Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2
1 Consumption with many periods 1.1 Finite horizon of T Optimization problem maximize U t = u (c t ) + β (c t+1 ) + β 2 u (c t+2 ) +... + β T u ( c t+t ) β < 1
subject to T + 1 flow budget constraints CA t = B t+1 B t = Y t + rb t C t G t I t CA t+1 = B t+2 B t+1 = Y t+b t+1 C t+1 G t+1 I t+1... CA t+t = B t+t +1 B t+t = Y t+t + rb t+t C t+t G t+t I t+t Present value budget constraint Add all current accounts, but retain individual bond terms from righthand side (must be substituted out)
Alternatively, subtract net foreign income to get trade balance B t+1 () B t = T B t = Y t C t G t I t B t+2 () B t+1 = T B t+1 = Y t+1 C t+1 G t+1 I t+1... B t+t +1 () B t+t = T B t+t = Y t+t C t+t G t+t I t+t Multiply second line by ( ) ( ) 1 1+r, third by 1 2 1+r, ending with multiplying final by ( ) 1 T 1+r to take present values Sum present values of trade surpluses, noting that all terms with
bonds except first and last drop out ( 1 Impose ) T B t+t +1 () B t = t+t (Y s C s G s I s ) ( ) 1 s t B t+t +1 = 0 since no one would hold assets going into the end of the world, and no other country would let this country hold debt (because they would be holding assets) t+t ( 1 ) s t t+t ( 1 (C s + G s + I s ) = () B t + Y s PV expenditure = net foreign assets + PV income ) s t
Write utility substituting flow budget constraint for consumption C t = Y t + rb t G t I t (B t+1 B t ) U t = t+t β s t u {() B s B s+1 + A s F (K s ) (K s+1 K s ) G s } First order conditions (Euler equations) Bonds u (C s ) u (C s+1 ) = β () Capital u (C s ) u (C s+1 ) = β [ 1 + A s+1 F (K s+1 ) ]
Together the two Euler equations imply () = 1 + A s+1 F (K s+1 )
1.2 Infinite Horizon Intertemporal budget constraint changes sum to infinity instead of to t + T = lim T ( 1 ) T B t+t +1 () B t (Y s C s G s I s ) ( ) 1 s t impose ( ) 1 T lim B t+t +1 = 0 T requiring that debt not grow faster than the interest rate
combines No Ponzi Game Condition (NPG) requiring that presentvalue assets be non-negative in the limit with optimality condition whereby agents would not choose to forego consumption so that present-value assets could be positive in the limit ( ) 1 s t ( ) 1 s t (C s + G s + I s ) = () B t + Y s PV expenditure = net foreign assets + PV income intertemporal budget constraint implies an upper bound on current debt since smallest present value of consumption is zero (Y s G s I s ) ( 1 ) s t () B t
requiring that debt [ () B t ] be less than the present value of net income (max country could repay if consumption were zero) Financial crisis Either government or households want to borrow more than the upper bound on debt Agents refuse to lend because they know the country cannot repay Measure of debt burden Since debt/gdp ( B s Y s ) cannot grow forever, fix it, requiring B s+1 = B s (1 + g), where g is rate of growth of output
From country flow budget constraint B s+1 B s = gb s = Y s + rb s G s I s C s = rb s + T B s gb s = rb s + T B s (g r) B s = T B s yielding trade balance surplus necessary to keep debt/gdp fixed Assume have debt so B s < 0. Either reduction in g or increase in r implies must run larger trade balance surplus to keep debt/gdp from growing
Consumption Assumptions Infinite horizon, requiring infinite-horizon intertemporal budget constraint β () = 1, requiring constant consumption from Euler equation Solve for constant consumption from intertemporal budget constraint (C s + G s + I s ) ( 1 ) s t = () B t + ( ) 1 s t Y s
( C r ) = () B t + C = r () B t + (Y s G s I s ) (Y s G s I s ) ( ) 1 s t ( 1 ) s t agent consumes the annuitized value of net wealth Benchmark consumption G = I = 0 Y = Ȳ C = rb t + Ȳ
