Macroeconomics (Fourth Edition)
4th Edition
ISBN: 9780393603767
Author: Charles I. Jones
Publisher: W. W. Norton & Company
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Question
Chapter 4, Problem 6E
To determine
Derive the per capita GDP and capital per person.
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Suppose the production function for widgets is given by KL – 0.5K2 – 0.1 L2 , where q represents the annual quantity of widgets produced, K represents annual capital input, and L represents annual labor input.
(a). Suppose K=5; what is the average productivity of labor (Average product of Labor, MPL)
(b). Suppose K=10; at what level of labor input does the total output reach the maximum?
Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at
a fixed rate: = g> 0
(a) Obtain the marginal product of capital algebraically, also discussing the second derivative.
(b) Transform the production function into efficiency-worker terms, showing how =ỹ depends
AL
on =k.
K
AL
100VKL, where q is
4. The production function for puffed rice is given by q =
the number of boxes produced per hour, K is the number of puffing guns used
each hour, and L is the number of workers hired each hour.
(a) Calculate the q = 1,000 isoquant for this production function and show it on a
graph.
(b) If K = 10, how many workers are required to produce q =1,000? What is the
average productivity of puffed-rice workers?
(c) Suppose technical progress shifts the production function to q = 200 VKL.
Answer parts (a) and (b) for this new situation.
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- Was there an error in calculation of average productivity? Since Function q = 10 L - 80 - 0.2L^2 then q/L = 10 - 80/L - 0.2 L, correct?Can you verify? Thanksarrow_forwardK 0.25 (AL) 0.75, where technology A grows at Given the following aggregate production function: Y = a fixed rate: = g> 0 (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing how =ỹ depends AL K on =k. ALarrow_forwardQ5 The rate of change of the function f(x) =x + 2 /1 − 8xwith respect to x when x = 1. (ii) The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled by Q(K) = 500 K^2/3. Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure,where K(t) =2t4 + 3t + 149 /t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?arrow_forward
- Output Y is produced according to Y = F(K, L), where K is the capital stock and Lis the number of workers. The production function F(K, L) has constant returns toscale and diminishing marginal returns to capital and labour individually. The levelof technology is assumed to be constant over time. Capital per worker is denotedby k = K/L and output per worker by y = Y/L, and the two are related according toy = f(k), where f(k) = F(k, 1) is the per-worker production function. Investment is equal to saving, which is a constant fraction s of income Y. Capitaldepreciates at rate δ and the number of workers grows at rate n. The change incapital per worker over time is ∆k = sf(k) − (δ + n)k. Consider a country that has reached its steady-state capital per worker. Then, inone year, the country suffers severe flooding that destroys part of its capital stock.Assume this is treated as a one-off event that is not expected to reoccur. explain what happens to incomeper worker in the short run and the…arrow_forwardThe Cobb - Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = COLc1Cc2 where c0, c1, and c2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a)In this example, assume c0 = 5, c1 = 0.25, and c2= 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. = 0 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0 < = L < = 3,000 and 0 <= C <= 1,000. Round your answers to the nearest integer when necessary.)________ units at (L, C) = Please use Excel to get answers. Show steps on Excel Please!arrow_forwardSuppose that the production function is given by Y=AK0.4N0.6. What is the percentage change in output if both capital and labor rise by 42%? Write the answer in percent terms with up to two decimals (e.g., 10.22 for 10.22%, or 2.33 for 2.33%).arrow_forward
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