Some economists believe that the U.S. economy as awhole can be modeled with the following productionfunction, called the O,bb-Douglas prod11ctio11 Ju11ctio11:Y = AK1/3 L 2/3,where Y is the amount ol outpu~ K is cl,e amount ofcapital, Lis the an10u.nt of labor, and A is a paran,eterthat measures the slate of technology. For this productionfunction, tl,e marginal product of labor isMPL = (2/3) A(K/ L)1/3 ·Suppose that the price of output Pis 2, A is 3, K is1,000,000, and Lis 1,000. The labor market is competitive,so labor is paid the value of its marginal product.a. Calcula te cl,e amount of output produced Y and thedollar value ol output PY.b. Calcula te cl,e wage Wand cl,e real wage W JP. (Note:11,e wage is labor compensation measured in dol•lars, whereas the real wage is labor compensationmeasured in units ol output.)c. Calcula te cl,e labor share (the fraction of the value ofoutpu t that is paid to labor), which is (Wl.)/(PY).d. Calcula te what happens to output Y, the wage W,the real wage W/ P,and cl,e labor share (WL)/(PY) ineach ol the following scenarios,i. Inflation increases I' from 2 to 3.ii. Technological progress increases A from 3 to 9.iii. Capital accumulation increases K from 1,000,000lo 8,000,000.iv. A plague decreases L from 1,000 to 125.e. Despite many changes in the U.S. economy overlime, the labor share has been relatively stable. Isthis observation consistent with the Cobb-Douglasproduction function? Explain.
Some economists believe that the U.S. economy as a
whole can be modeled with the following production
function, called the O,bb-Douglas prod11ctio11 Ju11ctio11:
Y = AK1/3 L 2/3,
where Y is the amount ol outpu~ K is cl,e amount of
capital, Lis the an10u.nt of labor, and A is a paran,eter
that measures the slate of technology. For this production
function, tl,e marginal product of labor is
MPL = (2/3) A(K/ L)1/3 ·
Suppose that the price of output Pis 2, A is 3, K is
1,000,000, and Lis 1,000. The labor market is competitive,
so labor is paid the value of its marginal product.
a. Calcula te cl,e amount of output produced Y and the
dollar value ol output PY.
b. Calcula te cl,e wage Wand cl,e real wage W JP. (Note:
11,e wage is labor compensation measured in dol•
lars, whereas the real wage is labor compensation
measured in units ol output.)
c. Calcula te cl,e labor share (the fraction of the value of
outpu t that is paid to labor), which is (Wl.)/(PY).
d. Calcula te what happens to output Y, the wage W,
the real wage W/ P,and cl,e labor share (WL)/(PY) in
each ol the following scenarios,
i. Inflation increases I' from 2 to 3.
ii. Technological progress increases A from 3 to 9.
iii. Capital accumulation increases K from 1,000,000
lo 8,000,000.
iv. A plague decreases L from 1,000 to 125.
e. Despite many changes in the U.S. economy over
lime, the labor share has been relatively stable. Is
this observation consistent with the Cobb-Douglas
production function? Explain.
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