Consumption which varies over time (β () 1) CRRA utility Euler equation simplifies to Taking this forward u (C) = C1 1/σ 1 1/σ C s+1 = C s β σ () σ C s+2 = C s+1 β σ () σ = C s [β σ () σ ] 2
Consumption term for present-value budget constraint Define ( ) 1 s t C s = C t + C t β σ () σ 1 [ + C t β σ () σ 1] 2 +... 1 = C t 1 β σ () σ 1 for β σ () σ 1 < 1 Substituting ν = 1 β σ () σ 1 β σ () σ 1 = ν + r = ν + r ( ) 1 s t C s = C t ν + r
Solving for consumption using IBC yields C t = ν + r () B t + (Y s G s I s ) ( 1 ) s t β () = 1, ν = 0, consumption path is flat (benchmark case) β () > 1, ν < 0, agent is patient, and consumption begins smaller than benchmark and grows forever (eat less than annuitized value and bonds increase) β () < 1, ν > 0, agent is impatient, and consumption begins larger than benchmark and shrinks forever (eat more than annuitized value and bonds decrease)
2 Current Account with Infinite Horizon 2.1 Perfect Foresight Useful representation when β () = 1 Define permanent value of a variable as ( 1 ) s t X t = ( 1 ) s t X s X t = r ( 1 ) s t X s the annuity value of the variable at the prevailing interest rate
Consumption C t = rb t + Ỹ t G t Ĩ t Current account CA t = rb t + Y t C t I t G t = Y t Ỹ t ( I t Ĩ t ) ( Gt G t ) Current account imbalance occurs when variables deviate from permanent values Temporary increase in income implies a surplus as agents smooth consumption Temporary increase in spending implies a deficit as agents smooth consumption
Consumption tilting when β () 1 Define W t = () B t + (Y s G s I s ) ( ) 1 s t Consumption becomes C t = ν + r W t = rb t + Ỹ t G t Ĩ t + ν W t Current account becomes Recall CA t = Y t Ỹ t ( I t Ĩ t ) ( Gt G t ) ν = 1 β σ () σ ν W t
For impatient (patient) agents ν > 0, (ν < 0), and have current account deficit (surplus) even if variables take on permanent values
2.2 Stochastic Current Account Consumers maximize expected utility U t = E t β s t u (C s ) subject to actual intertemporal budget constraint (C s + I s ) ( 1 ) s t = () B t + (Y s G s ) Budget constraint must be obeyed with probability one Typically implies an endogenous upper bound on debt ( 1 ) s t
Euler equation u (C t ) = () βe t u (C t+1 ) Quadratic utility u (C) = C a 0 2 C2 u (C) = 1 a 0 C u (C) = a 0 u (C) = 0 Substitute into the Euler equation 1 a 0 C t = () βe t (1 a 0 C t+1 )
When () β = 1, consumption is a random walk C t = E t C t+1 Take expected value of budget constraint and solve for consumption, C t = r () B t + E t (Y s G s I s ) ( 1 ) s t replacing perfect foresight of future net income with expectation of future net income Certainty equivalence: agents make decisions under uncertainty by acting as if the future stochastic variables were equal to their means
Problems with quadratic utility 1. Does not rule out C becoming so large that u (C) < 0 2. Does not rule out negative consumption 3. Ignores upper bound on debt
Consumption and the current account with quadratic utility, endowment economy, G s = I s = 0, and stationary output Stationary output has constant mean and variance Y t Ȳ = ρ ( Y t 1 Ȳ ) + ɛ t E t 1 ɛ t = 0; 0 ρ < 1 Expected future output shows that effect of disturbance on output falls over time E t ( Yt+1 Ȳ ) = ρ ( Y t Ȳ ) = ρ 2 ( Y t 1 Ȳ ) + ρɛ t 1 Disturbance does not affect expected value of long-run output lim s E ( t Yt+s Ȳ ) = lim ( s ρs Y t Ȳ ) = 0
Expected present value of output (Y t ) becomes Ȳ = Ȳ Consumption ( r ( r ) ) + ( Y t Ȳ ) ( 1 + ρ + + ( Y t Ȳ ) ρ ( ρ ) 2 +... ) E t C s = C t = rb t + Ȳ + ( Y t Ȳ ) r ρ = rb t + Ȳ + [ ρ ( Y t 1 Ȳ ) + ɛ t ] r ρ Increase in ɛ t raises consumption permanently due to random walk Output reverts back to Ȳ at rate ρ Increase in output creates increase in savings so B t rises to offset
