The Small-Open-Economy Real Business Cycle Model

The Small-Open-Economy Real Business Cycle Model

Comments Some Empirical Regularities Variable Canadian Data σ xt ρ xt,x t ρ xt,gdp t y 2.8.6 c 2.5.7.59 i 9.8.3.64 h 2.54.8 tb y.9.66 -.3 Source: Mendoza (AER, 99) Volatility ranking: σ tb/y <σ c <σ y <σ i. Consumption, investment, and hours are procyclical. The trade-balance-to-output ratios is countercyclical. All variables considered are positively serially correlated. Similar stylized facts emerge from other small developed countries (see, e.g., Aguiar and Gopinath, JPE, 26). 2

An RBC Model with Uzawa Preferences E t= θ t U(c t,h t ), θ =, θ t+ = β(c t,h t )θ t t, The Sequential Budget Constraint d t =(+r t )d t y t + c t + i t +Φ(k t+ k t ), with Φ() = Φ () =. Technology y t = A t F (k t,h t ), Evolution of the Capital Stock k t+ = i t +( δ)k t, No-Ponzi-Game Constraint lim j E t d t+j j s= ( + r s). 3

Optimality Conditions Define Ũ(c t,h t,η t )=U(c t,h t ) η t β(c t,h t ). Ũ c (c t,h t,η t )=λ t Ũ h (c t,h t,η t )=λ t A t F h (k t,h t ) λ t = β(c t,h t )( + r t )E t λ t+ λ t [ + Φ t] = β(c t,h t )E t λ t+ [A t+ F k (k t+,h t+ ) + δ +Φ t+] η t = E t U(c t+,h t+ )+E t η t+ β(c t+,h t+ ) Interpreting the multiplier η t η t = E t ( ) θt+j j= θ t+ U(c t+j,h t+j ) η t is next period s lifetime utility. 4

Evolution of Total Factor Productivity ln A t+ = ρ ln A t + ɛ t+ ; ɛ t+ NIID(,σ 2 ɛ ); t. Free Capital Mobility r t = r, where r is the world interest rate, assumed to be constant. 5

Functional Forms Period Utility Function [ c ω h ω] γ U(c, h) = γ Subjective Discount Factor β(c, h) = [ +c ω h ω] ψ Production Function F (k, h) =k α h α Adjustment Cost Function Φ(x) = φ 2 x2 ; φ>. 6

Calibration γ ω ψ α φ r δ ρ σ ɛ 2.455..32.28.4..42.29 Calibration Strategy ψ : Match Canadian trade balance-to-output ratio φ: Match Canadian investment volatility ρ: Match Canadian Output serial correlation σ ɛ : Match Canadian output volatility 7

Empirical and Theoretical Second Moments Variable Canadian Data Model σ xt ρ xt,x t ρ xt,gdp t σ xt ρ xt,x t ρ xt,gdp t y 2.8.6 3..6 c 2.5.7.59 2.3.7.94 i 9.8.3.64 9..7.66 h 2.54.8 2..6 tb.9.66 -.3.5.33 -.2 y ca.5.3.26 y Comments Parameters φ, σ ɛ, and ρ picked to match σ i, σ y, and ρ yy. So no real test here. The model matches the volatility ranking σ c <σ y <σ i. Empirical and theoretical trade-balance-tooutput ratios are countercyclical. The model overestimates the correlations of hours and consumption with output. 8

Response to a Positive Technology Shock Consumption Output 2.5.5.5 2 4 6 8 Investment.5 2 4 6 8 Hours.5 5.5 5 2 4 6 8 Trade Balance / GDP.5.5 2 4 6 8 2 4 6 8 Current Account / GDP.5.5 2 4 6 8 Source: Schmitt-Grohé and Uribe (JIE, 23) Comments: Output, consumption, investment, and hours expand. The trade balance deteriorates. 9

Adjustment Costs, Persistence of Shocks, and the Trade Balance-To-Output Ratio.6.4 benchmark high φ low ρ.2.2.4.6.8 2 3 4 5 6 7 8 9 Comment The more persistent the shock, the more countercyclical the response of the trade balance. The weaker the cost of adjusting capital, the more countercyclical the response of the trade-balance-to-output ratio.

