A Translog Cost Function and Derived Demands
/*--------------------------------------------------------------
SAS Sample Library
Name: modex21.sas
Description: Example program from SAS/ETS User's Guide,
The MODEL Procedure
Title: A Translog Cost Function and Derived Demands
Product: SAS/ETS Software
Keys: derived demand, systems of equations
PROC: MODEL
Notes:
--------------------------------------------------------------*/
data klems;
input year y k l e m s py pk pl pe pm ps vy vk vl ve vm vs;
datalines;
1949 26.5 67.1 179.3 50.1 39.2 26.4 43.8 23.7 9.4 17.1 21.8 13.6 6.978 1.047 3.271 0.198 2.279 0.183
1950 29.6 70 198.6 53.5 45.8 34.9 47 21.7 10 17.6 24.8 13.9 8.336 0.996 3.847 0.217 3.03 0.247
1951 30 72 192 54.5 48.7 36.5 52.9 30.3 10.1 16.6 29.6 14.7 9.506 1.432 3.743 0.208 3.85 0.272
1952 30.7 71.6 181.7 56.8 49.3 34.1 46.5 21.3 10.6 16.9 25.3 15.7 8.552 1.004 3.729 0.221 3.326 0.272
1953 31.5 69.3 180.4 56 52.7 35.2 45 18.1 11.1 18.1 23.3 16.6 8.495 0.824 3.853 0.234 3.287 0.297
1954 29 68.5 159.7 50.8 45.3 32 43 5.9 12.3 17.1 24.2 17 7.48 0.264 3.808 0.2 2.93 0.278
1955 32.7 68.1 167.9 56.4 52.7 37 43.2 19.9 11.3 16.7 23.8 17.5 8.461 0.889 3.67 0.217 3.355 0.329
1956 32.6 68.3 163.4 56.6 51.9 36.9 43 20.2 11.8 17.2 23.3 18.2 8.421 0.908 3.716 0.224 3.232 0.342
1957 31.8 67.1 152.5 53.2 49.1 37 42.8 18.6 12.3 17.8 24 18.7 8.165 0.82 3.633 0.218 3.142 0.353
1958 31.3 64.6 141.8 53 47.2 35.8 41.5 19.7 12.7 17.1 23 19.7 7.79 0.837 3.49 0.208 2.898 0.358
1959 34.6 63.4 152.1 55.8 53.4 34.7 42.4 25.1 13.2 16.8 23.2 20.1 8.794 1.047 3.875 0.216 3.301 0.355
1960 33.6 62.7 145.6 55.4 51.7 33.6 43 26.5 13.8 17 22.6 20.9 8.675 1.093 3.894 0.216 3.113 0.358
1961 35 62.1 141.9 55.5 54.2 32.5 41.8 26.2 13.9 16.9 22.9 21.5 8.774 1.07 3.823 0.216 3.31 0.355
1962 37.6 62.1 145.6 57.7 59.2 33.6 42.4 29.1 14.5 16.9 23.4 22.2 9.566 1.189 4.075 0.224 3.7 0.379
1963 39 62.5 143.4 60.1 62.8 34.1 42.3 30.6 14.9 16.4 23.2 22.8 9.89 1.256 4.115 0.228 3.896 0.395
1964 42.1 62.7 145.6 66.3 66.6 37.3 42.6 37.9 15.5 15.8 23.4 23.4 10.76 1.561 4.359 0.242 4.154 0.443
1965 45.4 64.5 154 69.2 67.9 38.1 42.8 43.8 16.1 15.6 23.7 24.1 11.656 1.858 4.78 0.248 4.302 0.467
1966 49 68.7 160.8 74 68.2 41.4 42.8 43.3 16.9 15.4 25.2 24.9 12.572 1.953 5.241 0.263 4.59 0.525
1967 50 72.3 156.5 81.4 67.9 47.1 42.3 37.7 17.9 15.5 25.3 25.6 12.68 1.79 5.405 0.292 4.581 0.613
1968 54 74.8 162.8 86.1 76.9 51 43.8 40.3 19.2 15.4 25.2 26.5 14.189 1.982 6.042 0.305 5.172 0.688
1969 56.2 77.3 161.9 87.8 77.9 54.4 44.1 40.8 20.5 15.8 25.4 27.5 14.867 2.071 6.421 0.32 5.293 0.762
