In the Name of God. Macroeconomics. Sharif University of Technology Problem Bank

In the Name of God Macroeconomics Sharif University of Technology Problem Bank 1 Microeconomics 1.1 Short Questions: Write True/False/Ambiguous. then write your argument for it: 1. The elasticity of demand for an input is smaller when complementary inputs are inelastically supplied. 2. In general, the demand for a good is more inelastic for the poor than for the rich. 3. Richer families are more likely in any given society to employ maids and other domestic help than poorer families. This implies that the importance of domestic help should increase over time as a country develops and per capita incomes grow. 4. If the demand for illegal drugs is inelastic, then an intervention that seizes and destroys 10% of drug production will raise the profits of drug suppliers. 5. Suppose an exogenous influx of immigrants into Tehran increases the market price of Tehran housing. Then those considering moving to Tehran are more affected by the price change than those considering leaving Tehran. 6. The rise in housing prices in many parts of Tehran has increased people s real wealth and therefore should increase their current and future consumption of income elastic goods. 7. (Optional) If we find that an individual s 1990 consumption bundle cost $20,000 in 1990 and $30,000 in 2000 and this same person s 2000 consumption bundle cost $45,000 in 2000 and less than $30,000 in 1990, we can reject the hypothesis that this individual has stable homothetic preferences. 1

1.2 Family Choice Often in economics we deal with data aggregated to the household level. Consider the case of two person households composed of individual M (Man) and W (Woman). Each individual has a utility function defined over the consumption of two sets of consumption goods. In particular you can assume individual M, has utility U M (Y M, XM) and individual W receives utility U W (Y W, Z W ), where the Y goods are goods consumed by both types of individuals and the type M individuals also consume the X goods while the type W individuals also consume the Z goods in addition to the Y goods. All of the goods, Y, X, and Z are sold in the market at prices P Y, P X and P Z respectively. For purposes of your analysis assume that all of the goods are normal. 1. Assume initially that the two individuals only care about themselves and not about the other party and that as a result each individual maximizes their individual utility subject to their own income constraint (i.e. M can only spend up to I M and W can spend only up to I W ). (a) Will household consumption of X, Y and Z satisfy the law of demand (i.e. will an increase in the price of one good, holding the prices of other goods and the income of all individuals constant lead to a decrease in the consumption of that good? (b) Is there a well defined household demand function in this case? What are its arguments? (c) If you had data on individuals with the same preferences that were part of single person households how could you use that to help you understand the consumption of households where you only observe household level (i.e. totaled across individuals) consumption of X, Y, and Z and household income? Could you use that information to learn about the incomes of the individuals? 2. Now assume that the individuals care about each other as much as they care about themselves so that they now choose Y M, X M, Y W and Z W to maximize the sum of their utilities (i.e. they maximize U M (Y M, X M ) + U W (Y W, Z W )) subject to their total spending being less than or equal to their total income I M + I W. You can assume that both U M and U W 2

are concave. (a) Will household consumption of X, Y and Z satisfy the law of demand (i.e. will an increase in the price of one good, holding the prices of other goods and the income of all individuals constant lead to a decrease in the consumption of that good? (b) Is there a well defined household demand function in this case? What are its arguments? (c) How would a decrease in the price of Y affect consumption and the utility of each of the parties in this case? Why? (d) How would a decrease in the price of X or Z affect consumption and the utility of each of the two parties in this case? Why? 2 General Equilibrium and Macroeconomics 2.1 Basics 2.1.1 GDP Calculations consider the following economy 1390 1391 1392 Agricultural Production 40 25 20 Industry Production 20 35 25 Services Production 65 60 55 Agricultural Prices 140 120 100 Industry Prices 180 130 100 Services Prices 125 110 100 1. Calculate the real GDP for each year using the method of "Constant Prices". 2. Calculate the real GDP for each year using the method of "Chained". 3. Calculate the Price Index for each year using three different methods. 3

2.2 Edgeworth Box 2.2.1 EdgeWorth box and the General Equilibrium In this problem we want to analyze how general equilibrium effects can be different from partial equilibrium ones. Consider two types of agents: N F number of Farmers (F) and N M number of Manufacturing workers (M). Each F owns Y 0 amount of Agricultural good and each M own X 0 amount of manufacturing good. Both types of agents have U (x, y) = α log x + (1 α) log y preferences over the two goods. These agents trade in a competitive market. 1. Setup the problem for each agent. Name income of the agents by I F and I M precisely determine both. 2. Solve each agent s problem in terms of their income. 3. Define the competitive equilibrium in this problem. Write down the equilibrium conditions. 4. Solve for the equilibrium allocations and prices: How much of each good, each agent consumes and what is the relative price of Y to X (q). 5. Comparative statics: Show how do changes in each of the following exogenous parameters affect the agents consumption of each good, their welfare, their income and the relative price q. Show how do partial and general equilibrium results differ. Explain your analysis in words in each case (write economical arguments on why we observe these equilibrium outcomes) (a) X 0 (b) Y 0 (c) N F (d) α 6. Suppose N F = N M = N. What would happen to the relative price q if each agent F or M owned the bundle (X 0 /2, Y 0 /2) of goods X, Y. Explain why do we observe such a property and what does it teach us? 4

