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Money Demand and Seigniorage

## Problem 1: Money Demand and Seigniorage

Suppose the solution to a 2-period householdís problem gives the following expressions for the Euler equation and the money demand:

denote the gross real interest rate, and Rn 0 = 1 + r n 0 denote the gross nominal interest rate. The parameter ! > 0 represents the elasticity of money demand to the gross nominal interest rate Rn 0 : Also let 0 = 1 + 0 denote the gross ináation rate in period 0, and 1 = 1 + 1 denote the gross ináation rate in period 1. Recall that R0 = R n 0 1 (see slides on "Monetary Policy: Part I").

Assume the central bank controls money supply according to the following money-growth policy rule: Mt = Mt1; for t = 0; 1 (3) where Mt denotes the nominal money stock in period t, and 1 is the gross growth rate of money supply. It is easy to show that the real money stock, mt = Mt Pt ; grows according to the following, in period 0 and period 1,

respectively: m0 = 0 m1 and m1 = 1 m0 (4) We are going to solve for the steady state of this simple model: A steady state is an equilibrium where all variables are constant: mt = m; t = ; ct = c; Rn t = Rn and Rt = R for t = 1; 0; 1: 1. Using the expressions in (1), (2) and (4) - assuming that c = 1 - Önd the steady state expressions for ; R; Rn; and m: How do they depend on the rate of money growth set by the central bank? Can you provide a brief economic intuition for your results? 2. Seigniorage (in nominal terms) in period 0 is deÖned as S0 = M0 M1: Using the notation from class, real seigniorage s0 = S0 P0 is s0 = M0 M1 P0 = m0 m1 0 From the latter, its steady state expression is (just set m0 = m1 = m and 0 = ) s = m 1 (5)

Using your results for and m in 1., and equation (5), answer to the followings. HINT: this exercise shares the same logic of the problem on taxes on labor and tax revenues given in the Midterm. i. Show that s is a bell-shaped (non-monotone) function of the money growth rate ; reaching a maximum for = 1+! ! : Why printing cash at an extremely large rate, say ! +1; generates no seigniorage for the government/central bank? ii. Calling = 1+! ! > 1; show that is a strictly decreasing function of !; and provide an economic intuition for why this is the case.

In class we derived a model of aggregate demand and aggregate supply, made of the following IS : y0 = y r0 + "0 (6) PC : 0 = +y0 + u0 (7) Real Rate : r0 = r n 0 0 = ( 1) 0 (8) We then solved for the equilibrium levels of ináation and output (in period 0) for the cases of a) no shocks, "0 = u0 = 0; and b) a positive cost push shock, u0 > 0: Following the same steps on the slides, solve for the equilibrium levels of ináation and output - both analytically and graphically - for the case of a negative demand-side shock, "0 < 0:

Provide an economic explanation for your results, that is, explain why the new equilibrium with a negative demand-side shock is di§erent from the baseline equilibrium where shocks are absent. In particular, discuss why the overall impact on output is stronger when the PC is áatte.