International Business Cycles Reduxvr0j/slides/slibcreduxjan16.pdf · 2013-01-16 · Punchline I Backus-Smith (Backus and Smith (1993)) puzzle: households consume more domestic goods

International Business Cycles Redux

Yan Bai and Jose-Vıctor Rıos-Rull

University of Rochester, University of Minnesota,Federal Reserve Bank of Minneapolis,

Mpls Fed, NBER

Wednesday 16th January, 2013

Punchline

I Backus-Smith (Backus and Smith (1993)) puzzle: householdsconsume more domestic goods when they are more expensive

I corr(RER, cH/cF ) > 0.I Yet standard models (e.g. RBC) predict the opposite.

I Literature: demand shocks, low elasticity between home andforeign goods (Corsetti, Dedola, and Leduc (2008)), non-tradablegoods (Engel and Wang (2011)), labor wedge from homeproduction (Karabarbounis (2012)).

I We pose an explanation based on demand shocks in thecontext of environments where expenditures shapeproductivity (a la Bai, Rıos-Rull, and Storesletten (2011)). We alsoobtain

I Countercyclical terms of trade.I Volatile net exports.I Lower international cons corr than output’s.

The two country two good model with shopping

I Two countries j = 1, 2 with representative agents in each.

I Build a top of stochastic growth model.

I There are incomplete markets (no insurance for preferenceshocks).

I There is perfect mobility of capital without impediments tocross country ownership.

Preferences: Current utility of Households in country j

u[Hc(

c jj , c jj∗), nj ,Hd

(d jj , d jj∗

), θj]

I Hc(c jj , c jj∗) is an (Armington) aggregator of the home (c jj)and the foreign (c jj∗) good.

I nj are hours worked in country j .

I Hd(d jj , d jj∗) is an aggregator (maybe linear) of searchshopping effort at home (d jj) and abroad (d jj∗).

I θj is a Markovian preference shock.

• Households cannot change their country of residence, whichmakes labor immobile. Like in Bai, Rıos-Rull, and Storesletten (2011)

consumption requires that the household finds the goods which inturn requires enough shopping effort to find them.

Production in each country• Measure one of firms–locations with installed capital k j

(depreciates at rate δ). Goods can be used for consumption orinvestment and capacity is

F (k j , (nF )j) = z(k j)γk ((nF )j

)γn• New capital consists of an (Armington) aggregator of home andforeign investment goods:

k ′j = (1− δ)k j + H i (i j , i j∗)

• New capjtal has to be purchased that requires shoppers lookingfor the home investment good (nj ,j)k and for the foreigninvestment good (nj ,j∗)k .

• Output in each country can be used by home consumers,foreign consumers, home investors and foreign investors.

• Unmatched capacity rots.

Households

Preferences are

u[Hc(

c jj , c jj∗), nj ,Hd

(d jj , d jj∗

), θj]

Again both consumptions require that they are shopped so

c jj = d jj Ψd(Qcjj) F cjj

c jj∗ = d jj∗ Ψd(Qcjj∗) F cjj∗

• Ψd(Qcj`) is the probability that a country j consumptionshopper has of matching a consumption firm from country ` thatcatters to j shoppers. Qcj` is market tightness in that consumptiongoods market and F cj` is its output capacity.

• Households own shares of a mutual fund that owns all firms.

A few lemmas alleviate notation

1. The state of the economy is S = θ1, θ2,K 1,K 2,B1 whereK j is capital installed in country j and B1 is the share of totalwealth held by country 1 households.

2. There are two active markets in consumption goods (one forlocals and the other for foreigners) and two markets ininvestment goods in each country for a total of 8 markets.

3. Firms that produce consumption and investment, for localbuyers and to export choose the same inputsF cjj = F i`j = .. = F (K j ,Nyj) = F j(S). We use n(k ,F ) todenote the inverse f (k , n).

