Applications

4 Applications 4.1 Consumption Smoothing 4.1.1 The Intertemporal Budget • For any given sequence of interest rates {Rt }∞ t=0 , pick an arbitrary q0 > 0 and define qt recursively by qt = qt−1 , 1 + Rt that is, qt = q0 . (1 + R1 )...(1 + Rt ) qt represents the price of period−t consumption relative to period−0 consumption. • The buget in period t is given by ct + at+1 ≤ (1 + Rt )at + yt where at denotes assets and yt denotes labor income. • Multiplying the period-t budget by qt and adding up over all t, we get T X t=0 [qt ct + qt at+1 ] ≤ T X [qt (1 + Rt )at + qt yt ] . t=0 Using the fact that qt (1 + Rt ) = qt−1 , we have T X qt (1 + Rt )at = q0 (1 + R0 )a0 + t=0 T X qt−1 at t=1 so that the above reduces to T T X X qt ct + qT aT +1 ≤ q0 (1 + R0 )a0 + q t yt t=0 t=1 • Assuming either that the agent dies at finite time without leaving any bequests, in which case aT +1 = 0, or that the time is infinite, in which case we impose qT aT +1 → 0 as T → ∞, we conclude that the intertemporal budget constraint is given by T X t=0 qt ct ≤ q0 (1 + R0 )a0 + T X qt yt , t=1 where T < ∞ (finite horizon) or T = ∞ (infinite horiozon). The interpretation is simple: The present value of the consumption the agent enjoys from period 0 and on can not exceed the value of initial assets the agent has in period 0 plus the present value of the labor income the agent receives from period 0 and on. • We can rewrite the intertemporal budget as T X t=0 qt ct ≤ q0 x0 where x0 ≡ (1 + R0 )a0 + h0 , h0 ≡ ∞ X qt yt . q 0 t=0 (1 + R0 )a0 is the household’s financial wealth as of period 0. h0 is the present value of labor income as of period 0; we often call h0 the household’s human wealth as of period 0. The sum x0 ≡ (1 + R0 )a0 + h0 represents the household’s effective wealth. • Note that the sequence of per-period budgets and the intertemporal budget constraint are equivalent. We can then write the household’s consumption problem as follows max s.t. T X β t U (ct ) t=0 T X t=0 qt ct ≤ q0 x0 • Note that the above is like a “static” consumption problem: Interpret ct as different consumption goods and qt as the price of these goods. This observation relates to the context of Arrow-Debreu markets that we discuss later. 4.1.2 Consumption Smoothing • The Lagrangian for the household’s problem is " # T T X X β t U (ct ) + λ q0 x0 − qt ct L= t=0 t=0 where λ is the shadow cost of resources for the consumer (that is, the Lagrange multiplier for the intertemporal budget constraint). • The FOCs give U 0 (c0 ) = λq0 for period 0 and similarly β t U 0 (ct ) = λqt for any period t. • Suppose for a moment that the interest rate equals the discount rate in all periods: Rt = ρ ≡ 1/β − 1. Equivalently, qt = β t q0 The FOCs then reduce to U 0 (ct ) = λq0 for all t, and therefore ct = c. for all t. That is, the level of consumption is the same in all periods. • But how is the value of c determined? From the intertemporal budget, using qt = β t q0 and ct = c, we infer q0 x0 = T X qt ct = t=0 and therefore 1 q0 c 1−β c = (1 − β)x0 = (1 − β) [(1 + R0 )a0 + h0 ] That is, the household consumes a fraction of his initial effective wealth. This fraction is given by 1 − β. 4.2 4.2.1 Arrow-Debreu Markets Arrow-Debreu versus Radner • We now introduce uncertainty... • Let q(st ) be the period-0 price of a unite of the consumable in period t and event st and w(st ) the period-t wage rate in terms of period-t consumables for a given event st . q(st )w(st ) is then the period-t and event-st wage rate in terms of period-0 consunambles. We can then write household’s consumption problem as follows max XX st t s.t. ¡ ¢ β t π(st )U cj (st ), z j (st ) XX t st q(st ) · cj (st ) ≤ q0 · xj0 where xj0 ≡ (1 + R0 )a0 + hj0 , hj0 ≡ ∞ X q(st )w(st ) t=0 q0 [lj (st ) − T j (st )]. (1 + R0 )aj0 is the household’s financial wealth as of period 0. T j (st ) is a lump-sum tax obligation, which may depend on the identity of household but not on its choices. hj0 is the present value of labor income as of period 0 net of taxes; we often call hj0 the household’s human wealth as of period 0. The sum xj0 ≡ (1 + R0 )aj0 + hj0 represents the household’s effective wealth. 