(a)
The nature of the production function.
(a)
Explanation of Solution
The production function exhibits constant returns to the scale when the output increases in a constant proportion to the increase in the factor inputs. In other words, a production function exhibits constant returns to the scale if a two times increase in the input factors of production increases the output by two times. A constant production function takes the following form, where z is any positive number.
The given production function can be written as follows:
Thus, it is evident that the given production function shows constant returns to the scale.
Constant returns to scale: Constant returns to scale implies that as the scale of production of the firm increases, the cost of production per unit remains unchanged.
(b)
The production function per worker.
(b)
Explanation of Solution
The per worker production function can be calculated using Equation (1) as follows:
The output per worker can be calculated by substituting the respective values in Equation (1) as follows:
The given production function exhibits constant returns to scale.
The ratio output per labor is
Thus, the production function per worker is
Output per worker: Output per worker is the ratio of the total output produced to the total workers employed.
(c)
The steady state level of capital per worker, income per worker and the consumption per worker.
(c)
Explanation of Solution
Given that the country faces a
In the steady state, the condition for the capital per worker is given as follows:
The steady state level of capital per worker in Country A is calculated by substituting the respective values in the Equation (2) as follows:
The steady state level of capital per worker in Country A is 0.35.
The steady state level of capital per worker in Country B is calculated by substituting the respective values in the Equation (2) as follows:
The steady state level of capital per worker in Country B is 1.84.
The steady state level of income can be calculated as follows:
The steady state level of income in Country A is calculated by substituting the respective values in the Equation (3) as follows:
The steady state level of income in Country A is 0.59.
The steady state level of income in Country B is calculated by substituting the respective values in the above equation.
The steady state level of income in country B is 1.36.
The consumption is the difference between income and savings. The consumption per worker can be calculated using the Equation (4) as follows:
The steady state level of consumption per worker in Country A is calculated by substituting the respective values in the Equation (4) as follows:
The steady state level of consumption per worker in Country A is 0.53.
The steady state level of consumption per worker in Country B is calculated by substituting the respective values in Equation (4) as follows:
The steady state level of consumption per worker in Country B is 0.95.
Output per worker: Output per worker is the ratio of the total output produced to the total workers employed.
(d)
The income per worker and the consumption per worker.
(d)
Explanation of Solution
If the capital per worker in both countries is equal to one, then it is given as follows:
The consumption per worker in Country A is calculated as follows:
The consumption per worker in Country A is 0.9.
The consumption per worker in Country B is calculated as follows:
The consumption per worker in Country B is 0.7.
(e)
The comparison of consumption in Country A and Country B.
(e)
Explanation of Solution
Table 1 shows the change in capital stock in Country A calculated using a spreadsheet.
Table 1
Year | k | y = k1/3 | c = (1 – sA)y | i = sAy | δk | Δk = i – δk |
1 | 1 | 1 | 0.9 | 0.1 | 0.2 | −0.10 |
2 | 0.9 | 0.97 | 0.87 | 0.1 | 0.18 | −0.08 |
3 | 0.82 | 0.93 | 0.84 | 0.09 | 0.16 | −0.07 |
4 | 0.75 | 0.91 | 0.82 | 0.09 | 0.15 | −0.06 |
5 | 0.69 | 0.88 | 0.79 | 0.09 | 0.14 | −0.05 |
6 | 0.64 | 0.86 | 0.78 | 0.09 | 0.13 | −0.04 |
7 | 0.6 | 0.84 | 0.76 | 0.08 | 0.12 | −0.04 |
Table 2 shows the change in capital stock in Country B calculated using a spreadsheet.
Table 2
Year | k | y = k1/3 | c = (1 – sA)y | i = sAy | δk | Δk = i – δk |
1 | 1 | 1 | 0.7 | 0.3 | 0.2 | 0.1 |
2 | 1.1 | 1.03 | 0.72 | 0.31 | 0.22 | 0.09 |
3 | 1.19 | 1.06 | 0.74 | 0.32 | 0.24 | 0.08 |
4 | 1.27 | 1.08 | 0.76 | 0.32 | 0.25 | 0.07 |
5 | 1.34 | 1.1 | 0.77 | 0.33 | 0.27 | 0.06 |
6 | 1.4 | 1.12 | 0.78 | 0.34 | 0.28 | 0.06 |
7 | 1.46 | 1.13 | 0.79 | 0.34 | 0.29 | 0.05 |
It is evident from the table values that it takes 5 years for Country B to have a higher consumption per worker than Country A. In the 6 th year the consumption per worker in Country B is 0.784 which exceeds the consumption per worker in Country A which is 0.775 in absolute terms.
Want to see more full solutions like this?