7. Now suppose the agents preferences is CES : U (x, y) = (αx ρ 1 ρ ) + (1 α) y ρ 1 ρ ρ 1 ρ. (a) Find the demand function for each good/agent and for the market. (b) Plot the demand function of Farmers for the agricultural goods. (c) Now solve for the competitive equilibrium again. How do your results for allocation and prices change quantitatively and qualitatively for each case ρ > 1 or 0 < ρ < 1? What about your comparative statics? Feel free to use software anywhere needed or use specific number for the parameters if you need. (d) How does parameter ρ play a role in the results you have found? How do results change whether ρ > 1 or 0 < ρ < 1? (e) What if ρ = 1? 8. Now suppose the preference functions of the agents are different: Farmers : U F (x, y) = α log x + (1 α) log y But Manufacturers: U M (x, y) = log y (This problem can be interpreted as one agent type is a worker of the economy who produces Non-Oil GDP and has preferences over Non-oil GDP good and Energy. the other agent is the government who owns Energy sector but needs to consume Non-Oil GDP.) (a) Re-solve for the competitive equilibrium (although the assumption of the competitiveness may not be relevant if we think of the government as one agent). How do your results change? Briefly explain the general equilibrium outcomes. 2.3 Labor-Leisure 2.3.1 A Representative Agent model with Labor-Leisure Decision Consider N infinite-lived households who consume c t and supplies 1 l t hours of work in each time period and received U (c t, l t ) = log c t + γ log l t utility and also receives w t (1 l t ) as wages. Each household supply k t amount of capital and receives rents v t. He invests i t and its capital depreciates with the rate δ. 5

Also, there is a representative firm which produces the final good with a production function as y t = F (K t, H t ) = AK α t H 1 α t where K t and H t are the total capital and total hours of work in the economy. 1. Setup the problem. 2. Define the Competitive Market Equilibrium. 3. Write down the FOC of the household and find the Euler Equation. 4. Find the labor supply equation. 5. Write down the FOC of the firm. 6. What are the market clearing conditions. 7. Solve for the steady state allocation. 8. Analyze the effects of changes in A, β and γ on the allocations and prices in the steady state equilibrium. 2.3.2 Crusoe s Preferences and Optimal Choice Suppose that Crusoe has a utility function of the following form: u(c, l) = (1 γ) log c + γ log (1 h) c and h are defined as in class (Consumption and hours of work). The production function of producing coconut is y = Ah α, where 0 < α < 1. 1. Solve for the optimal allocation using Lagrangian. 2. How does a change in A affect hours of work and total production. Explain in terms of income and substitution effects. 6

3. How does your result change if α = 1 Now consider the decentralized economy. Assume α = 1. 4. Setup the competitive equilibrium problem. 5. Find for the Robinson s labor supply function. 6. How does a change in w affect Robinson s labor supply and consumption. Explain your finding in terms of income and substitution effects. 7. How does a change in the profit of the firm to Robinson (π) affect Robinson s labor supply and consumption. Explain your finding in terms of income and substitution effects. 8. What if there is a tax on wages? How does it affect labor supply and consumption. Explain intuitively. 9. Find the representative firm s labor demand. 10. In a competitive equilibrium, find the real wage w/p. 11. Solve for the equilibrium hour of work and total production. 12. Intuitively, discuss about the effect of aggregate productivity on labor demand, labor supply, equilibrium hours of work and equilibrium total production. 13. (Optional- Extra Credit) How does your results change if 0 < α < 1. 14. (Optional) Now redo parts 1-6 with the following preferences: u(c, l) = c1 η 1 η γh1+φ 1 + φ 15. (Optional) In the class example, income and substitution effect canceled out each other when productivity term A changes. Now discuss the income and substitution effect in this case. What happens when A goes up? Which effect dominates? Be clear about the general equilibrium (GE) effect. Specifically, determine which effect comes from the GE? 16. (Optional) How does φ matter? 7

2.3.3 Crusoe Finds Oil! 1. Now, suppose Robinson (in our previous problem) finds oil in his island!!! He can sell oil to the neighbor island and receive consumption good. Total Oil revenue is O. You can assume α = 1 if needed. 2. Solve for the optimal allocation and find total hours of work and total production. 3. Explain intuitively, what happens to this economy? (Explain the effect of an increase in the oil revenue) 4. Now consider the competitive equilibrium and solve for the equilibrium real wages, hours of work and production. 5. How does oil revenues affect real wages? Explain mathematically and intuitively. 2.3.4 Gains from trade The unit labor requirements for the U.S. and China are given in the following table: Wheat Clothing Labor Endowment 1. US 5 1 200 China 6 3 500 2. Who has the absolute advantage in which good? Explain. 3. What is the opportunity cost of producing a bushel of wheat in the US? in China? 4. What is the opportunity cost of producing a piece of clothing in the US? in China? 5. Who has comparative advantage in which good? Explain. 6. What is the world relative price range that will allow for a free trade equilibrium? 7. Suppose the actual world relative price of wheat to clothing is 3. Who produces what? 8. Who gains from trade? Explain. 8

9. (Optional) Suppose both economies have preferences as below: U (W, C) = ln (W ) + ln (C) where W is Wheat and C is Clothing. (a) Suppose there s trade. What conditions do we need for the equilibrium prices? (b) Solve for the trade equilibrium prices and quantities. Is the condition above satisfied? (c) Are there gains from trade? (d) Now assume that China labor endowment is 2000. How would your answers change? 2.3.5 PPF 1. A country with 20 Million workers has 5 million high skilled workers (people with university degree) and the rest are low skilled workers. This country produces two type of good: Machines (M) and Clothes (C). Each high skilled worker, if works all the year can produce 10 machine or 3000 clothes. Each low skilled worker, if works all the year can produce 2 machine or 1000 clothes. Households have the following utility function: U (M, C) = 0.4 log M + 0.6 log C 2. Calculate and plot (Using MATLAB) the PPF (Production Possibility Frontier) for this economy. 3. Using the households preferences, find out how many machines and clothes will be produces in this economy (please prevent using zeros in your answer, instead use scientific format). 4. What is the relative price of Machines to clothes in this economy? 5. Now suppose price of a machine equals 20 clothes in the world and suppose this country opens up to trade with the world. Now calculate how many machines and clothes will be produces in this economy? How many of each will be exported or imported and how many will be consumed. 9

