Fixed  


Average cost function 

Marginal cost function 

L.R. Average cost function 

Decreasing economies of scale  
Now we get the traditional Ushaped average, short run cost curve, with a minimum to the left of q = 30.
Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
CES Production Function Data Decreasing economies of scale Fixed Capital 
q  L  K  M  cost  ave.cost  marg. cost  3 factor s_{LM}  2 factor s_{LM} 
5  2.44  32.82  2.09  456.3 
91.26  9.14  0.85  0.85 
10  7.34  32.82  6.29  515.74 
51.57  14.59  0.85  0.85 
15  14.45  32.82  12.38  602.09 
40.14  19.96  0.85  0.85 
20  23.81  32.82  20.39  715.64 
35.78  25.49  0.85  0.85 
25  35.48  32.82  30.39  857.36 
34.29  31.24  0.85  0.85 
30  49.58  32.82  42.46  1028.5 
34.28  37.26  0.85  0.85 
35  66.22  32.82  56.72  1230.5 
35.16  43.59  0.85  0.85 
40  85.54  32.82  73.26  1464.92 
36.62  50.23  0.85  0.85 
45  107.65  32.82  92.2  1733.41 
38.52  57.22  0.85  0.85 
50  132.72  32.82  113.67  2037.71 
40.75  64.56  0.85  0.85 
Note that q = 30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve.
Here I have listed two measures of the shortrun elasticity of substitution between L and M. The 3factor measure of s_{LM} uses the Allen partial elasticity of substitution formula. The 2factor measure of s_{LM} uses the standard shortrun formula, which assumes that only L and M can vary with output, with a fixed amount of capital present. Here again, these shortrun elasticities all equal .85, because the production function is CES.
2. Constant returns to scale:
A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = 1.0, sigma = .85.
factor prices: wL = 7, wK = 13, wM = 6.
Set capital
K = 24.42, the amount of capital associated with producing q = 30 units of product.
Graph of Average Cost and Marginal Cost CES Production Function  Capital Fixed  


Average cost function 

Marginal cost function 

L.R. Average cost function 

Constant economies of scale  
The traditional Ushaped average, short run cost curve has a minimum at q = 30.
Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
CES Production Function Data Constant economies of scale Fixed Capital 
q  L  K  M  cost  ave.cost  marg. cost  3 factor s_{LM}  2 factor s_{LM} 
5  2.27  24.42  1.94  344.96 
68.99  7.9  0.85  0.85 
10  6.31  24.42  5.4  394 
39.4  11.64  0.85  0.85 
15  11.82  24.42  10.13  460.93 
30.73  15.12  0.85  0.85 
20  18.76  24.42  16.07  545.1 
27.25  18.55  0.85  0.85 
25  27.11  24.42  23.22  646.44 
25.86  22  0.85  0.85 
30  36.89  24.42  31.59  765.17 
25.51  25.51  0.85  0.85 
35  48.13  24.42  41.22  901.63 
25.76  29.09  0.85  0.85 
40  60.87  24.42  52.13  1056.26 
26.41  32.77  0.85  0.85 
45  75.14  24.42  64.36  1229.55 
27.32  36.56  0.85  0.85 
50  91  24.42  77.94  1422.05 
28.44  40.46  0.85  0.85 
Note that q = 30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve.
3. Increasing returns to scale:
A = 1.0, alpha = .35, beta = .4, gamma = .25, rho = .17647, nu = 1.08, sigma = .85.
factor prices: wL = 7, wK = 13, wM = 6.
Set capital
K = 18.98, the amount of capital associated with producing q = 30 units of product.
Graph of Average Cost and Marginal Cost CES Production Function  Capital Fixed  


