Egwald Economics: Microeconomics

Production Functions

by

Elmer G. Wiens

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Duality: Production / Cost Functions:   Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES

Like the Translog production function, the Diewert production function has a flexible functional form, permitting the partial elasticities of substitution between inputs to vary. .

D. Diewert (Generalized Leontief) Production Function

The three factor Diewert production function is:

q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2   =   f(L,K,M)

where L = labour, K = capital, M = materials and supplies, q = product, and nu = elasticity of scale parameter.

I. To obtain estimates of the Diewert production function, let us use the CES production function to generate a sequence of observations relating the CES least cost factor inputs to factor prices and levels of output.

The three factor CES production function is:

q = A * [alpha * (L^-rho) + beta * (K^-rho) + gamma *(M^-rho)]^(-nu/rho)

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).

The estimated coefficients of the Diewert production and cost functions will vary with the parameters sigma, nu, alpha, beta and gamma of the CES production function.

Set the parameters below to re-run with your own CES parameters.

Restrictions: .7 < nu < 1.3; .5 < sigma < 2;
.25 < alpha < .45, .3 < beta < .5, .2 < gamma < .35
sigma = 1 → nu = alpha + beta + gamma (Cobb-Douglas)
sigma < 1 → inputs complements; sigma > 1 → inputs substitutes

 CES Production Function Parameters elasticity of scale parameter: nu elasticity of substitution: sigma alpha beta gamma

The CES production function as specified:

q = 1 * [0.35 * (L^- 0.17647) + 0.4 * (K^- 0.17647) + 0.25 *(M^- 0.17647)]^(-1/0.17647)

II. Estimating the Diewert production function using multiple regression yielded the following coefficient estimates:

QR Least Squares
Parameter Estimates
Parameter Coefficient std error t-ratio
L-0.1480530.001-183.569
K-0.181890-424.452
M-0.179970-380.444
LK0.6443770.001555.523
LM0.3466190.001322.232
KM0.519060.001465.723
 R2 = 1 R2b = 1 # obs = 182

nu = 1
aLL = -0.148053, aKK = -0.18189, aMM = -0.17997
bLK = 0.644377, bLM = 0.346619 bKM = 0.51906

aLL + aKK + aMM + bLK + bLM + bKM = 1

The estimated Diewert production function:

q^1/1 = -0.148053 * L + -0.18189 * K + -0.17997 * M + 0.644377 * L^1/2 * K^1/2 + 0.346619 * L^1/2 * M^1/2 + 0.51906 * K^1/2 * M^1/2     =     f(L,K,M)

Elasticity of Scale of Production:

Long Run: Capital Variable:

εLKM = ε(L,K,M) = nu * [fL(L,K,M) * L + fK(L,K,M) * K + fM(L,K,M) * M] / f(L, K, M),

= nu,
since f(L, K ,M) is linear homogeneous (see the Mathematical Notes).

Short Run: Captial Fixed: K = K:

εLKM = ε(L,K,M) = nu * [ fL(L,K,M) * L + fM(L,K,M) * M] / f(L, K, M).

Curvature:

The Diewert production function, q = F(L,K,M) = f(L,K,M)^nu, is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h1, h2, and h3:

 FLL FLK FLM FKL FKK FKM FML FMK FMM
=|h3|
 FLL FLK FKL FKK
=|h2|
 FLL
=|h1|

The Hessian of F(L,K,M) is negative definite if the determinants |h1|, |h2|, and |h3| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of F(L,K,M) is negative semidefinite, and F(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

The curvature of the estimated Diewert production function depends on the elasticity of substitution, sigma, of the CES production function.

sigma < 1 & nu = 1   →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),

sigma = 1   & nu = alpha + beta + gamma < 1 →   F(L,K,M) is concave to the origin of the 3-dimensional space of (L,K,M),
sigma = 1   & nu = alpha + beta + gamma = 1 →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),
sigma = 1   & nu = alpha + beta + gamma > 1 →   F(L,K,M) is NOT concave to the origin of the 3-dimensional space of (L,K,M),

sigma > 1 & nu = 1   →   F(L,K,M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M),

Check the displayed curvature conditions for other values of sigma, nu, alpha, beta, and gamma.