6. Write a MATLAB code to calculate to calculate these numbers for you. 7. Now use your code, and show (graphically) how your answers to the previous questions will vary if (a) Productivity of the high skilled workers varies from 10 to 20 in Machine production. (b) 50% percent of the workers become high skilled. (c) T(Optional- No credit) Intertemporal Consumption Choice: Heterogeneous Agents 2.4 Macro Dynamics 2.4.1 Intertemporal Consumption Choice: Identical Agents Consider the two-period Intertemporal consumption decision model that we discussed in class, where the household s utility function is given by: U(c 1 ; c 2 ) = c1 η 1 1 η + β c1 η 2 1 η where c 1 and c 2 denote period one and two consumption, respectively, and β < 1 is the discount factor and 0 < η. Suppose y 1 = y 2 = 1 is the household income in the two periods. Household can save s with returns r in period 1. 1. Write down the Euler Equation. 2. Solve for s, c 1 and c 2 with respect to r. 3. Explain how does r affect your results. Can you explain it intuitively? 4. Plot s, c 1 and c 2 versus r for r (0, 1) and η = 0.5 5. On the same graph but with a different color, plot s, c 1 and c 2 versus r for r (0, 1) and η = 1. 6. On the same graph but with a different color, plot s, c 1 and c 2 versus r for r (0, 1) and η = 2. 10

7. Explain how does η affect your results. Can you explain it intuitively? 8. We call 1/η the intertemporal elastictiy of substitution. Can you explain why? 9. Now for r = 0.1 plot s, c 1 and c 2 versus η (0.1, 10) 10. Now consider the general equilibrium and assume that all the agents are all the same and there are N agents in the economy. All households meet in the saving market to clear it. Solve for r with respect to η. 11. What are general equilibrium outcomes s, c 1 and c 2? Explain your finding. 12. Numerically solve for r for η = (0.1, 10). plot r versus η. Explain how does η affect r. Intuition? 13. Now suppose the government issues amount B of bonds. How does it affect r? Explain mathematcally and intuitively. 2.4.2 Intertemporal Consumption Choice: Heterogeneous Agents Consider the two-period Intertemporal consumption decision model that we discussed in class, where the household s utility function is given by: U(c 1 ; c 2 ) = c1 η 1 1 η + β c1 η 2 1 η where c 1 and c 2 denote period one and two consumption, respectively, and β < 1 is the discount factor and η < 1. Here we study a case where all the household are NOT identical; and there are two types of households. Type a is poor in the beginning and wealthy later but the other type b is wealthy today and poor tomorrow. More specifically, suppose that there are N a = 200 millions of households of type a whose income are y 1a = $10000 and y 2a = $50000 and there are N b = 100 millions of households of type b whose income are y 1b = $60000 and y 2b = $5000. You can take η = 1. 1. What is the aggregate borrowing of type a? (Write it as a negative number.) 11

2. What is the aggregate saving of type b? (As a positive number) 3. What is the net saving of the economy? 4. In equilibrium where the net saving is zero, what is the market value for the rate of return R? 5. (Optional) Solve for R in the general case where we don t have the assumption of η = 1. 6. What is the equilibrium b 1 for each type of household? Do we still have zero lending and borrowing? 7. Now suppose N a increases to N a = 400 millions. (a) What is the new equilibrium R? Explain and give an economic intuition for this change using a supply-demand analysis. (b) How does the consumption of each type would change? Then use your findings of the previous problem regarding the income and substitution effects to justify your answer. (c) How does the Saving/Borrowing of each type would change? Explain it using the supply and demand curves. 8. Now suppose y 1a increases to y 1a = $20000. (a) What is the new equilibrium R? Explain and give an economic intuition for this change using a supply-demand analysis. (b) How does the consumption of each type would change? Then use your findings of the previous problem regarding the income and substitution effect to justify your answer. (Hint: be careful! There are couple of effects) (c) How does the Saving/Borrowing of each type would change? Explain it using the supply and demand curves. 9. Now suppose the government issues B amount of bonds, meaning that they borrow B$. Show how does it affect r. Explain the economic intuition. 12

2.4.3 A simple infinite period model Consider an infinite-lived households who consume c t in period t and supplies fixed l hours of work in each time period and received U (c t ) = t=0 βt c 1 σ t 1 1 σ utility. He saves s t and receives the return s t 1 (1 + r t ) from savings in the last period. His wage income is w t thus his total income is y t = w t l. 1. Setup the household problem and find the Intertemporal Budget constraint. You can define R t such that 1 + R t = (1 + r 1 )... (1 + r t ). 2. Write the FOCs and find the Euler equation. Write the interpretation of the budget constraint. 3. Solve for the c t. 4. Suppose w t = w and r t = r for all t. Solve for the c t, s t. How does your response depend on β, σ, r, w. (if you need, you can also assume that σ = 1 but it s better to solve the problem for general σ) 5. Suppose r = 1 β 1 ρ. Recalculate c t,s t. Interpret your results. 6. Yet assume r t = r. Suppose the agent receives a bonus y at time zero. Find c t,s t and interpret your results. 7. Suppose the agent expects that his wage increases to 2 w forever from period t = 1.. Find c t,s t and interpret your results. 8. Now suppose the agent s wage has growth rate g so w t = w 0 (1 + g) t. Find c t,s t and interpret your results. 9. Assuming the growth rate g, suppose the are N similar households in the economy. Solve for the market value for r t. Explain how does it depend on the parameters of the model and interpret your results. 13

