EF5472 Advanced Macroeconomics Exercise 1
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EF5472 Advanced Macroeconomics
2022
Exercise 1
Question 1 (100 points) Classical model
Consider the classical model in Mankiw chapter 3 and Lecture Notes “Simple macro model” . Generalize the consumption function to allow it to depend negatively on the real interest rate: C(Y − T, r) with ∂C / ∂r < 0.
a) Use the calculus approach to analyze how the equilibrium interest rate would respond to changes in the exogenous variables Y , T, and G.
Note: Changes in Y are supply shocks. For example, the Covid pandemic can be interpreted as a negative supply shock.
b) Provide economic explanations to the comparative static results in (a) with the aid of graphical analysis in the loanable funds market. Note: The loanable funds supply curve should be upward sloping in this case. Why is the loanable funds supply curve vertical in Mankiw? Why is it upward sloping here?
c) Suppose that the government increases taxes and government purchases by equal amounts. What will happen to the equilibrium interest rate, consumption, and investment in response to this balanced-budget change? Does your answer depend on the marginal propensity to consume?
d) Add the monetary sector and the Fisher relationship to the classical model. Show that the full model can be condensed to two equations with two endogenous variables (r, P). Use the calculus approach to study the impact of fiscal policy (changes in G) and monetary policy (changes in M) on the equilibrium interest rate, consumption, investment, and the general price level.
M d / P = L(R, Y),
M s = M
M s = M d
R = r + πe
∂L / ∂R < 0, ∂L / ∂Y > 0 Demand for real balances
Money supply
Money market clearing (“All money willingly held”) Fisher relationship
where M and πe (inflationary expectations) are exogenously given.
e) What is meant by classical dichotomy? Make use of your findings in (d) to illustrate your answer. Hint: Read Mankiw chapter 4.7 for the concept of classical dichotomy.
Question 2 (100 points) Two-period consumption-saving model
Refer to the consumption-saving problem in handout "Appendix - Consumer theory from microeconomics", section 2.2. Mankiw chapter 17.2 and GLS chapter 9.1-9.3 talk about the same model.
a) Derive graphically the savings (i.e. loanable funds) supply curve of a saver, analogous to the labor supply curve in Figure 10 in the handout. Show that the savings supply curve could be backward bending. What explains the upward- sloping portion of the curve (i.e. interest rate and savings move in the same direction), and what explains the backward-bending portion of the curve?
b) Modify Mankiw Figure 17.7 to analyze the impact of an increase in the interest rate for a borrower ( Y1 < C1 at point A). Summarize the substitution effect, income effect, and overall effect on current and future consumption in a table similar to the one in the handout.
Hint: Choose Y2 a lot larger than Y1 in your picture. It will make your picture easier to draw and look better. GLS Figure 9.8 does the same analysis but it is hard to read, partly because their current endowment Yt and future endowment Yt+1 are too close to each other.
c) Derive graphically the borrowings or loans demand curve of a borrower, analogous to what you did in (a) for the savings supply curve. Is a forward- bending borrowings demand curve possible?
d) Suppose the utility function is U(C1 , C2 ) = ln C1 + βln C2 , 0 < β< 1. Apply the method of Lagrange multipliers to obtain analytical solution (C1 *, C2 *) to the two-period consumption-saving problem. Derive the savings supply function. Is a backward-bending savings supply curve possible in this case? What do you
conclude about the strengths of the substitution effect and the income effect in this case?
e) Suppose C1 and C2 are perfect complements -- indifference curves are L-shaped. Will there be any substitution effect? Show that the savings supply curve in this case must be downward sloping.
