International Monetary Theory: Mundell Fleming Redux

International Monetary Theory: Mundell Fleming Redux by Markus K. Brunnermeier and Yuliy Sannikov Princeton and Stanford University Princeton Initiative Princeton, Sept. 9 th, 2017

Motivation Global currency spillovers Flight to safety - Dollar appreciation when risk rises Local currency: good store of value/hedge for idiosyncratic risk Global currency: good hedge for international competitiveness risk When to peg to world currency? When to dollarize? MoPo space: Nuanced Mundell-Fleming Trilemma Local and global money have different risk profile (imperfect substitutes) increases MoPo space Too high inflation: local citizens substitute local currency for global currency limits MoPo space Reserve currency management Irrelevance theorem

Modelling Framework Closed economy Open Economy Static Hick s IS-LM Mundell-Fleming Dynamic Impulse response New Keynesian Obstfeld-Rogoff Risk & Dynamic financial frictions Samuelson Bewley I Theory of Diamond Aiyagari Money X

Modelling Framework \Friction OLG Incomplete Markets + idiosyncratic risk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond Aiyagari f k = r, Dynamic inefficiency r < r, K > K Inefficiency r < r, K > K Basic I Theory Pecuniary externality Inefficiency r > r, K < K

Frictions Incomplete markets Within country only w.r.t. idiosyncratic risk d Z t i (other risks can be shared within national economy) Across countries Only global money can be traded Money is a bubble Like in Samuelson, Bewley Price are fully flexible

International setting Small Economy Large Economy* Local currency Store of value Hedge against idiosyncratic risk Consumption basket Non-tradable local good tradable good 1 tradable good 2 Exchange rate ak t b 1,t K t Global currency* $ Hedge for SOE s citizens against international competitive risk Consumption basket* Non-tradable good* a K t tradable global good 1 b 1,t K t tradable global good 2 b 2,t K t

International setting Small Economy Large Economy* Local currency Store of value Hedge against idiosyncratic risk Consumption basket Non-tradable local good tradable good 1 tradable good 2 Exchange rate Global currency* $ Hedge for SOE s citizens against international competitive risk Consumption basket* Non-tradable good* a ak t K t tradable global good 1 b b 1,t K t 1,t K t b tradable global good 2 b 2,t K t b 2,t K 2,t t < b 2,t b 1,t b 1,t

Intuition Purchase good 2 in exchange of good 1 (depends on ToT) Hold global money as Net Foreign Asset Position Value of money money is safe asset Local money is store of value with nice hedge against idiosyncratic risk Global money ($) hedges better export risk (competitiveness = ToT + productivity) 2 money can coexist (even though both are bubbles ) Different return-risk profile

Overview Large country Portfolio choice between Physical capital k t US Dollar, $ bubble in positive net supply No state variable: due to scale invariance Small country Portfolio choice between Physical capital k t Peso hedge against idiosyncratic shocks US Dollar, $ hedge against ToT + export productivity shocks State variable ν t : Accumulation dynamics of foreign asset position (in $) μ t x = μ x ν t, σ t x = σ x (ν t )

Numeraire is nontradable local good Large country E න e ρt log c t dt 0 Consumption Cobb-Douglas preferences (over 1 non-tradable one 2 tradable goods) (1 α) c 0,t (c1,t ) β (c 2,t ) (1 β) Investment rate ι in terms of non-tradable local good Evolution of physical capital stock dk t k = Φ ι δ dt t Output shocks per unit of capital Determines relative prices has to be indifferent Idiosyncratic real cash flow shocks Net worth dynamics: dn t α a, b 1,t, or b 2,t Ito-processes σk t d Z t i n = θ t r M dt + (1 θ t )r K dt c t t n dt + σ k t t n d Z i t + τ t t n dt t Value of output of all goods produced a K t

Non-tradeable to consumption basket Tradeable, non-tradeable good 1 and 2 a units of non-tradeable good buy b 1,t of tradeable good, or a 1 α b 1,t αβ b2,t α(1 β) units of the aggregate good (consumption basket). Hence, production of consumption basket is a ι t b α t K t, with b t = b 1,t β b2,t a 1 β