2.4.4 Intertemporal Consumption Choice: Uncertainty Consider a two-period Intertemporal consumption decision model, where the household s utility function is given by: U(c 1 ; c 2 ) = E [log (c 1 ) + β log (c 2 )] where c 1 and c 2 denote period one and two consumptions, respectively, and β < 1 is the discount factor. The income in period 1 is y. But in period 2 the income is y(1 + ε) with probability 1 2 and it is y(1 ε) with probability 1 2 where ε > 0 and it is small relative to 1. Household saves (or borrows) b in period 1 and receives b (1 + r) in period 2. 1. Setup the household maximization problem and write down the FOCs accurately. 2. Solve the optimum household choice for c 1,c 2 and saving b for a given r.how does your answer vary with ε, r and y. Explain the economic intuition. 3. Solve the general equilibrium problem and find the market rate of return r. How does your answer vary with ε and y. Explain the economic intuition. [ 4. (optional) Now repeat the above exercises for the utility function: U(c 1 ; c 2 ) = E c 1 η 1 1 η + β c1 η 2 1 η ]. 2.4.5 Intertemporal Consumption Choice: Hetereogenenous Agents Consider the two-period Intertemporal consumption decision model that we discussed in class, where the household s utility function is given by: U(c 1 ; c 2 ) = c1 η 1 1 η + β c1 η 2 1 η where c 1 and c 2 denote period one and two consumption, respectively, and β < 1 is the discount factor and η < 1. Here we study a case where all the household are NOT identical; and there are two types of households. Type a is poor in the beginning and wealthy later but the other type b is wealthy today and poor tomorrow. More specifically, suppose that there are N a = 200 millions of households of type a whose income are y 1a = $10000 and y 2a = $50000 and there are N b = 100 14

millions of households of type b whose income are y 1b = $60000 and y 2b = $5000. You can take η = 1. 1. What is the aggregate borrowing of type a? (Write it as a negative number.) 2. What is the aggregate saving of type b? (As a positive number) 3. What is the net saving of the economy? 4. In equilibrium where the net saving is zero, what is the market value for the rate of return R? 5. (Optional) Solve for R in the general case where we don t have the assumption of η = 1. 6. What is the equilibrium b 1 for each type of household? Do we still have zero lending and borrowing? 7. Now suppose N a increases to N a = 400 millions. (a) What is the new equilibrium R? Explain and give an economic intuition for this change using a supply-demand analysis. (b) How does the consumption of each type would change? Then use your findings of the previous problem regarding the income and substitution effects to justify your answer. (c) How does the Saving/Borrowing of each type would change? Explain it using the supply and demand curves. 8. Now suppose y 1a increases to y 1a = $20000. (a) What is the new equilibrium R? Explain and give an economic intuition for this change using a supply-demand analysis. (b) How does the consumption of each type would change? Then use your findings of the previous problem regarding the income and substitution effect to justify your answer. (Hint: be careful! There are couple of effects) 15

(c) How does the Saving/Borrowing of each type would change? Explain it using the supply and demand curves. 9. Now suppose the government issues B amount of bonds, meaning that they borrow B$. Show how does it affect r. Explain the economic intuition. 2.4.6 Simulating an Infinite-Period Model Consider the infinite-period model of consumption choice, where the social planner solves: subject to max {c t,k t+1 } β t log c t t=0 c t + k t+1 = Ak α t + (1 δ) k t for all t = 0, 1, 2,...Again β < 1. 1. Write down the Euler Equation. 2. Solve for the steady state allocation. 3. Write a Matlab code to find the optimum c 0 for a given k 0. 2.4.7 Simulating a Representative Agent model with No Labor-Leisure Decision Consider the infinite-period model of consumption choice, where the social planner solves: subject to max {c t,k t+1 } β t log c t t=0 c t + k t+1 = Ak α t + (1 δ) k t for all t = 0, 1, 2,...Again β < 1. In this problem we d like to do the comparative statics using Dynare. 1. Calibrate your model: Replace the parameters of the model with some sensible numbers. 16

2. Setup the Dynare code to solve for the steady state of the problem. 3. Suppose the parameter A rises by 10%. Determine the new steady state and plot the transition paths for y t, k t, i t, c t, v t. Explain how the results make sense. 4. Now suppose the initial capital rises by 10%. Redo the previous part. 2.4.8 Dynamic Analysis of a Representative Agent model Consider the standard representative agent model with no labor supply decision. Suppose the economy is in the steady state. Analyze the following situations. Specifically explain (as we did in class) what would happen to each variable over time using supply-demand analysis in different markets. 1. A negative productivity shock arrives at time t = 0. 2. A positive productivity shock arrives at time t = T. 2.4.9 Representative Agent model Analysis Consider the standard representative agent model with no labor supply decision. There is a representative household who solves an infinite-period consumption and investment choices such that subject to max {c t,k t+1 } t=0 β t c1 σ t 1 1 σ c t + i t = w t 1 + v t k t + Π t k t+1 = (1 δ) k t + i t for all t = 0, 1, 2,...and β < 1. There is a representative firm which rents capital and employs workers to maximize its profit: 17

max Π t = Akt α lt 1 α w t l t v t k t k t,l t Markets clear such that l d t = l s t and k d t = k s t. The final good is the numéraire good with price one. (If needed, you can take σ = 1 (i.e. log utility)) 1. Write down the FOCs and the Euler equation. Solve for the steady state equilibrium. 2. Explain how does your S.S. results (all important macro variables like y, c, k, i) depend on A, β, δ and k 0. Suppose the economy is in the steady state at t = 0. Analyze the following situations using Dynare. Specifically explain (as we did in class) what would happen to each variable over time using supply-demand analysis in different markets. You can take β = 0.95, δ = 0.1. For σ try σ = 0.5, 1, 2 3. An unexpected negative productivity shock happens at time t = 2. 4. An expected positive productivity shock happens at time t = 2. 5. Vary σ from 0 to 10 and see how does your responses to c t vary. Explain intuitively. 2.4.10 Value Function Calculation 1. Consider the following Bellman equation: V (k) = max k { ln ( Ak a k ) + βv ( k )} We guess that the solution is of the following form: V (k) = c ln k + d k = eak α Verify this guess and solve for c, d, e. Explain the intuitions. 18

