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Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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Production Functions:   Cobb-Douglas | CES | Generalized CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | References and Links

Cost Functions:   Cobb-Douglas Cost | Normalized Quadratic Cost | Translog Cost | Diewert Cost | Generalized CES-Translog Cost | Generalized CES-Diewert Cost | References and Links

Duality: Production / Cost Functions:   Cobb-Douglas 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.

An online introduction to Neoclassical Theories of Production is provided by the New School.

A. Cobb-Douglas Production Function

If you have not already done so, look at how the parameters of a Cobb-Douglas production function can be estimated: Estimating a Cobb-Douglas production function.

The three factor Cobb-Douglas 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 least-cost 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 Cobb-Douglas 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 least-cost 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 Cobb-Douglas 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 Cobb-Douglas production function having decreasing returns to scale. Also, we see that these cost functions don't look like the U-shaped 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.

Cobb-Douglas Production Function Data
Decreasing Returns to Scale
qLKMcostave.costmarg. cost
5 7.16 5.07 4.9 145.46 29.0931.62
10 15.21 10.77 10.41 308.99 30.933.59
15 23.63 16.74 16.18 480.12 32.0134.79
20 32.31 22.89 22.12 656.38 32.8235.67
25 41.18 29.17 28.19 836.55 33.4636.37
30 50.2 35.56 34.37 1019.91 3436.95
35 59.36 42.05 40.64 1205.95 34.4637.45
40 68.63 48.61 46.98 1394.32 34.8637.89
45 78.01 55.25 53.4 1584.76 35.2238.28
50 87.47 61.96 59.88 1777.05 35.5438.63

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 Cobb-Douglas 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 Cobb-Douglas 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.

Cobb-Douglas Production Function Data
Increasing Returns to Scale
qLKMcostave.costmarg. cost
5 5.57 3.43 5.57 117.01 23.422.29
10 10.78 6.64 10.78 226.43 22.6421.56
15 15.86 9.76 15.86 333.14 22.2121.15
20 20.86 12.84 20.86 438.15 21.9120.86
25 25.8 15.88 25.8 541.9 21.6820.64
30 30.7 18.89 30.7 644.65 21.4920.47
35 35.55 21.88 35.55 746.6 21.3320.32
40 40.37 24.85 40.37 847.84 21.220.19
45 45.17 27.79 45.17 948.49 21.0820.07
50 49.93 30.73 49.93 1048.6 20.9719.97

If we splice together the two cases above, we do get something like the U-shaped 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 Cobb-Douglas 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
Cobb-Douglas Production Function - Capital Fixed
 
  Average cost function
  Marginal cost function
  L.R. Average cost function
Decreasing economies of scale 

Now we get the traditional U-shaped 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).

Cobb-Douglas Production Function Data
Decreasing economies of scale
Fixed Capital
qLKMcostave.costmarg. cost
5 1.42 35.56 0.98 478.1 95.626.29
10 5.65 35.56 3.87 525.06 52.5112.48
15 12.66 35.56 8.66 602.84 40.1918.63
20 22.42 35.56 15.35 711.32 35.5724.76
25 34.94 35.56 23.92 850.36 34.0130.86
30 50.2 35.56 34.37 1019.91 3436.95
35 68.21 35.56 46.69 1219.88 34.8543.03
40 88.95 35.56 60.89 1450.22 36.2649.1
45 112.41 35.56 76.95 1710.89 38.0255.16
50 138.61 35.56 94.88 2001.83 40.0461.21

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 Cobb-Douglas production function:

q = A * (L^alpha) * (K^beta) * (M^gamma) = f(L,K,M),

for the 3-dimensional 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 Cobb-Douglas 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 Cobb-Douglas 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 3-dimensional isoquant surface.

By fixing the amount of input for one factor, we obtain a 2-dimensional isoquant curve. As examples, fixing M = M in equation 2, and K = K in equation 3, we get:

2. → L-K Isoquant:   K = q^(1/beta) / (A * L^alpha * M^gamma)^(1/beta),

3. → L-M 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 L-K and L-M 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 L-K isocost line is mK = -wL / wK = -7 / 13; the slope of an L-M isocost line is mM = -wL / wM = -7 / 6.

For q = 30, the L-K isocost line has a K-intercept at (C(30) - wM * M) / wK = (1019.91 - 6 * 34.37)/13 = 62.59,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (1019.91 - 13 * 35.56)/6 = 92.94.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (50.2, 35.56), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (50.2, 34.37).

L-K Isoquants, M = ML-M 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 Cobb-Douglas 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 = fL = alpha * A*(L^(alpha-1)) * (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. Least-cost 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)]

  1. GL = wL - µ * fL = 0
  2. GK = wK - µ * fK = 0
  3. GM = wM - µ * fM = 0
  4. Gµ = q - f(L,K,M) = 0

From equations a., b., and c. we get:

  1. wL / wK = fL / fK = alpha * L / (beta * K) --> K = L * beta * wL / (alpha * wK)
  2. wL / wM = fL / fM = alpha * L / (gamma * M) --> M = L * gamma * wL / (alpha * wM)
  3. wK / wM = fK / fM = beta * K / (gamma * M)

Substituting equations e. and f. into the Cobb-Douglas production function:

q = A * L^alpha * (L*beta*wL/(alpha*wk))^beta * (L*gamma*wL/(alpha*wM))^gamma

Solving for L yields:

  1. 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 Cobb-Douglas production function.

VII. Cobb-Douglas 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 Cobb-Douglas production function.

The Cobb-Douglas production function is called homothetic, because the Cobb-Douglas 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 Cobb-Douglas 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 Cobb-Douglas production function:

H = fLLfLK fLM
fKLfKK fKM
fMLfMK fMM

H1 = fLL = -alpha * f * (1 - alpha) / L^2 < 0

H2 = fLLfLK
fKLfKK

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 Cobb-Douglas 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   =  
fLLfLK fLM
fKLfKK fKM
fMLfMK fMM
  =  
-0.002577113592520.00222135225158 0.00102523950073
0.00222135225158-0.00576760147814 0.00190401621564
0.001025239500730.00190401621564-0.00384582927923

H1 = fLL = -0.00257711359252 < 0

H2 = [alpha*beta/(L^2*K^2)] * f * (1 - alpha - beta) = 9.929358340000970e-006 > 0

H3 = det(H) = -1.410896418980941e-008 < 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 Cobb-Douglas (or CES) production function, and to ascertain the curvature of the production function and corresponding cost function.

 

 
   

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