III. The least-cost combination of inputs:

The entrepreneur, management, and employees of the profit maximizing firm choose the factor proportions and quantities, and output levels given the prices of factor inputs, and products. For any specified combination of positive factor prices, wL, wK, and wM, what combination of factor inputs, L, K, and M, will minimize the cost of producing any given level of positive output, q?

C(q; wL, wK, wM) = minL,K,M{ wL * L + wK * K + wM * M   :   q - f(L,K,M) = 0,   q > 0, wL > 0; wK > 0, and wM > 0 }

Equivalently: Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)^nu],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)^nu].

First Order Conditions:

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

To solve equations 0. to 3. numerically, for given q, wL, wK, and wM, I used the first order conditions 0. to 3. and the associated Jacobian.

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)^nu):

 Gµµ GµL GµK GµM GLµ GLL GLK GLM GKµ GKL GKK GKM GMµ GML GMK GMM
=
 0 FL FK FM FL GLL GLK GLM FK GKL GKK GKM FM GML GMK GMM
=|H3|
 0 FL FK FL GLL GLK FK GKL GKK
= |H2|

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered, principal minor determinant |H2| is positive, and the bordered determinant |H3| is negative. (See the table below, and the Mathematical Notes.)

IV. Long Run: Capital Variable.

Suppose the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 6 . Solving the least-cost problem, (and noting that µ = marginal cost,) we can compare the CES cost data with the estimated Diewert cost data as displayed in the following table. The terms sLK, sLM, and sKM are the Allen partial elasticities of substitution.

Diewert Long Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6
—   CES Data   ——   —   Estimated Diewert Data   —   —AESLagrangianDiewert Curvature
qLKMcost q LK Mcost ave. costmarg. costεLKMsLKsLMsKM|H2|*10|H3|*102|h1|*103|h2|*104|h3|*108
16 19.6713.02 16.85408.09 16 19.6813.02 16.86408.11 25.5125.511 0.8680.8480.856 1.94-10.77-10.741.840
18 22.1314.65 18.96459.1 18 22.1314.64 18.97459.12 25.5125.511 0.8680.8480.856 1.724-8.51-9.541.450
20 24.5916.28 21.06510.11 20 24.5916.27 21.08510.13 25.5125.511 0.8680.8480.856 1.552-6.893-8.591.18-0
22 27.0517.9 23.17561.13 22 27.0517.9 23.19561.15 25.5125.511 0.8680.8480.856 1.411-5.697-7.810.970
24 29.5119.53 25.27612.14 24 29.5119.52 25.3612.16 25.5125.511 0.8680.8480.856 1.293-4.787-7.160.820
26 31.9721.16 27.38663.15 26 31.9721.15 27.4663.18 25.5125.511 0.8680.8480.856 1.194-4.079-6.610.70
28 34.4322.79 29.49714.16 28 34.4322.78 29.51714.19 25.5125.511 0.8680.8480.856 1.109-3.517-6.140.6-0
30 36.8924.42 31.59765.17 30 36.8924.4 31.62765.2 25.5125.511 0.8680.8480.856 1.035-3.064-5.730.520
32 39.3526.04 33.7816.18 32 39.3526.03 33.73816.22 25.5125.511 0.8680.8480.856 0.97-2.693-5.370.460
34 41.8127.67 35.8867.19 34 41.8127.66 35.84867.23 25.5125.511 0.8680.8480.856 0.913-2.385-5.050.410
36 44.2729.3 37.91918.21 36 44.2729.28 37.94918.24 25.5125.511 0.8680.8480.856 0.862-2.127-4.770.36-0
38 46.7330.93 40.02969.22 38 46.7330.91 40.05969.26 25.5125.511 0.8680.8480.856 0.817-1.909-4.520.330
40 49.1832.55 42.121020.23 40 49.1932.54 42.161020.27 25.5125.511 0.8680.8480.856 0.776-1.723-4.290.29-0
42 51.6434.18 44.231071.24 42 51.6534.16 44.271071.28 25.5125.511 0.8680.8480.856 0.739-1.563-4.090.270
44 54.135.81 46.341122.25 44 54.1135.79 46.381122.3 25.5125.511 0.8680.8480.856 0.705-1.424-3.90.24-0
46 56.5637.44 48.441173.26 46 56.5737.42 48.481173.31 25.5125.511 0.8680.8480.856 0.675-1.303-3.730.22-0
48 59.0239.06 50.551224.27 48 59.0339.05 50.591224.32 25.5125.511 0.8680.8480.856 0.647-1.197-3.580.20

