Households: Preferences Continuum of households that live forever ( β ), are subject to uninsurable idiosyncratic and aggregate shocks. H’holds care about quantities and number of varieties of nontradables. �� I N � ρ 1 = c Ni I ρ ρ c N = c Ni di N 0 Households have to search for varieties, its number is a choice . I N = d Ψ d ( Q g ) Ψ d ( Q g ) : Probability (per search unit) of finding a variety. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 9 / 58
Households: Preferences Continuum of households that live forever ( β ), are subject to uninsurable idiosyncratic and aggregate shocks. H’holds care about quantities and number of varieties of nontradables. �� I N � ρ 1 = c Ni I ρ ρ c N = c Ni di N 0 Households have to search for varieties, its number is a choice . I N = d Ψ d ( Q g ) Ψ d ( Q g ) : Probability (per search unit) of finding a variety. Households also like tradables and housing and dislike goods searching u [ c A ( c N I ρ N , c T ) , h, d ] Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 9 / 58
Households: Endowments and Wealth Household skill type is ǫ , follows a Markov chain Γ ǫ,ǫ ′ . Moves slowly and accommodates opportunities to get rich. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 10 / 58
Households: Endowments and Wealth Household skill type is ǫ , follows a Markov chain Γ ǫ,ǫ ′ . Moves slowly and accommodates opportunities to get rich. Households either have a job e = 1 or not e = 0 . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 10 / 58
Households: Endowments and Wealth Household skill type is ǫ , follows a Markov chain Γ ǫ,ǫ ′ . Moves slowly and accommodates opportunities to get rich. Households either have a job e = 1 or not e = 0 . Type-dependent exogenous job destruction rate δ ǫ n . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 10 / 58
Households: Endowments and Wealth Household skill type is ǫ , follows a Markov chain Γ ǫ,ǫ ′ . Moves slowly and accommodates opportunities to get rich. Households either have a job e = 1 or not e = 0 . Type-dependent exogenous job destruction rate δ ǫ n . Job finding rate is type independent and depends on job creation by firms (workers are rationed, it is like no matching function in labor market but hiring costs) ([Fang and Nie(2013)]) . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 10 / 58
Households: Endowments and Wealth Household skill type is ǫ , follows a Markov chain Γ ǫ,ǫ ′ . Moves slowly and accommodates opportunities to get rich. Households either have a job e = 1 or not e = 0 . Type-dependent exogenous job destruction rate δ ǫ n . Job finding rate is type independent and depends on job creation by firms (workers are rationed, it is like no matching function in labor market but hiring costs) ([Fang and Nie(2013)]) . Households have assets a . These assets can be allocated to (frictionless) houses and/or to financial assets with a collateral constraint. The poor will have some housing wealth and a mortgage, the rich houses and shares of the economy’s mutual fund. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 10 / 58
Production: two sectors tradables and nontradables. Nontradables Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Tradables (standard) . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Tradables (standard) . Competitive. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Tradables (standard) . Competitive. (Large) Adjustment costs to both capital and labor. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Tradables (standard) . Competitive. (Large) Adjustment costs to both capital and labor. Its output is used for exports, investment, and (part of) consumption. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Production: two sectors tradables and nontradables. Nontradables Monopolistic firms, each one producing a different variety. Each firm/variety has many locations, and each location has its own production function. Labor can be partially reallocated to accommodate demand differences across locations. Firms post prices before the location is filled. Tradables (standard) . Competitive. (Large) Adjustment costs to both capital and labor. Its output is used for exports, investment, and (part of) consumption. Decreasing returns. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 11 / 58
