Endogenous growth with addictive habits

Transcription

1 Endogenous growth with addictive habits Warwick, 2012

2 Habit formation Habits may play a role in consumption behavior. Individual satisfaction can be affected by the change over time in customary consumption levels. (A. Smith, and later A. Marshall, Pigou, Duesenberry among others ) Customary consumption levels may refer to the past consumption levels (internal habit formation) or/and to the consumption of a reference group (external habit formation).

3 Role of the Habits in the Recent Literature Enhancing the agents desire to smooth consumption over time. This feature was found critical to Match in a RBC framework the high persistence in the U.S. output volatility (e.g. Boldrin et al., AER 2001) Resolve the equity premium puzzle (e.g. Constantinides, JPE 1990): Habit persistence smooths consumption growth over and above the smoothing implied by the life cycle permanent income hypothesis with time separable utility. (...) This illustrate the key role of habit persistence in resolving the puzzle (...).

4 Aim of the Paper Propose a definition of habit formation, which is general relative to the assumptions on the intensity, persistence, and lag structure, and unveil two mechanisms which point to the opposite direction: habits may reduce the desire of smoothing consumption over time. Habit formation always reduces the desire of consumption smoothing once the model is calibrated to match the average U.S. output and utility growth rates observed in the data (Easterlin s hypothesis - Blanchflower and Oswald, JPublicE2000).

5 Road Map 1. Propose a general definition of the habit formation. 2. Embed it in the simplest endogenous growth model (linear technology) with external addictive habits or catching-up-with-the-joneses utility function having subtractive form. 3. Explain the 2 mechanisms which points to a reduced desire in smoothing consumption. (Complete taxonomy of the dynamics) 4. Quantitative analysis (calibration of the model). 5. Discussion on modeling choices. 6. Conclusion.

6 A general definition This definition is only general relative to the assumptions on the intensity, persistence, and lag structure of the habits. t h(t) = ε c(u)e η(u t) du t τ where ε > 0, η 0, and τ > 0 indicate respectively the intensity, persistence and lag structure of the habits, while c(u) the average consumption.

7 Ryder and Heal (RES 1973): ε = η > 0 and τ +. Constantinides (JPE 1990): η > ε > 0, and τ +. Abel (AER 1990): discrete time and ε > 0, η = 0 and τ = 1. However our definition does not include the specification of Ravn et al. (RES 2006) Microfoundation and Identification/Estimation.

8 First Mechanism: It can be explained also (but not only) in a benchmark model, where τ. Focus on the role of the habits intensity and persistence. Second Mechanism: crucially depends on the lag structure of the habits. Then we need to study it in the general framework.

9 Benchmark General Benchmark Representative agent: (c(t) h(t)) 1 γ max e ρt dt c(t),k(t) 0 1 γ s.t. k(t) = (A δ)k(t) c(t) k(t) 0, c(t) h(t) 0 k(0) = k 0, h(0) = h 0 where t h(t) = ε c(u)e η(u t) du

10 Benchmark General A market equilibrium is described by any trajectory {φ(t), k(t), h(t)} t 0 which solves k(t) = (A δ)k(t) φ(t) h(t) (1) ḣ(t) = εφ(t) + (ε η)h(t) (2) φ(t) = φ 0e 1 γ (A δ ρ)t (3) subject to (i) the initial condition of capital, k(0) = k 0, (ii) the initial condition of habit, h(0) = h 0, (iii) the transversality condition lim t k(t)e (A δ)t = 0, and (iv) the inequality constraints, k(t) 0, and c(t) h(t) 0. φ(t) = ψ(t) 1 γ with ψ(t) the costate variable. c(t) = φ(t) + h(t) c(t) = c(t)

11 Benchmark General In a standard AK model, without external habits, all the aggregate variables jump immediately to the balanced growth path where they grow at the rate Γ = 1 (A δ ρ) γ then the economy faces positive growth if and only if: otherwise the economy shrinks over time. A Â = δ + ρ (4)

12 Benchmark General BGP and Transitional Dynamics An economy with external addictive habits, and linear production, has: i) a unique asymptotic and positive growth rate g = max(ε η, Γ) and ii) a unique general equilibrium path converging monotonically to the balanced growth path 1 if A Â, k 0 h 0 and g < A δ. A δ ε+η These results still hold when A < Â and ε > η.

