Habits and Multiple Equilibria

Transcription

1 Habits and Multiple Equilibria Lorenz Kuen and Eveny Yakovlev February 5, 2018 Abstract Abstract TBA. A Introduction Brief introduction TBA B A Structural Model of Taste Chanes Several structural models can ive rise to the persistent lon-run eects of public policies we identied in the main paper. In this section we propose one particular structural model of taste chanes under which even temporary policy interventions can lead to persistent eects in the lon run. This basic model is consistent with the consumption patterns documented in the paper. The model extends the habit formation model by Becker and Murphy (1988) to allow for two habit-formin oods, illustratin that in this situation several steady-state consumption patterns are possible even in the absence of any unobserved individual heteroeneity. A person's consumption shares in steady state depend solely on his initial consumption pattern. Moreover, it is hard to chane these consumption patterns even with very lare shocks once the stock of habit is suciently lare. Hence, policies aimed at increasin the relative price of one ood may not induce everybody or even many to reduce the consumption of this ood. Instead, due to the stock of habits already accumulated, people who are accustomed to this particular ood will still prefer it even after the policy chane. This implies that policies that inuence the initial choices of youner enerations can have lon-run consequences over their entire life spanintended or otherwise. Lorenz Kuen: Northwestern University, 2211 Campus Drive, Evanston, IL 60208, and National Bureau of Economic Research; l-kuen@northwestern.edu. Eveny Yakovlev: New Economic School, Department of Economics, 100A Novaya Street, Skolkovo, Moscow , Russia. eyakovlev@nes.ru. 1

2 B.1 Model Setup For simplicity we assume that consumers spend all of their budet on two habit-formin oods, beer and vodka. We also assume that consumers are myopic, i.e., that they maximize only current utility and do not save, that there are no outside oods, that income does not chane over time, and that there is no uncertainty. 1 The individual derives ow utility u(v t, b t, H v t, H b t ) from consumin vodka v t and beer b t and also from the correspondin stocks of habit H v t and H b t. The utility function has properties that are common in the literature, specically that u > 0, u < 0, and u H > 0 with {b, v}. These assumptions imply in particular that the marinal utilities of consumin beer or vodka are positive and increasin with the stock of habit of the correspondin ood. Assumin a common rate of depreciation of the two habit stocks, they evolve as H t+1 = (1 )H t + t, H 0 0, [0, 1]. (1) The budet constraint is p vt v t + b t = y t. Without loss of enerality, we focus on interior solutions. 2 The rst-order condition of this optimization problem is u v (v t, y t p vt v t, H v t, H b t ) p vt u b (v t, y t p vt v t, H v t, H b t ) = 0, (2) where u v and u b are the partial derivatives with respect to the rst and second aruments, respectively. Since we are interested in the lon-run eects of habit formation, we focus our analysis on the properties of the model's steady state. In the steady state where prices, income, and consumption are constant such that p vt = p v, y t = y, and t =, the expression for the stocks of habit is /. The rst-order condition that implicitly denes the steady state can then be rewritten as u v (v, y p v v, v/, (y p v v)/) p v u b (v, y p v v, v/, (y p v v)/) = 0. (3) In eneral, this is a non-monotonic function in the steady-state vodka consumption v. 3 Dependin on the parametrization of the utility function u, equation (3) may have a dierent number of solutions. Fiure A.6 illustrates that for certain parametrizations, there is a unique solution, but for many other parametrizations several steady states exist, up to a continuum of solutions. 4 These multiple equilibria are derived without any consumer heteroeneity except 1 Below we reach the same qualitative conclusions if consumers are forward lookin and solve a fully dynamic problem. 2 If there are corner solutions, there is always a symmetric specication with at least 3 equilibria where the two stable equilibria have a consumption share in each ood of either 1 or 0. 3 This condition can also be expressed as a function of the share of vodka, S v v+b, by usin the fact that y S v 1 (1 p v)s ; see below. v 4 See below for a proof. Similar results are obtained for the model with forward-lookin consumers because the steady-state Euler equation is also non-monotonic in the consumption levels. 2

