Housing in a two sector dynastic altruism model

Transcription

1 Housing in a two sector dynastic altruism model Xavier Raurich Universitat de Barcelona Fernando Sanchez-Losada Universitat de Barcelona December 2008

2 Introduction Aims: 1. Study the e ects of housing in the market of capital. 2. Analyze the long run e ects of housing taxes.

3 Introduction. Related literature - Finite horizon models: The introduction of housing reduces the long run stock of productive capital and favorable housing taxes also reduces this stock. References: Gahvari (1984), Skinner (1996), and Gervais (2002). - In nite horizon models: The introduction of housing reduces (increases) the stock of productive capital if the housing sector is less (more) capital intensive than the manufacturing sector. References: Goulder (1989), Turnovsky and Okuyama (1994) and Van Order (1990).

4 Introduction. Model We study a two-sector version of the dynastic altruism model, where one sector produces new housing capital. Housing has the following characteristics: 1. A durable consumption good with a depreciation rate lower than the depreciation rate of productive capital. 2. Consumers obtain utility from the services derived from housing 3. Housing is produced in a speci c sector

5 Introduction. Results 1. Housing increases (decreases) the demand of productive capital if the housing sector is the most (less) capital intensive sector. 2. Housing reduces the supply of capital if the bequests motive is not operative and, otherwise, has no e ect on this supply. 3. The introduction of housing makes more likely that altruistic parents leave positive bequests and reduces the dynamic ine ciency in the economy with zero bequests.

6 Introduction. Results 4. If the technologies in the two production sectors coincide, then housing taxes do not modify the stock of capital when bequests are positive, whereas they increase this stock when bequests are zero. A contribution of this paper is two study the e ects of taxes when they simultaneously modify the supply and the demand of capital. 5. The e ects of taxes on the stock of capital depend on the nature of housing, which is parametrized by the depreciation rate.

7 Consumers The economy. Consumers I N t consumers are born in period t. Consumers live for two periods I n 1 is the number of children per parent. I The utility function of a consumer belonging to generation t is where V t = U(c 1 t, c 2 t+1, h t ) + βv t+1, β 2 [0, 1) U(c 1 t, c 2 t+1, h t ) = u(c 1 t ) + ρu(c 2 t+1) + φu (h t ), ρ 2 [0, 1), φ > 0, and u (x t ) = ln x t, x t = c 1 t, c 2 t+1 and h t.

8 Consumers The economy. Consumers Consumers maximize V t subject to b t + w t = ct 1 + s t + p t h t, (1) p t+1 (1 δ) h t + R t+1 s t = ct nb t+1, (2) b t+1 0. (3)

9 Consumers The economy. Consumers Solution of the consumers maximization problem: u 0 c 1 t Note that (6) implies that = ρrt+1 u 0 ct+1 2, (4) ρnu 0 ct+1 2 βu 0 c 1 t+1, (5) p t R t+1 = p t+1 (1 δ) + φu0 (h t ) ρu 0. ct+1 2 (6) R t+1 > pt+1 p t (1 δ).

10 Production sectors The economy. Production sectors Manufacturing sector: F = A (v t K t ) α (u t L t ) 1 α α = u t L t A kt f, α 2 (0, 1), where kt f = v t K t u t L t. Housing sector: G = B [(1 v t ) K t ] γ [(1 u t ) L t ] 1 γ = (1 u t ) L t B (k g t ) γ, γ 2 (0, 1), where k g t = (1 v t )K t (1 u t )L t. Assumption. α γ

11 Production sectors The economy. Production sectors Sectoral mobility and perfect competition imply that p t bp (R t ) = A B α γ 1 γ α 1 γ (1 γ) Rt αa γ α 1 α, (7) w t bw (R t ) = (1 α αa 1 α α) A, (8) R t R t 1 αa kt f bk f 1 α (R t ) =, 1 kt g (1 α) γ αa bk g 1 α (R t ) =. (1 γ) α R t

12 Production sectors The economy. Production sectors The sectoral composition of GDP is measured by u t bu (R t, k t ) = k t k f t kt g kt g, and the per capita value of GDP is α y t by (R t, k t ) = u t A kt f + pt (1 u t ) B (kt g ) γ.

