Estimating a Derived Demand System from a Translog Cost Function

Overview

The transcendental logarithmic (translog) functional form can capture many of the attributes of a cost function that are implied by economic theory. Because of this flexibility, it has been widely used for studying production relationships. Before logarithms are taken, the function is

The function takes w and y as its inputs and returns minimum cost, where w is a vector of prices for the inputs to production and y is a single output. N is the total number of inputs and the α's are the parameters of the function. It may not be apparent from the equation above, but the translog form nests the Cobb-Douglas form and is equivalent if all α_{i j} and α_iy are zero.

One disadvantage of the previous form is that it is not linear in parameters. A standard technique when dealing with power functions like the translog cost function is to take logarithms. The resulting function is linear in parameters and standard statistical techniques can be used for estimation. After taking logarithms the function is

While it is possible to include terms to account for technological progress, the specification used here assumes that cost is independent of time. Using Shepherd’s lemma, the derived demand equations are

where is the cost share of the i^th input. The cost function is assumed to be continuous, so Young’s Theorem concerning symmetry of the second derivatives restricts α_{i j} = α_ji

for all i ≠j.

The result of this derivation is a system of N + 1 equations consisting of N derived demand equations and one cost function.

Homogeneity of the first degree implies

for all i and j.

It is also possible to impose constant returns to scale – equivalent to imposing homogeneity in y – and details of this procedure can be found in Diewert and Wales (1987). Global concavity can also be imposed on this specification by forcing the matrix [α_{i j}] to be negative semidefinite. A technique for accomplishing this can be found in Jorgenson (1986). This procedure tends to restrict the price elasticities in an undesirable way and destroys the flexibility of the form.

Analysis

This example shows the estimation of a system of nonlinear derived demand equations based on a translog cost function using iterated seemingly unrelated regression (ITSUR). Data pertains to the United States textiles manufacturing sector (standard industrial classification code 22). The series runs from 1949 to 2001 and contains real quantity indices, real price indices, and real cost measures for a single aggregate industry output and five aggregate inputs: capital (K), labor (L), energy (E), materials (M), and services (S). The original dataset, and information on other industrial sectors, can be obtained from the Multifactor Productivity Home Page of the Bureau of Labor Statistics at http://www.bls.gov/mfp/.

Derived demand equations are each expressed with a cost share as the endogenous variable. Because this dataset does not contain explicit information on cost shares, the shares must be formed by taking the ratio of the value of each input and the cost measure.

data klems;
set klems;
   array values {5} vk vl ve vm vs;
   array costshares {5} sk sl se sm ss;
   cost = sum(vk,vl,ve,vm,vs);
   do i = 1 to 5;
     costshares{i} = values{i}/cost;
   end;
run;

A plot of the cost shares over time is produced by the following statements.

proc sgplot data = klems;
   series x = year y = sk / markers markerattrs =(symbol=circle);
   series x = year y = sl / markers markerattrs =(symbol=square);
   series x = year y = se / markers markerattrs =(symbol=star);
   series x = year y = sm / markers markerattrs =(symbol=diamond);
   series x = year y = ss / markers markerattrs =(symbol=hash);
   title 'Factor Cost Shares';
   yaxis label = 'Cost Share';
run;

Figure 1: Change in Cost Shares

One benefit of the translog form is that the system of factor demands gives nearly the same information as the cost function. The only parameter of the cost function that is not captured by the derived demand system is the intercept term which is not used in calculating price or substitution elasticities. To be as parsimonious as possible, the cost function is typically not estimated.

In this example, the MODEL procedure is used to fit four derived demand equations. One of the equations (the derived demand equation for services) has been arbitrarily dropped from estimation; only N - 1 of the factor demands are linearly independent because the dependent variables are cost shares which must sum to one. The parameters of the dropped derived demand equation can be recovered after estimation through homogeneity and symmetry restrictions.

The following statements estimate the system of derived demand equations without imposing any restrictions from theory. Likelihood ratio tests are conducted to determine whether homogeneity and symmetry are satisfied both singularly and jointly.

 proc model data = klems;
      parameters a_k gkk gkl gke gkm gks gky
       	         a_l glk gll gle glm gls gly
	                 a_e gek gel gee gem ges gey
	                 a_m gmk gml gme gmm gms gmy;
      endogenous sk sl se sm;
      exogenous pk pl pe pm ps y;