Endogenous Discount Factor Without Internalization θ t+ = β( c t, h t )θ t t, θ =, where c t and h t denote per capita consumption and hours worked. λ t = β( c t, h t )( + r t )E t λ t+ λ t = U c (c t,h t ) U h (c t,h t )=λ t A t F h (k t,h t ) λ t [ + Φ t] = β( c t, h t )E t λ t+ [A t+ F k (k t+,h t+ ) + δ +Φ t+] In Equilibrium c t = c t and h t = h t

Debt-Elastic Interest Rate (external) r t = r + p( d t ), θ t = β t, λ t = β( + r t )E t λ t+ U c (c t,h t )=λ t, U h (c t,h t )=λ t A t F h (k t,h t ). λ t [+Φ t] = βe t λ t+ [A t+ F k (k t+,h t+ ) + δ +Φ t+] d t = d t. Functional Form for Country Spread ( ) p(d) =ψ 2 e d d, Calibration β d ψ 2 r.96.7442.742 β 2

Internal Debt-Elastic Interest Rate r t = r + p(d t ), The Euler equation becomes λ t = β[ + r + p(d t )+p (d t )d t ]E t λ t+ p(d) =ψ 2 ( e d d ), Calibration: Same as in the external case. Note that the steady-state value of debt is no longer equal to d. Instead, d solves ( + d)e d d = d =.44522. 3

Portfolio Adjustment Costs d t = (+r t )d t y t +c t +i t +Φ(k t+ k t )+ ψ 3 2 (d t d) 2 λ t [ ψ 3 (d t d)] = β( + r t )E t λ t+ Calibration β d ψ 3 r.96.7442.74 β 4

Complete Asset Markets E t r t+ b t+ = b t + y t c t i t Φ(k t+ k t ), lim E tq t+j b t+j, j q t = r r 2...r t, λ t r t+ = βλ t+. λ t r t+ = βλ t+. λ t+ λ t = λ t+ λ t λ t = ξλ t, λ t = ψ 4,. Calibration: Set ψ 4 so that steady-state consumption equals steady-state consumption in the model with Uzawa preferences. 5

Calibrations Debt-Elastic Interest Rate (internal external) and β d ψ 2 r.96.7442.742 β Portfolio Adjustment Costs β d ψ 3 r.96.7442.74 β Complete Asset Markets Set ψ 4 so that steady-state consumption equals steady-state consumption in the model with Uzawa preferences. 6

Impulse Response to a Unit Technology Shock in Models Through 5 2 Output.5 Consumption.5.5.5 2 4 6 8 Investment 2 4 6 8 Hours.5 5.5 5 2 4 6 8 Trade Balance / GDP.5.5 2 4 6 8 2 4 6 8 Current Account / GDP.5.5 2 4 6 8 Source: Schmitt-Grohé and Uribe (JIE, 23) Note. Solid line, endogenous discount factor. Squares, endogenous discount factor without internalization. Dashed line, Debt-elastic interest rate. Dashdotted line, Portfolio adjustment cost. Dotted line, complete asset markets. Circles, No stationarity inducing elements. 7

Observed and Implied Second Moments Data Model Model a Model 2 Model 3 Model 4 Standard Deviations y 2.8 3. 3. 3. 3. 3. c 2.5 2.3 2.3 2.7 2.7.9 i 9.8 9. 9. 9 9 9. h 2 2. 2. 2. 2. 2. tb/y.9.5.5.8.8.6 ca/y.5.5.5.5 Serial Correlations y.6.6.6.62.62.6 c.7.7.7.78.78.6 i.3.7.7.69.69.7 h.54.6.6.62.62.6 tb/y.66.33.32.5.5.39 ca/y.3.3.32.32 Correlations with Output c.59.94.94.84.85 i.64.66.66.67.67.66 h.8 tb/y -.3 -.2 -.3 -.44 -.43.3 ca/y.26.25.5.5 Source: Schmitt-Grohé and Uribe (JIE, 23) Note. Standard deviations are measured in percent per year. 8