Housing and the Business Cycle Morris Davis and Jonathan Heathcote - PowerPoint PPT Presentation

Housing and the Business Cycle Morris Davis and Jonathan Heathcote Winter 2009 Huw Lloyd-Ellis () ECON917 Winter 2009 1 / 21

Motivation Need to distinguish between housing and non–housing investment , ! produced using di¤erent technologies , ! di¤erent rates of depreciation , ! housing yields "home production" services not in National Accounts "Stylized facts" for models with heterogeneous capital goods: (1) comovement between consumption and investment in di¤erent assets (2) residential investment is 2x as volatile as business investment (3) residential investment leads cycle, business investment lags it Huw Lloyd-Ellis () ECON917 Winter 2009 2 / 21

Model Overview Households — consume goods and housing services, supply land and labour " Real estate developers — combine land and structures to build houses " Two …nal goods sector — one produces structures; the other C and K " Three intermediate sectors — construction, manufacturing and services " Labour, capital and productivity shocks Huw Lloyd-Ellis () ECON917 Winter 2009 3 / 21

Main Findings Purely neoclassical model can account for "puzzles" (1) and (2), but not (3) Also matches facts on relative volatility of sub-sectors Implies pro-cyclical house prices, but not volatile enough Why positive comovement and high volatility? , ! not due to correlated shocks , ! …nal goods sectors use all intermediates , ! housing requires land, which acts like an adjustment cost , ! residential investment is relatively labour intensive , ! low depreciation of housing Huw Lloyd-Ellis () ECON917 Winter 2009 4 / 21

Population Gross population growth: η > 1 All variables in per capita terms Huw Lloyd-Ellis () ECON917 Winter 2009 5 / 21

Intermediate sectors Intermediate …rms’ output: x it = k θ i it ( z it n it ) 1 � θ i i 2 f b , m , s g , ! rent capital at rate r t and hire labour at w t , ! output prices p it , ! productivity shocks: ^ z t + 1 = B^ z t + ε t + 1 ε t + 1 � N ( 0 , V ) where z st ] 0 ^ z t = [ ln ˜ z bt , ln ˜ z mt , ln ˜ ln ˜ z it = ln z it � t ln g zi � ln z i 0 Huw Lloyd-Ellis () ECON917 Winter 2009 6 / 21

Final goods sectors Final goods’ output: y jt = b B j jt m M j jt s S j j 2 f c , d g , S j = 1 � B j � M j jt , ! output prices, p ct = 1 (numeraire) and p dt = relative price of residential investment Huw Lloyd-Ellis () ECON917 Winter 2009 7 / 21

Land and real estate Each household sells one unit of land each period Developers combine new residential structures, x d , and new land, x l , to build houses: y ht = x φ lt x 1 � φ dt Structures depreciate at rate δ s Total stock of "e¤ective" housing: η h t + 1 = x 1 � φ x φ lt + ( 1 � δ s ) 1 � φ h t dt Let 1 � δ h = ( 1 � δ s ) 1 � φ Huw Lloyd-Ellis () ECON917 Winter 2009 8 / 21

Households Optimization problem: ∞ β t η t U ( c t , h t , n t ) ∑ max E 0 s.t. t = 0 c t + η k t + 1 + η p ht h t + 1 = ( 1 � τ n ) w t n t + [ 1 � ( 1 � τ k ) ( r t � δ k )] k t +( 1 � δ h ) p ht h t + p lt x lt + ξ t where � � 1 � σ 1 c µ c t h µ h t ( 1 � n t ) 1 � µ c � µ h U ( c t , h t , n t ) = 1 � σ Huw Lloyd-Ellis () ECON917 Winter 2009 9 / 21

First–order conditions: U c ( t ) = � U n ( t )( 1 � τ n ) w t U c ( t ) = β E t [ U c ( t + 1 ) ( 1 � ( 1 � τ k ) ( r t + 1 � δ k ))] U c ( t ) p ht = β E t [ U c ( t + 1 ) ( 1 � δ h ) p ht + 1 + U h ( t + 1 )] where � � 1 � σ c µ c t h µ h µ c c � 1 t ( 1 � n t ) 1 � µ c � µ h U c ( t ) = t � ( 1 � µ c � µ h ) ( 1 � n t ) � 1 � � 1 � σ c µ c t h µ h t ( 1 � n t ) 1 � µ c � µ h U n ( t ) = � � 1 � σ c µ c t h µ h µ h h � 1 t ( 1 � n t ) 1 � µ c � µ h U h ( t ) = t Huw Lloyd-Ellis () ECON917 Winter 2009 10 / 21

Market Clearing Conditions Final goods and real estate c t + η k t + 1 + g t = y ct + ( 1 � δ k ) k t η h t + 1 = y ht + ( 1 � δ h ) h t x dt = y dt x lt = 1 Intermediate goods b ct + b dt = x bt m ct + m dt = x mt s ct + s dt = x st Factor markets k bt + k mt + k st = k t n bt + n mt + n st = n t Huw Lloyd-Ellis () ECON917 Winter 2009 11 / 21

Government budget constraint: ξ t + g t = τ n w t n t + τ k ( r t � δ k ) k t Huw Lloyd-Ellis () ECON917 Winter 2009 12 / 21