3 for dierences in initial conditions. A person who initially consumes primarily beer will also prefer beer in the lon-run steady state, and vice versa for vodka. B.2 Model Properties and Extensions This section shows that the model above with two habit formin oods can have any number of equilibira. We then provide three numerical examples that enerate, respectively, one, three, and an innite number of equilibria. We also show how to map the steady state, which the model expresses in levels, to alcohol shares, which is the concept we use in our empirical analysis. Finally, we show that these insihts from be basic myopic model extend to a model with forward-lookin consumers. B.2.1 Number of Equilibria in the Model with Myopic Consumers The steady state rst-order condition (FOC) for myopic aents as a function of the level of vodka consumption, v, is F = u v (v, y p v v, [/(1 )]v, [/(1 )][y p v v]) p v u b (v, y p v v, [/(1 )]v, [/(1 )][y p v v]) = 0. Dierentiatin F with respect to v yields u vv p v u vb + /(1 )u vh v p v /(1 )u vh b p v [u bv p v u bb + /(1 )u bh v p v /(1 )u bh b]. Given the assumptions that u < 0, u H H < 0, and u H > 0, some terms in this expression are positive, e.., /(1 )u vh v, p 2 v/(1 )u bh b, and some are neative, e.., u vv, p 2 vu bb. Therefore, the sin of the overall sum is ambiuous. B.2.2 Numerical Examples One Equilibrium Let the utility function be u = ln(b) L b +ln(v) L v with L = ln(1.1+h ) for {b, v}so that the marinal utility is u = L. The FOC is 0 = u v p v u b = L v v p vl b b = L v p v v L b b = L v p v v L b y p v v. 3

4 Solvin for v we obtain L v y. L v + L b p v Three Equilibria Let the utility function be u = b L b + v L v with L = ln(1.1 + H ) for {b, v}so that the marinal utility is u x = L 2. Solvin for v we obtain R y 1 + R p v, with R = ( L v p v L b ) 2. Continuum of Equilibria marinal utility is u = H 2. Solvin for v we obtain Let the utility function be u = b H b + v H v, so that the R y 1 + R p v, with R = Hv. p 2 v H b B.2.3 Expressin the Model Solutions in Terms of Shares S = b+v, S b + S 1, p v v + b = y, and Sv S b S v S b b = = = v b. Hence, S v 1 S v (y p v v) y S v 1 (1 p v )S v. B.2.4 Allowin for Forward-Lookin Consumers We now relax the assumption of myopic behavior. Forward lookin aents maximize the present value of utility from consumin beer and vodka, U = u(v t, b t, Ht v, Ht b )+ i=1 β i [u(v t+i, b t+i, Ht+i, v Ht+i)]. b To keep the model simple, we follow Gruber and Köszei (2001) and assume no savins and that the stock of habits evolves as follows: H t+1 = (H t + t ). 4

5 The FOC for v t, after substitutin for b t usin the budet constraints, is The FOC for v t+1 is u vt p vt u bt + β i i (u H v t+i p vt u H b t+i ) = 0. i=1 u vt+1 p vt+1 u bt+1 + β i i (u H v t+i+1 p vt+1 u H b t+i+1 ) = 0. i=1 Combinin the two FOCs and analyzin the steady state we obtain the followin Euler equation: 0 = u v (v, y p v v, + β 1 β [u H v(v, y p vv, 1 v, 1 v, 1 v, 1 [y p vv]) p v u b (v, y p v v, 1 [y p vv]) p v u H b((v, y p v v, 1 [y p vv]) 1 v, Assumin that u as 0 uarantees the existence of a steady state. 1 [y p vv])]. To check the possibility of multiple steady states, we can analyze the monotonicity of the riht-hand side of the steady-state Euler equation by takin the rst derivative with respect to v, drhs(v)/d u vv 2p v u vb + p 2 vu bb + 1 [u vh v 2p vu vh b + p 2 vu bh b] + β 1 β [u vh v p vu bh v p v u vh v + p 2 vu bh b + 1 [u H v H v 2p vu H v H b + p2 vu H b H b]]. This expression can be both neative and positive. To see this, assume that the utility function is separable in the two oods and their stocks of habit. Then the expression above can be rewritten as drhs(v)/d [ u vv + p 2 vu bb + β + [ ( β 1 (u H v H v + p2 vu H b H b)] β 1 β )(u vh v + p2 vu bh b) ]. The terms in the rst square brackets are all neative, while the terms in the second square brackets are all positive. Thus, dependin on the relative manitude of these terms, the rst derivative can be positive or neative. The followin utility specications provide two examples, one with a unique and stable steady state and one with three steady states, two of which are stable and one is unstable. We aain set p y = 1 so that the consumption levels correspond to shares, and for simplicity we assume that β = 1 and = 0.5. Then the utility parametrization u = + H + H results in a one equilibrium, while u = + H + 5H yields three equilibria. References Becker, Gary S. and Kevin M. Murphy, A Theory of Rational Addiction, Journal of Political Economy, 1988, 96 (4),

6 Gruber, Jonathan and Botond Köszei, Is Addiction Rational? Theory and Evidence, Quarterly Journal of Economics, 2001, 116 (4),

7 Fiure 1: Potential Number of Steady States in the 2-Good Becker-Murphy Model share of vodka ae share of vodka ae share of vodka ae Notes: These fiures show the dynamic behavior of the share of vodka in the two-ood habit formation model, startin from different initial conditions, i.e., different initial consumption shares. The three fiures correspond to the three parametrizations specified in the text. The top panel has one stable steady state, the middle panel has three steady states, two stable and one unstable, and the bottom panel has an infinite number of steady states.