      /*System of Derived Demand Equations*/
      sk = a_k + gkk*log(pk) + gkl*log(pl) + gke*log(pe) + gkm*log(pm) + gks*log(ps)
               + gky*log(y);
      sl = a_l + glk*log(pk) + gll*log(pl) + gle*log(pe) + glm*log(pm) + gls*log(ps)
               + gly*log(y);
      se = a_e + gek*log(pk) + gel*log(pl) + gee*log(pe) + gem*log(pm) + ges*log(ps)
               + gey*log(y);
      sm = a_m + gmk*log(pk) + gml*log(pl) + gme*log(pe) + gmm*log(pm) + gms*log(ps)
               + gmy*log(y);

      fit sk sl se sm / itsur;

      test "Homogeneity"
         gkk+gkl+gke+gkm+gks=0,
         glk+gll+gle+glm+gls=0,
         gek+gel+gee+gem+ges=0,
         gmk+gml+gme+gmm+gms=0, / lr;

      test "Symmetry"
         gkl=glk,
         gke=gek,
         gkm=gmk,
         glm=gml,
         gle=gel,
         gem=gme, / lr;

      test "Joint Homogeneity and Symmetry"
         gkk+gkl+gke+gkm+gks=0,
         glk+gll+gle+glm+gls=0,
         gek+gel+gee+gem+ges=0,
         gmk+gml+gme+gmm+gms=0,
         gkl=glk,
         gke=gek,
         gkm=gmk,
         glm=gml,
         gle=gel,
         gem=gme, / lr;
   run;

Because the form of the elasticities is somewhat complicated, it can be difficult to interpret the values and signs of parameter estimates. The test results in Figure 2 show that both symmetry and homogeneity are rejected. A common practice is to assume that such restrictions hold and to impose them in estimation.

Figure 2: Tests of Symmetry and Homogeneity

The MODEL Procedure

Test Results
Test	Type	Statistic	Pr > ChiSq	Label
Homogeneity	L.R.	114.22	<.0001	gkk+gkl+gke+gkm+gks=0, glk+gll+gle+glm+gls=0, gek+gel+gee+gem+ges=0, gmk+gml+gme+gmm+gms=0
Symmetry	L.R.	109.72	<.0001	gkl=glk, gke=gek, gkm=gmk, glm=gml, gle=gel, gem=gme
Joint Homogeneity and Symmetry	L.R.	240.80	<.0001	gkk+gkl+gke+gkm+gks=0, glk+gll+gle+glm+gls=0, gek+gel+gee+gem+ges=0, gmk+gml+gme+gmm+gms=0, gkl=glk, gke=gek, gkm=gmk, glm=gml, gle=gel, gem=gme

The following code imposes both symmetry and homogeneity restrictions on the underlying model. A likelihood ratio test of constant returns to scale is introduced and a Chow test is added to the FIT statement.

 proc model data = klems;
       parameters a_k gkk gkl gke gkm gks gky
                  a_l glk gll gle glm gls gly
                  a_e gek gel gee gem ges gey
                  a_m gmk gml gme gmm gms gmy;
       endogenous sk sl se sm;
       exogenous pk pl pe pm ps y;
       restrict /*Homogeneity Restrictions*/
                gks=0-gkk-gkl-gke-gkm,
                gls=0-gkl-gll-gle-glm,
                ges=0-gke-gle-gee-gem,
                gms=0-gkm-glm-gem-gmm,
                /*Symmetry Restrictions*/
                gkl=glk, gke=gek, gkm=gmk, gle=gel, glm=gml, gem=gme;

       /*System of Derived Demand Equations*/
       sk = a_k + gkk*log(pk) + gkl*log(pl) + gke*log(pe) + gkm*log(pm) + gks*log(ps)
                + gky*log(y);
       sl = a_l + glk*log(pk) + gll*log(pl) + gle*log(pe) + glm*log(pm) + gls*log(ps)
                + gly*log(y);
       se = a_e + gek*log(pk) + gel*log(pl) + gee*log(pe) + gem*log(pm) + ges*log(ps)
                + gey*log(y);
       sm = a_m + gmk*log(pk) + gml*log(pl) + gme*log(pe) + gmm*log(pm) + gms*log(ps)
                + gmy*log(y);

       fit sk sl se sm / itsur chow = (24) outest=est;

       test "Constant Returns to Scale"
          gky=0,
          gly=0,
          gey=0,
          gmy=0, / lr;
   run;

The symmetry restriction shrinks the number of parameters of the model considerably. This is particularly useful in cases where the time series is not long and degrees of freedom need to be conserved. Figure 3 shows the parameter estimates of the MODEL procedure.