2. Consider the following Bellman equation: V (k) = max k {( Ak a k ) + βv ( k )} We guess that the solution is of the following form: V (k) = cak α + d k = k Verify this guess and solve for c, d, k. Explain the intuitions. 2.4.11 Financial sector and consumption smoothing One of the main results of long-run analysis is consumption smoothing behaiviour which is known as the permanent income hypothesis. It tells us that individuals consume based on their permanent income, not the current income. One of the critical assumptions that is necessary for this result to be achieved is that individuals be able to lend or borrow freely at any time. 1. Based on what you have learnt from long-run general equilibrium, discuss about why this assuption is necessary? 2. This condition is termed as complete financial markets in economics literature. In order to test the importance of the completeness of financial markets on consumption smoothing we have to look at the data and test our hypothesis. As there is no universal index to show the completeness of financial markets in a given country, we should find a proxy which reflects what is important for us in financial markets and see if it can explain the variation in the degree of consumption smoothing in different countries. Find such a proxy (or proxies) and explain why you think this is a good proxy for our question. Then test your hypothesis and discuss on the results. 19

3 Growth 3.1 The Solow Model 3.1.1 Review Consider an economy which has the production technology: Y t = K α t (A t L t ) 1 α Labor and Capital Share For simplicity, assume that there is one big competitive firm that operates this production function and that hires workers and capital for wage w t and rental rate r t. Formulate the profit maximization problem of this firm and calculate the FOCs. Verify that the labor share (w t L t /Y t) and capital share (r t K t /Y t) are constant in this economy. (Note: This is the primary reason why the Cobb-Douglas production function is so popular: the fraction of output paid to workers, i.e. the labor share, and the capital share have historically been pretty constant in the U.S.) The K-Change Rule Now assume that productivity and population are constant (A t = A and L t = L), that capital depreciates at rate δ, where 0 < δ < 1, and fraction s of output is invested in new capital each period. Assume that the law of motion for capital, K, is given my the following equation: K t+1 = (1 δ) K t + I t Using the above capital change rule, derive an expression for the initial growth rate of output: Y 1 Y 0 Y 0 as a function of initial capital K 0 (Hint: recall that I t = sy t ). Verify that the growth rate is decreasing in initial capital K 0. Steady State K We will look for a steady state where K t+1 = K t = K for all t. Derive the steady-state level of capital K as a function of the parameters. Use your solution for K 20

to derive expressions for output Y and consumption C in the steady state. (Hint: recall that C t + I t = Y t, where I t = sy t ). Optimal s Find the savings rate 0 < s < 1 that maximizes output at steady state. Also compute the golden rule savings rate s gold that maximizes consumption at steady state. 3.1.2 Solow Model with Population Growth Contrary to the initial assumption, assume that population grows at the constant rate n: L t+1 = (1 + n)l t, while A t continues to be constant. Assume that everything else is the same as above. Rewrite the law of motion in terms of capital per worker k t = K t /L t, and find the steady state level of k, y, c and i as a function of the model s parameters. What is the growth rate of output Y t in the steady state? 3.1.3 Code for Solow model 1. Consider the Solow model of growth with y = f (k) = Ak α and I t = sy t. (a) Write a Matlab code to calculate these variables for given parameters. (b) Calibrate your model with the following numbers: α = 0.5, δ = 0.1, A = 3, s = 0.15, g n = 0.01 and find the steady state. (c) Now use your code and show how do the steady state variables (c, k, i, y) respond to different values of parameters. To do so, pick one parameter and choose a range for it and run your code for different values of this parameter in this range. Then plot the variables with respect to this parameter. Explain your reasoning for each of the,. Why do we observe such a behavior? (d) Now write a code to calculate the transition path to the steady state. Then simulate your model for k 0 = k /2. (e) How does your transition path varies with α? How about s? (f) Define g 1 = y 1 y 0. Plot g 1 with respect to different ranges of the parameters. How does the growth rate in the first period respond to different parameters. Explain. 21

3.1.4 Solow with Government Let us introduce government spending in the basic Solow model. Consider the basic model without technological change and suppose that Y (t) = C(t) + I(t) + G(t) with G(t) denoting government spending at time t. Imagine that government spending is given by G(t) = σy (t). 1. Discuss how the relationship between income and consumption should be changed. Is it reasonable to assume that C(t) = sy (t)? 2. Suppose that government spending partly comes out of private consumption, so that C(t) = (s γσ)y (t), where λ [0; 1]. What is the effect of higher government spending (in the form of higher σ) on the equilibrium of the Solow model? 3. Now suppose that a fraction of G(t) is invested in the capital stock, so that total investment at time t is given by I(t) = (1 s (1 λ)σ + φσ)y (t): Show that if φ is suffi ciently high, the steady-state level of capital-labor ratio will increase as a result of higher government spending (corresponding to higher σ). Is this reasonable? How would you alternatively introduce public investments in this model? 3.1.5 Solow example Suppose that the world has two types of economies: industrial and agricultural. Both have a production function of the following form: Y = K α L 1 α Suppose, agricultural economies have α = 1 3, industrial economies have α = 2 3. All countries have the same population and there is no population growth. Assume that all countries have the same savings rate s = 1 2 and depreciations rate of capital δ = 1 8. An economy can only become 22