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, K, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), K = K(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the four functions, Gµ, GL, GK, GM, with respect to the choice variables, μ, L, K, and M:

J3   =

=
 0 -FL -FK -FM -FL -µ * FLL -µ * FLK -µ * FLM -FK -µ * FKL -µ * FKK -µ * FKM -FM -µ * FML -µ * FMK -µ * FMM

The Jacobian matrix of the four solution functions, Φ = {µ, L, K, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J3, yielding the comparative statics of the solution functions.

For example, at q = 30, wL = 7,   wK = 13, and   wM = 6 :

JΦ   =

 µq µwL µwK µwM Lq LwL LwK LwM Kq KwL KwK KwM Mq MwL MwK MwM
=
 -0 1.23 0.813 1.054 1.23 -3.005 1.022 1.293 0.813 1.022 -0.948 0.863 1.054 1.293 0.863 -3.377

The Uzawa Partial Elasticities of Substitution at these same values of q, wL, wK, and wM:

 uLK = C(q;wL,wK,wM) * LwK / (L * K)   =   0.868, uLM = C(q;wL,wK,wM) * LwM / (L * M)   =   0.848, uKL = C(q;wL,wK,wM) * KwL / (K * L)   =   0.868, uKM = C(q;wL,wK,wM) * KwM / (K * M)   =   0.856, uML = C(q;wL,wK,wM) * MwL / (M * L)   =   0.848, uMK = C(q;wL,wK,wM) * MwK / (M * K)   =   0.856,

The factor demand elasticities at these same values of q, wL, wK, and wM:

 εL,q εL,wL εL,wK εL,wM εK,q εK,wL εK,wK εK,wM εM,q εM,wL εM,wK εM,wM
=
 1 -0.57 0.36 0.21 1 0.293 -0.505 0.212 1 0.286 0.355 -0.641

where, for example, εL,wK = ∂ln(L(q; wL, wK, wM)) / ∂ln(wK) = (∂ln(L(q; wL, wK, wM)) / ∂wK) * (∂wK / ∂ln(wK)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wK) * wK = LwK * wK / L

The cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇2wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇2wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

2wwC(q;wL,wK,wM)   =

 LwL LwK LwM KwL KwK KwM MwL MwK MwM
=
 -3.005 1.022 1.293 1.022 -0.948 0.863 1.293 0.863 -3.377

The eigenvalues of ∇2wwC  are e1 = -4.4976, e2 = -2.8335, and e3 = 0.

V. Short Run: Capital Fixed: K = K = 24.4.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 6 . Set the level of capital at the least cost level for q = 30. .   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Diewert short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, sLM, on the translog production function page.)

The Lagrangian G(q;L,K,M,µ) obtains a maximum at q, L, K, and M if the the bordered determinant |H2| is positive. The Diewert production function, F(L, K, M), is concave to the origin of the 2-dimensional (L, M) space if |h1| < 0, and |h2| > 0. (See the table below, and the Mathematical Notes)

