Small Open Economy RBC Model

Uribe, Chapter 4

1 Basic Model

1.1 Uzawa Utility

E0

∞∑t=0

θtU (ct, ht)

θ0 = 1

θt+1 = β (ct, ht) θt; βc < 0; βh > 0.

• Time-varying discount factor

— With a constant discount factor, consumption and net foreign assetsare non-stationary

∗ Rise permanently with an increase in productivity

∗ Unit roots imply no steady state

∗ Cannot claim that the linearized model, which converges to a steadystate, behaves as the original model

∗ Cannot compute unconditional moments

— Discount factor is decreasing in wealth

∗ As wealth increases, c increases and h decreases reducing θt+1 belowθt

∗ Equivalently, agents become less patient as wealth increases

∗ Positive productivity shock raises wealth and consumption and re-duces hours

∗ Therefore, θ falls, increasing consumption, sending wealth back down

∗ Get a steady state value of wealth with this discount factor andtherefore a steady state

1.2 Household Budget Constraint

• Write in terms of household debt (d) instead of assets

dt = (1 + rt−1) dt−1 − yt + ct + it + Φ (kt+1 − kt)

• Cost of Investment includes capital adjustment costs Φ (kt+1 − kt)

— Adjustment cost reduce the volatility of investment

— Adjustment costs is increasing in the size of the change in capital,reducing the optimal size of investment

— Restrictions assure that when capital is fixed in the steady state, thereare no investment costs

Φ (0) = Φ′ (0) = 0

• Output

yt = AtF (kt, ht)

• Evolution of capital

kt+1 = it + (1− δ) kt

1.3 Optimization

• Households choose {ct, ht, kt+1, dt, θt+1}

— Subject to above constraints

— and NPG condition requiring the present value of debt in the limit tobe non-positive

limj→∞

Etdt+j

j∏s=1

(1 + rs)

≤ 0

— Substitute for investment in household budget constraint

dt = (1 + rt−1) dt−1− yt + ct + kt+1− (1− δ) kt + Φ (kt+1 − kt)

• Lagrange multipliers

— θtλt on household budget constraint

— ηt on evolution of discount factor

θt+1 = β (ct, ht) θt

1.4 First Order Conditions

• dtλt = β (ct, ht)Etλt+1 (1 + r)

— β (ct, ht) is the ratio of two discount factors

1 =β (ct, ht)Etλt+1 (1 + r)

λt

— if β (1 + r) is fixed at unity a wealth-reducing shock which raises λtraises Etλt+1 equally, hence permanently

— with β endogenous, the reduction in wealth reduces consumption whichraises λt and β (ct, ht)

∗ increase in Etλt+1 is less than increase in λt

∗ eventually λt returns to its steady-state value

• ctλt = Uc (ct, ht)− ηtβc (ct, ht)

— Marginal utility of wealth (λt) equals marginal utility of consumption[Uc (ct, ht)] less the marginal value of the reduction in the discountfactor (positive term)

— Unit decline in discount factor in turn reduces utility in period t by ηt

• htUh (ct, ht) + λtAtFh (kt, ht)− ηtβh (ct, ht) = 0

− [Uh (ct, ht)− ηtβh (ct, ht)] = AtFh (kt, ht) [Uc (ct, ht)− ηtβc (ct, ht)]

— marginal disutility of hours includes effect of increase in hours in in-creasing the discount factor

— adjusted marginal disutility of hours equals the wage times the adjustedmarginal utility of consumption

• θt+1

ηt = −EtU (ct+1, ht+1) + Etηt+1β (ct+1, ht+1)

— Solving forward

ηt = −Et∞∑j=1

θt+j

θt+1U(ct+j, ht+j

)

— ηt is the negative of the present discounted value of utility

• kt+1

λt[1 + Φ′ (kt+1 − kt)

]= β (ct, ht)Etλt+1

[At+1Fk (kt+1, ht+1) + 1− δ + Φ′ (kt+2 − kt+1)

]— cost of one additional unit of capital includes adjustment costs

— benefits are discounted utility value of capital’s marginal product plusits undepreciated value plus saved adjustment cost going forward sincecapital it higher

1.5 Assumptions

• Perfect capital mobility such that interest rate in small open economyequals fixed world rate

rt = r

• Productivity is stationary and AR(1)

lnAt+1 = ρ lnAt + εt+1 ρ ∈ (−1, 1) ε ∼ N(

0, σ2ε

)

