International Business Cycles Redux Yan Bai and Jos e-V ctor R - PowerPoint PPT Presentation

International Business Cycles Redux Yan Bai and Jos´ e-V´ ıctor R´ ıos-Rull University of Rochester, University of Minnesota, Federal Reserve Bank of Minneapolis, Mpls Fed, NBER Wednesday 16 th January, 2013

Punchline ◮ Backus-Smith (Backus and Smith (1993)) puzzle: households consume more domestic goods when they are more expensive ◮ corr ( RER , cH / cF ) > 0. ◮ Yet standard models (e.g. RBC) predict the opposite. ◮ Literature: demand shocks, low elasticity between home and foreign goods (Corsetti, Dedola, and Leduc (2008)) , non-tradable goods (Engel and Wang (2011)) , labor wedge from home production (Karabarbounis (2012)) . ◮ We pose an explanation based on demand shocks in the context of environments where expenditures shape productivity (a la Bai, R´ ıos-Rull, and Storesletten (2011)) . We also obtain ◮ Countercyclical terms of trade. ◮ Volatile net exports. ◮ Lower international cons corr than output’s.

The two country two good model with shopping ◮ Two countries j = { 1 , 2 } with representative agents in each. ◮ Build a top of stochastic growth model. ◮ There are incomplete markets (no insurance for preference shocks). ◮ There is perfect mobility of capital without impediments to cross country ownership.

Preferences: Current utility of Households in country j � H c � c jj , c jj ∗ � , n j , H d � d jj , d jj ∗ � , θ j � u ◮ H c ( c jj , c jj ∗ ) is an (Armington) aggregator of the home ( c jj ) and the foreign ( c jj ∗ ) good. ◮ n j are hours worked in country j . ◮ H d ( d jj , d jj ∗ ) is an aggregator (maybe linear) of search shopping effort at home ( d jj ) and abroad ( d jj ∗ ). ◮ θ j is a Markovian preference shock. • Households cannot change their country of residence, which makes labor immobile. Like in Bai, R´ ıos-Rull, and Storesletten (2011) consumption requires that the household finds the goods which in turn 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 or investment and capacity is ( n F ) j � γ n k j � γ k � F ( k j , ( n F ) j ) = z � • New capital consists of an (Armington) aggregator of home and foreign investment goods: k ′ j = (1 − δ ) k j + H i ( i j , i j ∗ ) • New capjtal has to be purchased that requires shoppers looking for the home investment good ( n j , j ) k and for the foreign investment good ( n j , 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 � H c � c jj , c jj ∗ � , n j , H d � d jj , d jj ∗ � , θ j � u Again both consumptions require that they are shopped so d jj Ψ d ( Q cjj ) F cjj c jj = d jj ∗ Ψ d ( Q cjj ∗ ) F cjj ∗ c jj ∗ = Ψ d ( Q cj ℓ ) is the probability that a country j consumption • shopper has of matching a consumption firm from country ℓ that catters to j shoppers. Q cj ℓ is market tightness in that consumption goods market and F cj ℓ 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 = { θ 1 , θ 2 , K 1 , K 2 , B 1 } where K j is capital installed in country j and B 1 is the share of total wealth held by country 1 households. 2. There are two active markets in consumption goods (one for locals and the other for foreigners) and two markets in investment goods in each country for a total of 8 markets. 3. Firms that produce consumption and investment, for local buyers and to export choose the same inputs F cjj = F i ℓ j = .. = F ( K j , N yj ) = F j ( S ). We use n ( k , F ) to denote the inverse f ( k , n ). 4. All firms in each country get the same expected revenue (but not necessarily the same price and market tightness).

The household problem � H c � c jj , c jj ∗ � , n j , H d � d jj , d jj ∗ � , θ j � v j ( S , b ) = v j ( S ′ , b ′ ) | � c .. , d .. , n , b ′ u max + β E j ∗ b ′ + p cj ℓ ( S ) c j ℓ = b [1 + R ( S )] + n j w j ( S ) � ℓ = j c j ℓ = d j ℓ Ψ d � � Q cj ℓ ( S ) F cj ℓ ( S ) ℓ = j , j ∗ , S ′ = G ( S )

The household problem — First Order Conditions Hholds’ FOC (and RA): for ℓ = j , j ∗ and m = j , j ∗ � p cj ℓ (1 + R ′ ) � � � u j d H dj u j ′ d H dj ′ u j c H cj u j ′ c H cj ′ m ℓ ℓ − = β E m − | θ Ψ cj ℓ Ψ cjm ′ p cjm ′ F cj ℓ F ckm ′ d d u j d H d p cj ℓ u j c H c ℓ ℓ − = u n Ψ cj ℓ w j F cj ℓ d � � � � u j d H dj u j d H dj 1 1 c H cj m u j ℓ u j c H cj ℓ − = m − p cj ℓ Ψ cj ℓ p cjm Ψ cjm F cj ℓ F cjm d d Define marginal utility of savings M ( S ): � � � � u j ′ d H dj ′ (1 + R ′ ) u j ′ c H cj ′ m M ( S ) = β E m − | θ p cjm ′ Ψ cjm ′ F ckm ′ d

