Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
ECON7230 Monetary Economics Summer 2024
Homework 1
Due Date: June 13 (before the end of the day, submit through Moodle)
1. The Goods Market: Consider the basic goods market model you have in the first class: C = co + c1 (y − T) and I, T and G are exogenous. The current equilibrium income y is 400, but the government’s goal is to increase it to 500. If a decrease of 25 in T can achieve that goal, then how much of an increase in government spending G can achieve the same goal? Explain your answer.
2. IS-LM Model The following equations describe the macro-economy:
Consumption function: C = 50 + 0.5(y − T) Taxes: T = 100
Government spending: G = 200 Investment: I = 200
Real money supply: P/M = 3600
Real money demand: P/Md = 2y + 10/i
a) Derive the IS and LM curves. Plot them on a graph with interest rate i on the Y-axis and output y on the X-axis carefully.
b) Calculate the equilibrium output and interest rate.
c) If the interest rate is 0.02 and output is 1200, is the economy in equilibrium? Do we have excess demand or supply in the goods market? Do we have excess demand or supply in the money market? Explain your answers.
The banking system has scarce reserve, so the money multiplier is a meaningful concept. The central bank has decided to increase monetary base by a real amount of 500. Calculate the impact of this monetary policy change for the following two scenarios:
d) Suppose the public keeps all money demand as currency and none as checkable deposit. What happens to the equilibrium output and interest rate? Show your answer both algebraically and graphically.
e) Suppose the private banks keep all checkable deposit as reserve. What happens to the equilibrium output and interest rate? Show your answer both algebraically and graphically.
3. ISLM Model and Stock Price In period t, the economy is described by the IS-LM model: IS : yt = 100 − 1000it
LM : yt = 60 + 1000it
That is, equilibrium output and interest rate for period t can be solved for by the above two equations. Company M&B’s stock pays dividends using the formula:
Dt = 0.5yt, Dt+1 = 0.5yt+1, …
That is, Company M&B pays an amount of dividend in each period equal to half of the output in that period. The stock price vt for Company A comes from the present value formula:
The discount rate rt is equal to the sum of the interest rate it and a risk premium p, which we assumed to be a constant of 1%. Assume that there is no inflation and there is no distinction between real and nominal values.
a) Suppose people do not expect the IS curve or LM curve to move anytime in the future. Calculate the stock price for Company M&B in period t.
b) Suppose people expect expansionary monetary policy starting from period t + 1, with the LM curve shifts to:
Yt+1 = 80 + 1000it+1
And, people expect the shift to be permanent, i.e. the LM curve is expected to be the one above for t + 2, t + 3, … . On the other hand, the IS curve is expected to be the same as the one used in a) for all future periods. Calculate the stock price for Company M&B in period t.
c) Suppose people do not expect any of the curve to shift, but people worry about the future and require a higher risk premium p of 2% instead, for all periods from t and beyond. Calculate the stock price for Company M&B in period t.
4. Choice of monetary policy tool and uncertainty
Consider the following ISLM model
Y = 5 − 2i + u, M = 5 + Y − i + v
The variance of the shock u is 1 and the variance of the shock v is 3, and the two shocks have zero mean and are not correlated.
a) To minimize loss for some target Y* , derive the optimal policy for i and M. Should the central bank choose M or i as the monetary policy tool?
b) Regardless of your answer to a), let’s say the central bank is choosing i instead of M. Instead of having a stable IS curve, it now has uncertainty
Y = 5 − ai + u
where a is a random variable that has a mean of 2 and variance of 1. How is the optimal policy for i different from that in a)?
5. Rational expectations and stock price Suppose stock price pt is related to dividend Dt as follows
where R is a constant discount rate that is larger than 1. The dividend process is AR(1), Dt = pDt 一1 + et where −1 < p < 1 and et is random shock with mean zero and uncorrelated over time.
a) Solve for the stock price pt under rational expectations using any method you want. (Hint: the solution for pt has the form of constant × Dt )
b) It turns out that there is also a “bubble” solution, where the solution is simply the pt you have just solved for in a) plus a “bubble” term Bt. Explain why we need the condition Et Bt+1 = RBt for the “bubble” solution to work.
c) To eliminate the “bubble”, we need the assumption that R一TEt pt+T goes to zero when T goes to infinity. Explain why the assumption can eliminate the “bubble” .
6. Cagan model with money growth that everybody knows
The money market equilibrium condition is similar to the one we have seen in class:
mt − pt = −Rpt(e)+1,t − ptS.
Money supply follows a simple formula of mt = 10(1.5t ) for t = 0,1,2, … . We assume that everybody knows about this formula and the central bank is never going to change it.
a) Suppose expectations are rational. Solve for the time path of pt for t = 0,1,2,3,4,5,6.
b) Suppose expectations are rational, but the money supply rule is mt = 10( 1.5t ) for t = 0,1,2,3,4 and mt = 11 (1.5t ) for t = 5,6, …. (again, everybody knows it and the central bank is not going to change it). Solve for pt for t = 0,1,2,3,4,5,6 and compare them to your answer in a).