Return on global money ($) In terms of non-tradable local good (as numeraire) (which is used for investment rate ι t ) r dt = Φ ι δ dt is risk free Change of numeraire In terms of tradable basket (change of numeraire) r t G r dt + db t b t Where price of non-tradable good in terms of tradable basket b t = b 1,t β b2,t a 1 β Special case: b 1,t β b2,t 1 β and hence bt is constant

Solving 1. Postulate Price processes dp t /p t = μ p t dt + σ p dz t, dq t /q t = Portfolio processes dθ t /θ t 2. Derive return processes dr K = Φ ι dr M = Φ ι δ dt + a ι dt + σ d Z q q t δ dt (μ M μ Mi )dt money supply growth rate that is NOT distributed via interest payment Set μ Mi = 0 3. Optimality conditions & Market clearing conditions 4. Solve undetermined coefficients (μ x s t, σ x (s t )) Solving ODE with boundary conditions For large country: simply solve for constants

Solving 1. Postulate Price processes Portfolio processes p t, q t θ t Simply constants for large country 2. Derive return processes dr K = Φ ι dr M = Φ ι δ dt + a ι dt + σ d Z q q t δ dt (μ M μ Mi )dt money supply growth rate that is NOT distributed via interest payment Set μ Mi = 0 3. Optimality conditions & Market clearing conditions 4. Solve undetermined coefficients (μ x s t, σ x (s t )) Solving ODE with boundary conditions For large country: simply solve for constants

Optimality (=) & market clearing (=) Investment rate, ι Tobin s q: Φ ι = 1 q (static problem) For Φ ι = 1 κ log(κι + 1) κι = q 1 Portfolio choice, θ E dr K dr M /dt = Cov[dr K dr M, 1 θ = E drk dr M /dt (σ /q ) 2 Dividend yield on capital must be ρ Consumption dn t ดn ] = (1 θ )( σ /q)2 t dr M +(1 θ ) dr K dr M Output market clearing Demand ρn t = ρ q + p K t = a ι K t = (a ι )/q +μ M (σ /q ) 2 = q q +p Capital market clearing Supply q = q q + p =1 θ (a ι )/ρ

Equilibrium Moneyless equilibrium p 0 = 0 q 0 = σ ρ+ μ M > Money equilibrium p = σ (1+κρ) ρ+ μ M (1+κa ) q = 1+κa κρσ ρ+ μ M where μƹ M = 1 θ portfolio share μ M (monotone transformation) Numeraire is local good

Optimal Monetary Policy Money growth μ affects inflation in two ways π = μ M (Φ ι μ M μ M,i δ) MoPo can correct pecuniary externality Citizens take real interest rate as given when choosing their portfolio between money & physical capital Money exists for σ > ρ Money growth > 0 is optimal for σ > 2 ρ (for κ = 0) g boosts growth like in Tobin (1965) MoPo improves insurance provided by safe asset Constrained optimal! Incentive compatible Money is neither neutral nor super-neutral (no price stickiness)

Small country Small country cannot produce tradable good 2 tradable basket can be traded for global good 1 at rate b t ดb 1,t Productivity b 2,t /b 1,t 1 β Terms of Trade = b 1,t b 1,t ( b 1,t Short-cut thinking: one unit of capital produces b t units of tradable basket (while actually it produces only good 1 at rate b 1 and trades some of them for tradable good 2) β b2,t 1 β ) db t b t = μ b dt + σ b dz t since all b 1,t, b 2,t, b 1,t Return on global money can be written as are (correlated) geometric Brownian. Ito product rule: d X t Y t = dx t Y t + X t dy t + σ X σ Y dt call prefer dr t G = μ G dt + σ G dz t + σ G, dz t Part of b t which is orthogonal to b t

Small country Same preferences: E 0 e ρt log c t dt c 0,t (1 α) β 1 β c 1,tc2,t α t K t devoted to produce tradable good 1 Can be traded for tradable basket since small county can t produce tradable good 2 itself ξ t b t K t ( α t ξ t )b t K t G t > 0 α consumption of tradable goods basket trade-imbalance (net export) Net foreign asset position (only global money) (in tradable goods basket) dg t G t = dr G t + ( α t ξ t )b t K t dt G t