Chapter 8 Solutions
MACROECONOMICS+SAPLING+6 M REEF HC>IC<
- Suppose you are given the aggregate production function for an economy and the amount of available technology increases for this economy. If labor and capital constant are held constant, increase in technology will causes labor productivity to decrease. True Falsearrow_forwardChoose the following opton: 1) In the very long run, across the next fifty years, a country’s economic standard of living as measured by Real GDP Per Person can grow steadily as a result of sustained growth in one of the following two values. The correct focus of growth policy will be to promote: (a) Labor Productivity (b) Total Labor Effort 2) The correct answer to Q#1 is a consequence of which feature of any country’s economy. In the very long run, fifty years and beyond: a) Total Labor Effort (Aggregate Labor Hours) can grow faster than a country’s Population. b) Total Labor Effort (Aggregate Labor Hours) cannot grow faster than a country’s Population.arrow_forwardAssume the production function takes the general form: Y=Z*F (K,L,A)where all marginal products are positive.Which 3 of the following statements are correct?a. If A is fixed, then population growth acts as a drag on growth of output per person.b. If A is fixed, then population growth acts as a drag on growth, and so Malthus was correct that populationgrowth will always reverse the impact of technological improvements.c. Both rises in z and rises in K/L (capital intensity) will boost output per worker.d. Growth in output per worker can occur due to rises in z (technology) or rises in K/L (capital intensity), orboth.arrow_forward
- The aggregate production function is y=3KL. If they are 30 units of capital and 40 units of labor, what is aggregate output? What is labor productivity? What is capital productivity?arrow_forward(1) In one country, its production function of national output is Y = F(K, L) = A * K(3/4) * L(1/5). Does the production function have a constant return to scale, an increasing return to scale or a decreasing return to scale? And why? (show your work)arrow_forwardIf techology grows at a rate of 1% per period- how fast does output per worker grow in the long run or steady state if the production function is Y = A*(K^0.5) *(L^0.5)arrow_forward
- Show graphically on the same x and y axis what happens to this production function if there is a technological advancement in this economy. Don,t copy from anywhere.arrow_forwardMa1. two countries have the same effectiveness of labor production function Y = F (K,LE) = K^0.5 (LE)^0.5 a. what is the per effective worker production function for these countries? b . what is the steady state value of y as a function of s, n, g, and δ only? c. countries A and b are identical in every way except rate of savings. countries A has a savings rate of 17 percent and country B has a savings rate of 10 percent. for both countries the rate of technical progress is g = 0.04 the birth rate is n = 0.05 and depreciation δ = 0.04. find the steady state value of y for each country. compare and commentarrow_forwardIf we have an aggregate production function of the form Y = AK, at what capital-labor ratio can a steady-state equilibrium be reached?arrow_forward
- Please no written by hand and graph Consider a small world that consists of two different countries, a developed and a developing country. In both countries, assume that the production function takes the following form: Y = F (K, LE) = K¹/4 (LE) 3/4, where Y is output, K is capital stock, L is total employment and E is labour augmenting technology. (a) Does this production function exhibit constant returns to scale in K and L? Explain. (b) Express the above production function in its intensive form (i.e., output per-effective worker y as a function of capital per effective worker k). (c) Solve for the steady-state value of y as a function of saving rate s, population growth rate n, technological progress g, and capital depreciation rate 6. (d) The developed country has a savings rate of 30% and a population growth rate of 2% per year. Meanwhile, the developing country has a savings rate of 15% and population growth rate of 5% a year. Technology evolves at the rate of 8% and 2% in…arrow_forwardSuppose that the per-worker production (labour productivity) function in South Korea is Y over L equals A open parentheses K over L close parentheses to the power of 0.4 end exponent open parentheses H over L close parentheses to the power of 0.6 end exponent. South Korea's labor productivity rises 6% per year, capital-labour ratio rises 5% per year, and human capital per worker rises 2% per year. This information suggests that total factor productivity grows at _________ per year.arrow_forward1.Many countries, including Pakistan, import substantial amounts of goods and services from other countries. However, economists claim that a country can enjoy a high standard of living only if it can produce a large quantity of goods and services itself. Can you reconcile these two facts? (Maximum 100 words). 2.Given the production function Y= AF (L, K, H, N), explain the determinants of productivity. ( Maximum100 words). 3.Population growth has a variety of effects on productivity. Explain this statement and justify your answer. (Maximum 200 words).arrow_forward
- Economics (MindTap Course List)EconomicsISBN:9781337617383Author:Roger A. ArnoldPublisher:Cengage LearningEssentials of Economics (MindTap Course List)EconomicsISBN:9781337091992Author:N. Gregory MankiwPublisher:Cengage Learning
- Brief Principles of Macroeconomics (MindTap Cours...EconomicsISBN:9781337091985Author:N. Gregory MankiwPublisher:Cengage Learning