1970 57 79.2 153.8 89.9 74.9 51.8 43.3 41.2 21.5 16.6 25.8 29 14.8 2.146 6.384 0.343 5.164 0.764
1971 60.2 81.5 153.1 94.6 81.3 57.2 43.2 37.6 22.8 18.4 25.7 30.4 15.591 2.011 6.728 0.401 5.566 0.885
1972 66.5 84.1 160.9 100.9 94.6 66.7 45.6 39.6 24.6 20 26.9 32.1 18.185 2.19 7.645 0.465 6.794 1.091
1973 65.7 87.4 162 102.1 90.5 65 50.5 36.2 27.4 22.8 31.3 33.7 19.896 2.081 8.588 0.537 7.574 1.115
1974 61.9 89.5 148.9 93.3 92.2 64 58.3 39.9 30.2 32.4 35.6 36.2 21.661 2.344 8.676 0.697 8.763 1.18
1975 60.6 89.7 133.4 93.2 84 65.8 56.3 36 31.5 40.2 36 39.1 20.468 2.124 8.109 0.863 8.062 1.31
1976 67.2 89.6 144.1 99.6 95.8 68.2 60.2 40.6 34.5 44.1 38.1 42.6 24.242 2.391 9.61 1.011 9.751 1.478
1977 73.8 90.3 143.8 98.7 100.6 73.4 61.6 51.4 37.4 50.2 40.8 46 27.261 3.052 10.382 1.143 10.968 1.716
1978 75 90.7 142.6 96.7 96.8 67.9 63.8 53.9 40.6 54.9 43.9 49.5 28.682 3.212 11.18 1.224 11.356 1.71
1979 76.4 90 139.6 93.4 94.1 67.4 67.3 53.6 44.1 62.8 50.1 53.1 30.828 3.169 11.894 1.352 12.591 1.821
1980 74.2 89.5 132.9 88 88.1 64.5 73.4 48.9 48.7 74.6 58.9 57.8 32.638 2.875 12.5 1.514 13.852 1.898
1981 73.1 89.5 127.7 85.4 86.6 58.1 80.4 52.5 53.1 87.1 67 63 35.246 3.089 13.102 1.715 15.478 1.862
1982 69.3 88.9 110.3 75.6 77.9 49.7 80.9 53.7 58.3 100.2 70.2 67.1 33.613 3.133 12.426 1.746 14.611 1.698
1983 78.3 88.7 117.4 81.7 89.5 56.7 82.3 68.1 61.2 105.6 70 71 38.62 3.969 13.881 1.987 16.736 2.047
1984 79.2 90.6 116.8 82.6 90.3 54.8 84.6 65.6 63.8 110.3 73.4 75 40.19 3.905 14.398 2.101 17.697 2.089
1985 77 90.8 109.9 79.5 87 53.2 84.3 64 65.9 107.8 73.4 78.2 38.931 3.818 13.983 1.974 17.04 2.117
1986 80.4 90.9 112.8 83 88.5 55.6 85 78.1 67.8 99.7 73.7 78.1 40.975 4.668 14.777 1.906 17.415 2.209
1987 86.5 91.1 119.1 91.3 95.1 61.2 86.8 77.4 71.1 97.9 76.6 81.6 45.052 4.633 16.35 2.059 19.468 2.542
1988 86.4 91.2 117.4 91.3 88 68.5 90.4 76 74.2 96.4 87.3 83.2 46.83 4.553 16.831 2.029 20.518 2.899
1989 88.3 91.5 115.5 92 90.7 77.9 92.3 70.5 78.6 99.4 89.7 83.2 48.895 4.238 17.526 2.106 21.725 3.3
1990 86.2 92 109.3 88.5 88.2 79.2 94.3 85.5 80.7 97.4 89.7 84.9 48.726 5.172 17.023 1.987 21.121 3.423
1991 85.8 92.1 107.6 90.5 86.6 84.2 95 89.3 82.4 95.4 89.5 85.6 48.884 5.406 17.134 1.989 20.69 3.664
1992 92.9 92.6 108.5 94.5 91.9 95.2 95.8 121.6 86.5 96 87.8 87.1 53.383 7.402 18.131 2.091 21.541 4.219
1993 97.6 93.9 109.8 98.8 97.4 98.3 95.6 110.6 89.8 97.9 89.9 90.2 55.964 6.819 19.041 2.229 23.364 4.512
1994 101.5 96.5 111.4 104.7 101.9 102.3 95.8 97.8 93.4 98.5 91.2 93.5 58.342 6.205 20.088 2.375 24.808 4.866