Figure 3: The Restricted Model

The MODEL Procedure

Nonlinear ITSUR Parameter Estimates
Parameter	Estimate	Approx Std Err	t Value	Approx Pr > \|t\|	Label
a_k	0.109996	0.0219	5.02	<.0001	SK Intercept
gkk	0.062014	0.00357	17.38	<.0001	SK K Price
gkl	-0.01898	0.00725	-2.62	0.0118	SK L Price
gke	-0.00179	0.000836	-2.15	0.0369	SK E Price
gkm	-0.03872	0.00610	-6.35	<.0001	SK M Price
gks	-0.00252	0.00342	-0.74	0.4648	SK S Price
gky	-0.00104	0.00496	-0.21	0.8354	SK Output
a_l	0.865473	0.0903	9.58	<.0001	SL Intercept
glk	-0.01898	0.00725	-2.62	0.0118	SL K Price
gll	-0.00999	0.0360	-0.28	0.7826	SL L Price
gle	-0.00106	0.00420	-0.25	0.8013	SL E Price
glm	-0.06766	0.0248	-2.73	0.0087	SL M Price
gls	0.097692	0.0214	4.57	<.0001	SL S Price
gly	-0.11488	0.0200	-5.74	<.0001	SL Output
a_e	0.012747	0.00984	1.30	0.2013	SE Intercept
gek	-0.00179	0.000836	-2.15	0.0370	SE K Price
gel	-0.00106	0.00420	-0.25	0.8013	SE L Price
gee	0.029876	0.00112	26.76	<.0001	SE E Price
gem	-0.01954	0.00275	-7.12	<.0001	SE M Price
ges	-0.00748	0.00325	-2.30	0.0257	SE S Price
gey	0.005808	0.00217	2.68	0.0101	SE Output
a_m	-0.08173	0.0701	-1.17	0.2492	SM Intercept
gmk	-0.03872	0.00610	-6.35	<.0001	SM K Price
gml	-0.06766	0.0248	-2.73	0.0087	SM L Price
gme	-0.01954	0.00275	-7.12	<.0001	SM E Price
gmm	0.146849	0.0222	6.62	<.0001	SM M Price
gms	-0.02092	0.0103	-2.03	0.0477	SM S Price
gmy	0.113729	0.0157	7.27	<.0001	SM Output
Restrict0	-563.453	242.8	-2.32	0.0187	gks=0-gkk-gkl-gke-gkm
Restrict1	82.23307	193.0	0.43	0.6747	gls=0-gkl-gll-gle-glm
Restrict2	321.7446	689.2	0.47	0.6455	ges=0-gke-gle-gee-gem
Restrict3	-279.71	211.0	-1.33	0.1879	gms=0-gkm-glm-gem-gmm
Restrict4	-261.228	196.3	-1.33	0.1860	gkl=glk
Restrict5	-1041.84	755.0	-1.38	0.1700	gke=gek
Restrict6	-27.855	219.3	-0.13	0.9005	gkm=gmk
Restrict7	-880.821	742.7	-1.19	0.2396	gle=gel
Restrict8	259.513	215.6	1.20	0.2326	glm=gml
Restrict9	1103.343	320.8	3.44	0.0003	gem=gme

The majority of the parameter estimates are significant. Insignificant parameters are statistically equivalent to zero and imply that corresponding elasticities of substitution are equal to the Cobb-Douglas value of one.

The TEST statement tests whether this industry exhibits constant returns to scale in the range of the sample. The CHOW option in the FIT statement performs a Chow test for a structural break at the twenty-fourth year of the sample. The twenty-fourth year of the sample is 1973; in October of that year OPEC declared an oil embargo. Markets were affected by significant shocks to oil prices and gasoline was rationed in the United States. The results of the two tests can be seen in Figure 4.

Figure 4: CRS and Chow Test Results

Test Results
Test	Type	Statistic	Pr > ChiSq	Label
Constant Returns to Scale	L.R.	74.27	<.0001	gky=0, gly=0, gey=0, gmy=0

Structural Change Test
Test	Break Point	Num DF	Den DF	F Value	Pr > F
Chow	24	23	2	0.28	0.9560

The null hypothesis of constant returns to scale is rejected. The null hypothesis of the Chow Test cannot be rejected. Even with the turmoil of the oil embargo, there is no evidence of a structural break in 1973.

Parameter estimates are stored in the est dataset and used in calculating price elasticities and elasticities of substitution in the example "Calculating Elasticities from a Translog Cost Function."

References

Blackorby, C., and Russell, R. R. (1989). “Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities).” American Economic Review 79:882–888.
Chambers, R. G. (1988). Applied Production Analysis: A Dual Approach. New York: Cambridge University Press.
Diewert, W. E., and Wales, T. J. (1987). “Flexible Functional Forms and Global Curvature Conditions.” Econometrica 55:43–68.
Jorgenson, D. (1986). “Econometric Methods for Modeling Producer Behavior.” In Handbook of Econometrics, edited by Z. Griliches, and M. D. Intriligator, 1841–1915. Amsterdam: North-Holland.