Question 3 (100 points) Money in the utility function
Refer to the money-in-the-utility-function model in Lecture Notes "Money and inflation". The consumer problem is
Choose (c1 , c2 , S, M d ) to maximize U(c1 , c2 , M d / p1 ) subject to (i) p1c1 + S + M d ≤ p1y1 (ii) p2 c2 ≤ p2 y2 + S(1+ R1 ) +M d (iii) M d ≥ 0, c1 ≥ 0, c2 ≥ 0 |
a) Let m 全 Md / p1 and s 全 S / p1 , money holdings and savings denominated in current goods. Show that the two period budget constraints, (i) and (ii), can be combined into one lifetime budget constraint that looks like
c + |
c2 1+ r |
≤ y1 + 1+ r1 − | 1+ R1 )| m |
(2) |
Hint: First rewrite the two period budget constraints in terms of goods, and then combine them to eliminate s. Keep in mind the Fisher relationship.
b) Suppose money does not yield utility -- the utility function depends only on
(c1 , c2 ) . Show by indifference curves diagram that the consumer will optimally choose real balances m to be zero, as long as the nominal interest rate is positive.
c) Consider the original setup as in (1) with money in the utility function. Rewrite the consumer problem in terms of choice variables (c1 , c2 , m) . Apply the method of Lagrange multipliers to characterize the solution.
d) Suppose U(c1 , c2 , m) = ln c1 + βln c2 +ψ ln m, β > 0,ψ > 0 . Solve the first-order conditions in (c) to obtain analytical solution. Show that the demand for real balances is inversely related to the nominal interest rate. What do you conclude about the strengths of the substitution effect and the income effect in this case?
e) Suppose the second period budget constraint (ii) is replaced by
p2 c2 ≤ p2 y2 + (S + M d )(1+ R1 ) . That is, money also pays interest like deposit. Derive the lifetime budget constraint and then apply the method of Lagrange multipliers to characterize the solution of (c1 , c2 , m) . Show that in this case the consumer will choose socially optimal money holdings, even though the nominal interest rate is positive.
Note: This problem demonstrates that the source of the social inefficiency is not the positive nominal interest rate per se; rather, it is the differential in returns between money and interest-bearing assets.
Question 4 (100 points) Cash-in-advance model
Refer to the cash-in-advance model in Lecture Notes "Money and inflation". Here we consider a simpler two-period version:
Consumer problem
Choose (c1 , c2 , S, M ) to maximize U(c1 , c2 ) subject to (i) p1c1 + S + M ≤ p1y1 (ii) p2 c2 ≤ M (iii) p2 c2 ≤ p2 y2 + S(1+ R1 ) +M (iv) M ≥ 0, c1 ≥ 0, c2 ≥ 0 |
a) Show that the consumer will always choose to borrow ( S < 0 ). How much will the consumer borrow? Hint: Examine (iii). Is the endowment income p2 y2 available for purchasing c2 ?
b) Suppose the consumer decides to use the full amount of borrowing in (a) for purchasing c2 . How many units of c2 can she buy? Argue that the consumer is effectively paying (1+ R1 )p2 (in period 2 dollars) per unit of c2 . That is, CIA creates distortion like a sales tax.
c) How much the consumer pays (in period 2 dollars) per unit of c1 ? With the help of (b), show that, from the consumer point of view, the relative price of c1 in terms of c2 is p1 / p2 = (1+ π2 )− 1 , the real return of money.
d) Show that (1) can be converted into real form as in (2), where
m 全 M / p1 , 1+ π2 全 p2 / p1 , and 1 + r1 全 (1+ R1 ) / (1+ π2 ) , the Fisher relationship. Apply the method of Lagrange multipliers to characterize the solution
(c1 *, c2 *, m*) . Show that (i) the consumer holds "too little" real balances from the society point of view, and (ii) how the Friedman rule can induce the consumer to hold real balances up to the socially efficient level.
Choose (c1 , c2 , m) to maximize U(c1 , c2 ) subject to (i) c1 + ≤ y1 + − m (ii) (1+π2 )c2 ≤ m (iii) m ≥ 0, c1 ≥ 0, c2 ≥ 0 |
2022-10-06