13 Equilibrium The economy. Market clearing conditions Capital market L t s t = K t+1. (9) Housing market L t h t = G [(1 v t ) K t, (1 u t ) L t ] + L t 1 h t 1 (1 δ). (10) Goods market L t ct 1 + ct 2 L t 1 + K t+1 = F (v t K t, u t L t ).

14 Equilibrium The economy. Equilibrium De nition. We de ne a competitive equilibrium as a path kt, ct 1, ct 2, h t, p t, b t t=0 that, given k 0 and h 0, solves the system of di erence equations (4), (5), (6), (9), (10) and satis es (1), (2), (3), (7), (8) and the transversality condition lim t! βt u 0 (c t )b t = 0. (11)

15 Steady State. The demand of capital De nition. A steady state is an equilibrium path along which the variables k t, c 1 t, c 2 t, h t, p t, and b t are constant. The demand of capital is the following function of the interest rate k d αw (R) (R) = (1 α) R Φ (R), where Φ (R) measures the composition e ect of GDP. If φ = 0 or α = γ then Φ (R) = 1 If φ > 0 or α > γ then Φ (R) is a function of the interest rate that modi es the demand of capital.

16 Steady State. The composition e ect An increase in R causes: 1. A substitution e ect that reduces h, because housing is an asset. 2. A wealth e ect that increases h, because housing is a consumption good. The net e ect on housing will depend on the nature of housing, which is parametrized by the depreciation rate.

17 Steady State. The composition e ect Figure 1. Housing demand Figure 2. Capital demand 0.30 h k R R Red δ = 1, Blue δ = 0.7, Yellow δ = 0.3 and Green δ = 0 Lemma. k d δ k d > (<) 0 if R < (>) n and φ < 0.

18 Steady State. The operativeness of the bequests motive The amount of bequests is the following function of the interest rate b = bb (R). Let R be such bb R = 0. Result. (Weil, 1987) The amount of bequests is positive if and only if β β = n R.

19 Positive bequests Steady State. Positive bequests Proposition. When β β, there is a unique steady state with positive bequests that satis es k = bk d (R ), b = bb (R ) and R = n β. Figure 3. Equilibrium k k * d k ( φ,δ ) n β R

20 Positive bequests Proposition. Assume that β β. a). If α = γ, then k k δ = 0 and φ = 0. b). If α > γ, then k k δ < 0 and φ < 0. Proposition. Assume that β β. Then, y y δ < 0 if α > γ. δ = 0 if α = γ and Remark. y = p B (k g ) γ + h A k f α p B (k g ) γi u (k (δ, φ)).

21 Positive bequests Steady State. Positive bequests Table 1. Economy with positive bequests Parameters Targets Variables k δ = % annual depreciation rate = h c+ d n n = % population growth rate y = 0.82 k β = % annual interest rate GDP = 1.73 B = h equals annual GDP u = 0.88 G φ = 0.29 GDP = 10% w GDP = 0.70 c ρ = d = 1 α = 0.31 γ = A = 1

22 Positive bequests Steady State. Positive bequests Table 2. Comparative statics when β > β δ β φ BM 0.5% 3% k h y b u R

23 Zero bequests Steady State. Zero bequests When bequests are zero, the equilibrium interest rate is such that the demand equals the supply of capital. The supply is k s ρ = 1 + ρ + φ φ 1 δ w (R), 1 + ρ + φ R (1 δ) n where α R α 1 w (R) = (1 α) A. αa Remark. If δ = 1 then the supply of capital is the following decreasing function of the interest rate: k s ρ w (R) =. 1 + ρ + φ n

24 Zero bequests Steady State. Supply of capital Remark. If δ < 1 then the supply of capital increases for small values of the interest rate and decreases for larger values. The increasing part of the supply is due to the substitution e ect k Figure 4. Supply of capital R Red δ = 1, blue δ = 0.7, green δ = 0.3, yellow δ = 0

25 Zero bequests Steady State. Supply of capital Lemma. k s δ k s > 0, φ < 0 and ks > 0 if φ + ρ R > (1 δ). ρ

26 Zero bequests Steady State. Equilibrium with zero bequests Proposition. When β < β, there is a unique steady state equilibrium with zero bequests. Let R, k,and h be, respectively, the steady state values of the interest rate, capital stock and housing capital. Then, R is such that k s = k d and k = k d R. Proposition. If α = γ, then the steady state with zero bequests is saddle path stable.

