Liquidity, Business Cycles, and Monetary Policy

Liquidity, Business Cycles, and Monetary Policy Nobuhiro Kiyotaki, John Moore Presenters: Pedram Nezafat, Ctirad Slavik

January 23, 2009

Liquidity, Business Cycles, and Monetary Policy

Motivation Build a canonical RBC model with essential money. To answer: 1

How does the economy fluctuate with shocks to productivity and liquidity?

2

What is the role of monetary policy?

Liquidity, Business Cycles, and Monetary Policy

Physical Environment Infinite horizon, discrete time 4 goods: nondurable general output labor equity (capital) fiat money

2 types of agents: unit measure of entrepreneurs unit measure of workers

Liquidity, Business Cycles, and Monetary Policy

Entrepreneurs All entrepreneurs have access to a production technology: yt = At ktγ lt1−γ yt , output good produced by entrepreneur At > 0, common stochastic productivity shock kt , capital good of the entrepreneur lt , labor hired by the entrepreneur

Liquidity, Business Cycles, and Monetary Policy

Entrepreneurs Fraction of π have access to a capital producing technology: produce new capital it , 1-1 from general good i.i.d. over time and agents law of motion for capital kt+1 = λkt + it Utility: E0

∞ X t=0

β t log ct

Liquidity, Business Cycles, and Monetary Policy

Borrowing Constraint can mortgage (borrow against) θ of the newly installed capital it θ exogenous arises endogenously in various models

Liquidity, Business Cycles, and Monetary Policy

Resalebility Constraint Entrepreneur can also hold: money mt equity of other entrepreneurs nt Resalebility Constraint: at t can sell at most φt nt φt stochastic, exogenous arises endogenously in Kiyotaki, Moore (2003)

Liquidity, Business Cycles, and Monetary Policy

Shocks Aggregate: At productivity shock φt liquidity shock At , φt follow a stationary Markov process around unconditional mean Idiosyncratic: investment opportunity with prob. π

Liquidity, Business Cycles, and Monetary Policy

Balance Sheet Assets own capital stock equity of others money

Liabilities own equity issued net worth

Note: all capital the same market price (no idiosyncratic shocks). Simplification: unmortgaged capital installed at t −→ can sell φ at t + 1. (depreciated) capital installed at t and nt+1 perfect substitutes at t + 1 capital installed at t ∼ it and nt NOT perfect substitutes at t

Liquidity, Business Cycles, and Monetary Policy

Constraints Summarized mt+1 ≥ 0 nt+1 ≥ (1 − θ)it + (1 − φt )λnt ct + it + qt nt+1 + pt mt+1 ≤ rt nt + qt it + qt λnt + pt mt nt+1 ≥ 0

Liquidity, Business Cycles, and Monetary Policy

Workers Utility max E0

∞ X t=0

· t

βU

c0t

ω − (l0 )1+η 1+η t

¸

Constraints: c0t + qt (n0t+1 − λn0t ) + pt (m0t+1 − m0t ) = wt lt0 + rt n0t n0t+1 ≥ 0 m0t+1 ≥ 0

Liquidity, Business Cycles, and Monetary Policy

Equilibrium Quantities for entrepreneurs (workers) {ct , it , kt+1 , nt+1 , mt+1 } ({c0t , lt0 , n0t+1 , m0t+1 }) and prices {pt , qt , wt } that solve entrepreneurs and workers maximization problems and markets clear.

Liquidity, Business Cycles, and Monetary Policy

Characterization labor supply:

wt ω

1 η

labor demand: Kt

h

(1−γ)At wt

i1 γ

Market clearing =⇒ η

γη

γ

wt = [(1 − γ)At ] η+γ Ktγ+η ω γ+η rt = at Ktα−1 ¶ 1−γ µ 1+η 1 − γ γ+η (At ) γ+η at = γ ω γ(1 + η) α = γ+η

Liquidity, Business Cycles, and Monetary Policy

Workers Problem Claim 0: (in a nbhd of the steady state) workers don’t hold money nor equity, i.e. c0t = wt lt0 . Argument for equity in no-money economy: in SS wt constant ⇒ lt0 constant rt + λ =