Average cost function 

Marginal cost function 

L.R. Average cost function 

Increasing economies of scale  
The traditional Ushaped average, short run cost curve has a minimum to the right of q = 30.
Because marginal cost is virtually a linear function of q, total cost (with capital fixed) is virtually a quadratic function of q (since its derivative, marginal cost, is linear).
CES Production Function Data Constant economies of scale Fixed Capital 
q  L  K  M  cost  ave.cost  marg. cost  3 factor s_{LM}  2 factor s_{LM} 
5  2.14  18.98  1.83  272.65 
54.53  6.96  0.85  0.85 
10  5.55  18.98  4.76  314.12 
31.41  9.55  0.85  0.85 
15  9.97  18.98  8.54  367.73 
24.52  11.87  0.85  0.85 
20  15.31  18.98  13.11  432.59 
21.63  14.07  0.85  0.85 
25  21.55  18.98  18.46  508.32 
20.33  16.22  0.85  0.85 
30  28.67  18.98  24.56  594.76 
19.83  18.36  0.85  0.85 
35  36.67  18.98  31.41  691.89 
19.77  20.5  0.85  0.85 
40  45.56  18.98  39.02  799.74 
19.99  22.65  0.85  0.85 
45  55.34  18.98  47.39  918.41 
20.41  24.82  0.85  0.85 
50  66.01  18.98  56.54  1048.02 
20.96  27.03  0.85  0.85 
Note that q = 30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, CES average cost curve.
V. Isoquants
A given level of output, q = q, can be produced by different combinations of factor inputs, L, K, and M.
Fixing the level of product output at q = q, we obtain an equation from the CES production function:
q = A * [alpha * (L^^{rho}) + beta * (K^^{rho}) + gamma *(M^^{rho})]^^{(nu/rho)} = f(L,K,M).
for the 3dimensional isoquant surface, when q = q.
The isoquant surface is tangent to the isocost plane:
C(q) = wL * L + wK * K + wM * M
at the cost minimizing combination of factor inputs, (L, K, M ) = (L, K, M).
Consider again the specific CES production function:
q = 1 * [0.3 * (L^^{0.17647}) + 0.4 * (K^^{0.17647}) + 0.3 *(M^^{0.17647})]^^{(1.08/0.17647)} = f(L,K,M).
When q = 30 and (wL, wK, wM) = (7, 13, 6), the cost minimizing inputs are:
(L, K, M) = (25.16, 18.99, 28.69), and
C(30) = 7 * 25.16 + 13 * 18.99 + 6 * 28.69 = 595.09.
Solving the CES equation for L, K, and M in turn, we get:
1. L = [(q / A)^^{(rho/nu)}  beta * K^^{rho}  gamma * M^^{rho}) / alpha]^^{(1/rho)},
2. K = [(q / A)^^{(rho/nu)}  alpha * L^^{rho}  gamma * M^^{rho}) / beta]^^{(1/rho)},
3. M = [(q / A)^^{(rho/nu)}  alpha * L^^{rho}  beta * K^^{rho}) / gamma]^^{(1/rho)},

three equations for the 3dimensional isoquant surface.
By fixing the amount of input for one factor, we obtain a 2dimensional isoquant curve. As examples, fixing M = M in equation 2, and K = K in equation 3, we get:
2. → LK Isoquant: K = [(q / A)^^{(rho/nu)}  alpha * L^^{rho}  gamma * M^^{rho}) / beta]^^{(1/rho)},
3. → LM Isoquant: M = [(q / A)^^{(rho/nu)}  alpha * L^^{rho}  beta * K^^{rho}) / gamma]^^{(1/rho)},

with K and M as functions of one variable, L. The following diagrams graph, in blue, the LK and LM isoquants for q = 24, 30, and 36.
The yellow lines represent the isocost lines, combinations of L, K, and M that can be purchased at a constant total cost at the prices wL = 7, wK = 13, and wM = 6.
The slope of an LK isocost line is m_{K} = wL / wK = 7 / 13; the slope of an LM isocost line is m_{M} = wL / wM = 7 / 6.
For q = 30, the LK isocost line has a Kintercept at (C(30)  wM * M) / wK = (595.09  6 * 28.69)/13 = 32.53, while the LM isocost line has a Mintercept at (C(30)  wK * K) / wM = (595.09  13 * 18.99)/6 = 58.04.
For q = 30, the LK isoquant is tangent to the LK isocost line at (L, K) = (25.16, 18.99), while the LM isoquant is tangent to the LM isocost line at (L, M) = (25.16, 28.69).
LK Isoquants, M = M  LM Isoquants, K = K 


The red rays emanating from the origin in the diagrams intersect each isoquant at the same angle. Consequently, any isoquant is a radial projection of each other isoquant. In particular, any isoquant is a radial projection of the unit isoquant, i.e. the isoquant for q = 1. Production functions with this property are called homothetic production functions.
VI. Formulae
The three factor CES production function is:
q = f(L,K,M) = A * [alpha * (L^^{rho}) + beta * (K^^{rho}) + gamma *(M^^{rho})]^^{(nu/rho)}.
a. Marginal Product of labour:
∂f(L, K, M)/∂L = f_{L} = alpha*nu*L ^^{(1+rho)}*A*[alpha * (L^^{rho}) + beta * (K^^{rho}) + gamma *(M^^{rho})]^^{(nu/rho)  1}
= alpha*nu*L ^^{(1+rho)} * A^^{(rho/nu)} * A^^{(1+rho/nu)}*[alpha*(L^^{rho}) + beta*(K^^{rho}) + gamma*(M^^{rho})]^^{(nu/rho)(1+rho/nu)}
= alpha*nu*L ^^{(1+rho)} * A^^{(rho/nu)} * q^^{(1+rho/nu)}
b. Marginal cost function: if (L,K,M) is the cost minimizing combination of inputs at prices (wL,wK,wM) for output q, then
C'(q) = ∂C/∂q = wL / (∂f(L,K,M)/∂L) = µ
VII. Leastcost combination of inputs
Find the values of L, K, M, and µ that minimize the Lagrangian:
G(q;L,K,M,µ) = wL * L + wK * K + wM* M + µ * [q  f(L,K,M)]
 G_{L} = wL  µ * f_{L} = 0
 G_{K} = wK  µ * f_{K} = 0
 G_{M} = wM  µ * f_{M} = 0
 G_{µ} = q  f(L,K,M) = 0
From equations a., b., and c. we get:
 wL / wK = f_{L} / f_{K} = (alpha/beta) * (K/L)^^{(1+rho)}
> K = L * (wL*beta/(wK*alpha))^^{(1/(1+rho))}
 wL / wM = f_{L} / f_{M} = (alpha/gamma) * (M/L)^^{(1+rho)}
> M = L * (wL*gamma/(wM*alpha))^^{(1/(1+rho))}
 wK / wM = f_{K} / f_{M} = (beta/gamma) * (M/K)^^{(1+rho)}
Substituting equations e. and f. into the CES production function and solving for L yields;