• Functional forms

U (c, h) =

[c− ω−1hω

]1−γ − 1

1− γ

β (c, h) =[1 + c− ω−1hω

]−ψ1

F (k, h) = kαh1−α

Φ (x) =φ

2x2 φ > 0

• Combining FO conditions on ct and ht, functional forms make labor supplyindependent of consumption (ratios of marginal utilities depend only onlabor)

At (1− α) kαt h−αt = hω−1

t

— lhs is wage at which firm is willing to hire labor, hence labor demandand rhs is wage as which household willing to supply hours, hence laborsupply

2 Calibration to Canadian Economy (Annual)

• ω = 1.455 implies a high labor supply elasticity of 1ω−1 = 2.2

— Totally differentiate second equation below with respect to wage andlabor supply

A (1− α) kαh−α = hω−1 = wt

dw = (ω − 1)hω−2dh

dh

dw=

1

ω − 1hω−2

wdh

hdw=

1

ω − 1

• δ = 0.1 implies that capital depreciates at 10% per year

• α = .32 implies a capital income share of 32%

• r = .04 is the average real rate of return of broad measures of the stockmarket in developed countries post WWII

• ψ1 is the elasticity of the discount factor with respect to the composite1 + c− ω−1hω

— match average Canadian trade-balance-to-GDP ratio of 2%

— steady state value of FO condition on bonds yields

β (c, h) (1 + r) = 1

— steady state value of FO condition on capital with A = 1 yields

r + δ = α

(h

k

)1−α

k

h=(

α

r + δ

) 11−α

implying that the steady-state capital labor ratio is independent of ψ1

— steady state value of hours is also independent of ψ1

h =

[(1− α)

(k

h

)α] 1ω−1

— steady-state values of capital, output, and investment (i = δk) will allbe independent of ψ1

— steady-state value of FO condition on bonds

β (c, h) (1 + r) =[1 + c− ω−1h

]−ψ1 (1 + r) = 1

∗ use resource constraint

c = y − i− tb

to substitute for c[1 + y − i− tb− ω−1h

]−ψ1 (1 + r) = 1

∗ solve for tby

tb

y= 1− i

y− (1 + r)

1ψ1 + ω−1h− 1

y

— since hours, capital, investment, and output are independent of ψ1,can calibrate ψ1 to match

tby

— find ψ1 is increasing intby

— dual role of ψ1

∗ determines steady-state trade balance/output ratio

∗ governs speed of convergence to steady state

∗ might want to separate these allowing speed of convergence to besmall enough as to not significantly affect business cycle dynamics

∗ add a parameter and respecify β (ct, ht)

β (ct, ht) = β[1 + (ct − c)− ω−1 (hωt − hω)

]−ψ1

where c and h are steady-state values

• φ = 0.028 matches standard deviation of investment of 9.8

• σε = 0.0129 matches standard deviation of output of 2.8

• ρ = 0.42 matches serial correlation of output of 0.61

3 Equilibrium

3.1 Definition

A competitive equilibrium is a set of processes {dt, ct, ht, yt, it, kt+1, ηt, λt, At}satisfying the resource constraint and the first order conditions, given the fixedworld interest rate, r, initial conditions A0, d−1, and k0 and the exogenousprocess {εt}

3.2 Approximation

• Solutions where endogenous variables fluctuate in small neighborhood aroundsteady state

— Debt will be bounded implying

limj→∞

Etdt+j

(1

1 + r

)j= 0

— Write system as

Etf (xt+1, xt) = 0

— Cannot solve non-linear system

• Linearize about steady state

— Express most variables as percent deviations about steady state

wt ≡ log (wt/w) ≈ wt − ww

— For variables that can take on negative values, like trade balance, orvariables already expressed as percent, like interest rates, just use firstdifference

wt ≡ wt − w

— Linearized system is expressed as

Axt+1 = Bxt

where A and B are conformable square matrices made up of knowncoeffi cients in the calibrated linearized model

• Ten variables in linearized system

— state variables

∗ variables whose t values are predetermined (determined before t)

