Lecture 3: Dynamics of small open economies Open economy macroeconomics, Fall 2006 Ida Wolden Bache September 5, 2006
Dynamics of small open economies Required readings: OR chapter 2. 2.3 Supplementary reading: Obstfeld, M and K. Rogo (995): The intertemporal approach to the current account in Grossman and Rogo (eds): Handbook of International Economics, vol 3 (or NBER Working Paper No. 4893, 994). 2
An in nite-horizon small open economy model Production function Y s = A s F (K s ) Capital accumulation (no depreciation) K s+ = I s + K s Intertemporally additive preferences U t = X s t u(c s ) = u(c t ) + u(c t+ ) + 2 u(c t+2 ) + 3
Deriving the intertemporal budget constraint: Period by period budget constraint CA t = B t+ B t = Y t + rb t C t I t G t or ()B t = C t + I t + G t Y t + B t+ (*) Forward by one period and divide by B t+ = C t+ + I t+ + G t+ Y t+ and substitute back in equation (*) + B t+2 ()B t = C t + I t + G t Y t + C t+ + I t+ + G t+ Y t+ + B t+2 Repeat proceso eliminate B t+2 B t+2 = C t+2 + I t+2 + G t+2 Y t+2 4 + B t+3
()B t = C t + I t + G t Y t Intertemporal budget constraint + C t+ + I t+ + G t+ Y t+ + C t+2 + I t+2 + G t+2 Y t+2 () 2 + B t+3 () 2 X (C s + I s ) = ()B t + X (Y s G s ) where we have imposed the transversality condition T lim B t+t + = 0 T! 5
Optimisation problem max X u(()b s Y s I s G s B s+ ) First-order condition with respect to B s+ u 0 (C s ) + s+ t u 0 (C s+ )() = 0 u 0 (C s ) = ()u 0 (C s+ ) Note that only if = =() is a constant steady-state consumption path optimal If < =() consumption shrinks forever If > =() consumption grows forever 6
How to get constant steady-state consumption with 6= =()? Make discount factor a function of consumption ( 0 (C t ) < 0) Overlapping-generations (OLG) model 7
Consumption functions: some special cases. = =() =) C t = C t+ = C t+2 = = C s X C t = ()B t + X (Y s G s I s ) +r C t = ()B t + 2 X (Y s G s I s ) r X C t = 4()B t + (Y s G s I s ) 5 r C t = W t i.e.; optimal consumption C t equalhe annuity value of total discounted wealth net of government spending and investment 3 8
2. Isoelastic utility u(c) = C Euler equation u 0 (C) = C Ct = ()Ct+ C t+ = () C t C t+2 = () C t+ = ( () ) 2 C t : : : C s = ( () ) C t 9
Insert into intertemporal budget constraint X C s = ()B t + X (Y s G s I s ) X ( () ) C t = ()B t + X (Y s G s I s ) C t X () = (+r)bt + X C t = ()B t + P +r (Ys G s I s ) P () (Y s G s I s ) 0
Noting that if () < then X C t = r + # () 2 4()B t + X = = () r + () {z } # (Y s G s I s ) 3 5
A fundamental current account equation The permanent level of a variable X t ihe hypothetical constant level of the variable that hahe same present value ahe variable itself X X s f X t = fx t r X t X s s t X s 2
Assume = +r CA t = B t+ B t = Y t + rb t C t G t I t = Y t + rb t G t I t r 2 4()B t + X (Y s G s I s ) {z } C t = Y t + rb t rb t G t I t r + r X G s + r = Y e t Y t Gt G e t It I e t X X Economy runs a current account de cit (surplus) if output is below (above) its permanent level, government spending is above (below) its permanent level or investment is above (below) its permanent level s s t Y s t I s 3 5 3
What if 6= +r? Assume isoelastic preferences CA t = Y t + rb t C t G t I t = Y t + rb t G t I t r + # 2 4()B t + X (Y s G s I s ) {z } C t = Y t + rb t G t I t r 2 4()B t + 2 X X (Y s G s I s ) (Y s G s I s ) # 4()B t + 5 {z } W t = Y e t Y t Gt G e t It I e # t W t where # = () 4 3 5 3 5 3
Current account driven by two distinct motives: pure smoothing motive (as in the case = +r ) and a tilting motive If # < 0 (country is relatively patient, > =()), current account balance is increased If # > 0 (country is relatively impatient, < =(+r)), current account balance is reduced 5
An aside: debt sustainability or when is a country bankrupt? Rearrange the intertemporal budget constraint ()B t = = X X where T B s ihe trade balance (Y s C s G s I s ) s Suppose that the economy grows at rate g t T B s Y s+ = + g; g > 0 Y s and that B s+ = + g B s so that the debt-to-output ratio (B s =Y s ) is constant 6
The current account balance is CA s = B s+ B s = gb s = rb s + T B s which implies T B s Y s = (r g)b s Y s that is; to maintain a constant debt-to-gdp ratio the country needo pay only the di erence between the real interest rate and the growth rate of the economy 7
A stochastic current account model Future levels of output, investment and government spending are stochastic Only asset is riskless bond which pays a constant real interest rate r Replace assumption of perfect foresight with assumption of rational expectations (Muth, 96): agents expectations are equal to the mathematical conditional expectation based on the economic model and all available information about current economic variables Let z t denote the information available to the agent at time t and let X e t+ be the agent s subjective expectation of a variable X t+ made at time t Muth s rational expectations hypothesis (REH) then implies X e t+ = E [X t+jz t ] 8
