The New Age firm produces clean coal. New Age's production function is given by q L³S3, where L denotes labor and S denotes its cutting-edge machinery. The price of labor is given by the wage, w, and price of machinery is denoted pc. Suppose that in the short run number of machines is fixed at 5: = 1 while L is variable. a. Write down the expression for the short-run production function. In no more than two sentences, describe the fundamental difference between the short-run and long-run time horizons in production. Invert the short-run production and derive variable costs, VC(q) = wL(q), where labor is a function of the level of output L(q). Next derive fixed costs and the short run cost function, C(q). b. c. Compute the marginal productivity of labor (MPL) and show that MP, is diminishing, i.e. ac амс d. Compute marginal cost (MC(q) = 3) and show that it is increasing, i.e., . > 0. 8q OMPL SL < 0. e. Show that a diminishing MPL is implies that MC is increasing. f. Show that the marginal productivity of labor (MPL) is less than the average product of labor (APL). And O APL show that MPL < APL implies that there is a diminishing APL, i.e. < 0. SL where average g. Find a formula for the level of output (q), which will depend on (be a function of) w and Ps, cost is minimized? What is the impact of an increase in fixed costs on the cost minimizing level of output (q) and why?
The New Age firm produces clean coal. New Age's production function is given by q L³S3, where L denotes labor and S denotes its cutting-edge machinery. The price of labor is given by the wage, w, and price of machinery is denoted pc. Suppose that in the short run number of machines is fixed at 5: = 1 while L is variable. a. Write down the expression for the short-run production function. In no more than two sentences, describe the fundamental difference between the short-run and long-run time horizons in production. Invert the short-run production and derive variable costs, VC(q) = wL(q), where labor is a function of the level of output L(q). Next derive fixed costs and the short run cost function, C(q). b. c. Compute the marginal productivity of labor (MPL) and show that MP, is diminishing, i.e. ac амс d. Compute marginal cost (MC(q) = 3) and show that it is increasing, i.e., . > 0. 8q OMPL SL < 0. e. Show that a diminishing MPL is implies that MC is increasing. f. Show that the marginal productivity of labor (MPL) is less than the average product of labor (APL). And O APL show that MPL < APL implies that there is a diminishing APL, i.e. < 0. SL where average g. Find a formula for the level of output (q), which will depend on (be a function of) w and Ps, cost is minimized? What is the impact of an increase in fixed costs on the cost minimizing level of output (q) and why?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Please help me answer d,e,f questions.
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1. The New Age firm produces clean coal. New Age's production function is given by q = L3S3, where L denotes labor
and S denotes its cutting-edge machinery. The price of labor is given by the wage, w, and price of machinery is
denoted ps. Suppose that in the short run number of machines is fixed at 5 = 1 while L is variable.
a. Write down the expression for the short-run production function. In no more than two sentences,
describe the fundamental difference between the short-run and long-run time horizons in production.
b. Invert the short-run production and derive variable costs, VC(q) = wL(q), where labor is a function of the
level of output L(q). Next derive fixed costs and the short run cost function, C(q).
c. Compute the marginal productivity of labor (MPL) and show that MPL is diminishing, i.e.
ac,
d. Compute marginal cost (MC(q)
C) and show that it is increasing, i.e., .
ƏMC
8q
əq
=
-> 0.
aMPL
SL
< 0.
e. Show that a diminishing MPL is implies that MC is increasing.
f. Show that the marginal productivity of labor (MPL) is less than the average product of labor (APL). And
O APL
show that MPL < APL implies that there is a diminishing APL, i.e. < 0.
SL
Ps¹
where average
g. Find a formula for the level of output (q), which will depend on (be a function of) w and
cost is minimized? What is the impact of an increase in fixed costs on the cost minimizing level of output
(q) and why?
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