∗ variables whose values are exogenous

∗ include kt, dt−1, At, and r

— co-state variables

∗ endogenous variables have values not predetermined in period t

• Initial conditions

— Three known initial conditions k0, d−1, A0

— Determine other initial conditions to satisfy boundedness condition

limj→∞

[Etxt+j

]= 0

4 Model Performance

4.1 Successes

• Model is calibrated to match some moments and does well here by con-struction

— volatility of output

— volatility of investment,

— serial correlation of output

• Volatility rankings

— investment volatility greater than output volatility

— consumption volatility less than output volatility

• trade balance is countercyclical

— productivity shock increases investment more than it reduces savingsdue to consumption smoothing

— result due to parameters φ, governing cost of investment, and ρ, per-sistence of productivity shock and these values were calibrated inde-pendent of trade balance performance

4.2 Failures

• too little countercyclicality in trade balance

— need investment and/or consumption to increase more in response topositive productivity shock

• correlation between

— consumption and output too high

— between hours and output is too high at unity

∗ due to functional form for the period utility index

∗ condition for equilibrium hours

At (1− α) kαt h−αt = hω−1

t

(1− α) yt = hωt

∗ log-linearized version is

yt = ωht

∗ implying a correlation of unity

5 Impulse responses

• Output, investment, hours, and consumption all respond positively to aproductivity innovation

• The trade balance/GDP and the current account/GDP both respond neg-atively

5.1 Countercyclicality of the trade balance

• Investment must respond strongly enough

— requires that adjustment cost not be too high

— persistence of productivity must be high enough

• Consumption must respond strongly enough

— requires persistence relatively high

6 Other Ways to Impose Stationarity

6.1 Debt-elastic interest rate

• Assume interest rate paid by small open economy is increasing in its ag-gregate external debt

rt = r + p(dt)

— Introduces a risk-premium on debt

• Assume agent does not take into account the effect of his change in debton the country risk premium implying that the Euler equation is

λt = β (1 + rt)λt+1

• Dynamics of the model: begin away from steady state and show that endup in steady state

— Assume β (1 + rt) > 1

— Agent are saving, wealth and consumption are rising and marginal util-ity (λ) is falling

— Increase in wealth implies a reduction in foreign debt and a reductionin rt, returning system to steady state

• Can calibrate the p parameter to be small enough that it yields stationaritywithout affecting business cycle dynamics

6.2 Portfolio Adjustment Costs

• Assume there is a cost to holding debt in quantity different from its long-run equilibrium value

dt = (1 + rt−1) dt−1−yt+ct+kt+1−(1− δ) kt+Φ (kt+1 − kt)+ψ3

2

(dt − d

)2

reducing the marginal utility of debt due to the adjustment cost

• Euler equation becomes

λt[1− ψ3

(dt − d

)]= β (1 + r)Etλt+1

• Dynamics of the model: shock model away from steady state and showthat end up in steady state

— Assume dt increases from steady state value such that dt > d[1− ψ3

(dt − d

)]= β (1 + r)

Etλt+1

λt

— lhs decreases below unity requiring

Etλt+1

λt< 1

— the increase in debt is a reduction in wealth such that consumptionfalls and marginal utility rises

— adjustment costs require that Etλt+1 rise by less than λt rises, requir-ing that current consumption fall by more than future consumption

— agent begins saving to replace wealth lost by shock returning systemto steady state

• Can calibrate ψ3 to be small enough that it yields stationarity withoutaffecting business cycle dynamics much

6.3 Precautionary Savings

• Euler equation with log utility1

ct= β (1 + r)Et

1

ct+1> β (1 + r)

1

Etct+1

• With β (1 + r) = 1

Etct+1 > ct

— Such that consumption is rising implying that wealth is growing

— Increase in wealth reduces magnitude of precautionary motive implyingwealth is rising at falling rate

— Eventually reach a steady state

— However, since world cannot have steady state at β (1 + r) = 1, reallyneed to consider β (1 + r) < 1 to reduce world savings

• Problem

— Precautionary saving requires a positive third derivative of utility

— Cannot use in a linearized model because linearization eliminates thethird derivative

7 Business Cycles in Emerging Markets (Aguiarand Gopinath 2007)

7.1 Model

• Utility per capita

E0

∞∑t=0

βt

[Cγt (1− ht)1−γ]1−σ − 1

1− σ

• Budget constraint per capita

Dt+1

Rt= Dt+Ct+Kt+1−(1− δ)Kt+

φ

2

(Kt+1

Kt− g

)2

Kt−AtKαt (Xtht)