State variable Equilibrium is a map Histories of shocks prices q t, p t, allocation Z τ, Z τ, 0 τ t α t, ι t, ξ t & portfolio (1 θ t ζ t, θ t, ζ t ) net foreign asset position to tradable production potential Evolution ν t = G t b t K t

Portfolio choice & Asset pricing Portfolio share (processes) Local money Global money dθ t θ t =μ t θ dt+σ t θ dz t +σ t θ, dzt dζ t ζ t = μ t ζ dt + σt ζ dzt + σ t ζ, dzt Returns expressed with country networth, N t, as numeraire Return on individual networth dr n t = ρdt + (1 θ t ζ t ) σ(q t ) σ n Return on local money Return on global money ($) Asset pricing equation (due to log utility) E dr t n dr t MG dr ML t = dθ t θ t dr MG t = α t ξ t ν t dt + dζ t ζ t = Cov[dr t n dr t MG, dr t n ] ρ α t ξ t ν t μ t ζ = σ n 2 E dr t n dr t ML = Cov[dr t n dr t ML, dr t n ] ρ μ t θ = σ n 2

Portfolio choice & Asset pricing Portfolio share (processes) Local money Global money dθ t θ t =μ t θ dt+σ t θ dz t +σ t θ, dzt dζ t ζ t = μ t ζ dt + σt ζ dzt + σ t ζ, dzt Returns expressed with country net worth N t as numeraire Return on individual net worth dr t n = ρdt + (1 θ t ζ t ) σ n σ(q t ) Return on local money Return on global money ($) Asset pricing equation (due to log utility) E dr t n dr t MG dr ML t = dθ t θ t dr MG t = ξ t α t ν t Money worth θ net worths dt + dζ t ζ t = Cov[dr t n dr t MG, dr t n ] ρ α t ξ t ν t μ t ζ = σ n 2 E dr t n dr t ML = Cov[dr t n dr t ML, dr t n ] ρ μ t θ = σ n 2

Portfolio choice & Asset pricing Portfolio share (processes) Local money Global money dθ t θ t =μ t θ dt+σ t θ dz t +σ t θ, dzt dζ t ζ t = μ t ζ dt + σt ζ dzt + σ t ζ, dzt Returns expressed with country net worth N t as numeraire Return on individual net worth Return on local money Return on global money ($) Asset pricing equation (due to log utility) dr t n = ρdt + 1 θ t ζ t σ n dr t ML = dθ t θ t dr t MG = ξ t α t ν t dt + dζ t ζ t σ q t d Z t E dr t n dr t ML = Cov[dr t n dr t ML, dr t n ] ρ μ t θ = σ n 2 E dr t n dr t MG E[net worth money return] = Cov[dr t n dr t MG, dr t n ] ρ ξ t α t ν t μ t ζ = σ n 2 Price of risk risk

Consumption & Investment Consumption Demand ρ ณ G t ζ t wealth in trad.basket = ξ t b t K t α Supply Cobb-Douglas constant consumption expenditure shares Consumption of tradables Production allocation ξ t b t K t α P g t = [ 1 α t a ι t ] l P 1 α t Output of non-tradable Nontradable Tradeable (incl. net exports) Investment rate ι t Depends on q t

(Co-)Existence of Money Proposition 1: If σ 2 > ρ and M Φ ι ν=0, then local money has value and ν = 0 (no NFAP) is absorbing state Otherwise, if σ 2 ρ + M Φ ι ν=0 > 0, then global money has value for citizens in small country (and local money may or may not have value) M Φ ι = μ G μ b + σ b σ G σ b Φ ι + δ attractiveness of global money GLOBAL MONEY + possibly LOCAL MONEY ρ ONLY LOCAL MONEY in the long run σ 2 attractiveness of local money

Numerical Example ρ = 5%, σ =.3, α =.2, μ b = 1%, σ b =.15, μ G = 2.2%, a =.13, δ = 2%, σ G = σ G, = 0, κ = 2 M =.0545, σ ν =.15

