Optimal Monetary Policy Lecture 3 Winter School 2019 Delhi School of Economics

Optimal Monetary Policy Lecture 3

Winter School 2019 Delhi School of Economics

V.V. Chari

University of Minnesota Federal Reserve Bank of Minneapolis

December 2019

Outline

Develop cash-credit goods model

Provide sufficient conditions for Friedman rule to be optimal

Setting up the Cash-Credit Goods Model

Resource constraint

c1t+c2t+gt ≤lt

Preferences

∞

X

t=0

β^tu(c1t,c2t,lt)

Timing

At

PeriodtStarts

At=Mt+Bt

SM Opens

{nt,c1t,c2t} Shop/Work/Cash

{RtBt,ptnt,Tt} Returns and Pay Credit

At+1

Periodt+ 1

Markets and Constraints

At =Mt+Bt. (1)

p_tc_1t≤M_t (2)

A_t+1+p_tc_1t+p_tc_2t =M_t+p_t(1−τ_t)n_t+R_tB_t (4) Mt+1+Bt+1+ptc1t+ptc2t =Mt+pt(1−τt)nt+RtBt (4^∗)

Bt

p_t =bt ≥ −B (5)

RtB_t^s+ptτtnt=B_t+1^s + (M_t+1^s −M_t^s). (GBC)

Competitive Equilibrium

Definition: A competitive equilibrium is defined by the allocations

{(c1t,c2t,lt)}^∞_t=0, quantities{(Mt,Bt)}^∞_t=0 and prices{(pt,Rt)}^∞_t=0 such that households maximize (3) subject to constraints (1), (2), (4^∗), (5) and the initial asset conditions

markets clear:

1 c_1t+c_2t+g_t =n_t ∀t

2 R_tB_t^s+p_tτ_tn_t =B_t+1^s + (M_t+1^s −M_t^s) ∀t

3 M_t^s =M_t ∧ B_t^s =B_t ∀t.

Implementability Constraint

Present value budget constraint for households

∞

X

t=0

q_t

c_1t+c_2t−(1−τ_t)n_t+ (R_t−1)Mt

P_t

=q₀A0R0

P₀

Implementability constraint

∞

X

t=0

β^t[c1tu1t+c2t+ntunt] = u_c0A₀ P0

u_1t u2t

≥1 Initial wealth restriction

u_c0A₀ p0

≥ W₀

Ramsey Problem

max

∞

X

t=0

β^tu(c_1t,c_2t,n_t)

subject to IC, nonnegativity of interest rate, RC, and initial wealth restriction

Optimality of Friedman Rule

Proposition: Supposeu(c1,c2,n) =u(G(c1,c2),n) andG is homogeneous of degree 1. Then optimal policy rule is to have the Friedman rule,Rt = 1.

Proof sketch: Logic same as uniform commodity taxation. See lecture 1.

Extension to Uncertainty

Allocation nowc1(s^t),c2(s^t),n(s^t),M(s^t),B(s^t).

Resource constraint

c₁(s^t) +c₂(s^t) +g(s^t)≤A(s_t)n(s^t) Preferences

∞

X

t=0

X

s^t

β^tµ(s^t)u c1(s^t),c2(s^t),n(s^t)

Extension to Uncertainty

Prices: q(s^t),w(s^t),R(s^t),P(s^t).

Policies: τ(s^t),M(s^t),B(s^t).

Budget constraint

∞

X

t=0

X

s^t

q(s^t)

c1(s^t) +c2(s^t)−(1−τ(s^t))n(s^t) + (R(s^t)−1)M(s^t) P(s^t)

=q0(s0)R₋₁A0

P0(s0) Implementability constraint

∞

X

t=0

X

s^t

β^tµ(s^t)

c₁(s^t)u₁(s^t) +c₂(s^t)u₂(s^t) +n(s^t)u_n(s^t)

= R₋₁A0

P₀(s₀)u_c(s₀)

Ramsey Problem

Maximize utility subject to IC,

u1(s^t) u₂(s^t) ≥1 and resource constraint.

Result: ifuis homothetic in consumption goods and separable, then R(s^t) = 1 is still optimal.

Here money growth need not be to insure deflation

Still optimal to satiate the economy by setting opportunity cost of holding money to zero

Extension to Other Monetary Models

Money-in-the-utility function, shopping time mode of money

Similar assumptions about homotheticity and separability give similar results.