L = (q/A)^^{1/nu} * [alpha + beta*(wL*beta/(wK*alpha))^^{(rho/(1+rho))} + gamma(wL*gamma/(wM*alpha))^^{(rho/(1+rho))}]^^{(1/rho)}
= (q/A)^^{1/nu} * (alpha / wL)^^{(1/(1+rho))} * [alpha^^{(1/(1+rho))} * wL^^{(rho/(1+rho))} + beta^^{(1/(1+rho))} * wK^^{(rho/(1+rho))} + gamma^^{(1/(1+rho))} * wM^^{(rho/(1+rho))}]^^{(1/rho)}
Finally, substituting e., f. and h. into the cost function:
C(q) = wL * L + wK * K + wM * M
yields the cost function, as a function of output, depending on the input prices and the parameters of the CES production function.
VIII. CES Cost Function
If we actually solve explicitly for C(q):
C(q;wL,wK,wM) = h(q) * c(wL,wK,wM)
where the returns to scale function is:
h(q) = (q/A)^^{1/nu}
a continuous, increasing function of q (q >= 1), with h(0) = 0 and h(1)=1,
and the unit cost function is:
c(wL,wK,wM) = [alpha^^{(1/(1+rho))} * wL^^{(rho/(1+rho))} + beta^^{(1/(1+rho))} * wK^^{(rho/(1+rho))} + gamma^^{(1/(1+rho))} * wM^^{(rho/(1+rho))}]^^{((1+rho)/rho)}
The unit cost function c(wL, wK, wM) looks, interestingly, like its parent  the CES production function.
The CES production function is called homothetic, because the CES cost function can be separated (factored) into a function of output, q, times a function of input prices, wL, wK, and wM.
IX. Factor demand functions:
If we take the derivative of the cost function with respect to an input price, we get the factor demand function for that input:
∂C/∂wL = h(q) * [(alpha / wL) * c(wL,wK,wM)]^^{(1/(1+rho))} = L
∂C/∂wK = h(q) * [(beta / wK) * c(wL,wK,wM)]^^{(1/(1+rho))} = K
∂C/∂wM = h(q) * [(gamma / wM) * c(wL,wK,wM)]^^{(1/(1+rho))} = M
X. Properties of the unit CES Cost Function, c(wL,wK,wM).
a. c is linear homogeneous in factor prices.
b. c is concave in factor prices.
XI. Elasticity of substitution between inputs (sigma).
From equation e. of Part VI we get:
K/L = [(beta / alpha)* (wl / wK)]^^{1/(1+rho)} → ln(K/L) = (1/(1+rho))*ln(beta/alpha) + (1/(1+rho))*ln(wL/wK)
sigma = d(ln(K/L))/d(ln(wL/wK)) = 1/(1+rho)
a. K and L substitutes:
1 < rho < 0, then 1 < sigma < infinity
b. K and L complements:
0 < rho < infinity, then 0 < sigma < 1
XII. Allen partial elasticity of substitution
Writing the production function as q = F(L,K,M), let the bordered Hessian be:
F = 
0  F_{L}  F_{K} 
F_{M} 
F_{L}  F_{LL}  F_{LK} 
F_{LM} 
F_{K}  F_{KL}  F_{KK} 
F_{KM} 
F_{M}  F_{ML}  F_{MK} 
F_{MM} 
If F is the determinant of the bordered Hessian and F_{LK} is the cofactor associated with F_{LK}, then the Allen elasticity of substitution is defined as:
s_{LK} = ((F_{L} * L + F_{K} * K + F_{M} *M) / (L * K)) * (F_{LK}/F)
XIII. Two factor elasticity of substitution
Let the production function be q = F(L,K,M), where K is underlined to indicate it is constant in the shortrun. Then the two factor (L and M) bordered Hessian is:
F = 
0  F_{L}  F_{M} 
F_{L}  F_{LL}  F_{LM} 
F_{M}  F_{ML}  F_{MM} 
The 2factor elasticity of substitution between L and M is:
s_{LM} =  (F_{L} * L + F_{M} * M) / (L * M ) * (F_{L} * F_{M}) / F
XIII. Further examples:
The web page, "The Duality of Production and Cost Functions," permits one to specify the parameters of the CES (or CobbDouglas) production function, and to ascertain the curvature of the production function and corresponding cost function.