Consumption (or invt) firms in a ( p cj ℓ , F cj ℓ , Q cj ℓ ) submarket n k .. , k ′ , i jj , i jj ∗ Ψ T ( Q cj ℓ ) p cj ℓ F cj ℓ − w j ( S ) � n ( k , F cj ℓ ) + n kjj + n kjj ∗ � Ω j ( S , k ) = max � Ω j ( S ′ , k ′ ) − p ijj ( S ) i jj − p ijj ∗ ( S ) i jj ∗ + E � � � � θ � 1 + R ( S ′ ) i j ℓ = ( n kj ℓ ζ ) Ψ d [ Q ij ℓ ( S )] F ℓ ( S ) for ℓ = j , j ∗ s.t. k ′ = (1 − δ ) k + H i � i jj , i jj ∗ � S ′ = G ( S ) with FOC (and RA condition) for ℓ = j , j ∗ � Ω j � � S ′ , K j ′ � � w j ( S ) � H ij k ζ Ψ d [ Q ij ℓ ( S )] F ℓ ( S ) + p ij ℓ ( S ) ℓ E � θ = � 1 + R ( S ′ ) �

The household problem � H c � c jj , c jj ∗ � , n j , H d � d jj , d jj ∗ � , θ j � v j ( S , b ) = v j ( S ′ , b ′ ) | � c .. , d .. , n , b ′ u max + β E j ∗ b ′ + p cj ℓ ( S ) c j ℓ = b [1 + R ( S )] + n j w j ( S ) � ℓ = j c j ℓ = d j ℓ Ψ d � � Q cj ℓ ( S ) F cj ℓ ( S ) ℓ = j , j ∗ , S ′ = G ( S )

The household problem — First Order Conditions Hholds’ FOC (and RA): for ℓ = j , j ∗ and m = j , j ∗ p cj ℓ (1 + R ′ ) � � � � u j d H dj u j ′ d H dj ′ u j c H cj ℓ u j ′ c H cj ′ m ℓ − = β E m − | θ Ψ cj ℓ p cjm ′ Ψ cjm ′ F cj ℓ F ckm ′ d d u j d H d p cj ℓ u j c H c ℓ ℓ − = u n Ψ cj ℓ w j F cj ℓ d � u j d H dj � � u j � d H dj 1 1 c H cj u j ℓ u j c H cj m ℓ − = m − p cj ℓ Ψ cj ℓ p cjm Ψ cjm F cj ℓ F cjm d d We define marginal utility of savings M ( S ) as � � � � u j ′ d H dj ′ (1 + R ′ ) m u j ′ c H cj ′ M ( S ) = β E m − | θ Ψ cjm ′ p cjm ′ F ckm ′ d

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 given expected revenue for sellers. • The equilibrium objects (44) are functions of ( S ) for � � Q cj ℓ , Q ij ℓ , N kj ℓ , C j ℓ , I j ℓ , p cj ℓ , p ij ℓ , T cj ℓ , T ij ℓ , w j , N yj , N j , B ′ , R j ∈{ 1 , 2 } ,ℓ ∈{ 1

Competitive Search in Consumption Market cj ℓ The rewards for the hhold to send a shopper to a ( p cj ℓ , F cj ℓ , Q cj ℓ ) market is ℓ ( S )+Ψ d ( Q cj ℓ ) F cj ℓ � � Q cj ℓ , p cj ℓ , F cj ℓ − u j d ( S ) H dj u j c ( S ) H cj ℓ ( S ) − p cj ℓ M ( S ) Φ = max ς cj ℓ ≤ p cj ℓ Ψ d ( Q cj ℓ ) F cj ℓ − w j ( S ) n ( k , F cj ℓ ) (1) Q cj ℓ Solving p cj ℓ from equation (1) and plugging it into the objective function, we have (and ignore the sunk shopping cost) ℓ ( S ) − ς cj ℓ + w j ( S ) n ( k , F cj ℓ ) � � Q cj ℓ , p cj ℓ , F cj ℓ Ψ d ( Q cj ℓ ) F cj ℓ u j c ( S ) H cj max M ( S ) Ψ d ( Q cj ℓ ) F cj ℓ / Q cj ℓ

Competitive Search in Consumption Market cj ℓ First order condition over Q cj ℓ : ς cj ℓ + w j ( S ) n ( k , F cj ℓ ) (1 − α ) A ( Q cj ℓ ) − α F cj ℓ u j c ( S ) H cj � � ℓ ( S ) − M ( S ) = 0 First order condition over F cj ℓ : ℓ ( S ) − w j ( S ) dn ( k , F cj ℓ ) c ( S ) H cj A ( Q cj ℓ ) 1 − α u j Q cj ℓ M ( S ) = 0 dF cj ℓ Thus, two equations characterize the equilibrium in market cj ℓ , c ( S ) H cj (1 − α ) u j ℓ ( S ) p cj ℓ = M ( S ) w j ( S ) Ψ d ( Q cj ℓ ) 1 = f n ( S ) p cj ℓ Q cj ℓ 1 − α

Competitive Search in Investment Market ij ℓ The rewards for a firm to send a shopper to a ( p ij ℓ , F ij ℓ , Q ij ℓ ) market is Q ij ℓ , p ij ℓ , F ij ℓ − w j ( S )+ ζ Ψ d ( Q ij ℓ ) F ij ℓ � ℓ ( S ) E { Ω( S ′ , k ′ )Π( S , S ′ ) } − p ij ℓ � H ij Φ F = max ς ij ℓ ≤ p ij ℓ Ψ d ( Q ij ℓ ) F ij ℓ − w j ( S ) n ( k , F ij ℓ ) (2) Q ij ℓ Solving p ij ℓ from equation (2) and plugging it into the objective function, we have (and ignore the sunk wage cost) ℓ ( S ) E { Ω( S ′ , k ′ )Π( S , S ′ ) } − ζ ij ℓ + w j ( S ) n ( k , F ij ℓ ) � � H ij Q ij ℓ , p ij ℓ , F ij ℓ ζ Ψ d ( Q ij ℓ ) F ij ℓ max Ψ d ( Q ij ℓ ) F ij ℓ / Q ij ℓ