13 Benchmark General What pins down a g = ε η? Intuitively these results depend on the inequality ε > η and on the habit addiction which trigger a positive growth rate in the habit stock and in the aggregate consumption.

14 Benchmark General In fact, the representative consumer faces two constraints: ḣ(t) = (ε η)h(t) + ε( c(t) h(t)) and c(t) h(t) At the equilibrium, the solution path of c(t), obtained by solving the representative agent problem, coincides with the given path, c(t): ḣ(t) = (ε η)h(t) + ε(c(t) h(t)) (ε η)h(t) and then the habit stock, as well as aggregate consumption, grows at least at the rate ε η even if c(t) h(t) shrinks over time at the rate Γ < 0.

15 Benchmark General When is the growth rate sustainable? Under two conditions: g < A δ = r (standard requirement) h 0 (A δ ε + η)k 0

16 Benchmark General The last condition implies that h 0 rk 0 if ε = η; Suppose does not hold, then c 0 h 0, only if an initial net disinvestment, i 0 < 0, takes place. If this happens a positive growth rate of consumption and habits is clearly not sustainable over time because financed with a continuous selling of assets which leads to zero capital stock in a finite time. The condition becomes even more restrictive when ε > η.

17 Benchmark General The First Mechanism: Emerge when g = ε η: transitional dynamics led by a save now or regret it later mechanism: the representative agent decides to consume, till the very beginning, less than in the standard AK model in order to save enough and accumulate the capital necessary to keep over time the consumption level higher than the habits stock avoiding, in this way, an infinite disutility. The faster accumulation of capital will imply a higher consumption in the long run than the one in the AK model without habits. Habits reduce the desire to smooth consumption.

18 Benchmark General 30 Capital Dynamics 0.8 Saving Rate Dynamics 0.5 Consumption Dynamics AK with ε η>γ 2 >0 AK with no habit AK with ε=η t t t Figure: Capital, saving rate and consumption dynamics when g = ε η.

19 Benchmark General General Same model setup but now the habits are t h(t) = ε c(u)e η(u t) du t τ or differentiating it ḣ(t) = ε ( c(t) c(t τ)e ητ ) ηh(t) In this framework the second mechanism can be studied. Before moving on it, we come back quickly on the first mechanism.

20 Benchmark General First mechanism with a finite lag structure: The presence of a finite τ may reduce or even break down the mechanism for a positive growth rate in the habits and then the save now or regret it later mechanism. The reason is that at the market equilibrium we have that ḣ(t) = ε ( c(t) c(t τ)e ητ ) ηh(t) (ε η)h(t) εc(t τ)e ητ

21 Benchmark General On the second mechanism: At the equilibrium c(t) = c(t) and c(t) h(t) = φ 0 e Γt, then the habits formation equation becomes: ḣ(t) = (ε η)h(t) + εh(t τ)e ητ + g(t) with g(t) = εφ 0 ( 1 e (Γ+η)τ ) e Γt. Delay differential equation with forcing term. habits and consumption. Endogenous fluctuations in the Habits revised periodically and risk adverse agents.

22 Benchmark General To analyze formally the first and second mechanism, we proceed as follows: 1. Describe the market equilibrium; 2. Find the asymptotic growth rate of consumption and habits; 3. Use it together with the transversality condition and the capital accumulation equation to completely describe the transitional dynamics of the economy by finding the general equilibrium path of the main aggregate variables;

23 Benchmark General Market Equilibrium A market equilibrium is described by any trajectory {φ(t), k(t), h(t)} t 0 which solves k(t) = (A δ)k(t) φ 0 e Γt h(t) (5) t [ h(t) = ε h(u) + φ 0e Γu] e η(u t) du (6) t τ φ(t) = φ 0 e Γt (7) subject to (i) the initial condition of capital, k(0) = k 0, (ii) the past history of habit, h 0(t), t [ τ, 0], and (iii) the transversality condition lim t k(t)e (A δ)t = 0 and (iv) satisfying k (t) 0 and c(t) h(t) 0.