4. All firms in each country get the same expected revenue (butnot necessarily the same price and market tightness).

The household problem

v j(S , b) = maxc..,d ..,n,b′

u[Hc(

c jj , c jj∗), nj ,Hd

(d jj , d jj∗

), θj]

+ βEv j(S ′, b′)|θ

b′ +

j∗∑`=j

pcj`(S) c j` = b [1 + R(S)] + nj w j(S)

c j` = d j` Ψd

(Qcj`(S)

)F cj`(S) ` = j , j∗,

S ′ = G (S)

The household problem — First Order Conditions

Hholds’ FOC (and RA): for ` = j , j∗ and m = j , j∗

ujc Hcj

` −ujd Hdj

`

Ψcj`d F cj`

= βE

pcj` (1 + R ′)

pcjm′

[uj′c Hcj′

m −uj′d Hdj′

m

Ψcjm′d F ckm′

]|θ

ujc Hc

` −ujd Hd

`

Ψcj`d F cj`

= unpcj`

w j

1

pcj`

[ujc Hcj

` −ujd Hdj

`

Ψcj`d F cj`

]=

1

pcjm

[ujc Hcj

m −ujd Hdj

m

Ψcjmd F cjm

]

Define marginal utility of savings M(S):

M(S) = βE

(1 + R ′)

pcjm′

[uj′c Hcj′

m −uj′d Hdj′

m

Ψcjm′d F ckm′

]|θ

Consumption (or invt) firms in a (pcj`,F cj`,Qcj`)submarket

Ωj(S , k) = maxnk..,k′,i jj ,i j j∗

ΨT (Qcj`) pcj` F cj` − w j(S)[n(k ,F cj`) + nkjj + nkjj∗

]−pijj(S) i jj − pijj∗(S) i jj

∗+ E

Ωj(S ′, k ′)

1 + R(S ′)

∣∣∣∣ θs.t. i j` = (nkj` ζ) Ψd [Q ij`(S)] F `(S) for ` = j , j∗

k ′ = (1− δ)k + H i(

i jj , i jj∗)

S ′ = G (S)

with FOC (and RA condition) for ` = j , j∗

H ij` E

Ωj

k

(S ′,K j′)

1 + R(S ′)

∣∣∣∣∣ θ

=w j(S)

ζ Ψd [Q ij`(S)] F `(S)+ pij`(S)

The household problem

v j(S , b) = maxc..,d ..,n,b′

u[Hc(

c jj , c jj∗), nj ,Hd

(d jj , d jj∗

), θj]

+ βEv j(S ′, b′)|θ

b′ +

j∗∑`=j

pcj`(S) c j` = b [1 + R(S)] + nj w j(S)

c j` = d j` Ψd

(Qcj`(S)

)F cj`(S) ` = j , j∗,

S ′ = G (S)

The household problem — First Order Conditions

Hholds’ FOC (and RA): for ` = j , j∗ and m = j , j∗

ujc Hcj

` −ujd Hdj

`

Ψcj`d F cj`

= βE

pcj` (1 + R ′)

pcjm′

[uj′c Hcj′

m −uj′d Hdj′

m

Ψcjm′d F ckm′

]|θ

ujc Hc

` −ujd Hd

`

Ψcj`d F cj`

= unpcj`

w j

1

pcj`

[ujc Hcj

` −ujd Hdj

`

Ψcj`d F cj`

]=

1

pcjm

[ujc Hcj

m −ujd Hdj

m

Ψcjmd F cjm

]

We define marginal utility of savings M(S) as

M(S) = βE

(1 + R ′)

pcjm′

[uj′c Hcj′

m −uj′d Hdj′

m

Ψcjm′d F ckm′

]|θ

Competitive Search in Goods Markets

• Markets are now indexed by good type (country of production),quantity, price, and market tightness.

• There are four possible purchasers of goods (home and foreign,consumption and investment. A total of 8 markets.

• We get additional conditions from the FOC of shoppers givenexpected revenue for sellers.

• The equilibrium objects (44) are functions of (S) forQcj`,Q ij`,Nkj`,C j`, I j`, pcj`, pij`,T cj`,T ij`,w j ,Nyj ,N j ,B ′,R

j∈1,2,`∈1,2

.

