Question

1. Suppose that you have a standard Solow model with production given by Cobb-Douglas function. Assume A = 1, s = 0.1, α = 1/3, and δ = 0.1.
   1. Solve for the steady-state level of capital per worker, k* (Hint: use dynamic formula for capital stock.).
   2. Create an Excel spreadsheet to compute the dynamics of the capital stock. Plot the evolution of capital stock for 10 periods (i.e., t = 1, 2, … , 10) using your result in part (a).
   3. Suppose that in period 11 the savings rate increases to 0.2 and stays there permanently. Use Excel to compute the dynamic adjustment in the capital stock per worker as a result of the change in the savings rate for periods t = 11, 12, …. , 150).
   4. Plot adjustment in the capital stock per worker for periods 1 through 150. What is the new steady-state level of k, which the capital stock approaches asymptotically? How many periods will it take for capital stock per worker to reach its new steady state level?

Answer 1

Introduction

Develop a simple framework for the proximate causes and the mechanics of economic growth and cross-country income di§erences. Solow-Swan model named after Robert (Bob) Solow and Trevor Swan, or simply the Solow model Before Solow growth model, the most common approach to economic growth built on the Harrod-Domar model. Harrod-Domar mdel emphasized potential dysfunctional aspects of growth: e.g, how growth could go hand-in-hand with increasing unemployment. Solow model demonstrated why the Harrod-Domar model was not an attractive place to start. At the center of the Solow growth model is the neoclassical aggregate production function. Daron Acemoglu (MIT) Economic Growth Lectures.

a)

The production function in the Solow growth model is Y = f(K,L), or expressed in terms of output per worker, y = f(k). If a war reduces the labor force through casualties, the L falls but Capital-labor ratio k = K/L rises. The production function tells us that total output falls because there are fewer workers. Output per worker increases, however, since each worker has more capital.

b)

The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady state prior to the war, then after the war the economy has a capital stock that is higher than the steady-state level. This is shown in the figure below as an increase in capital per worker from k1 to k2. As the economy returns to the steady state, the capital stock per worker falls from k2 back to k1, so output per worker also falls.

Suppose the economy begins with an initial steady-state capital stock below the Golden Rule level. The immediate effect of devoting a larger share of national output to investment is that the economy devotes a smaller share to consumption; that is, “living standards” as measured by consumption fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise.

The productivity of workers is the average amount produced by each worker – that is, output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises. k = K/L y = Y/L y = f(k) sy (n+d)k k1 k2 y2 y1 In the new steady state, output grows at rate n+g, while output per worker grows at rate g. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher Thus, an increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth, but not the sacrifice required to achieve it.

c)

To solve this problem, it is useful to establish what we know about the U.S. economy: a A Cobb-Douglas production function has the form y = k is capital’s sharea, where   = 0.3, so we k now that the productionaof income. The question tells us that 3.function is y = k In the steady state, we know that the growth rate of output equals 3%, so we know that (n+g) = .03 = .04d The depreciation rate .The capital-output ratio K/Y = 2.5. Because k/y = [K/(LxE)]/[Y/(LxE)] = K/Y, we also know that k/y = 2.5. (That is, the capital-output ratio is the same in terms of effective workers as it is in levels.) + n + g)k.

Begin with the steady-state condition, sy = ( to a formula for saving in the steady state: + n + g(k/y)ds = ( Plugging in the values from above: s = (0.04 + 0.03)(2.5) = .175 The initial saving rate is 17.5%. Cobb-Douglas production function, capital’s = MPK(K/Y). Rewriting, we have:ashare of income /(K/Y)aMPK = Plugging in the values from above: MPK = 0.3/2.5 = .12

We know that at the Golden Rule steady state: MPK = (n + g + d) Plugging in the values from above: MPK = (.03 + .04) = .07 At the Golden Rule steady state, the marginal product of capital is 7%, whereas it is 12% in the initial steady state. Hence, from the initial steady state we need to increase k to achieve the Golden Rule steady state.

d)

We know from Chapter 3 that for a Cobb-Douglas production function, (Y/K). Solving this for the capital-output ratio, we find:aMPK = (Y/K). Solving this for the capital-output ratio, we find:

/MPKaK/Y = We can solve for the Golden Rule capital-output ratio using this equation. If we plug in the value 0.07 for the Golden Rule steady-state MPK, and the value 0.3 for a, we find: K/Y = 0.3/0.07 = 4.29 In the Golden Rule steady state, the capital-output ratio equals 4.29, compared to the current capital-output ratio of 2.5.

We know from part that in the steady state )(k/y)ds = (n + g + where k/y is the steady-state capital-output ratio. In the introduction to this answer, we showed that k/y = K/Y, and in part we found that the Golden Rule K/Y = 4.29. Plugging in this value and those established above: s = (0.04 + 0.03)(4.29) = 0.30 To reach the Golden Rule steady-state, the saving rate must rise from 17.5% to 30%.

conclusion

Simple and tractable framework, which allows us to discuss capital accumulation and the implications of technological progress. Solow model shows us that if there is no technological progress, and as long as we are not in the AK world, there will be no sustained growth. Generate per capita output growth, but only exogenously: technological progress is a blackbox. Capital accumulation: determined by the saving rate, the depreciation rate and the rate of population growth. All are exogenous. Need to dig deeper and understand what lies in these black boxes.