27 Zero bequests Steady State. Equilibrium with zero bequests Proposition. Assume that β < β and α γ. Then R k δ > 0, and β δ > 0. Figure 5. δ < 0, β k β b * >0 b=0 δ b * >0 b=0 δ

28 Zero bequests Proposition. Assume that β < β and α γ. Then R φ > 0, k β φ < 0, and φ < 0. Proposition. The introduction of housing makes more likely that altruistic parents leave positive bequests and reduces the over accumulation of capital due to the dynamic ine ciency.

29 Zero bequests Steady State. Equilibrium with zero bequests Table 3. Comparative static analysis when β < β δ φ BM δ = 0.5% δ = 3% φ = 0.2 φ = 0.4 k h y u R β

30 Tax policy. The economy with taxes Adult budget constraint (1 τ b ) b t + (1 τ w ) w t + t y t = (1 + τ c ) c 1 t + s t + (1 + τ h ) p t h t, Old budget constraint p t+1 (1 δ) h t + (1 τ k ) R t+1 s t + t o t = (1 + τ c ) c 2 t+1 + nb t+1, Government budget constraints: t y t = τ b b t + τ w w t + τ c c 1 t + τ h p t h t, t o t+1 = τ k R t+1 s t + τ c c 2 t+1.

31 Tax policy. The economy with taxes Proposition. If α > γ then k d τ c < 0, k d τ k < 0, k d τ h > 0 and = 0. If α = γ then taxes do not modify the demand of capital. k d τ b Proposition. k s τ k < 0, k s τ c where br = (1 δ) < 0, k s τ h > 0 and k s > 0 if R > br,. φ (1 + τc ) + ρ ρ (1 + τ h ) (1 τ k ) Proposition. The bequests motive is operative if β β n R (1 τ k ) (1 τ b ).

32 Positive bequests Tax policy. Positive bequests Proposition. When β β, there is a unique steady state with positive bequests that satis es k = k d (R ) and R n = β(1 τ k )(1 τ b ). Figure 6. Equilibrium k k * ( ) β ( 1 τ )( 1 τ ) k n b k d τ, τ, τ k c h R

33 Positive bequests Tax policy. Positive bequests Proposition. Assume that β β. Then, a). If α = γ then k τ h = 0, k τ c = 0, k τ k < 0 and k τ b < 0. b). If α > γ then k τ h > 0, k τ c < 0, k τ k < 0 and k τ b < 0.

34 Positive bequests Tax policy. Positive bequests Table 4. Comparative static analysis when β >β BM τ c = 0.3 τ k = 0.3 τ b = 0.05 τ h = 0.3 k h y b u R

35 Zero bequests Tax policy. Zero bequests Proposition. When β < β, there is a unique steady state equilibrium with zero bequests that satis es k = k d R and R is such that k d R = k s R.

36 Zero bequests Tax policy. Zero bequests Figure 7. E ects of an increase in τ h k k s k k d k d k s δ small R δ large R

37 Zero bequests Tax policy. Zero bequests Proposition. Assume that β < β and α γ. Then, k a). τ k k < 0, τ c < 0, and k τ h > 0. b). R τ k > 0, R τ c > 0, and R τ h < 0. β c). τ k > 0, < 0, and β τ h > 0. β τ c

38 Zero bequests Tax policy. Zero bequests Table 5. Comparative static analysis when β < β δ = 0.5% BM τ c = 0.3 τ k = 0.3 τ h = 0.3 k h y δ = 3% BM τ c = 0.3 τ k = 0.3 τ h = 0.3 k h y

39 Concluding Remarks 1. If the two sectors use di erent technologies, then the introduction of housing reduces the demand of capital. 2. If bequests are positive, the introduction of housing does not modify the supply of capital, whereas reduces this supply when bequests are zero. 3. The introduction of housing makes more likely that altruistic parents leave positive bequests and reduces the dynamic ine ciency in the economy with no-bequests. 4. A larger depreciation rate reduces the stock of capital if bequests are positive, whereas increases this stock when bequests are zero. A larger depreciation rate makes more likely that altruistic parents do not leave positive bequests.

40 Concluding Remarks 5. The tax on housing increases the stock of capital. In the economy with positive bequests, this result arises because this tax increases the demand of capital and in the economy with zero bequests this result arises because this tax rises the supply of capital. 6. The e ects of the housing tax on the stock of capital depend on the depreciation rate of housing.