Diewert Short Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6; K = 24.4
—   CES Data   ——   —   Estimated Diewert Data   —   —AES
qLKMcostq LK Mcost ave. costmarg. costεLKMεLKM3 factor
sLM
2 factor
sLM
|H2|*102|h1|*103|h2|*104
1613.124.4 11.22476.2916.01 13.1724.4 11.37477.69 29.86 15.5110.646 1.311.116.992-22.755.17
1815.8224.4 13.55509.2818 15.8424.4 13.65509.99 28.33 16.9510.632 1.231.0612.321-17.73.11
2018.7624.4 16.07545.0120 18.7524.4 16.14545.38 27.27 18.410.62 1.161.029.195-14.081.96
2221.9324.4 18.79583.522 21.9124.4 18.84583.65 26.53 19.8410.61 1.090.987.036-11.431.28
2425.3324.4 21.7624.7424 25.324.4 21.74624.79 26.03 21.2710.602 1.020.945.5-9.430.87
2628.9624.4 24.8668.7526 28.9324.4 24.83668.77 25.72 22.6910.596 0.960.914.379-7.890.6
2832.8124.4 28.1715.5528 32.7924.4 28.12715.48 25.55 24.1110.59 0.90.883.544-6.690.43
3036.924.4 31.6765.1730 36.8824.4 31.62765.12 25.5 25.5110.585 0.850.852.906-5.730.31
3241.2224.4 35.3817.6332 41.2124.4 35.3817.54 25.55 26.910.581 0.80.832.413-4.950.23
3445.7824.4 39.21872.9534 45.7724.4 39.18872.72 25.67 28.2810.578 0.750.82.025-4.320.17
3650.5824.4 43.31931.1636 50.5524.4 43.26930.67 25.85 29.6510.575 0.70.781.717-3.80.13
3855.6124.4 47.63992.2938 55.5724.4 47.52991.34 26.09 31.0210.572 0.660.761.468-3.360.1
4060.8924.4 52.151056.3840 60.8124.4 51.971054.74 26.37 32.3710.57 0.620.741.265-30.08
4266.4224.4 56.881123.4642 66.2724.4 56.621120.83 26.69 33.7210.567 0.580.721.098-2.680.06
4472.1924.4 61.831193.5644 71.9524.4 61.451189.61 27.04 35.0510.566 0.540.70.96-2.420.05
4678.2224.4 66.991266.7246 77.8624.4 66.461261.05 27.41 36.3810.564 0.50.690.844-2.190.04
4884.524.4 72.371342.9748 83.9924.4 71.661335.14 27.82 37.710.562 0.470.670.746-1.990.03

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, εLKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, εLKM. The short run elasticity of scale with capital fixed at K = 24.4 is a decreasing function along the short run average cost curve, since sigma is less than 1.

The Lagrangian method for obtaining the least-cost combination of inputs, yields values for the solution variables µ, L, and M for specified values of q, wL, wK, and wM. That is, the solution variables are functions:

µ = µ(q; wL, wK, wM), L = L(q; wL, wK, wM), M = M(q; wL, wK, wM).

The Lagrangian method also produces the Jacobian matrix (bordered Hessian of G = -G) of the three functions, Gµ, GL, GM, with respect to the choice variables, μ, L, and M:

J2   =

=
 0 -FL -FM -FL -µ * FLL -µ * FLM -FM -µ * FML -µ * FMM

The Jacobian matrix of the three solution functions, Φ = {µ, L, M}, with respect to the variables, q, wL, wK, and wM equals the negative of the matrix inverse of J2, yielding the comparative statics of the solution functions.

For example, at q = 30,   K = K = 24.4,   wL = 7,   wK = 13,  and   wM = 6 :

JΦ   =

 µq µwL µwM Lq LwL LwM Mq MwL MwM
=
 0.698 2.106 1.794 2.106 -1.904 2.222 1.794 2.222 -2.592

The factor demand elasticities at q = 30, wL = 7,   wK = 13, and   wM = 6 :

 εL,q εL,wL εL,wM εM,q εM,wL εM,wM
=
 1.713 -0.361 0.361 1.702 0.492 -0.492

where, for example, εL,wM = ∂ln(L(q; wL, wK, wM)) / ∂ln(wM) = (∂ln(L(q; wL, wK, wM)) / ∂wM) * (∂wM / ∂ln(wM)) = (1 / L(q; wL, wK, wM)) * (∂L(q; wL, wK, wM) / ∂wM) * wM = LwM * wM / L

The short run cost function C(q;wL,wK,wM) is a concave in factor prices if the Hessian matrix ∇2wwC(q;wL,wK,wM) of second order partial derivatives with respect to factor prices is negative semidefinite. The matrix ∇2wwC(q;wL,wK,wM) is negative semidefinite if its eigenvalues are nonpositive.

2wwC(q;wL,wK,wM)   =

 LwL LwM MwL MwM
=
 -1.904 2.222 2.222 -2.592

The eigenvalues of ∇2wwC  are e1 = -4.4964, and e2 = -0 .