Goods markets Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Random search. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Random search. Richer people consume and search more. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Random search. Richer people consume and search more. Cuts in consumption cut search which cuts productivity. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Goods markets Search frictions in the markets for nontradables: Households look for varieties. Random search. Richer people consume and search more. Cuts in consumption cut search which cuts productivity. Perfect competition and frictionless markets for tradables. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 12 / 58
Labor market Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Labor market Workers are rationed. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Labor market Workers are rationed. Firms hire as many workers as they wish paying hiring costs. (like a vacancy filling probability of 1, with hiring costs). Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Labor market Workers are rationed. Firms hire as many workers as they wish paying hiring costs. (like a vacancy filling probability of 1, with hiring costs). Employment: N = N N + N T . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Labor market Workers are rationed. Firms hire as many workers as they wish paying hiring costs. (like a vacancy filling probability of 1, with hiring costs). Employment: N = N N + N T . Same job finding probability across types: Φ e = V 1 − N . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Labor market Workers are rationed. Firms hire as many workers as they wish paying hiring costs. (like a vacancy filling probability of 1, with hiring costs). Employment: N = N N + N T . Same job finding probability across types: Φ e = V 1 − N . Wages are determined via the following formula � � log w − log w = ε w log Y − log Y It simplifies things. [Gornemann, Kuester, and Nakajima(2012)]. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 13 / 58
Assets markets: Financial assets and houses Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Its interest rate 1 q is predetermined at borrowing time, � 1 , if b ≥ 0 q ( θ, b ′ ) = 1 1+ r ∗ − ς ( θ ) , if b < 0 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Its interest rate 1 q is predetermined at borrowing time, � 1 , if b ≥ 0 q ( θ, b ′ ) = 1 1+ r ∗ − ς ( θ ) , if b < 0 Mortgages have to be collateralized by housing: if b < 0 then q ( θ, b ) | b | ≤ [1 − λ ( θ )] p h ( S ) h Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Its interest rate 1 q is predetermined at borrowing time, � 1 , if b ≥ 0 q ( θ, b ′ ) = 1 1+ r ∗ − ς ( θ ) , if b < 0 Mortgages have to be collateralized by housing: if b < 0 then q ( θ, b ) | b | ≤ [1 − λ ( θ )] p h ( S ) h Positive financial assets ( b > 0 ) are shares of a mutual fund. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Its interest rate 1 q is predetermined at borrowing time, � 1 , if b ≥ 0 q ( θ, b ′ ) = 1 1+ r ∗ − ς ( θ ) , if b < 0 Mortgages have to be collateralized by housing: if b < 0 then q ( θ, b ) | b | ≤ [1 − λ ( θ )] p h ( S ) h Positive financial assets ( b > 0 ) are shares of a mutual fund. Its return is stochastic. Possible capital gains and loses. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