industrial if it can finance a lot of research and development. Assume that, in this world, this happens suddenly. Specifically, assume that an economy becomes industrial (its changes) once it has a level of 27 units of capital per capita. 1. What is convergence? Would we observe convergence in this world? Explain and show graphically. 2. Suppose that the World Bank wants to help poor economies develop. It decides to give a gift to any economy with an output per capita of less than 3. The gift consists of 2 units of capital per capita. Will this make any help to the convergence? What if the gift is 19 units of capital per capita? 3. Suppose that there are no World Bank subsidies and that Agricultural countries have a government that can set the savings rate of the economy. Is there a savings rate that will allow Agricultural economies to become industrialized? What if the world bank subsidizes the economy by donating a dollar for every dollar that the country saves? 3.2 Neoclassical Growth Model 3.2.1 A Simple Neoclassical Growth Model Consider the infinite-period model of consumption choice, where each representative household solves: subject to max {c t,k t+1 } t=0 β t c1 σ t 1 σ c t + i t = Ak α t + (1 δ) k t k t+1 = i t + (1 δ) k t for all t = 0, 1, 2,...Again β < 1. 1. Solve the model. 23

(a) Write down the Euler Equation. (b) Write the recursive equations that can solve for c t, k t+1 (c) Write the transversality condition. (d) Solve for the steady state allocation. (e) Explain intuitively how the macro variables depend on the exogenous parameters of the model. 2. Simulation (a) Write a Matlab code to find the optimum k 1 for a given k 0 (We call this the policy function for k 1 ). Plot k 1 versus k 0. Find the optimum c 0 for a given k 0. Plot c 0 versus k 0. (To do so, your code receives k 0. It makes a guess for c 0. then it goes forward using the resource constraints and the Euler equation and solves for c t and k t for t = 1...T (large enough). Then you should check whether you are violating any boundary conditions or assumptions. If so, you should update your guess in the right direction (for example change c 0 to c 0 + ε) and try again. Iterate this process until you converge to a specific value of c 0.(Note that You can also make your guesses for k 1 instead of c 0 )) (b) Now plot the path of k t, c t, y t, i t, s t = it y t for a given k 0. Plot three time paths of capital,consumption, output and investment for at least three values for k 0. Use standard parameter values for the exogenous parameters. (See Cooley 95: "The calibrated Model" paper). 3. Convergence Analysis (a) How long does it take the economy to reach close to 90% of the steady state; i.e. find the smallest t such that k t > 0.9k SS. Compare this with a Solow model. (b) Do we have conditional convergence in this model? Why? 4. Suppose the economy is in the steady state for t < 0. At time t = 0, productivity increases from A to A > A. How do the policy functions for k 1 and c 0 change? Show how does the economy respond (plot different macro variables). Explain intuitively. 24

5. Suppose the economy is in the steady state for t < 0. At time t = 0, capital per capita drops to k 0 < k SS (This can happen through a disaster like earthquake or with a sudden increase in population like a baby boom or emigration. Show how does the economy respond. Explain intuitively. 3.2.2 Neoclassical growth model: Productivity Growth Consider the standard discrete neoclassical model as in the class with inelastic labor. Now suppose the productivity grows with rate g such that A t = Ā(1 + g)t. In period, household s utility function is U (c) = c1 σ 1 σ 1. Setup the problem. 2. Write the First order conditions and find the Euler equation. 3. Solve for the Balanced Growth Path (BGP) equilibrium : Determine the allocations and prices in the long run. 4. Do a comparative statics for Ā and g. 5. Explain intuitively how does the rental price v depends on g in the long run. How about w, K, I and Y 6. Explain how your results differ for the aggregate variables and per capita variables with the one we discussed in class. 7. Suppose the growth rate increases to g > g. How does the economy converge to the new BGP? 3.2.3 A Neoclassical Growth Model with Oil Consider a simple neoclassical growth model where the economy has a natural resource O t = Ō for t 0. 1. Setup the problem. (You should think deeply here on how to incorporate the oil in to your model. There is not necessarily one way to do it.) 25

2. Find the steady state allocation (capital, output, consumption, investment and marginal product of capital) and discuss your results. 3. Suppose the economy can save its oil revenue at the international market at a fixed rate r. What would happen to your results? 4. (Optional) Simulate the transition paths. 3.2.4 A Neoclassical Growth Model with External Finance Consider a simple neoclassical growth model where the economy has access to foreign finance at a fixed interest rate r. 1. Setup the problem. (You should think deeply here on how to incorporate the oil in to your model. There is not necessarily one way to do it.) 2. Find the steady state allocation and discuss your results. Does the economy borrow a positive value in the long run? 3. Think deeply on how does the transition occurs. You may find it counter-intuitive at the beginning. 4. Now suppose you can have foreign direct investment but the return is at the marginal rate for capital. Now resolve the problem. 3.2.5 A Neoclassical Growth Model with Uncertainty Consider the two-period model, where the household s utility function is given by: U(c 1 ; c 2 ) = log (c 1 ) + E [β log (c 2 )] where c 1 and c 2 denote the consumption in periods one and two, respectively, and β < 1 is the discount factor. Household income is y t = A t kt α. He starts with and endowment k 1 in period 1 and decides how much to consume c 1 or invest. For simplicity assume that the depreciation rate is δ = 1. Household can also save s at a fixed rate r. 26