Graph of Average Cost and Marginal Cost
Diewert Production / Cost Functions - Capital Fixed

Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 6.
 Average cost function Marginal cost function L.R. Average cost function

VI. Long Run: Capital Variable.

Let us increase the price of M, materials and supplies. Now the firm buys its inputs at the prices: wL = 7   wK = 13   wM = 7 . Solving the least-cost problem, (and noting that µ = marginal cost,) we can compare the CES cost data with the estimated Diewert cost data as displayed in the following table. The terms sLK, sLM, and sKM are the Allen partial elasticities of substitution.

Diewert Long Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7
—   CES Data   ——   —   Estimated Diewert Data   —   —AESLagrangianDiewert Curvature
qLKMcost q LK Mcost ave. costmarg. costεLKMsLKsLMsKM|H2|*10|H3|*102|h1|*103|h2|*104|h3|*108
16 20.3313.45 15.27424.12 16 20.3413.46 15.25424.13 26.5126.511 0.830.890.89 1.776-12.239-10.131.580
18 22.8715.14 17.18477.13 18 22.8915.14 17.15477.15 26.5126.511 0.830.890.89 1.578-9.67-91.250
20 25.4116.82 19.09530.15 20 25.4316.83 19.06530.17 26.5126.511 0.830.890.89 1.421-7.833-8.11.010
22 27.9518.5 21583.16 22 27.9718.51 20.97583.18 26.5126.511 0.830.890.89 1.291-6.473-7.370.840
24 30.4920.18 22.91636.18 24 30.5220.19 22.87636.2 26.5126.511 0.830.890.89 1.184-5.439-6.750.70
26 33.0321.86 24.82689.19 26 33.0621.87 24.78689.22 26.5126.511 0.830.890.89 1.093-4.635-6.230.60
28 35.5723.55 26.73742.21 28 35.623.56 26.68742.24 26.5126.511 0.830.890.89 1.015-3.996-5.790.520
30 38.1225.23 28.64795.22 30 38.1425.24 28.59795.25 26.5126.511 0.830.890.89 0.947-3.481-5.40.450
32 40.6626.91 30.54848.24 32 40.6926.92 30.5848.27 26.5126.511 0.830.890.89 0.888-3.06-5.060.40
34 43.228.59 32.45901.25 34 43.2328.6 32.4901.29 26.5126.511 0.830.890.89 0.836-2.71-4.770.35-0
36 45.7430.27 34.36954.27 36 45.7730.29 34.31954.3 26.5126.511 0.830.890.89 0.789-2.418-4.50.310
38 48.2831.96 36.271007.28 38 48.3231.97 36.211007.32 26.5126.511 0.830.890.89 0.748-2.17-4.260.280
40 50.8233.64 38.181060.29 40 50.8633.65 38.121060.34 26.5126.511 0.830.890.89 0.71-1.958-4.050.250
42 53.3635.32 40.091113.31 42 53.435.34 40.031113.35 26.5126.511 0.830.890.89 0.676-1.776-3.860.23-0
44 55.937 421166.32 44 55.9537.02 41.931166.37 26.5126.511 0.830.890.89 0.646-1.618-3.680.210
46 58.4438.68 43.911219.34 46 58.4938.7 43.841219.39 26.5126.511 0.830.890.89 0.618-1.481-3.520.190
48 60.9940.36 45.821272.35 48 61.0340.38 45.741272.4 26.5126.511 0.830.890.89 0.592-1.36-3.380.180

VII. Short Run: Capital Fixed: K = K = 25.24.

Let the firm buy its inputs at the same prices: wL = 7   wK = 13   wM = 7 . Set the level of capital at the least cost level for q = 30: then .   Solving the least-cost problem holding capital fixed, we can compare the short-run CES cost data with the estimated Diewert short-run cost data as displayed in the following table. (See the discussion of the short-run elasticity of substitution, sLM, on the translog production function page.) The Diewert production function, F(L, K, M), is concave to the origin of the 2-dimensional (L, M) space if |h1| < 0, and |h2| > 0.