fall in Y t MP C = r 1+r ρ < 1 Current account CA t = rb t + Y t C t = Y t Ȳ [ ρ ( Y t 1 Ȳ ) ] r + ɛ t ρ = [ ρ ( Y t 1 Ȳ ) ] ( ) r + ɛ t 1 ρ = [ ρ ( Y t 1 Ȳ ) ] ( ) 1 ρ + ɛ t ρ Current account is independent of B t since wealth affects CA and C identically
Output shock has a transitory effect on the current account, which disappears once output has returned to its mean
Let output growth be stationary, giving output a unit root Y t Y t 1 = ρ (Y t 1 Y t 2 ) + ɛ t 0 < ρ < 1 E t 1 ɛ t = 0 To simplify, set Y t 1 Y t 2 = 0, yielding Y t = Y t 1 + ɛ t Now, take time t expectations of future output E t Y t+1 = Y t + ρ (Y t Y t 1 ) + E t ɛ t+1 = Y t 1 + ɛ t + ρɛ t = Y t 1 + (1 + ρ) ɛ t E t Y t+2 = E t Y t+1 + ρe t (Y t+1 Y t ) = Y t 1 + (1 + ρ) ɛ t + ρ 2 ɛ t = Y t 1 + ( 1 + ρ + ρ 2) ɛ t
Effect of current output shocks to future output grows over time If a shock raises output today, expect it to raise future output even more Consumption smoothing implies that consumption increases more than output creating CA deficit Note, additionally that E t Y t+j = ρ j ɛ t
Expression for the current account Current account is the difference between income and permanent income Recall permanent income is CA t = Y t Ỹ t Ỹ t = r E t j=0 Y t+j () j, CA t = Y t r E t j=0 Y t+j () j.
Recognize that Y t+1 = Y t + Y t+1 Y t+2 = Y t+1 + Y t+2 = Y t + Y t+1 + Y t+2 Therefore, the sum we need can be written as yielding j=0 Y t ( ) Y t r We can write () j + j=1 + Y t+1 ( 1 r Y t+1 () j + j=2 Ỹ t = Y t + E t j=1 ) + Y t+2 ( Y t+j () j, Y t+2 () j +..., 1 r () ) +...
Substituting, the current account becomes CA t = E t j=1 Y t+j () j = j=1 ( ρ ) j ɛt = ρ ɛ t
3 Current Account with Production and Uncertainty 3.1 Investment Optimization problem for household = E t max U t B s+1,k s+1 β s t u [() B s B s+1 + A s F (K s ) (K s+1 K s ) G s ]
First order condition on bonds yields bond Euler equation with uncertainty U { t = E t u (C t ) ( 1) + βu (C t+1 ) [] } = 0 B t+1 u (C t ) = E t { βu (C t+1 ) [] } 1 = E t { βu (C t+1 ) u (C t ) [] } First order condition on captial yields capital Euler equation with uncertainty U { t = E t u (C t ) ( 1) + βu (C t+1 ) [ A t+1 F (K t+1 ) + 1 ]} = 0 K t+1 u (C t ) = E t { βu (C t+1 ) [ A t+1 F (K t+1 ) + 1 ]}
1 = E t { βu (C t+1 ) u (C t ) [ At+1 F (K t+1 ) + 1 ]} Take expectation using E (xy) = E (x) E (y) + cov (x, y) 1 = 1 E t [ At+1 F (K t+1 ) + 1 ] ( βu (C +cov t+1 ) u ; (C t ) [ At+1 F (K t+1 ) ]) 1 cov ( βu (C t+1 ) u ; (C t ) [ At+1 F (K t+1 ) ]) = E t [ At+1 F (K t+1 ) ] + 1 ( r βu cov (C t+1 ) u ; (C t ) [ At+1 F (K t+1 ) ]) = E t [ At+1 F (K t+1 ) ] Expected future marginal product of capital equals interest rate adjusted by a covariance term Differs from certainty equivalence (CEQ) by this covariance term
Sign of covariance term is likely negative because when A t+1 is high, the marginal productivity of capital is high, Y t+1 and C t+1 are high, making u (C t+1 ) low. Negative covariance adds a risk premium to capital because capital returns are high when consumption is already high
Effect of productivity shock on current account with investment and stationary productivity Assume A t+1 Ā = ρ ( A t Ā ) + ɛ t+1 = ρ ( A t 1 Ā + ɛ t ) + ɛt+1 Taking expectations E t ( At+1 Ā ) = ρ ( A t 1 Ā + ɛ t ) implying that a current increase in productivity increases expected future productivity Current account response depends on response of consumption and investment CA t = A t F (K t ) C t I t
Consumption increases less than output because output is temporarily high raising the current account Investment also increases since expected future productivity is high, reducing the current account Empirically current account is countercyclical, implying that the investment response must be large enough to reverse the response predicted by consumption alone
