1. Following Kydland and Prescott (1977), assume that the monetary authority chooses its inflationary policy in order to maximize the social objective function:

S(x_i, u_t) = -u_t - x_t²/2

subject to the Phillips Curve constraint:

u_t = u* - λ (x_t - x_t^e),

where ut is the unemployment rate in period t, λ is a positive constant, x_t is the inflation rate, u* the expected rate of inflation, and to the natural rate of unemployment

(a) Draw the indifference curves map corresponding to welfare function (1) with the rate of unemployment on the horizontal axis and the inflation rate on the vertical axis.

(b) Derive the pre-commitment (or non-discretionary) equilibrium (x_t^P - u_t^P). Use the Phillips Curve (2) to illustrate this equilibrium on the in¬difference curve map from Part (a). Calculate the resulting welfare S^p (x_t^P, u_t^P)
(c) Derive the fooling (or discretionary) equilibrium (x1, /4). Illustrate this equilibrium on the indifference curves map from Part (b). Calculate the resulting welfare S^f (x_t^f - u_t^f). What is the welfare gain from "cheating"? Why is the fooling equilibrium not sustainable?

(d) Derive the time-consistent equilibrium (x_t^d - u_t^d) Illustrate this equi¬librium on the indifference curves map from Part (c). Calculate the resulting welfare S^d (x_t^d - u_t^d)and compare it to S^p (x_t^p - u_t^p)

2. "Refusing to negotiate with airplane hijackers is a time inconsistent but optimal policy." Comment on this statement.

3. Please visit the Bank of Canada website:

Choose the "Rates and Statistics" group on the toolbar menu and retrieve the available data on the following variables: (i) the inflation
rate (irt), measured as the 12-month growth rate in the total Consumer Price Index (CPI), (ii) the target overnight rate (4), and (iii) the
actual overnight rate (it). Note that the target inflation rate (e) is the midpoint of the 1%-3% target range: in other words, 7r. = 2%.
Next, access the CANSIM database and retrieve series V1992067. This series contains quarterly data for real GDP from 1961 until the present.

(a) Perform the necessary transformations to ensure that the ac-quired data for all variables cover the same period and have the same frequency (quarterly).

(b) Let n denote the total number of observations for each variable. Regress real GDP on a linear time trend:

Y_t = β_o + β₁t + u_t, where t = 1, 2, ... , n.

Use the fitted values of this regression as your estimate of "po-tential" GDP at time t, y_t.

(c) Estimate an equation representing the Taylor Rule as follows: di_t = adΠ_t + bdy_t + e_t, where di_t = i_t - i;, dΠ_t = Π_t - Π_t* and dyt = yt - yt*. Let a and b denote the OLS estimates for coefficients a and b. Calculate the overnight rate implied by the "policy rule" as:

i_t = i_t* + adΠ_t + bdyt.

Plot i_t and it against time on the same graph. Does the Taylor Rule explain the behavior of the Bank of Canada reasonably well?