24 Benchmark General Asymptotic Growth Rate of Consumption and Habits To find it the first step is to use a change of variable to rewrite the habits equation (6) as an autonomous algebraic equation. Then to study the spectrum of roots of its characteristic equation, which is: when Γ > 0. 0 ( λ) = 0 with ( λ) = 1 + ε e (Γ+ λ+η)u du (8) τ

25 Benchmark General The spectrum of roots of (8) has a leading real root, α, which is positive if and only if ε η > Γ and τ > τ with τ = log ( ) ε Γ η ε Γ + η Moreover α is an increasing function of τ and it converges to ε Γ η, as τ. From now α = α + Γ. We can now write the solution of the habits and consumption.

26 Benchmark General The consumption dynamics is described by the equation c(t) = σ 0e α t + (1 + θ)φ 0e Γt + }{{} χ 2(t)e Γt }{{} trend component oscillatory component (9) with lim t χ 2 (t) e (Γ α )t = 0 and σ 0 = σ 0(h 0(t), φ 0). Then the asymptotic positive growth rate of consumption and the habit stock is i) g = α if ε η > Γ and τ > τ, and σ 0 0, or ii) g = Γ if ε η < Γ or ε η > Γ and τ < τ, and φ 0 0 where χ 2 (t) is the oscillatory component, coming from all the (infinite) roots having real part lower than α.

27 Benchmark General Balanced Growth and Transitional Dynamics A market economy with a finite lag structure in the habits, and a high level of technology (Γ > 0) has i) a unique asymptotic and positive growth rate g = max(γ, α ); ii) a unique competitive equilibrium path which oscillatory converges over time to the balanced growth path: h(t) = ˆσ 0 e α t + θ ˆφ 0 e Γt + χ 2 (t)e Γt c(t) = ˆσ 0 e α t + (1 + θ) ˆφ 0 e Γt + χ 2 (t)e Γt k(t) = ˆσ 0 t A δ α eα + ˆφ 0(1 + θ) A δ Γ eγt + if A δ > α, and (k 0, h 0 (t)) such that ˆφ 0 = φ 0 (k 0, h 0 (t)) 0. t χ 2 (u)e (A δ Γ)(u t) du

28 Benchmark General The explicit form of the last condition is h(0) k(0) A δ + η ε + εe (A δ+η)τ A δ + η ε + εe (A δ+η)τ εe (A δ+η)τ which converges to the condition in the benchmark model as τ. It is also quite straightforward to observe that this condition is less restrictive than in the benchmark model since 0 0 rk 0 h 0 = ε c(u)e ηu du > ε c(u)e ηu du τ 0 τ e (A δ)u c (u)

29 Benchmark General Moreover we have also found the conditions on the parameters which allow us to distinguish two types of the oscillatory behavior: Damping fluctuations around the balanced growth path; Small fluctuations smoothed out quickly; convergence is monotonic. The taxonomy of the dynamics can be summarized in a figure under the normalization t t τ e η(u t) du = 1 t 0

30 Benchmark General Low Technology Case (Γ<0) High Technology Case (Γ>0) α*>0 and damping fluctuations around the BGP α*>0 and no damping fluctuations around the BGP g=α* >0 and no damping fluctuations around the BGP ε 1 ε 1 g=γ >0 and no damping fluctuations around the BGP α*<0 and damping fluctuations around the BGP α*<0 and no damping fluctuations around the BGP g=γ >0 and damping fluctuations around the BGP 0 0 Γ 1 η η Figure: Taxonomy of the dynamics in the space (η, ε)

31 Benchmark General calibration Calibration of the benchmark model: along the balanced growth path where g = ε η; ε and η can be calibrated to match the performance of the average U.S. annual output and utility growth rates as observed in the data: g = g u.s. = 0.02 and g u = 0 In fact, the main aggregate variables evolve along the balanced growth path as follows: h(t) = h 0 e gt, c(t) = h(t), k(t) = c(t) r g