3.2 Precautionary savings Bond Euler equation with uncertainty 1 = () βe t { u (C t+1 ) u (C t ) To have a precautionary motive, the third derivative of utility must be positive, implying that marginal utility is a convex function of consumption u > 0, u < 0, u > 0 }
Illustrate with log utility u = 1 C > 0 u = 1 C 2 < 0, u = 1 C 4 > 0 Euler equation with log utility 1 = () βe t { Ct C t+1 } { } 1 = () βc t E t C t+1 In the absence of uncertainty () β = 1, yields constant consumption over time However, with uncertainty, () β = 1 does not imply that consumption is expected to be constant over time E t { 1 C t+1 } > 1 E t C t+1
If () β = 1, { } 1 1 = E t C t C t+1 > 1 E t C t+1, implying that E t C t+1 > C t, such that consumption is expected to rise over time: precautionary saving In a closed economy with uncertainty, the a stationary equilibrium requires that () β < 1 to assure that consumption is expected to be constant
Wealth diminishes the precautionary motive as marginal utility flattens out as wealth and consumption increase Precautionary savings yields a role for bonds (wealth) in determining the current account As bonds increase with a current account surplus the precautionary motive weakens increasing spending and reducing the current account surplus With precautionary savings, lose permanent effects of transitory shocks (through permanent effect on bonds) Models linearized about the steady state lose the positive third derivative, lose the precautionary motive, and have the implication that a shock creates a permanent change in bonds and consumption
3.3 Current Account with Consumer Durables and No Uncertainty Utility U t = β s t [γ log (C s ) + (1 γ) log D s ] Budget constraint with p s the relative price of durables in terms of consumption B s+1 B s = rb s +Y s C s p s [D s (1 δ) D s 1 ] (K s+1 K s ) G s
Maximization problem using budget constraint to substitute for consumption U t = β s t (1 γ) log D s + β s t γ log { () Bs B s+1 + Y s p s [D s (1 δ) D s 1 ] (K s+1 K s ) G s } First order condition with respect to bonds is bond Euler equation C s+1 = () βc s First order condition with respect to durables γp s C s = 1 γ D s + β (1 δ) γp s+1 C s+1 MU cost of acquiring durables = MU of immediate use + discounted MU of selling what remains in one period
Combine FO conditions to eliminate C s+1 γp s C s = 1 γ D s + (1 δ) γp s+1 () C s MRS (1 γ) C s γd s = p s (1 δ) p s+1 = ι s user cost where user cost is the price less the resale value of depreciated durables Intertemporal budget constraint with durables
Durables term p t [D t (1 δ) D t 1 ] Collecting terms on each D t +p t+1 [D t+1 (1 δ) D t ] +p t+2 [D t+2 (1 δ) D t+1 ] ( 1 ( 1 ) ( ) 1 s t ι s D s p t (1 δ) D t 1 + lim p t+nd t+n N ) 2 ( 1 ) t+n
Setting limit to zero, IBC becomes (C s + ι s D s ) ( ) 1 s t = () B t + p t (1 δ) D t 1 + (Y s G s I s ) ( ) 1 s t Consumption and Durables when β () = 1 C t = D t = γr () B t + p t (1 δ) D t 1 + (1 γ) r () B t + p t (1 δ) D t 1 + ι t () (Y s G s I s ) If ι s is constant, then C t and D t are proportionate ( 1 (Y s G s I s ) ( 1 ) s t ) s
However, expenditures on the two, C t and p t [D t (1 δ) D t 1 ] are not with p and ι fixed, and δ = 0, consumer buys all durables in beginning and never purchases them again with δ > 0, replaces durables as they wear out when there are shocks to p and ι, get large changes in durables expenditure as move immediately to new equilibrium a change in demand for durables can lead to a large current account imbalance as durables expenditures change, yielding high current account volatility
4 Firms Distinct from Households with Certainty 4.1 Assumptions Production is homogenous of degree one, yielding CRS Y = AF (K, L) L is fixed V t is the price of a claim to a firm s entire stream of future profits beginning on date t + 1
