Egwald Economics: Microeconomics
Production Functions
by
Elmer G. Wiens
Egwald's popular web pages are provided without cost to users. Please show your support by joining Egwald Web Services as a Facebook Fan:
Follow Elmer Wiens on Twitter:
Production Functions:
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
In the oligopoly / government enterprise model, I derived each firm's short run cost function from a quadratic, long run, average cost function. Now I will show how cost functions look when they are obtained from a production function. Recall that a production function produces levels of output for combinations of inputs. A profit maximizing firm will try to use a combination of inputs that will minimize its cost of producing a given level of output.
A. CobbDouglas Production Function
If you have not already done so, look at how the parameters of a CobbDouglas production function can be estimated: Estimating a CobbDouglas production function.
The three factor CobbDouglas production function is:
q = A * (L^^{alpha}) * (K^^{beta}) * (M^^{gamma}) = f(L,K,M).
where L = labour, K = capital, M = materials and supplies, and q = product. The symbol "^" means "raise to the power," i.e. L^^{alpha} means "raise the value of L to the power of the value of alpha."
Production functions need to have certain properties, to ensure that we can solve the leastcost problem: Check any of the many textbooks. If for given values of L,K, and M, the Hessian of the production function f is negative definite, then its isoquants at that point are concave to the origin.
I. Decreasing returns to scale: alpha + beta + gamma < 1
With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant.
When we estimated the CobbDouglas production function, we found that:
A = 1.01278, alpha = .317, beta = .417, and gamma = .186.
alpha + beta + gamma = .317 + .417 + .186 = .92 < 1
Then, q = A * (L^^{alpha}) * (K^^{beta}) * (M^^{gamma}) >
q = 1.01278 * (L^{^.317}) * (K^{^.417}) * (M^{^.186})
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
Then to produce 35 units of product at minimum cost, it should use:
L = 59.36, K = 42.05, and M = 40.64 units of inputs.
Notes:
1.
35 = 1.01278 * (59.36^{^.317}) * (42.05^{^.417}) * (40.64^{^.186})
2.
c(q) = 7 * L + 13 * K + 6 * M >
1205 = 7 * 59.36 + 13 * 42.05 + 6 * 40.64
3. Average cost = c(q)/q > 1205.95 / 35 = 34.46
Other combinations of factor inputs will also produce 35 units of product, like L = 74.01, K = 37.44, and M = 36.19. But, these combinations will be more costly at the given factor prices.
With these inefficient input combinations:
1. 35 = 1.01278 * (74.01^{^.317}) * (37.44^{^.417}) * (36.19^{^.186})
2. 1221.9 = 7 * 74.01 + 13 * 37.44 + 6 * 36.19
3. Average cost = 1221.9 / 35 = 34.91
The inputs L = 59.36, K = 42.05, and M = 40.64 are the leastcost combination of inputs that will produce q = 35 units of product at the input prices wL = 7, wK = 13, and wM = 6.
If for each feasible amount of product, we compute the cost of producing the product using the cost minimizing combination of inputs, we obtain the cost function, from which the average cost and marginal cost functions can be obtained.
Graph of the Average Cost and Marginal Cost for a CobbDouglas Production Function  


Average cost function 

Marginal cost function 

Decreasing returns to scale  
From the graphs, we see that both average cost and marginal cost are increasing, and marginal cost is greater than average cost. Both of these results are the consequence of our CobbDouglas production function having decreasing returns to scale. Also, we see that these cost functions don't look like the Ushaped cost functions I used in the oligopoly model.
It is always a good idea to look at some numbers, to get an understanding for the beast at hand.
CobbDouglas Production Function Data Decreasing Returns to Scale 
q  L  K  M  cost  ave.cost  marg. cost 
5  7.16  5.07  4.9  145.46 
29.09  31.62 
10  15.21  10.77  10.41  308.99 
30.9  33.59 
15  23.63  16.74  16.18  480.12 
32.01  34.79 
20  32.31  22.89  22.12  656.38 
32.82  35.67 
25  41.18  29.17  28.19  836.55 
33.46  36.37 
30  50.2  35.56  34.37  1019.91 
34  36.95 
35  59.36  42.05  40.64  1205.95 
34.46  37.45 
40  68.63  48.61  46.98  1394.32 
34.86  37.89 
45  78.01  55.25  53.4  1584.76 
35.22  38.28 
50  87.47  61.96  59.88  1777.05 
35.54  38.63 
II. Increasing returns to scale: alpha + beta + gamma > 1
With increasing returns to scale, a proportional increase in all inputs will increase output by more than the proportional constant. Our CobbDouglas production function might now have the form:
q = A * (L^{^.35}) * (K^{^.4}) * (M^{^.3})
where A = 1 and alpha + beta + gamma = .35 + .4 + .3 = 1.05 > 1
With the same factor prices as before, we compute the cost of producing the product using the cost minimizing combination of inputs, obtaining the cost function, and the average cost and marginal cost functions.
Graph of the Average Cost and Marginal Cost for a CobbDouglas Production Function  