4.2.2 The Consumption Problem with CEIS • Suppose for a moment that preferences are separable between consumption and leisure and are homothetic with respect to consumption: U(c, z) = u(c) + v(z). c1−1/θ u(c) = 1 − 1/θ • Letting µ be the Lagrange multiplier for the intertemporal budget constraint, the FOCs imply ¢ ¡ β t π(st )u0 cj (st ) = µq(st ) for all t ≥ 0. Evaluating this at t = 0, we infer µ = u0 (cj0 ). It follows that β t π(st )u0 (cj (st )) q(st ) = = β t π(st ) j 0 q0 u (c0 ) µ cj (st ) cj0 ¶−1/θ . That is, the price of a consumable in period t relative to period 0 equals the marginal rate of intertemporal substitution between 0 and t. £ ¤−1/θ • Solving qt /q0 = β t π(st ) cj (st )/cj0 for cj (st ) gives j t c (s ) = cj0 £ · ¸ ¤θ q(st ) −θ . β π(s ) q0 t t It follows that the present value of consumption is given by XX t st q(st )cj (st ) = q0−θ cj0 ∞ X £ t=0 ¤θ β t π(st ) q(st )1−θ Substituting into the resource constraint, and solving for c0 , we conclude cj0 = m0 · xj0 where 1 m0 ≡ P∞ £ t . ¤ t ) θ [q(st )/q ]1−θ β π(s 0 t=0 Consumption is thus linear in effective wealth. m0 represent the MPC out of effective wealth as of period 0. 4.2.3 Intertemporal Consumption Smoothing, with No Uncertainty • Consider for a moment the case that there is no uncertainty, so that cj (st ) = cjt and q(st ) = qt for all st . • Then, the riskless bond and the Arrow securities satisfy the following arbitrage condition qt = q0 . (1 + R0 )(1 + R1 )...(1 + Rt ) Alternatively, h i−t e0,t qt = q0 1 + R e0,t represents the “average” interest rate between 0 and t. Next, note that m0 where R is decreasing (increasing) in qt if and only if θ > 1 (θ < 1). It follows that the marginal propensity to save in period 0, which is simply 1 − m0 , is decreasing (increasing) in e0,t , for any t ≥ 0, if and only if θ > 1 (θ < 1). R • A similar result applies for all t ≥ 0. We conclude Proposition 22 Suppose preferences are seperable between consumption and leisure and homothetic in consumption (CEIS). Then, the optimal consumption is linear in contemporaneous effective wealth: cjt = mt · xjt where xjt ≡ (1 + Rt )ajt + hjt , hjt ≡ ∞ X qt τ =t mt ≡ P∞ τ =t qt β [wτ lτj − Tτj ], 1 θ(τ −t) (qτ /qt )1−θ . mt is a decreasing (increasing) function of qτ for any τ ≥ t if and only θ > 1 (θ < 1). That is, the marginal propensity to save out of effective wealth is increasing (decreasing) in future interest rates if and only if the elasticity of intertemporal substitution is higher (lower) than unit. Moreover, for given prices, the optimal consumption path is independent of the timining of either labor income or taxes. • Obviously, a similar result holds with uncertainty, as long as there are complete ArrowDebreu markets. • Note that any expected change in income has no effect on consumption as long as it does not affect the present value of labor income. Also, if there is an innovation (unexpected change) in income, consumption will increase today and for ever by an amount proportional to the innovation in the annuity value of labor income. • To see this more clearly, suppose that the interest rate is constant and equal to the discount rate: Rt = R = 1/β − 1 for all t. Then, the marginal propensity to consume is m = 1 − β θ (1 + R)1−θ = 1 − β, the consumption rule in period 0 becomes £ ¤ cj0 = m · (1 + R)a0 + hj0 and the Euler condition reduces to cjt = cj0 Therefore, the consumer choose a totally flat consumption path, no matter what is the time variation in labor income. And any unexpected change in consumption leads to a parallel shift in the path of consumption by an amount equal to the annuity value of the change in labor income. This is the manifestation of intertemporal consumption smoothing. • More generally, if the interest rate is higher (lower) than the discount rate, the path of consumption is smooth but has a positive (negative) trend. To see this, note that the Euler condition is log ct+1 ≈ θ(R − ρ) + log ct . 4.2.4 Incomplete Markets and Self-Insurance • The above analysis has assumed no uncertainty, or that markets are complete. Extending the model to introduce idiosyncratic uncertainty in labor income would imply an Euler condition of the form u0 (cjt ) = β(1 + R)Et u0 (cjt+1 ) Note that, because of the convexity of u0 , as long as V art [cjt+1 ] > 0, we have Et u0 (cjt+1 ) > u0 (Et cjt+1 ) and therefore Et cjt+1 > [β(1 + R)]θ cjt This extra kick in consumption growth reflects the precautionary motive for savings. It remains true that transitory innovations in income result to persistent changes in consumption (because of consumption smoothing). At the same time, consumers find it optimal to accumulate a buffer stock, as a vehicle for self-insurance.