1 β

⇒ ct = ct+1 by EE

n00 = m00 = 0 ⇒ c = wl0 rt + λ <

1 β

⇒ borrow/dissave

Liquidity, Business Cycles, and Monetary Policy

Claim 1 Suppose that θ and φ satisfy (1 − λ)θ + πλφ > (1 − λ)(1 − π) Then: 1

the allocation of resource is the first best

2

pt = 0, qt = 1

3

1 + rt '

1 β

+1−λ

Liquidity, Business Cycles, and Monetary Policy

Claim 1 argument: assume qt = 1, pt = 0, solve entrepreneurs problem in SS w/o liquidity constraint use the fact that log utility + linear returns ⇒ consumption = (1 − β) of wealth verify LC doesn’t bind (⇐⇒ Assumption 1) use continuity to extend to nbhd of SS (problematic?)

Liquidity, Business Cycles, and Monetary Policy

Claim 1 ⇒ if all entrepreneurs have access to investment opportunities, π = 1, then money does not have any value, liquidity constraints do not affect the economy. Note: M P Kt + λ = β1 = = return on equity at t (NO investment opportunity at t + 1) = = return on equity at t (investment opportunity at t + 1).

Liquidity, Business Cycles, and Monetary Policy

Claim 2 Suppose that θ and φ satisfy Φ(θ, φ) > 0 where, Φ is a derived so that the liquidity constraints bind. A sufficient condition for inequality (1): (1 − λ)θ + πλφ < (β − λ)(1 − π)} A necessary condition for inequality (1): (1 − λ)θ + πλφ < (1 − λ)(1 − π) Then: 1 2

pt > 0, qt > 1 mit = 0, the money holding of an investing entrepreneur

(1)

Liquidity, Business Cycles, and Monetary Policy

The idea: assume pt > 0, qt > 1, solve entrepreneurs problem aggregate using market clearing (linear policies, distributions don’t matter) reduce to 4 equations with 4 unknowns (Kt+1 , It , pt , qt ) assume SS, show p > 0, q > 1 use continuity for nbhd of SS Note: qt > 1 ⇒ LC binding, mit = 0

Liquidity, Business Cycles, and Monetary Policy

Claim 3 restated In the neighborhood of the steady state monetary economy: 1 rt+1 + λqt+1 > Et > β qt R rt+1 + φt+1 λqt+1 + (1 − φt+1 )λqt+1 > Et qt Et (rt+1 + λ) >

Et

pt+1 pt

Implications: the stock of capital Kt+1 is less than the first-best economy, workers don’t save.

Liquidity, Business Cycles, and Monetary Policy

Dynamics - Deterministic Productivity Shift

No liquidity shock and the aggregate productivity fluctuates between high and low deterministically every period.

Liquidity, Business Cycles, and Monetary Policy

Dynamics - Deterministic Productivity Shift

Liquidity, Business Cycles, and Monetary Policy

Dynamics - Two State Markov Liquidity Shock

No aggregate productivity shock and a two state Markov liquidity shock.

Liquidity, Business Cycles, and Monetary Policy

Dynamics - Two State Markov Liquidity Shock

Liquidity, Business Cycles, and Monetary Policy

Government How does the equilibrium change in response to exogenous government policies Government can hold equity but cannot produce new capital Government can buy and sell equity by issuing money

Liquidity, Business Cycles, and Monetary Policy

Government OMO against Productivity Shifts

Liquidity, Business Cycles, and Monetary Policy

Government OMO against Liquidity Shifts

Liquidity, Business Cycles, and Monetary Policy

Summary RBC model with money to analyze: 1

How does the economy fluctuate with shocks to productivity and liquidity?

2

What is the role of monetary policy? What can the government do to smooth consumption?

Liquidity, Business Cycles, and Monetary Policy

Our project Calibrate to US economy in RBC fashion. Optimal government response to a drop in φ. Problems to be addressed: rate of return on equity less than rate of time preference, non-standard utility functions.

Liquidity, Business Cycles, and Monetary Policy

Jan 23, 2009 - Liquidity, Business Cycles, and Monetary Policy. Liquidity, Business .... Assumption 1) use continuity to extend to nbhd of SS (problematic?)
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