solow growth homework solution

solow growth homework solution

Analyzing Solow Growth Model: A Comprehensive Homework Solution

1. Introduction to the Solow Growth Model

In the WWII era, economists nurtured two new branches of traditional inquiries: shadows of poverty and wealth. The great models of “classical” growth were engendered under the combined stimulation of Harrod’s 1939 “An Essay in Dynamic Theory,” Domar’s “Capital Expansion and Rate of Growth,” and Solow’s 1956 “A Contribution to the Theory of Economic Growth.” These models state that the leap from Marx’s and Walras’s analyses of relationships among capital, interest, wages, saving, and investment, which are all short-run but designate significantly different dynamics, to immediate long-run modelings of general dynamics was initiated by capitalized-value aggregate production functions.

Solow’s 1956 model explains the transitions of nations to long-run growth paths with associated capital intensities and productivities. While it generalizes “classical” models by important topological properties, Solow’s 1956 model exhibits questionable behavior when tested against the facts. Unlike “neoclassical” models, which depict perpetual consumer welfare growth, Solow’s 1956 model admits only one of both Marglin’s critique directions. Given that Solow’s 1956 entropy analysis maps Solow’s statement news, the teaching of Solow’s 1956 model could be greatly enhanced by intertwining the “neoclassical” foundation of instantaneous optimality with “classical” policy analyses.

2. Key Concepts and Assumptions

In this paper, we examine the Solow growth model in a closed economy. The model is a celebration of the classical tradition, particularly Smith’s dynamical method of analysis, Phelps Brown’s pure income theory, and Domar’s decisional approach to growth problems. We begin by stating the key concepts and assumptions of the model. Next, we analyze the equilibrium dynamics, that is, we carry out the continuous time analysis of the model. Lastly, we show some implications of the dynamics, including the convergence properties of the model.

We analyze the model both for the original case when the aggregate production function is assumed to have an elasticity of substitution and a constant returns to scale, and when the aggregate production function is generalized to include non-constant returns to scale. The capital share of national income is measured to investigate the nature of income distribution, to test the hypotheses that the capital share is constant and that the rate of growth of the capital stock is independent of income distribution, and to assess the equality issue. Finally, we compare the neoclassical assumptions on income distribution with the empirical counterparts.

3. Deriving the Solow Diagram

3. Deriving the Solow Diagram. In this section of the homework, you will use the Solow model to analyze how the Solow growth model explains the sustained growth in the East Asian economies (Japan, South Korea, Taiwan, and the like) relative to G-7 economies in the 1970s and 1980s, but also presents the argument for stagflation in these economies in the 1990s in accordance with the Solow model. For the purpose of this question, assume that all countries are the same in all respects other than average income per person, so that the aggregate production function and the transition diagram work equally well for any country in the world you want to choose.

a) Identify the growth rates of income per person and capital per person in the steady state. Explain what forces move these variables to their steady-state levels. Support your explanation with words, graphs, and algebra.

b) Compute the steady-state value of the savings rate, s*, for each country, assuming that the depreciation rate, d, and the population growth rate, n, are a quarter of the growth rate of the per capita income difference between these countries, g.

c) On the same graph, plot the actual saving rates for each country in 1970 and 1993 as one point and its corresponding steady-state value of the saving rate that you computed in (b) as another point, and briefly describe what had actually happened between these years.

d) Now, model the Crown Plaza accord of 1987 that was successful in doubling the saving rate of Japan for the years after 1987 by drawing a second set of the same graph of each country that has Japan’s saving rate in 1988 and Japan’s saving rate in 1993 as one point and Japan’s steady state saving rate (calculated from the specified growth rate in the background material) as another point.

4. Analyzing Steady State and Economic Growth

1. In this question, we are going to analyze the Solow Model with technological growth. Oi = O < 1. In order to compare several steady states, we take the following math expectation: suppose conditional on an initial period capital stock y0 = (K/L)0 and a draw of some initial level of technology, n0, y1 = (K/L)1 are independent and identically distributed. Our first task is to solve for the steady state. Denote this steady state as Ks. Use the math you learned in macro to write the growth rate of output per effective unit of labor as a function of these parameters, (S, A, O, δ). Be sure to explain all of your steps and why we receive these results. Are the equilibrium growth rates increasing, constant, or decreasing in A, O, S, δ? Only provide general relationships between the growth rates for each case.

2. In this second question, we are going to further analyze the Solow model with technological growth and, in particular, focus on A and the contours for technological improvements. These types of characterizations are nice because they are easy to test. For example, you could just code up measures of technology to see if there is a prediction in the data that supports these relationships. Start by proving that the ratio of the true growth rate of output per effective unit of labor to the true growth rate and δ = 1 is:

(1 + g) = (S/A)(1 − δ) + δ Which parameter(s) of the Solow model (s, A, O, or δ) pin down the fraction of true to potential growth? What about the rate of technology growth? If we use data on the fraction of true to potential growth in the Solow model and solve this equation for A, what functions will we be drawing in A and S? Continue the derivation of the output per effective unit of labor with respect to TFP.

3. For this question, consider the case where the Solow Model with the population growth rate equals the output growth rate. This type of model is often used to characterize the income distribution across different nations. To simplify the math, assume O = 1. Suppose that, upon observing capital-output ratios across countries, you find that capital-output ratios are reduced by less than one for one. In particular, a decrease in the steady state where the capital-output ratio is 6 to 3 increases the level of steady state output per effective unit of labor by 0.1, which in turn increases steady state output per worker by 0.1. Given this information, what story do you have in the back of your mind for why this happens in this simple model? (HINT: what cost must increase to explain this fact)

5. Empirical Applications and Policy Implications

5.1. The Solow Model and the International Data: Conditioning on Labor Hours What will happen to the growth rate in countries like China and India with a lot of poor people with a lower level of per capita output? In other words, do we forecast that, at a “steady state growth” (that is, the income per head converges to Chinese and Indian levels), the growth rate will be higher if the level of technology in China or India did not differ from the technology level in developed countries? The answer is “no.”

5.2. Is the World “Closed” or “Open”? Balanced growth implies that, when TFP and population growth are exogenous (the Solow model as reviewed in section 1 analysis of Sec. 1 is a special case of such), the per capita output in an economy will converge to the “leading” economy. Moreover, this convergence is realized without any transfer of technologies from the leaders (recognizing that the sources of the technology are not Egyptians in the ancient past but activities occurring inside the country and that these activities involve activities or goods produced elsewhere).

5.3. The Mankiw-Romer-Weil Model What might be more difficult to accept in the Romer’s model is the assumption that Eisenhower, Molotov, and Murdoch are valueless. For the model in the context analyzed here works if these activities are “pure technology,” that is, creating no efficiency for the followers. In contrast, these activities must create monopolistic market power in the full Romer model. Any analysis of the Romer model should recognize that the accumulation of enormous wealth (think of Bill Gates) and competitive markets are not consistent with the ‘full’ model of ‘monopolistic competition’.