Diewert Short Run Cost Data
Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7; K = 25.24
—   CES Data   ——   —   Estimated Diewert Data   —   —AES
qLKMcostq LK Mcost ave. costmarg. costεLKMεLKM3 factor
sLM
2 factor
sLM
|H2|*102|h1|*103|h2|*104
1613.5825.24 10.2494.6216 13.9125.24 10.1496.22 31.01 16.1510.651 1.341.1220.011-20.95.81
1816.3925.24 12.31529.0318 16.6625.24 12.17529.89 29.44 17.6510.635 1.271.0814.499-16.353.5
2019.4325.24 14.6566.2720 19.6525.24 14.43566.71 28.34 19.1510.623 1.191.0410.814-13.072.2
2222.725.24 17.05606.3422 22.8725.24 16.9606.51 27.57 20.6410.613 1.1318.264-10.651.44
2426.225.24 19.68649.2624 26.3425.24 19.55649.31 27.05 22.1210.605 1.060.976.455-8.820.97
2629.9325.24 22.49695.0326 30.0325.24 22.39695.06 26.73 23.5910.598 10.945.135-7.410.68
2833.925.24 25.47743.6828 33.9625.24 25.41743.73 26.56 25.0510.592 0.940.914.15-6.30.48
3038.125.24 28.63795.2230 38.1125.24 28.63795.31 26.51 26.5110.587 0.890.883.402-5.410.35
3242.5525.24 31.96849.6832 42.5125.24 31.99849.62 26.55 27.9610.583 0.830.852.826-4.690.26
3447.2325.24 35.48907.0834 47.1225.24 35.57906.94 26.67 29.410.579 0.780.832.371-4.10.2
3652.1525.24 39.18967.4536 51.9525.24 39.34967.12 26.86 30.8210.576 0.740.82.008-3.610.15
3857.3225.24 43.061030.8238 5725.24 43.291030.15 27.11 32.2410.573 0.690.781.716-3.210.12
4062.7425.24 47.131097.2240 62.2725.24 47.421095.99 27.4 33.6510.571 0.650.761.478-2.860.09
4268.4125.24 51.391166.6842 67.7725.24 51.741164.64 27.73 35.0510.569 0.610.741.283-2.570.07
4474.3225.24 55.841239.2444 73.4825.24 56.231236.08 28.09 36.4410.567 0.580.731.12-2.320.06
4680.525.24 60.481314.9345.99 79.4125.24 60.91310.28 28.48 37.8210.565 0.540.710.984-2.10.05
4886.9325.24 65.311393.7847.99 85.5525.24 65.761387.25 28.9 39.1910.563 0.510.690.87-1.910.04

We get a U-shaped, short run average cost curve, with capital fixed. The short run average cost curve is (approx.) tangent to the long run average cost curve, at q = 30. The elasticity of scale, εLKM, is constant along the short run average cost curve, and equals the long run elasticity of scale, εLKM. The short run elasticity of scale with capital fixed at K = 25.24 is a decreasing function along the short run average cost curve, since sigma is less than 1.

Graph of Average Cost and Marginal Cost
Diewert Production / Cost Functions - Capital Fixed

Returns to Scale = 1, Elasticity of Substitution = 0.85
wL = 7, wK= 13, wM = 7.
 Average cost function Marginal cost function L.R. Average cost function

VIII. 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 Diewert production function:

q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2   =   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 the Diewert production function as estimated above:

q^1/1 = -0.148053 * L + -0.18189 * K + -0.17997 * M + 0.644377 * L^1/2 * K^1/2 + 0.346619 * L^1/2 * M^1/2 + 0.51906 * K^1/2 * M^1/2

When q = 30 and (wL, wK, wM) = (7, 13, 6), the cost minimizing inputs are:

(L, K, M) = (36.89, 24.4, 31.62), and

C(30) = 7 * 36.89 + 13 * 24.4 + 6 * 31.62 = 765.2.

Solving the Diewert equation for L, K, and M in turn, we get three equations for the 3-dimensional isoquant surface. By fixing the amount of input for one factor in these equations, we obtain three 2-dimensional isoquant curves.

Firstly, consider the L-K Isoquant. Set M = M, and let:

a = aKK, b = bLK * L^(1/2) + bKM * M^(1/2),
c = q^(1/nu) - aLL * L - aMM * M - bLM * L^(1/2) * M^(1/2),

The L-K isoquant expressed as K as a function of L:

K = -(1/2/a*(-b + sqrt(b * b + 4*a*c))*b-c)/a.

