Intermediate macro problem set
ECON-UA 12 — Intermediate Macroeconomics Professor: Corina Boar TA: Benjamin Castiglione Spring 2024 Problem Set 4 Due Friday, February 23 1 Understanding the experiments in the Solow growth model Remember that we reduced the Solow growth model to two equations Yt = AKtα L1−α ∆Kt+1 = sYt − δKt where A, s, δ, L are parameters whose changes we will analyze. We want to analyze what happens to the level of output (and other quantities) immediately after the change in parameters, and what is the total percentage change in the level of output from the original steady state to the new steady state. Question 1.1 What do the parameters A, s, δ, L and α represent? Question 1.2 What are the ranges of plausible values for these parameters? Question 1.3 Consider first the production function model Yt = AKtα L1−α in isolation. We want to express the percentage change of output as a function of percentage changes of TFP, capital and labor inputs. Denote Yold the output before the change, and Ynew the output after the change. Notice that we define the rate of change (or percentage change) of output as Ynew ≡ 1 + gY . Yold Analogously, define the rate of change of A, K and L as gA , gK and gL , respectively. Express gY as a function of gA , gK and gL . Notice that you will obtain the same answer as we derived in the model of the production function in Chapter 4. Question 1.4 Imagine thef following experiments (each of them in isolation). Derive the result both for a general α as well as assuming α = 1/3. 1 1. TFP increases by 10%. 2. Immigration increases the population size by 1%. 3. The saving rate decreases by 20% (notice that a decrease in the saving rate by 20% from say s = 0.2 implies a new saving rate of s0 = 0.16)! 4. An earthquake suddenly decreases the capital stock by 10%. By how much does output increase immediately after the changes listed above? We will now study the total percentage change in output from the old steady state to the new steady state. The crucial difference is that we now need to allow for capital accumulation toward the new steady state, and the impact of this capital accumulation on the level of output. In order to answer this question, we need to compute the steady state level of capital as a function of the parameters of the model. Question 1.5 What condition does the capital accumulation equation have to satisfy in steady state? Denote Y ∗ and K ∗ the steady state level of output and capital. Question 1.6 Use the production function to substitute the steady state level of output out of this equation, and express the steady state level of capital as a function of the parameters of the model. Question 1.7 Denote Kold and Knew the old and new steady state level of capital (before and after the policy change). Denote the percentage change in capital between these steady states as gK , defined as Knew ≡ 1 + gK . Kold Express gK as a function of percentage changes of the parameters of interest, gs , gA , gδ and gL . Question 1.8 What is the percentage change in the steady state level of capital in response to the following policy changes (each of them in isolation)? Derive the result both for a general α as well as assuming α = 1/3. 1. A 10% increase in the saving rate (say, from 0.2 to 0.22). 2. A 1% increase in TFP. 3. A 10% increase in the depreciation rate (say from 0.1 to 0.11). 4. A 10% increase in population size. 2 Summarizing what we have derived so far, we have a pair of growth rate decompositions: gY gK = gA + αgK + (1 − α) gL 1 1 1 = gS + gA − gδ + gL . 1−α 1−α 1−α (1) (2) Equation (1) represents the percentage increase in output in response to change in TFP, capital, and labor. Notice that when we analyze the immediate impact of a policy change, then, in this equation, gK = 0 (capital stock cannot change immediately through accumulation, unless we assume that it is suddenly destroyed or brought from outside of the economy, e.g., from abroad). When we analyze the percentage increase in output from the old steady state to the new steady state, we need to take into account also the increase in the stock of capital generated by endogenous capital accumulation. This increase in stock of capital is captured by equation (2). Question 1.9 What is the total percentage change in output per capita in response to the following policy changes (each considered in isolation). Answer the question for a general α as well as for α = 13 . 1. A 10% increase in population. 2. A 10% increase in TFP. What is the contribution to this total change arising from the initial change, as well as the contribution from endogenous capital accumulation? Finally, we want to analyze the impact of the policy changes on capital per capita and output per capita. In principle, we could re-derive everything from scratch but this is not necessary. Question 1.10 Recall the exercises from Chapter 3 and 4. Assume that output Y increases by gY percent and labor L increases by gL percent. By how much does the output per capita, defined as y = Y /L, increase? Denote this percentage increase as gy (notice the distinction between gy and gY ). Answer the same question also for the growth rate of capital per capita gk , expressed as a function of the growth rate of total capital gK and the growth rate of labor gL (again notice the distinction between gk and gK ). Question 1.11 Use these results and equations (1) and (2) to express the percentage change of output per capita gy and capital per capita gk in response to the changes in the parameters of the model. 3 Question 1.12 Imagine that the population increases by 10%. What is the initial impact on output per capita and capital per capita. What is the total impact on output per capita and capital per capita from the old steady state to the new steady state. Provide an economic interpretation of the results. 4
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