1−α

• First order conditions for household letting Λt denote multiplier

— Ct [Cγt (1− ht)1−γ]−σ γCγ−1

t (1− ht)1−γ − Λt = 0

— ht

−[Cγt (1− ht)1−γ]−σ (1− γ)C

γt (1− ht)−γ+Λt (1− α)AtK

αt X

1−αt h−αt = 0

— Dt+1

Λt1

Rt− EtΛt+1β = 0

— Kt+1

Λt

(1 + φ

(Kt+1

Kt− g

))= βEtΛt+1[1− δ + αAt+tK

α−1t+1 (Xt+1ht+1)1−α

+φ(Kt+2Kt+1

− g)Kt+2Kt+1

− φ2

(Kt+2Kt+1

− g)2

]

• Stationary productivity shock

lnAt = ρa lnAt−1 + σaεat 0 < ρa < 1

• Non-Stationary productivity shock

— Define

gt =Xt

Xt−1

— Growth is stationary

ln (gt − g) = ρg ln (gt−1 − g) + σgεgt 0 < ρg < 1 g > 1

— Unit root in productivity gives other variables unit root, so equilibriumis not stationary

• Find a way to transform the model to make transformed variables stationary

7.2 Transformation to make the model stationary

• Divide trending variables byXt−1 and define resulting variables, likeCtXt−1

≡ct

• In FO condition on Ct, expressing in terms of ct requires multiplying equa-tion by X1+γ(σ−1)

t−1

— Therefore, define

λt = ΛtX1+γ(σ−1)t−1

• Detrended FO conditions

— ct [cγt (1− ht)1−γ]−σ γcγ−1

t (1− ht)1−γ − λt = 0

— ht multiply equation by Xγ(σ−1)t−1[

cγt (1− ht)1−γ]−σ (1− γ) c

γt (1− ht)−γ = λt (1− α)Atk

αt g

1−αt h−αt

— Using the two equations together

(1− γ) ct

γ (1− ht)= (1− α)Atgt

(kt

gtht

)α

— Dt+1 multiply equation by X1+γ(σ−1)t−1

λt1

Rt− βEtΛt+1X

1+γ(σ−1)t−1 = 0

λt1

Rt− βEtλt+1g

−1−γ(σ−1)t = 0

— Kt+1

Kt+2

Kt+1=

Kt+2Xt

Xt+1Xt+1

Kt+1Xt

=gt+1kt+2

kt+1

λt

(1 + φ

(gtkt+1

kt− g

))

= βg−1−γ(σ−1)t Etλt+1[1− δ + αAt+t

(kt+1

gt+1ht+1

)α−1

+φ(gt+1kt+2kt+1

− g)gt+1kt+2kt+1

− φ2

(gt+1kt+2kt+1

− g)2

]

— Linearized system has the property that a transitory shock will createa permanent increase in consumption if Rt is exogenous

• Risk premium on interest rate increasing in debt above some minimumlevel, d

Rt = R∗ + ψ

[edt+1−d − 1

]

7.3 Equilibrium

7.4 Properties of equilibrium

• Stationary in detrended variables

• Actual variables inherit the unit root in Xt since

Ct = ctXt−1

• Stationary distribution of wealth around its mean due to assumption aboutrisk-premium on interest rate

7.5 Calibration

• Calibrate β = 0.98, γ = 0.36, ψ = 0.001, α = 0.68, σ = 2, δ = 0.05

• Estimate parameters for stochastic processes and adjustment cost φ atquarterly horizon using Mexico 1980Q1 to 2003Q1.

— σg = 0.0213 ρg = 0.00 g = 1.0066

— σa = 0.0053 ρa = 0.95

— φ = 1.37

• What fraction of variance of Solow residual is determined by the permanentshock?

— Detrended output

Atkαt (gtht)

1−α

— Solow Residual

SRt = Atg1−αt

— Fraction of variance of Solow residual accounted for by non-stationaryshock is 88%

— Compares to 40% for Canada

— Implies that importance of non-stationary shock is major difference inrich and emerging market countries

• Potential problem

— Need long time series to confidently identify a data series as non-stationary

7.6 Compare moments to data

• Since data is non-stationary, need to make it stationary to compute mo-ments

— Unconditional mean and variance do not exist in non-stationary data

— Use HP filter

— Theoretically should estimate the non-stationary trend and deflate bythat as in model

• Consumption is more volatile than output

— Importance of non-stationary output

• Trade balance is countercyclical

• Other matches also good