Secondly, consider the L-M Isoquant. Set K = K, and let:

a = aMM, b = bLM * L^(1/2) + bKM * K^(1/2),
c = q^(1/nu) - aLL * L - aKK * K - bLK * L^(1/2) * K^(1/2),

The L-M isoquant expressed as M as a function of L:

M = -(1/2/a*(-b + sqrt(b*b + 4*a*c))*b-c)/a.

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 = (765.2 - 6 * 31.62)/13 = 44.27,
while the L-M isocost line has a M-intercept at (C(30) - wK * K) / wM = (765.2 - 13 * 24.4)/6 = 74.66.

For q = 30, the L-K isoquant is tangent to the L-K isocost line at (L, K) = (36.89, 24.4), while the L-M isoquant is tangent to the L-M isocost line at (L, M) = (36.89, 31.62).

Diewert Production Function Isoquants
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.

It can be profitable to compare the Diewert isoquants with the isoquants of the CES production function, from which the Diewert's isoquants were derived.

When q = 30 and (wL, wK, wM) = (7, 13, 6), the CES cost minimizing inputs are:

(L, K, M) = (36.89, 24.42, 31.59), and

C(30) = 7 * 36.89 + 13 * 24.42 + 6 * 31.59 = 765.17.

CES Production Function Isoquants
L-K Isoquants, M = ML-M Isoquants, K = K

Mathematical Notes

I. The Diewert (Generalized Leontief) Production Function

 q^1/nu = aLL * L + aKK * K + aMM * M + bLK * L^1/2 * K^1/2 + bLM * L^1/2 * M^1/2 + bKM * K^1/2 * M^1/2   =   f(L,K,M) where L = labour, K = capital, M = materials and supplies, q = product, and nu = elasticity of scale parameter.

Elasticity of Scale of Production:

Long Run: Capital Variable:

εLKM = ε(L,K,M) = nu * [fL(L,K,M) * L + fK(L,K,M) * K + fM(L,K,M) * M] / f(L, K, M),

= nu,
since f(L, K ,M) is linear homogeneous.

Short Run: Captial Fixed: K = K:

εLKM = ε(L,K,M) = nu * [ fL(L,K,M) * L + fM(L,K,M) * M] / f(L, K, M).

Curvature:

The Diewert production function, q = F(L,K,M) = f(L,K,M)^nu, is concave to the origin of the 3-dimensional (L,K, M) space if its Hessian is negative (semi)definite. Define the matrices h1, h2, and h3:

 FLL FLK FLM FKL FKK FKM FML FMK FMM
=|h3|
 FLL FLK FKL FKK
=|h2|
 FLL
=|h1|

The Hessian of F(L,K,M) is negative definite if the determinants |h1|, |h2|, and |h3| alternate in sign, starting with negative. If one or more the determinants have a zero value, then the Hessian of F(L,K,M) is negative semidefinite, and F(L, K, M) is quasi-concave to the origin of the 3-dimensional space of (L,K,M).

II. The partial derivatives of f(L,K,M)

 fL(L,K,M) = aLL + (1/2) * [bLK*L^-1/2*K^1/2 + bLM*L^-1/2*M^1/2], fK(L,K,M) = aKK + (1/2) * [bLK*L^1/2*K^-1/2 + bKM*K^-1/2*M^1/2], fM(L,K,M) = aMM + (1/2) * [bLM*L^1/2*M^-1/2 + bKM*K^1/2*M^-1/2], fLL = -(1/4) * [bLK*L^-3/2*K^1/2 + bLM*L^-3/2*M^1/2],   fLK = (1/4) * bLK*L^-1/2*K^-1/2,   fLM = (1/4) * bLM*L^-1/2*M^-1/2,   fKK = -(1/4) * [bLK*L^1/2*K^-3/2 + bKM*K^-3/2*M^1/2],   fKL = (1/4) * bLK*L^-1/2*K^-1/2,   fKM = (1/4) * bKM*K^-1/2*M^-1/2, fMM = (-1/4) * [bLM*L^1/2*M^-3/2 + bKM*K^1/2*M^-3/2],   fML = (1/4) * bLM*L^-1/2*M^-1/2,   fMK = (1/4) * bKM*K^-1/2*M^-1/2.

III. The least-cost combination of inputs:

Find the values of L, K, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)^nu],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)^nu].