32 Benchmark General Alternatively: along the balanced growth path with g = Γ; any calibration of the parameters usually used in the literature to induce a higher desire to smooth consumption, violates the Easterlin s hypothesis. g = g u.s. but g u > 0 Is this violation quantitatively relevant? The main aggregate variables evolve along the balanced growth path as follows: h(t) = h 0 e gt, c(t) = (Γ + η)h(t), k(t) = c(t) ε r g

33 Benchmark General This implies the following path of the instantaneous discounted utility ũ(t) = [(g + η ε)h 0] 1 γ e (g r)t (10) ε 1 γ (1 γ) The two main parameters to be calibrated are γ and ρ. We set ρ = and γ = 1.1 or 2 or 2.8. (e.g. Campbell and Cochrane JEP 1999). Implied annual real interest rate is 3.9%, 5.7% and 7.3% respectively. Discounted utility (10) always negative as well as the argument at the exponent. Wellbeing increases over time since the discounted disutility ( ũ) grows at g ũ = g ũ = g ũ = 0.053

34 Benchmark General Among the three values of the interest rate, the latter implies the highest consumption-output ratio: r = δ = 0.1 c y 1 3 This value is still quite low when compared with the empirical evidences (around two-thirds). This difference depends on how capital is defined in an AK model: capital should be viewed broadly to include not only physical but also human capital. However Jones et al. argue that investment in human capital are counted as private consumption in national income accounts and for this reason a discrepancy between the predicted and actual consumption-output ratio naturally emerge.

35 Benchmark General 0.03 Discounted inst. disutility growth rate on the BGP model prediction when g= α model prediction when g= Γ Easterlin s hypothesis Quarters Figure: predictions and the Easterlin s hypothesis

36 Benchmark General What happens when habits are characterized by a finite lag structure? Calibration of the parameters ε and η in order to match the growth rate g = α = To do so we need first to set a value for τ and then use the characteristic equation to find the values of ε and η, if any, which solve it and respect also the other two sign conditions introduced when we look at the consumption dynamics. According to Crawford s test on microeconomic panel data (see Crawford RES 2010): τ = 2 quarters improves the agreement between theory and data with respect to the one lag case without reducing too much the power of the test. Then we set τ = 2 quarters and we find numerically that ε = and η = 0.1 are a good choice.

37 Benchmark General More precisely τ = 2, ε = and η = 0.1 implies ε e (g+η)u du = 0 τ ε Γ η log( ε ) and τ = = when g = 0.005, γ = 2, τ = 2, ε = , Γ+η η = 0.1, r = 0.01, and ρ = at quarterly frequency. This calibration matches both the average growth rate of the U.S. economy and the Easterlin s hypothesis; Similar results emerge when we have tried for different value of r and τ.

38 Benchmark General What about the second mechanism? The positive value of Γ implies that the economy is always in the region of the parameter space where small deviations from the balanced growth path imply no damping fluctuations around it. On the other hand if we calibrate the model along the balanced growth path where g = Γ the same conclusions as in the benchmark model hold but now deviations from the balanced growth path implies convergence toward it by damping fluctuations.

39 Benchmark General In fact for the values of ε in the range (0.093, 0.492) - Constantinides JEP we have found that damping fluctuations raise as long as and τ < quarters when ε = τ < 2.02 quarters when ε = 0.492

40 Benchmark General Damping fluctuations Save now or Matched around the BGP regret it later moments Benchmark model (τ = ) g = ε η no yes average output growth no utility growth g = Γ no no average output growth General model (τ = 1, 2 or 3) g = α no yes average output growth no utility growth g = Γ yes no average output growth The oscillatory component is smoothed out quickly and convergence to the BGP is monotonic. Table: Summary of the Results

41 Benchmark General Conclusion Taxonomy of the dynamics reveal a more complex relation between habits and consumption than usually stressed by the literature. Calibration Exercise: first mechanism, holds independently by the lag structure as soon as the targets to be matched are the average output and utility growth rate. second mechanism not relevant under this calibration but it becomes relevant, when the model is calibrated as usually done in the literature, and habits lag structure is finite.