Assets markets: Financial assets and houses Total housing H is in fixed supply. Negative financial assets ( b ′ < 0 ) are (undefaultable) mortgages. Its interest rate 1 q is predetermined at borrowing time, � 1 , if b ≥ 0 q ( θ, b ′ ) = 1 1+ r ∗ − ς ( θ ) , if b < 0 Mortgages have to be collateralized by housing: if b < 0 then q ( θ, b ) | b | ≤ [1 − λ ( θ )] p h ( S ) h Positive financial assets ( b > 0 ) are shares of a mutual fund. Its return is stochastic. Possible capital gains and loses. The return is � 1 + r ( S, S ′ ) , if b ≥ 0 R ( S, S ′ , b ) = 1 , if b < 0 . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 14 / 58
State variables A household is characterized by { ǫ, e, a } . Let X denote the measure over types x = { ǫ, e, a } . The vector of aggregate state variables is S = { θ, B, K N , K T , N N , N T , X } Here B is the net foreign asset position. K and N are predetermined factor inputs. Hence either we do Krusell-Smith or the transition after an unforeseen shock. Today, we do the latter. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 15 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ subject to � I N p i ( S ) c N,i + c T + p h ( S ) h + q ( θ, b ) b = a + 1 e =1 w ( S ) ǫ + 1 e =0 w BC 0 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ subject to � I N p i ( S ) c N,i + c T + p h ( S ) h + q ( θ, b ) b = a + 1 e =1 w ( S ) ǫ + 1 e =0 w BC 0 a ′ ( S ′ , b, h ) = p h ( S ′ ) h + R ( S, S ′ , b ) b AA Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ subject to � I N p i ( S ) c N,i + c T + p h ( S ) h + q ( θ, b ) b = a + 1 e =1 w ( S ) ǫ + 1 e =0 w BC 0 a ′ ( S ′ , b, h ) = p h ( S ′ ) h + R ( S, S ′ , b ) b AA q ( θ, b ) b ≥ − λ ( θ ) p h ( S ) h FC Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ subject to � I N p i ( S ) c N,i + c T + p h ( S ) h + q ( θ, b ) b = a + 1 e =1 w ( S ) ǫ + 1 e =0 w BC 0 a ′ ( S ′ , b, h ) = p h ( S ′ ) h + R ( S, S ′ , b ) b AA q ( θ, b ) b ≥ − λ ( θ ) p h ( S ) h FC I N = d Ψ d [ Q g ( S )] SC Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Households’ problem V ( S, ǫ, e, a ) = c N,i ,c T , I N ,h,d u ( c A , h, d )+ max � Π θ θ,θ ′ Π w e ′ | e,ǫ ( S ′ ) Π ε ǫ,ǫ ′ V [ S ′ , ǫ ′ , e ′ , a ′ ( S ′ , b, h )] β ǫ ′ ,e ′ ,θ ′ subject to � I N p i ( S ) c N,i + c T + p h ( S ) h + q ( θ, b ) b = a + 1 e =1 w ( S ) ǫ + 1 e =0 w BC 0 a ′ ( S ′ , b, h ) = p h ( S ′ ) h + R ( S, S ′ , b ) b AA q ( θ, b ) b ≥ − λ ( θ ) p h ( S ) h FC I N = d Ψ d [ Q g ( S )] SC S ′ = G ( S, θ ′ ) RE Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 16 / 58
Nontradable firms’ problem At each location, the production function is F N ( k, ℓ 1 , ℓ 2 ) = z N k α 0 ℓ α 1 1 ℓ α 2 2 k and ℓ 1 are pre-installed. ℓ 2 is variable to meet different demands. � � ρ 1 − ρ c N ( S, x ) p i The demand function is given by c ( p i , S, x ) = p ( S ) When a shopper wants to buy c units of goods at a location, the amount of variable labor ℓ 2 needed to produce c is � � − 1 c − 1 z N k α 0 ℓ α 1 f ℓ ( c, k, ℓ 1 ) = α 2 1 At the posted price p i , the total variable labor needed is � f ℓ [ c ( p i , S, x ) , k, ℓ 1 ] d ( x, S ) ℓ 2 ≥ Ψ f [ Q g ( S )] D ( S ) Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 17 / 58
Nontradable firms’ problem � Ω N ( S, k, n ) = max Ψ f [ Q g ( S )] p i c ( p i , S, ǫ, e, a ) dx − w ( S ) ℓ − i − κv i,v,p i ℓ 1 ,ℓ 2 � θ,θ ′ Ω N ( S ′ , k ′ , n ′ ) Π θ + 1 + r ∗ θ ′ subject to � f ℓ [ c ( p i , S, x ) , k, ℓ 1 ] d ( x, S ) ℓ 2 ≥ Ψ f [ Q g ( S )] DC D ( S ) ℓ 1 + ℓ 2 = nǫ ( S ) SL k ′ = (1 − δ k ) k + i − φ N ( k, i ) LMK n ′ = [1 − δ n ( S )] n + v LML S ′ = G ( S, θ ′ ) RE Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 18 / 58