Competitive Search in Consumption Market cj`

The rewards for the hhold to send a shopper to a (pcj`,F cj`,Qcj`)market is

Φ = maxQcj`,pcj`,F cj`

−ujd(S)Hdj

` (S)+Ψd(Qcj`)F cj`(

ujc(S)Hcj

` (S)− pcj`M(S))

ςcj` ≤ pcj`Ψd(Qcj`)

Qcj`F cj` − w j(S)n(k ,F cj`) (1)

Solving pcj` from equation (1) and plugging it into the objectivefunction, we have (and ignore the sunk shopping cost)

maxQcj`,pcj`,F cj`

Ψd(Qcj`)F cj`

(ujc(S)Hcj

` (S)− ςcj` + w j(S)n(k,F cj`)

Ψd(Qcj`)F cj`/Qcj`M(S)

)

Competitive Search in Consumption Market cj`

First order condition over Qcj`:

(1− α)A(Qcj`)−αF cj`ujc(S)Hcj

` (S)−[ςcj` + w j(S)n(k,F cj`)

]M(S) = 0

First order condition over F cj`:

A(Qcj`)1−αujc(S)Hcj

` (S)− w j(S)dn(k ,F cj`)

dF cj`Qcj`M(S) = 0

Thus, two equations characterize the equilibrium in market cj`,

pcj` = (1− α)ujc(S)Hcj

` (S)

M(S)

w j(S)

pcj`=

1

1− αΨd(Qcj`)

Qcj`fn(S)

Competitive Search in Investment Market ij`

The rewards for a firm to send a shopper to a (pij`,F ij`,Q ij`)market is

ΦF = maxQ ij`,pij`,F ij`

−w j(S)+ζΨd(Q ij`)F ij`[H ij` (S)E Ω(S ′, k ′)Π(S ,S ′) − pij`

]

ς ij` ≤ pij`Ψd(Q ij`)

Q ij`F ij` − w j(S)n(k ,F ij`) (2)

Solving pij` from equation (2) and plugging it into the objectivefunction, we have (and ignore the sunk wage cost)

maxQ ij`,pij`,F ij`

ζΨd(Q ij`)F ij`

[H ij` (S)E Ω(S ′, k ′)Π(S ,S ′) − ζ ij` + w j(S)n(k ,F ij`)

Ψd(Q ij`)F ij`/Q ij`

]

Competitive Search in Investment Market ij`

First order condition over Qij`:

(1−α)A(Q ij`)−αF ij`H ij` (S)E Ω(S ′, k ′)Π(S ,S ′)−

[ς ij` + w j(S)n(k,F ij`)

]= 0

First order condition over Fij`:

A(Q ij`)1−αH ij` (S)E Ω(S ′, k ′)Π(S ,S ′) − w j(S)

dn(k,F ij`)

dF ij`Q ij` = 0

Thus, two equations characterize the equilibrium in market ij`,

pij` = (1− α)H ij` (S)E Ω(S ′, k ′)Π(S ,S ′)

w j(S)

pij`=

1

1− αΨd(Q ij`)

Q ij`fn(S)

Lemma: All firms with k = K j in country j choose thesame labor

The revenue of a firm in country j to produce m goods (x = c , i) forcountry ` = j , j∗ is given by

ςxj` = pxj`A[Qxj`(S)]−αF xj` − w j(S)n(K j ,F xj`)

= w j(S)

[pxj`A[Qxj`(S)]−α

w j(S)F xj` − n(K j ,F xj`)

]Using the equilibrium condition from competitive search,w j (S)pxj` = 1

1−αA[Qxj`(S)]−αfn(S), we can rewrite firm’s revenue

ςxj` = w j(S)

[(1− α)

F xj`

fn(K j , n(K j ,F xj`))− n(K j ,F xj`)

].

In equilibrium, all firms in country j have the same revenue,ςcjj = ςcjj

∗= ς ijj = ς ijj

∗, and thus have the same output and the same

labor.

Recursive Equilibrium

1. Households and firms solve their problems (12 in households,4 in firms).

2. Competitive Search Conditions. (16).

3. Representative Agent Conditions

4. Market Clearing Conditions for j : (12)

N j = Nyj + Nkjj + Nkjj∗ ,

X j` = T xj` ΨT (Qxj`) F (K j ,Nyj) for X = C , I, ` = j , j∗

1 =

j∗∑`=j

(T cj` + T ij`

)for j ∈ 1, 2

5. Value of all firms is 1.

Putting the model to work• We want a clear version of this model. So separable utility withconstant Frisch elasticiy and Cobb-Douglas technology. We willplace shocks on preferences and on the investment shoppingtechnology.

