ECON 6022B Problem Set Suggested Solutions Fall 20 September 5, 20 Shocking the Solow Model Consider the basic Solow model in Lecture 2. Suppose the economy stays at its steady state in Period 0 and there is an one-time increase in the depreciation rate at the end of Period 0, from d to d 2.. Show in a Solow Diagram, how the economy converges to a new steady-state. Solutions: The following figure shows how the economy converges to a new steady-state: In the case with an one-time increase in the depreciation rate, the economy will converge to a new steady state with lower k and y. Before Period t = 0, the investment equals depreciation and the economy is at the steady state. At Period t = 0, the depreciate rate is higher, so that the investment is lower than the depreciation, which means the capital stock starts declining. So does the income y. The economy keeps declining until the point where it rests in the new steady state. 2. Plot the trajectory for output per capita, y t, from Period t = 0.
Solutions: The trajectory for output y(t) from Period t = 0 is shown below, and the reasons are discussed in Part. 3. Plot the trajectory for log(k), from Period t = 0. Solutions: The trajectory for log(k) from Period t = 0 is as follows: Since K = k N, and in a steady state, k is equal to k and N grows at a constant rate n, log(k) should approximately grow at a constant rate n, i.e., log(k t ) log(k t ) ( + n). Thus, in a steady state, the slope of log(k) should be a constant (n). In the case with an one-time increase in the depreciation rate, the slope of log(k) should first drop, and then gradually return to n when the economy converges to a new steady state. 2 The Solow Model with Exogenous Productivity Growth In Lecture 2, we have seen a version of Solow model without productivity growth. Now assume that the growth rate of TFP is constant, g > 0. Specifically, A A = g 2
. Rewrite the capital accumulation function with k, where K k = A α L Solution: We start with the aggregate capital accumulation equation, K = sa K α N α d K Then we rearrange the equation, K = s K α (A α N) α d K The term A α N refers to effective labor and therefore, k K = refers to capital per unit A α N of effective labor. We want to rewrite the equation with k. Divide both sides with A α N and we have the following, Or, K K = s ( ) α K d ( ) A α N A α N A α N K = s k α d k A α N Note that K k. And we know, A α N k k = K K ( α A A + N N ) Multiplying both sides by K A α N, we have, K k = A α N k A A α n k Therefore, we can replace the left hand side of the accumulation equation with k + n k + k g so k + n k + k g α = s k α d k α, Rearrange it, we have, k = s k α (d + n + g α ) k 3
2. What s the proper definition of steady state in this model? Explain why? Solution: For the steady state in this model, the capital per unit of effective labor and income per unit of effective labor rather than per unit of labor should be steady. Specifically, in the steady state, k = 0. We can solve for k ss : Given the definition of y, we can easily show that k ss s = ( d + n + g ) α α ȳ ss = ( k ss ) α That is because the productivity grows at a rate of g, which can be regarded as an equivalent growth in total labor force, and capital per unit of effective labor takes into account the extra growth in the productivity. Note, the similarity in the definition of the steady state between this model and the basic Solow model in Lecture 2 is that the capital-output ratio ( K ) is constant, which is also consistent with the reality. Y 3. Does capital (or output) per capita grow over time when the economy stays at its steady state? If so, what s the growth rate for capital per capita at steady state, k k. Solution: In the steady state, k does not grow. However, the capital per worker k = K/N, does grow over time in the long run. To see that, we take log of both sides of the definition of capital per unit of effective labor, and take time derivatives, We also know in the long run k k ln( k) = ln(k) ln(a) ln(n) α k k = K K A α A N N = 0, therefore, k k = K K N N = A α A = α g A Therefore, the growth rate for capital per capita at steady state is 3 Simulate the Solow model α g A > 0. Consider the Solow model in lecture 2. We parametrize the model with the following data, α = /3; d = 0.08; s = 0.0; A = ; N =. 3. Simulation : Rising saving rate We want to understand how the increase in saving rate leads to higher growth rates during the transition path in the Solow model. Suppose the economy stays at the steady state before Period 0. The saving rate 4