1995 100.5 98.8 105.7 106.2 99.7 102.7 98.8 88.8 96.5 94.4 100.7 96.3 59.593 5.765 19.701 2.31 26.782 5.035
1996 100 100 100 100 100 100 100 100 100 100 100 100 59.979 6.572 19.316 2.305 26.697 5.089
1997 100.7 101.8 99.7 95.9 99 97.2 100.4 98.8 102.4 101.8 101.7 104.2 60.601 6.605 19.713 2.248 26.879 5.155
1998 99.6 105.6 95.7 94.1 101.5 92.2 100 94.3 106.8 95.5 97.3 107.6 59.763 6.548 19.739 2.07 26.358 5.049
1999 96.9 106.9 88.5 91.7 93.7 85.2 97.4 86.3 113.1 97.2 97.3 110.6 56.602 6.061 19.337 2.054 24.354 4.796
2000 95.2 106.2 85.6 90.3 87.9 80.3 97.3 68 116.3 112.1 104.9 114.5 55.599 4.748 19.231 2.332 24.611 4.676
2001 85.3 103.5 74.5 81.1 76.3 68.3 97 71.5 118.5 115.4 105.5 117.3 49.627 4.865 17.042 2.156 21.49 4.074
;
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;
data klems;
set klems;
label y="Output"
k="Capital"
l="Labor"
e="Energy"
m="Materials"
s="Services"
pk="Capital Price"
pl="Labor Price"
pe="Energy Price"
pm="Materials Price"
ps="Services Price"
vk="Capital Value"
vl="Labor Value"
ve="Energy Value"
vm="Materials Value"
vs="Services Value"
sk="Capital Share"
sl="Labor Share"
se="Energy Share"
sm="Materials Share"
ss="Services Share";
run;
proc sgplot data = klems;
series x = year y = k / markers markerattrs =(symbol=circle);
series x = year y = l / markers markerattrs =(symbol=square);
series x = year y = e / markers markerattrs =(symbol=star);
series x = year y = m / markers markerattrs =(symbol=diamond);
series x = year y = s / markers markerattrs =(symbol=hash);
title 'Factor Quantities';
yaxis label = 'Quantity';
run;
proc sgplot data = klems;
series x = year y = pk / markers markerattrs =(symbol=circle);
series x = year y = pl / markers markerattrs =(symbol=square);
series x = year y = pe / markers markerattrs =(symbol=star);
series x = year y = pm / markers markerattrs =(symbol=diamond);
series x = year y = ps / markers markerattrs =(symbol=hash);
title 'Factor Prices';
yaxis label = 'Price';
run;
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;
goptions reset=title;
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;
label a_k = "SK Intercept"
a_l = "SL Intercept"
a_e = "SE Intercept"
a_m = "SM Intercept"
gkk = "SK K Price"
gkl = "SK L Price"
gke = "SK E Price"
gkm = "SK M Price"
gks = "SK S Price"
gky = "SK Output"
glk = "SL K Price"
gll = "SL L Price"
gle = "SL E Price"
glm = "SL M Price"
gls = "SL S Price"
gly = "SL Output"
gek = "SE K Price"
gel = "SE L Price"
gee = "SE E Price"
gem = "SE M Price"
ges = "SE S Price"
gey = "SE Output"
gmk = "SM K Price"
gml = "SM L Price"
gme = "SM E Price"
