Read all the instructions and carefully follow them. You are allowed to work with at most three other classmates. You can use your notes, the ebook, class notes, homeworks, and slides, and other material from the web. You will upload two documents: a .pdf with answers and an Excel file. – Excel: include your name and ID number in the first sheet. If you choose to work in a group, all their names must be included in the excel files. Use separate spreadsheets for different questions/items within a question, and make sure you label the sheets appropriately so we can figure out what questions they refer to. When working with plots, place the plots on separate sheets (not in the same sheet in which you did the computations). Include any explanations in the PDF piece of paper and make sure to cite your sources. – PDF: Please write your answers clearly and neatly on blank paper, and specify which question you are answering by including the question number. Make sure you write your name and ID number at the top of the page. If you choose to work in a group, all their names must be included in the .pdf. Make sure that all pages are numbered before you submit and that you have converted the file into one big pdf. When discussing a result from the excel file, please include clarify which sheet you are referring to so we can track it back easily. Good luck! 1 Solow Model Consider a Solow growth model like the one discussed in class, but incorporating the possibility of less than full employment. Let t denote a period. The production function is Yt = ztKα t [etL] 1−α , so that given the current TFP zt , the capital input Kt and the labor input Nt = etL, firms produce output Yt in period t. Here et ∈ [0, 1] denotes the proportion of the labor force L that is employed at each point in time. If et = 1 the economy is at full employment, with e = 0, on the other hand, everyone is unemployed. Total resources satisfy Yt = Ct + It where Ct denotes aggregate consumption, It aggregate investment, and Yt is GDP. Assume that people invest a constant proportion s of their income, so that aggregate investment satisfies It = sYt and capital evolves according to Kt+1 = Kt(1 − δ) + It where δ denotes depreciation. This function tells us how capital next period is related to current capital, investment, and depreciation. In the book, you have seen the expression written as ∆K = I − δK, note that the two are equivalent. 1. Assuming that zt = z and et = e are constant, compute the steady state value of capital per capita k ∗ for this economy as a function of all the parameters of the model. Remember that capital per capita satisfies k = K L , and that you will need to transform all variables into per capita terms. 2. Let zt = 1.3, s = 0.3, α = 0.35, et = 1 and δ = 0.07. What is the value of capital in the steady state? Denote it by k ∗ .? 3. Does this value of s satisfy the golden rule? If yes, explain. If not, compute the value of s that would. 2 4. Using Excel, create a plot that has k in the x-axis, and output, investment, and total depreciation on the y-axis (all in per-capita terms). Show the steady state in your graph. Make sure you label all the axis, and include the legend so the different lines plotted can be identified as y, Inv, and Dep. You will need to upload this into the ‘Excel upload’ folder, so make sure you save your changes and that each spreadsheet is properly labeled. 5. Show what happens to investment and depreciation, as functions of k when employment declines to e = 0.5. Do this using a graph that has k in the x-axis and both, investment and depreciation, in the y-axis. You can do this graph in Excel or simply drawing it in a piece of paper. Show the old steady state value and the new steady state value. What is the value of the new k ∗ new? 6. Compute the evolution of economic variables over time (capital, output, consumption, investment, depreciation, all in per-capita terms), for 100 periods, assuming that the initial level of capital is k1 = 1 in the first period and that et = 1 for every t. Create a graph that has time in the x-axis and the evolution of the variables output, consumption, investment, depreciation (all in per-capita terms) in the y-axis. Do variables grow or decline over time? 7. Assume that the economy starts at k1 = k ∗ , that is, initial capital equals the steady state value of capital found in (2) for e = 1. Compute the evolution of variables assuming the following: et = 1 between periods 1 and 12, et = 0.5 for periods 13 to 16, and et = 1 between periods 17 and 100. Plot output, consumption, investment, and depreciation— all in per-capita terms—over time (that is, using time in the xaxis and the evolution of the variables in the y-axis). Explain how what you see in this plot can help you understand the dynamics of the economy during the Pandemic.