increases from 0% to 40% at Period 0. Compute the new steady state capital level. And plot the growth rates after period zero against time. Solution: In the discrete time case, the capital accumulation function is k t+ k t = s A k α t d k t () In the steady state, new investment equals depreciation and capital stays constant, that is, k t+ = k t. By substituting both k t and k t+ in equation () by k ss, we have k ss k ss = s A k ssα d k ss Solving for k ss, ( ) s A k ss α = d When the saving rate s is 0., the steady state capital level is ( ) s A k α =.40 d ( ) s A With a new saving rate s = 0.4, the new steady state capital level k α =.8. d Plotting the growth rate against time, The graph shows that the growth rate increases abruptly at Period 0, and then gradually decreases to 0. It means the rise in saving rate could lead to economic growth for a long period of time (but not forever). 3.2 Simulation 2: Convergence (Optional) We want to evaluate quantitatively to what extent the Solow model helps us understand the convergence pattern we observe in the data. (You are advised to take a speadsheet to simulate the model.) We choose multiple initial capital values, each of which represents a country s capital stock in 960. k 0 = N k, where N =.5, 2.5, 3.5,..., 0.5 and k is the steady state level of capital per capita. 5
. Compute the transition path for k t, by iterating over the capital accumulation function for 00 periods. Plot the transition path for the case where k 0 = 2.5 k. Solution: By iterating over the accumulation funciton, we can obtain a sequence of capital, k 0, k, k 2,... k 0 = 2.5 k = 2.5.40 k k 0 = 0. k 3 0 0.08 k 0 k 2 k = 0. k 3 0.08 k k 00 k 99 = 0. k 3 99 0.08 k 99 Plot the sequence of capital against time, we can observe how the capital stock converges to the steady state level. 2. Compute the output growth rates for the case where k 0 = 2.5 k. And plot the growth rates against time. Comment on your findings. Solution: To compute the output growth rates, note that y t = A k tα L t α = k t α, and that g Y t = (y t+ y t )/y t, for all t = 0,, 00. Plotting the growth rates, 6
We observe that the growth rate of output is really high when the economy is far below the steady state level. And the growth rate declines over time when the economy grows. In other words, the speed of growth is positively related to the gap between the capital in current period and the steady state level. 3. Compute the average growth rate of first 40 periods of each case and plot them again their initial income levels. Compare your finding with the graph (Evidence: OECD) presented in Lecture 2. Comment on your findings. Solution: The average growth rates of the first 40 periods against y 0 /y can be plotted as in the following graph: This graph indicates that in the short run, countries grow faster when their capital stocks are low, which is consistent with evidence from OECD countries presented in Lecture 2. 4 The Solow Model with Saving Dependent on Income Level (optional) In Lecture 2, we assume that the saving rate (or investment rate) is exogenous and constant over time. We now consider feedbacks from GDP to saving rates. For example, poor countries can t afford to save, 7
because they are living at the margin of subsistence, so that they need all their income for consumption. Or poor countries are more impatient because being poor makes you care less about the future. Let us extend the model, and assume: where, s < s 2.. Show that there is a capital level, k, such that γ = { s, if y < y s 2, if y y γ = { s, if k < k s 2, if k k Solution: Since y = A k α, output monotonically increases in capital. Given the cut-off value of GDP, y, there is a k = (y /A) α, such that γ = { s, if k < (y /A) α s 2, if k (y /A) α 2. Show graphically (in a Solow Diagram) that there are more than one steady states in this model. Are they stable steady states? Why or why not? (Hint: Graph two curves for investment, given different saving rates.) Solution: There are two non-trivial steady states in this model. Both of them are (locally) stable. If the initial capital is below (above) k, the economy converges towards the steady state with lower (higher) capital. 3. Comparing with the Solow model, can the extended model explain more of the differences in income among countries? (Hint: Do we observe a self-reinforcing behaviour in the extended model?) Solution: The steady state with lower capital (or income level) features a poverty trap. Think of the case where the initial capital k 0 is slightly below the cut-off value k. The low income level leads to a 8
lower saving rate, and therefore the economy gets stuck in the steady state with low income (or capital). It does help to explain more of the difference in long run income gap between countries. Suppose we compare two countries, with initial income levels slightly higher and lower than y, respectively. In this model economy, we do observe that the income gap between the two countries gets bigger over time. In the Solow model, however, two countries with slightly different initial income levels converge to the exactly same steady state. 4. Based on this model, can you recommend any government policies, which help increase the income levels of poor countries. Solution: Policies of the Big-Push type are helpful in this model. Suppose that economy starts out from somewhere slightly below the cut-off value. The government could rely on foreign aids to increase the national income by a little bit so that the national income will be higher than the cut-off value y. And the saving rate in this economy will be higher (s 2 ) and the economy converges to the better steady state so that the poverty trap can be avoided. 9