x s+1 is share of domestic firm owned by the representative consumer at end of the period s d s is dividends per share on date s
4.2 Household Problem Consumer budget constraint B s+1 B s + V s (x s+1 x s ) = rb s + d s x s + w s L C s G s Optimization problem using budget constraint to substitute for consumption U t = β s t u [ B s+1 + () B s V s (x s+1 x s ) + d s x s + w s L G s ] First order condition with respect to x s+1 U t x s+1 = { u (C s ) ( V s ) + βu (C s+1 ) (d s+1 + V s+1 ) } = 0
= u (C s ) βu (C s+1 ) = d s+1 + V s+1 V s returns on equity are dividend plus value of asset relative to initial value of the asset in equilibrium with no uncertainty returns on all assets must be equal
Reformulate individual budget constraint Define Q s+1 as total financial wealth going into period s + 1 Q s+1 = B s+1 + V s x s+1 FO condition, and equivalently arbitrage across asset returns, implies () V s = d s+1 + V s+1 Solving for dividends d s+1 = () V s V s+1
Copy agent budget constraint from above and rewrite B s+1 B s + V s (x s+1 x s ) = rb s + d s x s + w s L C s G s Q s+1 Q s V s x s +V s 1 x s = rq s rv s 1 x s +d s x s +w s L C s G s collect terms on x s d s [() V s 1 V s ] = 0 use the definition of dividends from above Q s+1 Q s = rq s + w s L C s G s for s > t holds only if arbitrage allows equality of returns to stocks and bonds does not hold in event of unanticipated shock which invalidates equality
Sum present values of budget constraints + Q t+1 [() B t + d t x t + V t x t + w t L C t G t ] = 0 + 1 {Q t+2 () Q t+1 + w t+1 L C t+1 G t+1 } = 0 ) 2 {Q t+3 () Q t+2 + w t+2 L C t+2 G t+2 } = 0 ( 1 to yield ( 1 ) N 1 Q t+n +... () B t + d t x t + V t x t + N 1 (w s L C s G s ) ( 1 ) t s take limit as N and set the limit term to zero implying PV consumption equals PV labor income net of taxes plus initial assets
with interest and dividends ( ) 1 t s C s = () B t +d t x t +V t x t + (w s L G s ) ( 1 ) t s
4.3 Value of the firm and stock prices Asset pricing equation is the FO condition with respect to x t+1 V t = d t+1 + V t+1 Solve forward V t = V t+1 = d t+2 + V t+2 V t = d t+1 + 1 +1 [ dt+2 + V t+2 ( ) 1 s t ( ) 1 T d s + lim V t+t T ]
Impose limit term equals zero requiring that stock value not grow faster than interest rate Value of stocks today is the present value of future dividends
4.4 Behavior of the Firm Dividends d s = A s F (K s, L s ) w s L s (K s+1 K s ) Value of the firm V t = ( 1 +1 ) s t [A s F (K s, L s ) w s L s (K s+1 K s )] Firm maximizes present value of dividends V t + d t = ( 1 ) s t [A s F (K s, L s ) w s L s (K s+1 K s )]
FO condition with respect to K s+1 (V t + d t ) K t+1 = 1 + ( 1 ) [A t+1 F K (K t+1, L t+1 ) + 1] = 0 A t+1 F K (K t+1, L t+1 ) = r FO condition with respect to L s (V t + d t ) L t = A t F L (K t, L t ) w t = 0
4.5 Alternative Intertemporal Budget Constraint CRS implies AF (K, L) = AF K K + AF L L = rk + wl Using this in the value of stock expression ( ) 1 s t V t = [rk s (K s+1 K s )] +1 ( ) 1 s t = [() K s K s+1 ] = +1 ( 1 ) = K t+1 lim T [() K t+1 K t+2 ] + ( 1 ) t+t K t+t +1 ( 1 ) 2 [() K t+2 K t+3 ] +.
Impose limit equal to zero Ex dividend market value of the firm is the value of capital in place for production next period Country s financial wealth is the sum of its net foreign assets plus capital Q = B + K IBC can be written ( ) 1 t s C s = () Q t + (w s L G s ) ( ) 1 t s
When β () = 1 C t = rq t + r = rq t + w t L G t Alternative current account expression (w s L G s ) S t = rq t + w t L G t C t ( 1 ) t s CA t = S t I t = w t L w t L ( G t G ) I t Positive future productivity shock raises I t for one period and w t permanently CA deficit for one period Deficit could last longer if investment takes more than one period