Equilibrium prices Factor prices ( z it n it ) 1 � θ i p it θ i k θ i � 1 r t = i 2 f b , m , s g it z it p it ( 1 � θ i ) k θ i it ( z it n it ) � θ i w t = Prices of intermediates y ct p dt y dt p bt = B c = B d b ct b dt y ct p dt y dt p mt = M c = M d m ct m dt y ct p dt y dt p st = S c = S d s ct s dt Prices of structures and land ( 1 � φ ) p ht y ht p dt = x dt φ p ht y ht p lt = x lt Huw Lloyd-Ellis () ECON917 Winter 2009 13 / 21

Implication for House prices Relative price of residential investment can be written as ln p dt = ( B c � B d ) ( 1 � θ b ) ln z bt + ( M c � M d ) ( 1 � θ m ) ln z mt + ( S c � S d ) ( 1 � θ s ) ln z st + other terms , ! a positive shock in sector i will reduce p dt if residential investment is relatively intensive in input i ) implications for comovement Price of new housing ln p ht = � ln ( 1 � φ ) + φ ln y dt + ln p dt Huw Lloyd-Ellis () ECON917 Winter 2009 14 / 21

Mapping between model and NIPA In NIPA private consumption includes imputed value for rents from owner–occupied housing: PCE t = c t + q t h t where q t = U h ( c t , h t , n t ) U c ( c t , h t , n t ) In NIPA, raw land is not part of GDP ) should only include value of residential investment, not of new houses: GDP t = y ct + p dt y dt + q t h t Real private consumption and GDP de…ned using balanced growth prices ) does not capture short-run price movements Huw Lloyd-Ellis () ECON917 Winter 2009 15 / 21

Balanced Growth Path Although each sector has di¤erent growth rates, a BGP exists due to Cobb-Douglas assumptions All variables are made stationary by dividing by gross growth rate x t = x t ˆ g t x Model is solved using Klein (2000) Huw Lloyd-Ellis () ECON917 Winter 2009 16 / 21

Table 1: Growth Rates on Balanced Growth Path (growth rates gross, variables per-capita) n b , n m , n s , n, r 1 [ ] k b , k m , k s , k, c, i k , g, y c , w 1 ( ) ( ) ( ) = − θ − θ − θ B 1 M 1 S 1 g g g g − θ − θ − θ c b c m c s 1 B M S c b c m c s k zb zm zs b c , b h , x b θ − θ = 1 g g g b b b k zb θ − θ m c , m h , x m = 1 g g g m m m k zm s c , s h , x s = θ − θ 1 g g g s s s k zs x d = B M S g g g g h h h d b m s x l = η − 1 g l y h , h = φ − φ 1 g g g h l d p h y h , p d x d , p l x l , p b x b , p m x m , p s x s g k Table 2: Tax Rates, Depreciation Rates, Adjustment Costs, Preference Parameters Davis Heathcote Grenwood Hercowitz (GH) Tax rate on capital income: τ k 0.3788 0.50 Tax rate on labor income: τ n 0.2892 0.25 0.179* 4 Govt. cons. to GDP 0.0 Transfers to GDP 0.076* Depreciation rate for capital: δ k 0.0557* 0.078 Depreciation rate for res. structures: δ s 0.0157* 0.078 Land’s share in new housing: φ 0.106 Population growth rate: η 1.0167* 0.0 Discount factor: β 0.9512 0.96 Risk aversion: σ 2.00* 1.00 Consumption’s share in utility: µ c 0.3139 0.2600 Housing’s share in utility: µ h 0.0444 0.0962 Leisure’s share in utility: 1- µ c - µ h 0.6417 0.6438 4 Starred parameter vales are chosen independently of the model.

Calibration Period is one year µ c and µ h chosen so that ˆ n = 0 . 3 and value of stock of residential structures = GDP τ k and τ n chosen so that non-residential capital stock = 1.5 x annual output and ξ / GDP = 0.076 Shock processes estimated as VAR , ! little evidence of spillovers — weak correlation of shocks , ! shocks to construction and manufacturing much more volatile input shares based on input–output tables Huw Lloyd-Ellis () ECON917 Winter 2009 17 / 21

Table 3: Production Technologies Con. Man. Ser. GH Input shares in cons/inv production B c, M c, S c 0.0307 0.2696 0.6997 Input shares in res. structures B d, M d, S d 0.4697 0.2382 0.2921 Capital’s share by sector θ b , θ m, θ s 0.132 0.309 0.237 0.30 Trend productivity growth (%) g zb , g zm , g zs -0.27 2.85 1.65 1.00 Autocorrelation coefficient see table 4 ρ = 1.0 Std. dev. innovations to logged productivity see table 4 0.022 Table 4: Estimation of Exogenous Shock Process ~ ~ = + ε System estimated: z B z + + t 1 t t 1 ~ ε     log z     bt bt ~ ~ = ε = ε ε z log z ~ N ( 0 , V ) . 5 where   ,   and t mt t mt t     ~ ε log z     st st Autoregressive coefficients in matrix B (Seemingly unrelated regression estimation method: standard errors in parentheses) ~ ~ ~ + + + log z log z log z b, t 1 m, t 1 s, t 1 ~ 0.707 -0.006 0.003 log z bt (0.089) (0.078) (0.038) ~ log z 0.010 0.871 0.028 mt (0.083) (0.073) (0.036) ~ log z -0.093 -0.150 0.919 st (0.098) (0.087) (0.042) R 2 0.551 0.729 0.903 Correlations of innovations Standard deviation of innovations ε b ε m ε s ε b ε b 1 0.089 0.306 0.041 ε m 1 0.578 ε m 0.036 ε s 1 ε s 0.018 5 All variables are linearly detrended prior to estimating this system.