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- Suppose that total factor productivity, A, fluctuates in the Solow model. In particular: if t = 0,3, 6, ... if t = 1,4, 7,... if t = 2, 5, 8, ... A = Ã = 1 A, - An >1 A - AL <1 Also suppose that the saving and depreciation rate are some positive constants s and 6, re- spectively, and that population and labor force remain constant over time. Finally, the aggregate production function is given by: Y; = A, Kª N}-a a) Solve for GDP per worker, y := Y;/N. Like we did in last class, show in a dynamic diagram how the economy will experience fluctuations in real income, i.e., "business cycles". What do these artificial cycles look like and in what ways do they resemble real macroeconomic fluctuations? b) Show how the values (Ỹ , Y1, Yu) depend on the parameters (a, 8, 6, Ā, AL, Aµ). Explain carefully. c) Solve for the capital to output ratio and show how it depends on the parameter space (a, s, 6, Ā, AL, Au). Explain carefully.Let the production function be Q = K0.®L0.2 Solow's assumptions are K = sQ – 6K The symbols s represents a (constant) marginal propensity to save, n, a (constant) rate of growth of labor and 8 constant depreciation rate. (a) Derive the fundamental equation of Solow growth model for given production function. (b) Sketch graph of with k on the vertical axis and k on the horizontal take n = 0.01, s = 0.3, 8 = 0.1.5. Consider the Solow growth model. The production function is: Y = Kª(AN)!-«, the is the capital depreciation rate is & per year, the population and technology savıng rate grow at annual rates of n and g respectively. Calculate K/N, K/(AN), Y /N and Y/(AN) in the steady state. Does the change in s affect the growth rate? Explain. S, 6. Suppose that the model of the economy is given by Y = C +I+G + X 99 sunny hp 10 f8 144 14 is & 6. 7. 8. 24
- Review of the Solow model. Suppose that the production function is given by Y(t) = AK(t)ª L(t)'-«, where A is a fixed technological parameter. There is no technical progress over time. Assume a fix rate of depreciation 8, a fixed savings rate s, and a constant rate of growth of population n. a. Write down the equation that describes the accumulation of capital stock over time in terms of per capita capital stock k(t) and per capita income y(t). %3D b. Given your answer to part a., use graph(s) to explain why there exists a steady state in the Solow model. c. Recall that the Harrod-Domar has the following production function Y = K. Given your answer to part a., use graphs to explain why there does not exist a steady state in the H-D model. d. Now let us go back to the Solow model. Recall that in the steady state, k(t) = k(t + 1) = k*. Given your answer to part a., explicitly solve for the steady state value of the per capita capital stock k* and per capita income y* (i.e., solve for k*…In the Solow growth model, if investment exceeds depreciation, the capital stock will and output will until the steady state is attained. O increase; increase increase; decrease decrease; decrease decrease; increaseA7. Consider the basic Solow model with production function Y - Ka N¹-a and no employ- = ment growth. Which of the following must be false if the saving rate increases? (a) consumption per worker decreases in the long run (b) total capital stock increases in the long run (c) the growth rate of capital per worker increases in the long run (d) the growth rate of capital per worker increases in the short run
- 11. Alter the Solow growth model so that the pro- duction technology is given by Y= zK, where Y is output, K is capital, and z is total factor pro- ductivity. Thus, output is produced only with саpital. (a) Show that it is possible for income per person to grow indefinitely. (b) Also show that an increase in the savings rate increases the growth rate in per capita income.Consider the basic Solow model. Assume that Country A has a production function as following. Y = A√K Where A represents the technology available in the country and & the aggregate capital. Let the national saving rate be equal to 30%, s = 0.3. Also, assume that capital depreciates at a constant rate of 3%, delta = 0.03. a) Now, the economy experiences a positive shock to the idea/technology and A becomes 2 which was originally 1. Relative to the previous steady state level of capital and output, what will happen?Consider the following production function that does not exhibit a diminishing marginal product of capital: y=Ak. a. In the standard Solow model, the per-worker level of capital converges toward a steady state. Show that this does not occur in this model. b. Show that a higher saving rate leads to a permanently higher growth rate. Explain. c. Why does this conclusion differ from that in the Solow model?
- Assume a production function in the Solow model is given by= AK,"L, A-Z– 1, the depreciation rate is 0.05, and the investment rate is 0.1. Then the steady-state level of capital is about: Select one: O a. 2.8 O b. 0.8 1.6 O d. 0.3 O e. 1.31. Define Capital Accumulation Equation under the Solow Growth ModelPopulation Growth and Technological Progress-Work It Out An economy has a Cobb-Douglas production function: Y = K (LE)¹- The economy has a capital share of 0.20, a saving rate of 49 percent, a depreciation rate of 4.00 percent, a rate of population growth of 1.50 percent, and a rate of labor-augmenting technological change of 4.0 percent. It is in steady state. a. At what rates do total output and output per worker grow? Total output growth rate: Output per effective worker is constant in the steady state and does not change. increases in the steady state. declines in the steady state. % Output per worker growth rate: %