Exchange rate dynamics - UIP i t i t = E t ΔE + ψ t ψ t = ถ σ θ (σ ζ σ θ ) >0 >0 (risk premium in terms of Peso) For i t = i t (= 0) foreign currency is expected to appreciate relative to local currency (whenever it is held in positive quantity). Local currency is a hedge, it appreciates relative to net worth when ν drops. Global currency is risky, so to be held in positive amount it must earn a risk premium UIP violation, ψ t, depends whether money is printed to pay interest μ Mi No real changes (portfolio choice is not affected) Higher inflation π = μ Mi (φ ι δ), E t ΔE = μ Mi (dollar appreciates) to generate seignorage (redistributed wealth share) (μ M μ Mi ) Affects portfolio choice, q, investment rate ι, growth rate risk premium ψ t

Flight to safety (into dollar) Unanticipated increase in σ b E.g. ToT becomes more volatile Portfolio share held in dollars increases Dollar valuation is higher (increase in volatility of ν) Transition Start with current (dollar holding) G Recalculate new state variable ν t Our full dynamics also includes transition dynamics

Spillover from lower μ G Higher money supply growth μ M in large country Lower growth Φ ι δ in large country Loss of competitive edge in global tradable basket

Higher Peso inflation π t seniorage μ M μ Mi Store of value is less attractive Pricing equation now ρ + π t μ θ = σ N 2 is distributed capital holding higher investment ι t boosts growth, but higher idio risk

Mundell-Fleming Trilemma Trilemma: Can only pick a 2 desiderata out of 3 1 side Autonomous Monetary Policy Fixed exchange rate Free Capital Flow Dilemma : Pick only 1

Floating Exchange Rate With floating exchange rate & open capital account Still range of Monetary policy, since local and global money are imperfect substitutes Inflation boosts growth, but only possible up to a limit തπ(μ G ). Beyond തπ(μ G ) monetary policy has little bite Global money becomes too attractive Range is higher with higher inflation in large country (global money) Large country s MoPo determines range for small country Policy range is larger if local money is backed by taxes (തπ depends on distribution of seignorage)

Closed capital account Range of Monetary policy is much larger, up ധπ = σ 2 ρ 4% (physical capital is risky store of value) Total money holding is larger with closed capital account Global money would be a better hedge for export risk

Fixed exchange rate regimes & no MoPo Dollarization = (fully backed) Currency Board Xx Exchange rate peg Requires strong fiscal backing (since no backing through holding of global reserves) After a string of adverse shocks, government must tax and use to proceeds to remove some of the local currency in circulation

Foreign Currency Reserves Irrelevance Theorem: If central bank holds global money (reserves) Citizens in small country will hold accordingly less Remark: If central banks holds more $-reserves than citizens would like to hold, then agents borrow foreign currency from abroad. If local money is worthless (without foreign reserves), then the value of local money with reserves only derives from the latter (currency board) With fiscal backing of the local money, complicates analysis

Optimal Monetary Policy LARGE COUNTRY MoPo can correct pecuniary externality Citizens take real interest rate as given when choosing their portfolio between money & physical capital Money exists for σ > ρ Inflation is optimal for σ > 2 ρ MoPo improves insurance provided by safe asset Constrained optimal! Incentive compatible (For κ = 0, no adjustment costs) SMALL COUNTRY Additional savings decision due open capital account Generally, optimal monetary policy depends on control social planner has

Conclusion Endogenous value of money (safe asset) in 2 countries Local currency: better hedge for idiosyncratic risk (non-tradable consumption) Global currency: hedge against ToT + export productivity shocks Spillover effects from US monetary policy Flight to safety When to peg? When to dollarize? Nuanced Mundell-Fleming Trilemma Local and global money have different risk profile (imperfect substitutes) increases MoPo space Too high Peso inflation: local citizens substitute local currency for global currency limits MoPo space Central Bank s foreign reserves holding: Irrelevance Result Optimal Monetary Policy Idiosyncratic risk correct pecuniary externality (real interest rate) International savings