I Preferences

u(Hc , n, d j , d j∗ , θ) = θ(Hc)1−σ

1− σ− χ

(nj)1+ψ

1 + ψ− (d j + d j∗)

I Aggregator

Hc(c j , c j∗) =[µ(c j)η + (1− µ)(c j∗)η

] 1η

I Production function

F (k , ny ) = z kγk (ny )γn

I Shocks

log(θt) = ρθ log(θt−1) + vt , vt ∼ N(0,Σ2)

Calibration

Targets Value Parameter ValueFirst Group: Parameters Set Exogenously

Risk aversion 2. σ 2.Real interest rate 4% β 0.99Frisch elasticity 0.72 1

ν0.72

Armington elasticity 1.1 η 0.09α 0.25

Second Group: Standard TargetsFraction of time spent working 30% χ 2.38Physical capital to output ratio 2.75 δ 0.07Consumption share of output 0.80 γk 0.31Labor share of income 0.67 γn 0.45Steady-state output 1 z 1.10Import share 0.10 µ 0.88

Third Group: Targets Specific to This EconomyCapacity utilization of consumption sector 0.81 A 1.66Capacity utilization of investment sector 0.81 ζ 0.38

Implications over Other Aggregate VariablesShare of production workers 0.90

Results

Data: for US and EU15

Quantities Data IRBC Shopping ModelY Z Y c Y uc Z c Zuc

A. Variance (US)Output 1.76 1.76 0.57 1.75 1.76 2.16 2.05Consumption 0.98 0.06 0.02 1.52 1.77 2.48 2.06Investment 13.74 37.43 12.43 21.88 26.02 37.32 30.25Employment 1.16 0.14 0.05 1.44 1.52 1.97 1.76Net exports 0.16 0.01 0.006 0.44 0.62 0.98 0.72TFP 0.63 1.17 0.38 0.29 0.28 0.31 0.32Terms of trade 4.98 0.15 0.06 6.39 8.79 13.84 10.22

B. International comovementOutput 0.40 0.40 0.22 0.40 0.16 -0.10 0.16Consumption 0.25 0.38 0.20 0.15 -0.11 -0.35 -0.11Investment 0.23 0.26 0.08 -0.65 -0.80 -0.89 -0.80Employment 0.21 0.45 0.28 0.15 -0.12 -0.37 -0.12TFP 0.30 0.39 0.21 0.73 0.57 0.36 0.57

C. Co-movement within a countryNet exports, Output -0.49 -0.50 -0.56 -0.51 -0.61 -0.71 -0.62Terms of trade, Output -0.33 0.46 0.52 -0.51 -0.61 -0.71 -0.61RER, cH/cF -0.71 0.99 0.99 -0.96 -0.96 -0.96 -0.96

Sensitivity analysis

Data: for US and EU15

Quantities Data Shopping ModelBenchmark Low Armington (0.55)

A. Variance (US)Output 1.76 1.75 1.76Consumption 0.98 1.52 1.14Investment 13.74 21.88 16.76Employment 1.16 1.44 1.40Net exports 0.16 0.44 0.16TFP 0.63 0.29 0.30Terms of trade 4.98 6.39 5.37

B. International comovementOutput 0.40 0.40 0.41Consumption 0.25 0.15 0.28Investment 0.23 -0.65 -0.35Employment 0.21 0.15 0.23TFP 0.30 0.73 0.65

C. Co-movement within a countryNet exports, Output -0.49 -0.51 -0.52Terms of trade, Output -0.33 -0.51 -0.51RER, cH/cF -0.71 -0.96 -0.95

Summary

With demand shocks only, our shopping model can account forpuzzles in the international economics:

I Backus-Smith puzzle: corr(RER, cH/cF ) < 0

I Countercyclical terms of trade

I Volatile net exports

I International consumption correlation smaller than outputcorrelation

References

Backus, D.K. and G.W. Smith. 1993. “Consumption and real exchange rates in dynamic economies withnon-traded goods.” Journal of International Economics 35 (3):297–316.

Bai, Yan, Jose-Vıctor Rıos-Rull, and Kjetil Storesletten. 2011. “Demand Shocks as Productivity Shocks.” WorkingPaper, Federal Reserve Bank of Minneapolis.

Corsetti, G., L. Dedola, and S. Leduc. 2008. “International risk sharing and the transmission of productivityshocks.” Review of Economic Studies 75 (2):443–473.

Engel, C. and J. Wang. 2011. “International trade in durable goods: Understanding volatility, cyclicality, andelasticities.” Journal of International Economics 83 (1):37–52.

Karabarbounis, L. 2012. “Home Production, Labor Wedges, and International Real Business Cycles.” Tech. rep.,National Bureau of Economic Research.