Suppose A 1 = Ā. but A 2 = Ā with probability p and A 2 = 0 with probability 1 p.(in other words, someone steals all of his capital or the government takes it) 1. Setup the household maximization problem and write down the FOCs accurately. 2. Solve the optimum household choice for c 1,c 2, k 1 and saving s. 3. How do the household s decisions depend on p? 3.2.6 A Neoclassical Growth Model with Leisure Consider the infinite-period model of consumption choice, where each representative household solves: subject to max {c t,k t+1 } β t u (C t, l t ) t=0 C t + I t = AK α t l 1 α t K t+1 = I t + (1 δ) K t + (1 δ) K t for all t = 0, 1, 2,...And β < 1. Assume that u (C t, l t ) = (Ct(1 lt)γ ) 1 σ 1 σ 1. Write down the Euler Equation. 2. Write the transversality condition. 3. Solve for the steady state allocation (capital, leisure and labor, output, rate of return on capital, marginal product of labor (wage)) 4. How do the reults differ with a case with no leisure in the preferences. 5. Explain intuitively how the macro variables depend on the exogenous parameters of the model. 1 1 σ 6. How do your responses differ if u (C t, l t ) = C1 σ t γ l1+φ t 1+φ. 27

3.2.7 Neoclassical growth model: Continous Time Consider the continuous time neoclassical growth model as in the class where the social planner maximizes W = 0 e ρt U (c t ) dt where U (C) = C1 σ 1 1 σ, subject to the resource constraint c t + k t δk t = A t k α t 1. Write the two differential equations in terms of c and k that determines the equilibrium. 2. Suppose A t = Ā. Solve for the steady state values of c and k. 3. Find kt y t in the steady state. Calibrate your model to US data using typical numbers (ρ = 0.04, δ = 0.1, α = 1 3 ) and to Iran data (α = 2 3 ). 4. Go to Iran s data from cbi.ir and calculate kt y t number? How should you change ρ in the model so that k y state? for the years 1370-1389. What is the average matches the data in the steady 5. (Optional) Find k t k t k in the steady state. This shows how fast we are converging to the steady state. Use typical numbers to calculate the number of years to get as half path to the steady state. 6. Now suppose A t = Āegt. Calculate the growth rate of consumption and capital ( g = g k = g c ). 7. Suppose we are on the Balanced Growth Path such that all the variables grow with constant rate. What is the Euler Equation? Solve for the marginal product of capital. 8. Define k t = k t e gt and c t = c t e gt and setup the problem using these variables. Do we have steady state for this problem? What are the two differential equations that solve for them? What are the steady state values if exist? This will show the transition to the 28

balanced growth path. Draw a diagram that shows how capital or consumption converges to the BGP. 3.2.8 Matlab: Neoclassical growth model Consider the discrete neoclassical model as in the class. 1. Write a code in Matlab to solve for the steady state values of the variables given the parameter values. 2. Write a code in Matlab to solve for the transition path to the steady state given the parameter values. To do so, your code receives k 0. It makes a guess for c 0. then it goes forward using the resource constraints and the Euler equation and solves for c t and k t for t = 1...T (large enough). Then you should check whether you are violating any boundary conditions or assumptions. If so, you should update your guess in the right direction (for example change c 0 to c 0 + ε) and try again. Iterate this process until you converge to a specific value of c 0.(Note that You can also make your guesses for k 1 instead of c 0 ) 3. Now use your code and make a plot of k 1 versus k 0 for different values (at least 100 data points) of k 0 between [0, k]. What does this graph show? What does it mean? 3.3 Endogenous Growth Models 3.3.1 Simple AK Model 1. Setup a simple AK growth model with consumtion and income taxes. Sove the model and find the growth rates and the levels. Show how do these taxes affect the growth rates and the levels. 3.3.2 Simple AK Model Consider a simple AK growth model where household preferences are U = 0 ρt c1 σ t 1 e 1 σ dt 29

and the representative household decides on consumption and capital with depreciation rate of 100% such that: k t = (1 τ k ) v t k t (1 + τ c ) c t where v t is the return on capital and τ c, τ k are the tax rates on consumption and capital. The representative firm produces with a CRS production function y t = Ak t. The government collects taxes to provide public goods. 1. Suppose there is no taxes and no government purchases. Solve for the BGP competitive equilibrium. Find economy s growth rates in output. 2. Suppose τ c = 0, τ k = τ; i.e. the government finances through the capital taxes. Solve for the BGP competitive equilibrium. Find economy s growth rates in output. Solve for y 0 and c 0. Solve for the total welfare. Interpret your results. 3. Suppose τ k = 0, τ c = τ; i.e. the government finances through the consumption taxes. Solve for the BGP competitive equilibrium. Find economy s growth rates in output. Solve for y 0 and c 0. Solve for the total welfare. 4. Write the Solve the model and find the growth rates and the levels. Show how do these taxes affect the growth rates and the levels. 3.3.3 Lucas 1988 In this problem, we want to replicate the Lucas s 1988 paper (the human capital endogenous growth part). Consider the Lucas 1988 economy with endogenous human capital growth in continuos time. 1. Setup the problem for the social planner case. 2. Write down the FOCs. 3. Following Lucas s method, solve for the growth rates of capital, output, consumption and human capital. You can first take u as given and solve everything in terms of the growth rate of human capital and then solve for the optimal u. 30