Tradable firms’ problem Ω T ( S, k, n ) = max i,v F T ( k, ℓ ) − w ( S ) ℓ − i − κv − φ T,n ( n ′ , n ) θ,θ ′ Ω T ( S ′ , k ′ , n ′ ) � Π θ + 1 + r ∗ θ ′ subject to k ′ = (1 − δ k ) k + i − φ T,k ( k, i ) ℓ = nǫ ( S ) n ′ = [1 − δ n ( S )] n + v S ′ = G ( S ) . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 19 / 58
Mutual fund Financial wealth in the economy is � L + = b ( S, ǫ, e, a ) dx b> 0 Mortgages in the economy are � L − = − b ( S, ǫ, e, a ) dx b< 0 Net foreign asset position of the country (the mutual fund owns all firms) � � 1 Ω N ( S ) − π N ( S ) + Ω T ( S ) − π T ( S ) + B = L + − 1 + r ∗ L − The realized rate of return is 1 + r ( S, S ′ ) = Ω N ( S ′ ) + Ω T ( S ′ ) + (1 + r ∗ ) B + L − L + Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 20 / 58
Equilibrium An equilibrium is a set of decision rules and values for households, firms’ values and decision rules, and a set aggregate variables of aggregate states, such that: Households’ and firms’ policy functions and value functions solve the corresponding program problems. Aggregate searching consistence � D ( S ) = d ( S, ǫ, e, a ) dx, Nontradable prices satisfies p ( S ) = p i ( S, K N , N N ) dx, Housing market clears � h ( S, ǫ, e, a ) dx = H . Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 21 / 58
Equilibrium Average separation probability and labor force quality � � ǫ δ n ( ǫ ) n ( ǫ ) ǫ ǫn ( ǫ ) δ n ( S ) = ǫ ( S ) = , N N Rate of return to the mutual fund satisfies � 1 + r ( S, S ′ ) = Ω N ( S ′ ) + Ω T ( S ′ ) + (1 + r ∗ ) B + b< 0 b ( S, x ) � b> 0 b ( S, x ) Wage satisfies � � log w ( S ) − log w = ε w log Y ( S ) − log Y The law of motion G ( S ) is consistent with households’ decisions and employment dynamics. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 22 / 58
Mapping the Model to Data Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 23 / 58
Functional forms Preferences � � 1 − σ c d 1+ γ 1 u ( c A , h, d ) = c A − ξ d + v ( h ) 1 − σ c 1 + γ where there is an Armington aggregator for consumption � � η η − 1 η − 1 η − 1 ω ( c N I ρ η c A = N ) η + (1 − ω ) c T and houses are inferior goods as a proxy for segmentation of housing markets � h ( h + h 1 ) 1 − σ 1 h , ξ h if h < � h 1 − σ 1 v ( h ) = h ( h + h 2 ) 1 − σ 2 h , if h ≥ � ξ h h. 1 − σ 2 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 24 / 58
−0.1 3.5 Housing −0.15 3 −0.2 2.5 −0.25 2 −0.3 1.5 −0.35 1 Consumption −0.4 0.5 −0.45 Housing function with less curvature Housing function with more curvature −0.5 0 1 2 3 4 5 0 0.5 1 1.5 2 Housing utility function Engel Curve: consumption vs housing Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 25 / 58
Functional forms Production function F T ( k, ℓ ) = z T k θ 0 ℓ θ 1 F N ( k, ℓ 1 , ℓ 2 ) = z N k α 0 ℓ α 1 ℓ α 2 2 , 1 Capital adjustment cost in the nontradable goods sector � i � 2 φ N ( i, k ) = ε N k − δ k k 2 Capital and employment adjustment cost in the tradable goods sector � i � 2 � n ′ � 2 φ T,k ( i, k ) = ε T,k φ T,n ( n ′ , n ) = ε T,n k − δ k n − 1 k, n 2 2 Matching technology M ( D, T ) = νD µ T 1 − µ Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 26 / 58
Exogenously determined parameters A period is half a quarter. Parameter Value Risk aversion for consumption, σ c 2.0 Risk aversion for housing, σ 1 2.0 h Risk aversion for housing, σ 2 10.0 h Curvature of shopping, γ 1.5 Elasticity of substitution bw tradables and nontradables, η 0.80 Cutoff value for housing utility, � h 1.4 Price markup, ρ 1.1 Loan to value ratio, λ 0.75 Interest rate for international bonds, r ∗ 4% Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 27 / 58