De ne the rational expectations forecast error According to the REH then t+ = X t+ E [X t+ jz t ] E [ t+ jz t ] = 0 Note: t+ has zero expected value and is uncorrelated with any information available to the agents at time t In the absence of uncertainty REH correspondo the perfect foresight assumption. 9
Representative consumer maximises expected value of lifetime utility U t = E t 2 4 X u(c s ) 3 5 Intertemporal budget constraint with transversality condition imposed X (C s + I s ) = ()B t + X (Y s G s ) 20
Optimisation problem max E t 2 4 X u(()b s Y s I s G s + B s+ ) 3 5 First-order condition with respect to B s+ E t h u 0 (C s ) i = ()E t h u 0 (C s+ i ) for s = t u 0 (C t ) = ()E t h u 0 (C s+ ) i 2
The linear-quadratic permanent income model Assume that the period utility function is Impliehat marginal utility is linear u(c) = C a 0 2 C2 ; a 0 > 0 u 0 (C) = a 0 C Stochastic Euler equation assuming = =() a 0 C t = a 0 E t [C t+ ] C t = E t [C t+ ] 22
REH implies C t+ = E t [C t+ ] + " t+ and thus C t+ = C t + " t+ that is; consumption follows a random walk (Hall, 978) 23
Consumption function 2 X E t 4 C s 3 5 = E t 2 4()B t + From the Euler equation: E t C s = C t C t = r 2 4()B t + X X E t (Y s I s G s ) (Y s I s G s ) Certainty equivalence: people make decisions under uncertainty as if the future stochastic variables were sure to turn out equal to their conditional expectations (that is; uncertainty about future income has no impact on current consumption). 3 5 3 5 24
The linear quadratic model and the current account Consider an endowment economy without government spending Output follows a rst-order autoregressive (AR) process Y t Y = Y t Y + " t where 0 is a measure of the persistence of the process and E [" t ] = 0 and E [" t " s ] = 0 for s 6= t. Impulse response function (or dynamic multiplier) @Y s @" t = s t When = 0, a shock has purely transitory e ects on Y t When =, a shock has permanent e ect on Y t (unit root process) See Hamilton (994): Time Series Analysis, chapter for details. 25
By the law of iterated conditional expectations (the current expectation of a future expectation of a variable equalhe current expectation of the variable, e.g., E t [E t+ [" t+2 ]] = E t [" t+2 ]) E t h Ys Y i = Y t Y 26
Impulse response functions for first order AR process for different values of ρ ρ = 0 0.8 ρ = 0.5 ρ = 0.9 0.6 ρ = 0.99 ρ = 0.4 0.2 0 0 2 4 6 8 270 2 4 6 8 20
Insert into consumption function C t = = r r 2 4()B t + 2 = rb t + r + r 4()B t + 2 X 4 X X X s = rb t + Y + r t Y E t Y s 3 Y t Y 3 5 2 4 X 2 = rb t + Y + r 4 r = rb t + Y + +r Yt 3 5 Y t Y + Y 3 5 3 5 Y t Y Y 5 Y t Y 28
For < a one unit increase in output increases consumption by leshan one unit ( Keynesian consumption function). Substitute in for Y t Y = Y t Y + " t r C t = rb t + Y + Yt Y + " t r = rb t + Y + Yt Y r + " t Implications for the current account CA t = rb t + Y t C t = rb t + Y t rb t + Y + = Yt Y r Yt Y! 29
Substitute in for Y t Y = Y t Y + " t CA t = Yt Y + " t = Yt Y + " t An unexpected shock to output (" t > 0) causes a rise in the current account surplus as long ahe shock iemporary ( < 0) Graphs show the impulse responses of Y t, C t and CA t to a one unit shock to output in the linear quadratic model for di erent values of when B = 0; Y 0 = Y = 0; = 0:99; r = ( )=. 30
Impulse responseo one unit shock to output. ρ=0.9. ε t 0.8 Y t C t 0.6 CA t 0.4 0.2 0 0 5 0 5 320 25 30 35 40
Impulse responseo one unit output shock. ρ=0. 0.8 ε t Y t 0.6 C t CA t 0.4 0.2 0 32 2 3
Impulse responseo one unit output shock. ρ=. 0.8 ε t 0.6 C t =Y t 0.4 CA t 0.2 0 0 2 4 6 8 330 2 4 6 8 20
04.09.06 9:22 C:\Programfiler\MATLAB704\work\Lectures.m of clear all %Set parameters P = %Impulse response horizons beta = %Discount factor r = (-beta)/beta %Exogenous real interest rate YMEAN = % Mean output level rho = % Persistence of output process B() = % Initial level of debt %Shock process (one unit shock in period ) e = zeros(,p) e() %Output process (assuming Y(0)=YMEAN) Y() = YMEAN+e() for s = 2:P Y(s)=rho*(Y(s-)-YMEAN)+YMEAN+e(s) end %Solution for consumption, net foreign assets and current account balance for s = :P C(s) = r*b(s)+ymean+(r/(+r-rho))*(y(s)-ymean) %Consumption function B(s+)=(+r)*B(s)+Y(s)-C(s) % Period s budget 34 constraint CA(s) = B(s+)-B(s) % Definition of current account balance end
Does a positive output innovation always imply a current account surplus? NO! Assume that output growth is positively serially correlated Y t Y t = (Y t Y t 2 ) + " t Impulse responses @Y t @" t = ; @Y t+ @" t = + ; @Y t+2 @" t = + + 2 @Y s ; : : : ; lim = s! @" t Implication: Future output rises more than " t! permanent output uctuates more than current output! unexpected increase in output leado a larger increase in consumption and a current account de cit 35
Deaton s paradox: In the data, output growth is positively serially correlated, yet consumption does not respond more than proportionally to output changes. Homework on Deaton s paradox: Exercise 4 in OR ch.2 (answers will be provided next week) 36