gmm = "SM M Price"
gms = "SM S Price"
gmy = "SM Output";
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;
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+gkk+gkl+gke+gkm=0,
gls+gkl+gll+gle+glm=0,
ges+gke+gle+gee+gem=0,
gms+gkm+glm+gem+gmm=0,
/*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;
label a_k = "SK Intercept"
a_l = "SL Intercept"
a_e = "SE Intercept"
a_m = "SM Intercept"
gkk = "SK K Price"
gkl = "SK L Price"
gke = "SK E Price"
gkm = "SK M Price"
gks = "SK S Price"
gky = "SK Output"
glk = "SL K Price"
gll = "SL L Price"
gle = "SL E Price"
glm = "SL M Price"
gls = "SL S Price"
gly = "SL Output"
gek = "SE K Price"
gel = "SE L Price"
gee = "SE E Price"
gem = "SE M Price"
ges = "SE S Price"
gey = "SE Output"
gmk = "SM K Price"
gml = "SM L Price"
gme = "SM E Price"
gmm = "SM M Price"
gms = "SM S Price"
gmy = "SM Output";
test "Constant Returns to Scale"
gky=0,
gly=0,
gey=0,
gmy=0, / lr;
run;
proc means data = klems noprint mean;
variables sk sl se sm ss;
output out = meanshares mean = sk sl se sm ss;
run;
proc iml;
/*Read in parameter estimates*/
use est;
read all var {gkk gkl gke gkm gks};
read all var {gll gle glm gls};
read all var {gee gem ges};
read all var {gmm gms};
close est;
/*Calculate S parameter based on homogeneity constraint*/
gss=0-gks-gls-ges-gms;
/*Read in mean cost shares and construct vector*/
use meanshares;
read all var {sk sl se sm ss};
close meanshares;
w = sk//sl//se//sm//ss;
print w;
/*Construct matrix of parameter estimates*/
gij = (gkk||gkl||gke||gkm||gks)//
(gkl||gll||gle||glm||gls)//
(gke||gle||gee||gem||ges)//
(gkm||glm||gem||gmm||gms)//
(gks||gls||ges||gms||gss);
print gij;
nk=ncol(gij);
mi = -1#I(nk); /*Initialize negative identity matrix*/
eos = j(nk,nk,0); /*Initialize Marshallian EOS Matrix*/
mos = j(nk,nk,0); /*Initialize Morishima EOS Matrix*/
ep = j(nk,nk,0); /*Initialize Price EOD Matrix*/
/*Calculate Marshallian EOS and Price EOD Matrices*/
i=1;
do i=1 to nk;
j=1;
do j=1 to nk;
eos[i,j] = (gij[i,j]+w[i]#w[j]+mi[i,j]#w[i])/(w[i]#w[j]);
ep[i,j] = w[j]#eos[i,j];
end;
end;
/*Calculate Morishima EOS Matrix*/
i=1;
do i=1 to nk;
j=1;
do j=1 to nk;
mos[i,j] = ep[i,j]-ep[j,j];
end;
end;
/*Print Elasticity Matrices*/
factors = {"Capital" "Labor" "Energy" "Materials" "Services"};
print
ep[label="Price Elasticities of Demand" rowname=factors colname=factors
format=d7.3],
eos[label="Hicks-Allen Elasticities of Substitution" rowname=factors colname=factors
format=d7.3],
mos[label="Morishima Elasticities of Substitution" rowname=factors colname=factors
format=d7.3];
run;
