Part 1 (50 Points): Rational Expectations Model

Part 1 (50 Points): Rational Expectations Model

Professor Erdinç

ECO 402

Homework #4

(Due April 27th at 7:00 p.m. Office 233)

Part 1 (50 Points): Rational Expectations Model

Consider the following model.

1. Lucas Supply

2. Aggregate Demand

3. Money Market Equilibrium

where ,, are positive constants. In this model, AD is subject to real shocks with known mean and variance, i.e.). At the same time, the money demand in equation (3) is subjected to velocity shocks,.

  1. (10 Points) Derive the solution values for the endogenous variables,, and under rational expectations.
  2. (10 Points) Show that in the presence of velocity (nominal) and demand (real) shocks, it is possible for the output to deviate from its natural level (normalized here to zero for simplicity).
  3. (10 Points)If thepolicy makerswish to stabilize output around its natural level via a constant money supply rule, what is the variance of output under this policy? (Hint: Set ==).
  4. (10 Points)If thepolicy makerswish to stabilize output around its natural level via interest rate targeting, what is the variance of output under this policy? (Hint: Set = and take it exogenous. Now you should treat as endogenous.)
  5. (10 Points)Which of the two policies, fixed money supply orfixed interest policy regime, will be optimal in stabilizing he output around its natural level? Does your answer depend on the relative magnitude of these shocks?

Part 2 (50 Points): Dornbusch “Overshooting” Model

Consider a simple open economy á la Dornbusch (1976) with exchange rate overshooting. In this model, there is perfect capital mobility and stands for full-employment output. The model shows that even with flexible prices (is endogenous), there is possibility for the exchange rate to overshoot its long-run value in the short-run in response to an unanticipated expansion in the money supply.

1. (Money Market Equilibrium) MM Line

2. (Aggregate Demand)

3. (Price adjustment in the Goods Market) GM line

This is a reduced form model with Uncovered Interest Parity Condition is given as: The expectations formation mechanism is such that if is less than its long-run value, , then agents expect a depreciation and rational expectations holds:. The variables,, ,, are the logs of money, output, expenditure, the price level and the nominal exchange rate respectively. is the nominal interest rate, is the foreign nominal interest rate and is inflation.

Exogenous Variables:,, , (steady state value)

Endogenous Variables: and

Constants (all positive):,,,, , , with

a)(10 Points)Derive the slopes of the GM and the MM lines with on the vertical and on the horizontal axis first after substituting equation (2) into (3). (Hint: First find the slope of MM line without imposing equal to zero (i.e. ) , then set equal to zero at the steady state and derive an expression from the MM equation for and plug it into GM equation before taking the slope of GM- we need to assume that domestic interest satisfies MM equilibrium before we consider the goods market). Impose the equal to zero condition along with =0 condition for deriving the slope of GM. Show that the slope of GM line is less than 1 but positive and slope of MM is negative.

b)(10 Points)Find the steady state values of the endogenous variables, and as a function of all exogenous variables and constant terms.

c)(10 Points)Draw a Phase Diagram with on the vertical and on the horizontal axis. Phase Diagram should be based on two lines: GM line (Goods market Line) and the MM line (Money Market Equilibrium).

d)(10 Points)Show thatthese two lines shift as money supply expands i.e. goes up. Show also that the MM shifts more than the GM.

e)(15 Points)Show graphically with the help of arrows that as money supply expands, the exchange rate may overshoot its long-run equilibrium value (under the assumption that prices do not adjust immediately). Provide an economic interpretation of the reasons behind this overshooting.

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