4. How does u depend on the exogenous parameters. Explain the intuitions. 5. How do growth rates of human capital and output depend on the exogenous parameters of the model. Explain the intuitions. 6. Explain why γ shows up in the growth rates. 7. Explain how A does not show up in the growth rates. Any intuiton? 8. In one paragraph, write dew how this model explains the endogenous growth of the economies and why it is a model of development. 3.3.4 Lucas 1988: A variation In this problem, we want to setup the Lucas s 1988 paper (the human capital endogenous growth part) in discrete time. Consider the Lucas 1988 economy with endogenous human capital growth in discrete time where h t+1 = h t (1 + δ (1 u t )) and c t + k t+1 = y t = Ak α t (u t N t h t ) 1 α and W = β t c1 σ 1 1 σ 1. Setup the problem for the competitive equilibrium case. 2. Write down the FOCs. Use λ t, µ t in as the Lagrange Multipliers. 3. Eliminate λ t, µ t from the FOCs and then explain the intuiton of the two equations. One of them is the Euler equation and the other is a similar one for human capital accumulation and hours of work. Explain how does a household decide about one more hour of work. 3.3.5 Lucas 1988: With Taxes 1. Consider the Lucas 1988 economy with endogenous human capital growth and focus on the competitive equilibrium case. Suppose the government impose the following taxes: τ c consumption tax, τ k capital tax (on the rental capital), and τ l labor income tax on wages. Then it rebates all the taxes to households as lump-sum subsidies. (a) Write down the household budget constraint. (b) Write down the model FOCs. 31

(c) First, suppose the all taxes are zero. Solve for the growth rates of consumption, capital, output and human capital. Also solve for the interest rates and the fraction of time that HH works. What is the fraction of total consumption to output? (d) Now, under the following scenarios: Solve for the BGP s equilibrium endogenous growth rate. Then solve for the relationship between the levels (K and H). Explain how government can affect the growth and levels through these taxes. i. Suppose only consumption tax is non-zero. ii. Suppose only labor income tax is non-zero. iii. Suppose only capital tax is non-zero. (e) (Optional) Can you design a tax system (τ c, τ l, τ k ) to achieve the first best allocation (Social planer). Note that negative taxes mean subsidies. 3.3.6 Romer 1990 1. Based on Romer 1990, (a) Explain why subsidy to the capital investment cannot affect growth. What type policy can have some effects? (b) Write how this model can explain the growth rate heterogeneity across world economies. Especially, why some countries stagnate? (c) Now set up the social planner problem for this model and solve for the optimum solution. (d) Explain the sources that the decentralized economy deviates from the optimum. 4 Real Business Cycles 4.1 Benchmark RBC 4.1.1 RBC model (1) 1. Setup an RBC model with inelastic labor (no labor supply decision: l = 1) where household per period utility function is u (c) = c1 η 1 η and the production function is y = f (k, l) = 32

Ak α l 1 α and a persistent productivity shock (z t log A t = ρ log A t 1 + ε t ). (the social planner problem) 2. Write down the FOCs and find the Euler equation. 3. Solve for the steady state. 4. Log linearize the model, simplify to get an equation which has only capital and the productivity shock. 5. Solve for the Recursive Law of Motion to find γ 1, γ 2 such that k t+1 = γ 1 k t + γ 2 z t. 6. Explain how do γ 1, γ 2 depend on the exogenous parameters. Can you come up with some intuitions? 7. Now solve for c t, i t, y t, v t, w t in terms of k t, z t. 8. Calibrate your model using typical numbers we use in class. 9. use the parameters you found in the previous part and plot all of the endogenous responses (listed above) to a 1% productivity shock at time 0. Explain the intuitions behind these responses. How large are the spot effects and how large are the persistency? In one paragraph, explain how does a productivity shock is affecting different variables over time. 10. Simulate your model using the calibrated model and estimate the standard deviations of important variables and their correlations with GDP. 11. Analytically, solve for the conditional and unconditional mean and variance of z t, k t+1 and c t. Now compare it with the simulated ones in the previous part. 12. Analytically, solve for the unconditional correlation of consumption and investment with output. How does it compare with the simulated ones? 4.1.2 RBC model (2) 1. Setup an RBC model (like the one in class) with capital and labor and a persistent productivity shock in a competitive equilibrium environment (do not setup the social planner 33

problem). Household per period utility function is u (c) = log (c) γ h1+φ 1+φ and the production function is y = f (k, h) = Ak α h 1 α 2. Find the Euler equation, the labor supply and the labor and capital demand functions. 3. Define the equilibrium and then simplify the equation by clearing the markets. 4. Solve for the steady state. 5. Log linearize the model (you should have at most three equations: one for labor, one for capital and one for consumption) 6. Calibrate your model to US data using Cooley Prescott paper (Read the paper carefully). How do you treat government purchases? How about government investment? How do you treat the change in the inventory data? 7. Use Dynare to solve and simulate your model. Explain your finding on how the model match with the data (compare the moments of the data and your model like the variances of variables and their correlations with GDP). What fraction of output volatility can your model explain? What is the ratio of the output volatility to Solow residual volatility in the data and in your model. 8. Plot the impulse responses using Dynare. How do consumption, output, labor and capital respond to a 1% shock to productivity. Explain how the amplification process (spot effect) and the transmission mechanism (persistency) is working? Specifically, pay attention on how a productivty shock is affecting employment and investment. Write a story for how a productivity shock is affecting the economy and then propagates over time. 9. How does your impulse responses change when you change ρ the persistence parameter of the productivity? 10. How does your impulse responses change when you change σ 2 e the variance the productivity innovation ε t? 11. (optional-no credit) How does your responses change if the utility function is u (c, h) = (1 γ) log c + γ log (1 h)? How about u (c, h) = (c1 γ (1 h) γ ) 1 η 1 η? 34