Endogenously determined parameters: aggregate Target Value Parameter Value Wealth to output ratio 4.70 0.98 β Housing value to output ratio 1.67 0.95 ξ h Debt to output ratio 0.75 30.77 ǫ 4 Share of tradables 0.30 0.95 ω Occupancy Rate 0.81 ν 0.81 Capital to output ratio 2.00 δ k 0.01 Labor Share in nontradables 0.64 α 0 0.27 α 1 = α 2 —— α 1 0.36 Labor Share in tradables 0.66 θ 1 0.66 1 . 4 θ 0 + θ 1 = 1 —— θ 0 0.23 Vacancy cost to output ratio 0.02 κ 0.42 Home production to lowest earning ratio 0.50 w 0.07 Units Parameters Output 1 z N 0.93 Relative price of nontradables 1 z T 0.48 Market tightness in goods markets 1 ξ d 0.03 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 28 / 58
Endogenously determined parameters: cross-section Lorenz Target Value Parameter Value δ 1 Job duration for type 1 1.5 year 0.083 n δ 3 Job duration for type 3 5 year 0.025 n δ 4 Job duration for type 4 5 year 0.025 n δ 2 Unemployment rate 6% 0.048 n Π ǫ Wealth Gini index 0.82 0.0007 1 , 4 Π ǫ Earnings Gini index 0.64 0.0156 4 , 1 Π ǫ Earning autocorrelation 0.91 0.9660 1 , 1 Π ǫ Earning stdev 0.20 0.9774 2 , 2 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 29 / 58
Lorenz Curve Return Data Model 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Net worth Housing asset Financial asset Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 30 / 58
Experiments: once and for all set of surprises in the environment Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 31 / 58
Experiments: once and for all set of surprises in the environment Over the next 4.5 months the down payment changes from 25% to 1 27.5% to 30% to 32.5% (to avoid having households with empty choice set). Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 31 / 58
Experiments: once and for all set of surprises in the environment Over the next 4.5 months the down payment changes from 25% to 1 27.5% to 30% to 32.5% (to avoid having households with empty choice set). The borrowing interest rate’s surcharge goes from zero to .3%. 2 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 31 / 58
Experiments: once and for all set of surprises in the environment Over the next 4.5 months the down payment changes from 25% to 1 27.5% to 30% to 32.5% (to avoid having households with empty choice set). The borrowing interest rate’s surcharge goes from zero to .3%. 2 Both at the same time. 3 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 31 / 58
Experiments: once and for all set of surprises in the environment Over the next 4.5 months the down payment changes from 25% to 1 27.5% to 30% to 32.5% (to avoid having households with empty choice set). The borrowing interest rate’s surcharge goes from zero to .3%. 2 Both at the same time. 3 The inverse process. Credit expansion. 4 • All of these with fixed and flexible wages. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 31 / 58
Long Run Properties • Typically like in all [Aiyagari(1994)] - [Bewley(1986)] - [Huggett(1993)] - [Imrohoro˘ glu(1989)] type models, in the long run output and wealth end up being higher. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 32 / 58
Long Run Properties • Typically like in all [Aiyagari(1994)] - [Bewley(1986)] - [Huggett(1993)] - [Imrohoro˘ glu(1989)] type models, in the long run output and wealth end up being higher. • But in our economies the transition is associated to a recession. Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 32 / 58
Experiment : gradual worsening of both λ and borrowing cost 12 1 11 0 10 −1 9 −2 8 −3 7 −4 6 −5 5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Real output Unemployment 0 5 −1 0 −5 −2 −10 −3 −15 −4 −20 −5 −25 −6 −30 −7 −35 −40 −8 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Investment Consumption Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 33 / 58