First Order Conditions:

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

Jacobian of Second Order Conditions (Bordered Hessian with F(L,K,M) = f(L,K,M)^nu):

 Gµµ GµL GµK GµM GLµ GLL GLK GLM GKµ GKL GKK GKM GMµ GML GMK GMM
=
 0 FL FK FM FL GLL GLK GLM FK GKL GKK GKM FM GML GMK GMM
=|H3|
 0 FL FK FL GLL GLK FK GKL GKK
= |H2|

where:

 (0,0) Gµµ = 0,   (0,1) GµL = fL * f^nu-1 * nu,   (0,2) GµK = fK * f^nu-1 * nu,   (0,3) GµM = fM * f^nu-1 * nu (1,0) GLµ = GµL,   (1,1) GLL = µ * [fLL * f^nu-1 + fL * fL * f^nu-2 * (nu - 1)] * nu,   (1,2) GLK = µ * [fLK * f^nu-1 + fL * fK * f^nu-2 * (nu - 1)] * nu,   (1,3) GLM = µ * [fLM * f^nu-1 + fL * fM * f^nu-2 * (nu - 1)] * nu,   (2,0) GKµ = GµK,   (2,1) GKL = GLK,   (2,2) GKK = µ * [fKK * f^nu-1 + fK * fK * f^nu-2 * (nu - 1)] * nu,   (2,3) GKM = µ * [fKM * f^nu-1 + fK * fM * f^nu-2 * (nu - 1)] * nu,   (3,0) GMµ = GµM,   (3,1) GML = GLM,   (3,2) GMK = GKM,   (3,3) GMM = µ * [fMM * f^nu-1 + fM * fM * f^nu-2 * (nu - 1)] * nu,

IV. The least-cost combination of inputs: Capital fixed:

With the value of the capital value fixed at K = K, find the values of L, M, and µ that minimize the Lagrangian when the factor prices are wL, wK, and wM:

G(q;L,K,M,µ) = wL * L + wK * K + wM * M + µ * [q - f(L,K,M)^nu],

or equivalently, that maximize the Lagrangian G(q;L,K,M,µ) = - G(q;L,K,M,µ).

G(q;L,K,M,µ) = -(wL * L + wK * K + wM * M) - µ * [q - f(L,K,M)^nu].

First Order Conditions: K = K:

 0. Gµ = f(L,K,M) ^nu - q = 0 1. GL = -wL + µ * fL * f^nu-1 * nu = 0 2. GM = -wM + µ * fM * f^nu-1 * nu = 0

Jacobian of Second Order Conditions: K = K: (Bordered Hessian with F(L,K,M) = f(L,K,M)^nu):

 Gµµ GµL GµM GLµ GLL GLM GMµ GML GMM
=
 0 FL FM FL GLL GLM FM GML GMM
=|H2|

where:

 (0,0) Gµµ = 0,   (0,1) GµL = fL * f^nu-1 * nu,   (0,2) GµM = fM * f^nu-1 * nu (1,0) GLµ = GµL,   (1,1) GLL = µ * [fLL * f^nu-1 + fL * fL * f^nu-2 * (nu - 1)] * nu,   (1,2) GLM = µ * [fLM * f^nu-1 + fL * fM * f^nu-2 * (nu - 1)] * nu,   (2,0) GMµ = GµM,   (2,1) GML = GLM,   (2,2) GMM = µ * [fMM * f^nu-1 + fM * fM * f^nu-2 * (nu - 1)] * nu,

V. Allen partial elasticity of substitution

Writing the production function as q = F(L,K,M), let the bordered Hessian be:

 F = 0 FL FK FM FL FLL FLK FLM FK FKL FKK FKM FM FML FMK FMM

If |F| is the determinant of the bordered Hessian and |FLK| is the cofactor associated with FLK, then the Allen elasticity of substitution is defined as:

sLK = ((FL * L + FK * K + FM *M) / (L * K)) * (|FLK|/|F|)

VI. 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 short-run. Then the two factor (L and M) bordered Hessian is:

 F = 0 FL FM FL FLL FLM FM FML FMM

The 2-factor elasticity of substitution between L and M is:

sLM = - (FL * L + FM * M) / (L * M ) * (FL * FM) / |F|