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Egwald Economics: Microeconomics Duality and the Translog Production / Cost Functions
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Duality: Production / Cost Functions: Cobb-Douglas Duality | CES Duality | Theory of Duality | Translog Duality - CES | Translog Duality - Generalized CES | 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 Production Functions: Cobb-Douglas | CES | Generalized CES | Translog | Diewert | Translog vs Diewert | Diewert vs Translog | Estimate Translog | Estimate Diewert | References and Links S. Translog (Transcendental Logarithmic) Duality and the Generalized CES Technology I. Profit (Wealth) Maximizing Firm. Production and cost functions (and profit functions) can be used to model how a profit (wealth) maximizing firm hires or purchases inputs (factors), such as labour, capital (structures and machinery), and materials and supplies, and combines these inputs through its production process to produce the products (outputs) that the firm sells (supplies) to its customers. The theory of duality links the production function models to the cost function models by way of a minimization or maximization framework. The cost function is derived from the production function by choosing the combination of factor quantities that minimize the cost of producing levels of output at given factor prices. Conversely, the production function is derived from the cost function by calculating the maximum level of output that can be obtained from specified combinations of inputs. II. The Production and Cost Functions. The Translog production function:
ln(q) = ln(A) + aL*ln(L) + aK*ln(K) + aM*ln(M) + bLL*ln(L)*ln(L) + bKK*ln(K)*ln(K) + bMM*ln(M)*ln(M) an equation in 10 parameters, A, aL, aK, aM, bLL, bKK, bMM, bLK, bLM, bKM, where L = labour, K = capital, M = materials and supplies, and q = product. The Translog (total) cost function:
an equation in 18 parameters, c, cq, cL, cK, cM, dqq, dLL, dKK, dMM, dLK, dKL, dLM, dML, dKM, dMK, dLq, dKq, and dMq, where wL, wK, and wM are the factor prices of L, K, and M respectively. III. Duality between Production and Cost Functions. Mathematically, the duality between a production function, q = F(L, K, M), and a cost function, C(q; wL, wK, wM), is expressed: 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, L > 0, K > 0, M > 0} (*) F*(L, K, M) = maxq {q : C(q; wL, wK, wM) <= wL * L + wK * K + wM * M, L > 0, K > 0, M > 0, for all wL >= 0 , wK >= 0, wM >= 0} (**), with the questions, F == F*, and C == C*? The dual functions C*, and F* will be derived from the estimated functions F, and C. If the functions F(L, K, M), and C(q; wL, wK, wM) obey sufficient conditions, the above minimization and maximization problems can be solved by nonlinear optimization techniques, such as Newton's Method. Furthermore, the implicit function theorem can be exploited to facilitate such calculations. IV. Constrained Optimization. The (*) minimization problems are solved for the Generalized CES production function, the Translog production function, and for the Diewert (Generalized Leontief) production function. The maximization problem (**) is somewhat more difficult than the minimization problem (*), since as specified, the requirements — for all wL >= 0 , wK >= 0, wM >= 0 — imply an infinite number of constraints. This difficulty is overcome with cost functions that factor: C(q; wL, wK, wM) = q^1/nu * c(wL, wK, wM) by solving the reduced dimension problem: F*(L, K, M)^1/nu = 1 / maxwL,wK,wM {c(wL, wK, wM) : wL * L + wK * K + wM * M = 1, wL >= 0, wK >= 0, wM >= 0) However, with cost functions that do not factor, such as the Translog cost function, the following method can be used to reduce the dimensionality of the constraints of (**), if the cost function is linear homogeneous in factor prices. Cost Function Linear Homogeneous in Factor Prices. When (L, K, M) is the least cost combination of inputs at the specific combination of factor prices (wL, wK, wM): C(q; wL, wK, wM) = wL * L + wK * K + wM * M. With the cost function, C(q; wL, wK, wM), linear homogeneous in factor prices: C(q; wL / wM, wK / wM, 1) = wL / wM * L + wK / wM * K + M, assuming that wM > 0. Writing vL = wL / wM, and vK = wK / wM, the maximization problem (**) becomes the minimization problem: F*(L, K, M) = minq, vL, vK {q : C(q; vL, vK, 1) >= vL * L + vK * K + M, L > 0, K > 0, M > 0, for all vL >= 0 , vK >= 0} (***), an optimization problem with one constraint. V. Cost Function to Production Function: The Method of Langrange. Using the Method of Lagrange, define the Langrangian function, H, of the minimization problem (***): H(L, K, M, q, vL, vK, λ) = q + λ * (vL * L + vK * K + M - C(q; vL, vK, 1)), where the new variable, λ, is called the Lagrange multiplier. a. First Order Necessary Conditions:
b. Solution Functions: We want to solve, simultaneously, these four equations for the variables λ, q, vL, and vK as functions of the variables L, K, and M, and the parameters of the cost function.
λ = λ(L, K, M) c. Jacobian Matrices: The Jacobian matrix (bordered Hessian of H) of the four functions, Hλ, Hq, HvL, HvK, with respect to the choice variables, λ, q, vL, vK :
The bordered principal minor of the bordered Hessian of the Langrangian function, H:
d. Second Order Necessary Conditions: The second order necessary conditions require that the Jacobian matrix (bordered Hessian of H) be positive definite at the solution vector W = (L, K, M, q, vL, vK, λ). The Jacobian matrix, J3, is a positive definite matrix if the determinant of J2 is negative, and the determinant of J3 is negative. e. Sufficient Conditions: If the second order necessary conditions are satisfied, then the first order necessary conditions are sufficient for a minimum at W. f. The Solution Functions' Comparative Statics. The Jacobian matrix of the four functions, Hλ, Hq, HvL, HvK, with respect to the variables, L, K, M:
The Jacobian matrix of the four solution functions, Φ = {λ, q, vL, vK}, with respect to the variables, L, K, and M:
From the Implicit Function Theorem:
JL, K, M; + Jλ, q, vL, vM * JΦ = 0 (zero matrix) → for points (L, K, M) in a neighborhood of the point (L, K, M), and wL = wL(L, K, M), wK = wK(L, K, M), wM = wM(L, K, M), and λ = λ(L, K, M), with:
VI. Question: F == F*? The obtained values for F*(L, K, M), F*L(L, K, M), F*K(L, K, M), and F*M(L, K, M) can be compared with the values of F(L, K, M), FL(L, K, M), FK(L, K, M), and FM(L, K, M), and with the corresponding values from the Generalized CES production function. VII. Duality. The Plan:
1. Specify the parameters of a Generalized CES production function, and obtain the derived Generalized CES cost function using Newton's Method. VIII. Generate Generalized CES production / cost data. The three factor Generalized CES production function is: q = A * [alpha * L^-rhoL + beta * K^-rhoK + gamma *M^-rhoM]^(-nu/rho) = f(L,K,M). where L = labour, K = capital, M = materials and supplies, and q = product. The parameter nu permits one to adjust the returns to scale, while the parameter rho is the geometric mean of rhoL, rhoK, and rhoM: rho = (rhoL * rhoK * rhoM)^1/3. If rho = rhoL = rhoK = rhoM, we get the standard CES production function. If also, rho = 0, ie sigma = 1/(1+rho) = 1, we get the Cobb-Douglas production function. The estimated coefficients of the Translog production and cost functions and will vary with the parameters nu, rho, rhoL, rhoK, rhoM, alpha, beta and gamma of the Generalized CES production function. Set the parameters below to re-run with your own Generalized CES parameters. The restrictions ensure that the least-cost problems can be solved to obtain the underlying Generalized CES cost function, using the parameters as specified.
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