Experiment: gradual worsening of both λ and borrowing cost 0 0 −1 −5 −2 −10 −3 −4 −15 −5 −20 −6 −25 −7 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Debt Wealth 0 −5 −10 −15 −20 0 1 2 3 4 5 6 7 8 9 10 Housing price Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 34 / 58
Experiment: gradual worsening of both λ and borrowing cost 0.5 2 1.5 0 1 0.5 −0.5 0 −1 −0.5 −1 −1.5 −1.5 −2 −2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 TFP with total hours Labor Productivity 2.5 0.5 2 0 1.5 −0.5 1 −1 0.5 −1.5 0 −0.5 −2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Labor quality TFP with total labor inputs Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 35 / 58
Experiment : gradual worsening of both λ and borrowing cost Change of labor quality in both pools when wages are flexible 2.5 Employed Unemployed 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 36 / 58
Experiment: gradual improvement of λ from 0.75 to 0.825 6.4 1.2 6.2 1 6 0.8 5.8 0.6 5.6 0.4 5.4 0.2 5.2 0 5 −0.2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Real output Unemployment 14 2.5 12 2 10 1.5 8 1 6 4 0.5 2 0 0 −0.5 −2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Consumption Investment Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 37 / 58
Experiment: gradual improvement of λ from 0.75 to 0.825 12 3 2.5 10 2 8 1.5 6 1 4 0.5 2 0 −0.5 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Wealth Debt 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Housing price Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 38 / 58
Experiment: gradual improvement of λ from 0.75 to 0.825 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 TFP with total hours Labor Productivity 0.1 0.6 0.5 0 0.4 −0.1 0.3 0.2 −0.2 0.1 −0.3 0 −0.4 −0.1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Labor quality TFP with total labor inputs Flexible wage Fixed wage Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 39 / 58
Experiment 5: More flexible wage schedule 9 0.5 8.5 0 −0.5 8 −1 7.5 −1.5 7 −2 6.5 −2.5 6 −3 5.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Real output Unemployment 1 5 0 0 −1 −5 −2 −10 −3 −15 −4 −20 −5 −25 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Investment Consumption Flexible wage ǫ w = 0 . 45 Flexible wage ǫ w = 1 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 40 / 58
Experiment 5: More flexible wage schedule 1 0 0 −2 −1 −4 −2 −6 −8 −3 −10 −4 −12 −5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Debt Wealth 0 −2 −4 −6 −8 −10 −12 −14 0 1 2 3 4 5 6 7 8 9 10 Housing price Flexible wage ǫ w = 0 . 45 Flexible wage ǫ w = 1 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 41 / 58
Experiment 5: More flexible wage schedule 0.4 0.5 0.2 0 0 −0.2 −0.4 −0.5 −0.6 −0.8 −1 −1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 TFP with total hours Labor Productivity 1.2 0.2 1 0 0.8 −0.2 0.6 −0.4 0.4 −0.6 0.2 −0.8 0 −1 −0.2 −1.2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Labor quality TFP with total labor inputs Flexible wage ǫ w = 0 . 45 Flexible wage ǫ w = 1 Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 42 / 58
Results: a boom and bust cycle 0.76 0.75 0.74 0.73 0.72 0.71 0.7 0.69 0.68 0.67 0 10 20 30 40 50 60 70 80 Loan to value ratio λ Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 43 / 58
Results: a boom and bust cycle 8.5 1.5 1 8 0.5 7.5 0 7 −0.5 6.5 −1 6 −1.5 5.5 −2 −2.5 5 −3 4.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Real output Unemployment 3 15 10 2 5 1 0 0 −5 −1 −10 −2 −15 −3 −20 −25 −4 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Investment Consumption Huo & R´ ıos-Rull, UMN, Mpls Fed, CAERP Financial Frictions, Asset Prices, & the Great Recession LBS Nov 13 2014 44 / 58
Recommend
More recommend