Average cost function 

Marginal cost function 

Increasing returns to scale  
Now we see that both average cost and marginal cost are decreasing, with marginal cost below average cost, a consequence of increasing returns to scale.
CobbDouglas Production Function Data Increasing Returns to Scale 
q  L  K  M  cost  ave.cost  marg. cost 
5  5.57  3.43  5.57  117.01 
23.4  22.29 
10  10.78  6.64  10.78  226.43 
22.64  21.56 
15  15.86  9.76  15.86  333.14 
22.21  21.15 
20  20.86  12.84  20.86  438.15 
21.91  20.86 
25  25.8  15.88  25.8  541.9 
21.68  20.64 
30  30.7  18.89  30.7  644.65 
21.49  20.47 
35  35.55  21.88  35.55  746.6 
21.33  20.32 
40  40.37  24.85  40.37  847.84 
21.2  20.19 
45  45.17  27.79  45.17  948.49 
21.08  20.07 
50  49.93  30.73  49.93  1048.6 
20.97  19.97 
If we splice together the two cases above, we do get something like the Ushaped average and marginal costs that I used in the oligopoly model.
III. Short run  capital fixed  decreasing returns to scale
We usually assume that capital is fixed in the short run. Suppose our firm is to operate efficiently (using the cost minimizing combination of inputs) producing product in the 25  35 unit range (using the decreasing returns CobbDouglas production function). It might set its capital K = 35.56, which, from the table above, is the amount of capital associated with producing q = 30 units of product.
Graph of Average Cost and Marginal Cost CobbDouglas 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).
CobbDouglas Production Function Data Decreasing economies of scale Fixed Capital 
q  L  K  M  cost  ave.cost  marg. cost 
5  1.42  35.56  0.98  478.1 
95.62  6.29 
10  5.65  35.56  3.87  525.06 
52.51  12.48 
15  12.66  35.56  8.66  602.84 
40.19  18.63 
20  22.42  35.56  15.35  711.32 
35.57  24.76 
25  34.94  35.56  23.92  850.36 
34.01  30.86 
30  50.2  35.56  34.37  1019.91 
34  36.95 
35  68.21  35.56  46.69  1219.88 
34.85  43.03 
40  88.95  35.56  60.89  1450.22 
36.26  49.1 
45  112.41  35.56  76.95  1710.89 
38.02  55.16 
50  138.61  35.56  94.88  2001.83 
40.04  61.21 
Note that q=30 is the point at which the short run (capital fixed), average cost curve is tangent to the long run, Cobb Douglas average cost curve.
IV. 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 CobbDouglas production function:
q = A * (L^^{alpha}) * (K^^{beta}) * (M^^{gamma}) = 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 CobbDouglas production function:
q = 1.01278 * (L^^{0.317}) * (K^^{0.417}) * (M^^{0.186}).
When q = 30 and (wL, wK, wM) = (7, 13, 6), the cost minimizing inputs are:
(L, K, M) = (50.2, 35.56, 34.37), and
C(30) = 7 * 50.2 + 13 * 35.56 + 6 * 34.37 = 1019.91.
Solving the CobbDouglas equation for L, K, and M in turn, we get:
1. L = q^^{(1/alpha)} / (A * K^^{beta} * M^^{gamma})^^{(1/alpha)},
2. K = q^^{(1/beta)} / (A * L^^{alpha} * M^^{gamma})^^{(1/beta)},
3. M = q^^{(1/gamma)} / (A * L^^{alpha} * K^^{beta})^^{(1/alpha)},

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^^{(1/beta)} / (A * L^^{alpha} * M^^{gamma})^^{(1/beta)},
3. → LM Isoquant: M = q^^{(1/gamma)} / (A * L^^{alpha} * K^^{beta})^^{(1/alpha)},

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 = (1019.91  6 * 34.37)/13 = 62.59, while the LM isocost line has a Mintercept at (C(30)  wK * K) / wM = (1019.91  13 * 35.56)/6 = 92.94.
For q = 30, the LK isoquant is tangent to the LK isocost line at (L, K) = (50.2, 35.56), while the LM isoquant is tangent to the LM isocost line at (L, M) = (50.2, 34.37).
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.
V. Formulae
The three factor CobbDouglas production function is:
q = A * (L^^{alpha}) * (K^^{beta}) * (M^^{gamma}) = f(L,K,M).
a. Marginal product of labour: ∂f(L,K,M)/∂L = f_{L} = alpha * A*(L^(alpha1)) * (K^beta) * (M^gamma) = (alpha/L) * f(L,K,M)
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)
VI. 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 * L / (beta * K)
> K = L * beta * wL / (alpha * wK)
 wL / wM = f_{L} / f_{M} = alpha * L / (gamma * M)
> M = L * gamma * wL / (alpha * wM)
 wK / wM = f_{K} / f_{M} = beta * K / (gamma * M)
Substituting equations e. and f. into the CobbDouglas production function:
q = A * L^^{alpha} * (L*beta*wL/(alpha*wk))^^{beta} * (L*gamma*wL/(alpha*wM))^^{gamma}
Solving for L yields:

L = {q / [A* (beta*wL/(alpha*wK))^^{beta} * (gamma*wL/(alpha*wM))^^{gamma}]}^^{(1/(alpha+beta+gamma))}
= q^{(1/(alpha+beta+gamma))} * (alpha / wL) * [ wL^^{alpha} * wK^^{beta} * wM^^{gamma}/ (A * alpha^^{alpha} * beta^^{beta} * gamma^^{gamma})] ^^{(1/(alpha+beta+gamma))}
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 CobbDouglas production function.
VII. CobbDouglas 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^^{(1/(alpha+beta+gamma))}
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) = B * [ wL^^{alpha} * wK^^{beta} * wM^^{gamma}] ^^{(1/(alpha+beta+gamma))}
with B = (alpha + beta + gamma) / [A * alpha^^{alpha} * beta^^{beta} * gamma^^{gamma}] ^^{(1/(alpha+beta+gamma))}
The unit cost function c(wL, wK, wM) looks, interestingly, like its parent — the CobbDouglas production function.
The CobbDouglas production function is called homothetic, because the CobbDouglas cost function can be separated (factored) into a function of output, q, times a function of input prices, wL, wK, and wM.
VIII. 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) / (alpha + beta + gamma) = L
∂C/∂wK = h(q) * (beta / wK) * c(wL,wK,wM) / (alpha + beta + gamma) = K
∂C/∂wM = h(q) * (gamma / wM) * c(wL,wK,wM) / (alpha + beta + gamma) = M
IX. Properties of the unit CobbDouglas Cost Function, c(wL,wK,wM).
a. c is linear homogeneous in factor prices:
c(t*wL, t*wK, t*wM) = B*[(t*wL)^^{alpha} * (t*wK)^^{beta} * (t*wM)^^{gamma}]^^{(1/(alpha+beta+gamma))}
= t * c(wL, wK, wM)
b. c is concave in factor prices.
Check that the Hessian for the function c is negative (semi)definite.
X. Elasticity of substitution between inputs (sigma).
From equation e. of Part V we get:
K/L = (beta / alpha) * (wl / wK) → ln(K/L) = ln(beta/alpha) + ln(wL/wK)
sigma = d(ln(K/L))/d(ln(wL/wK)) = 1
XI. Negative definite:
The Hessian, H, of a function, f is negative definite, if the principal minors of H alternate in sign, starting with negative. If one (or more) principal minors have a zero value, f is negative semidefinite.
For the CobbDouglas production function:
H = 
f_{LL}  f_{LK} 
f_{LM} 
f_{KL}  f_{KK} 
f_{KM} 
f_{ML}  f_{MK} 
f_{MM} 
H1 = f_{LL} = alpha * f * (1  alpha) / L^^{2} < 0
H2 = 
f_{LL}  f_{LK} 
f_{KL}  f_{KK} 
H2 = 
alpha*f*(1  alpha)/L^^{2} 
alpha*beta*f/(L*K) 
alpha*beta*f/(L*K) 
beta * f * (1  beta)/ K^^{2} 
H2 = [alpha*beta/(L^^{2}*K^^{2})] * f * (1  alpha  beta) > 0
H3 = determinate of H <(=) 0
Example:
Consider the specific CobbDouglas production function:
q = 1.01278 * (L^^{0.317}) * (K^^{0.417}) * (M^^{0.186}).
At L = 50.20, K = 35.56, K = 34.366,
q = 30 = 1.01278 * (50.20^^{0.317}) * (35.56^^{0.417}) * (34.366^^{0.186})
H = 
f_{LL}  f_{LK} 
f_{LM} 
f_{KL}  f_{KK} 
f_{KM} 
f_{ML}  f_{MK} 
f_{MM} 

= 
0.00257711359252  0.00222135225158  0.00102523950073 
0.00222135225158  0.00576760147814  0.00190401621564 
0.00102523950073  0.00190401621564  0.00384582927923 

H1 = f_{LL} = 0.00257711359252 < 0
H2 = [alpha*beta/(L^^{2}*K^^{2})] * f * (1  alpha  beta) = 9.929358340000970e006 > 0
H3 = det(H) = 1.410896418980941e008 < 0
The Hessian of the defined production function is negative definite at L = 50.20, K = 35.56, K = 34.366.
XII. Further examples:
The web page, "The Duality of Production and Cost Functions," permits one to specify the parameters of the CobbDouglas (or CES) production function, and to ascertain the curvature of the production function and corresponding cost function.
