ECON 581: Growth with Overlapping Generations Instructor: Dmytro Hryshko
Readings Acemoglu, Chapter 9.
Motivation Neoclassical growth model relies on the representative household. OLG models allow for the arrival of new households into the economy: realistic (takes account of life cycle), but also introduces a range of new economic interactions. Decisions by older generations affect the prices faced by younger generations.
The baseline OLG model: demographics, preferences, and technology Time is discrete and runs to infinity. Each individual lives for 2 periods: those born at t live at t and t + 1 only. Utility function for an individual born at t: U t (c 1 (t), c 2 (t + 1)) = u(c 1 (t)) + βu(c 2 (t + 1)), where u C 2 and concave, and c 1 (t) is the consumption of an individual born at t when young, c 2 (t + 1) is the consumption of the same individual when old (at t + 1).
The baseline OLG model Factor markets are competitive. Individuals work in the first period of their lives only, supply one unit of labor inelastically, and earn w(t). Exponential population growth. The size of generation t is L(t) = (1 + n) t L(0).
The baseline OLG model Production side: a set of competitive firms, aggregate technology is CRS in K and L: Y (t) = F (K(t), L(t)). Employment at t is equal to the size of the young group at t. F (K, L) = LF ( K L, 1) = Lf(k), where k K L. The rental price of capital, or the gross return to saving is The wage rate is: [ ] k R(t) = F K (K, L) = Lf (k) = f (k). (1) K w(t) = F L (K, L) = f(k) Lf (k) K L 2 = f(k) f (k)k. (2)
The baseline OLG: consumption decisions An individual of generation t solves: max u [c 1(t) + βu(c 2 (t + 1))] c 1 (t),c 2 (t+1),s(t) s.t. c 1 (t) + s(t) w(t) c 2 (t + 1) R(t + 1)s(t). Young individuals rent their savings as capital to final good producers at the end of t and receive the return at t + 1, after production is carried out. Savings should be positive since c 2 (t + 1) 0, and the gross return on savings is R(t + 1) = 1 + r(t + 1) > 0.
The basic OLG: consumption decisions The budget constraint in present value terms is The Euler equation is: c 1 (t) + c 2(t + 1) R(t + 1) w(t). u (c 1 (t)) = βr(t + 1)u (c 2 (t + 1)), or u [w(t) s(t)] = βr(t + 1)u [R(t + 1)s(t)]. (3) Thus, the optimal savings per person is Total savings in the economy is s(t) = s (w(t), R(t + 1)). (4) S(t) = s(t)l(t).
The law of motion of aggregate capital Capital depreciates fully after its use in production, and all savings are invested into capital: K(t + 1) = L(t)s (w(t), R(t + 1)). (5)
Equilibrium A competitive equilibrium can be represented by sequences of aggregate capital stocks, household consumption, and factor prices, {K(t), c 1 (t), c 2 (t), R(t), w(t)} t=0 such that the factor price sequence is given by (1) and (2), individual consumption decisions {c 1 (t), c 2 (t)} t=0 are given by (3) and (4), and the aggregate capital stock {K(t)} t=0 evolves according to (5). A steady-state equilibrium is defined as an equilibrium when k K L is constant.
From (5), k(t + 1) = L(t)s(w(t), R(t + 1)) L(t + 1) Utilizing equilibrium prices, = s(w(t), R(t + 1)). 1 + n k(t + 1) = s (f(k(t)) k(t)f (k(t)), f (k(t + 1))). (6) 1 + n A steady-state is given by k = k(t + 1) = k(t): k = s (f(k ) k f (k ), f (k )). 1 + n
Introduction to Modern Economic Growth Various equilibria in the baseline OLG model k(t+1) 45 0 k 1 * k 3 * k 2 * k 4 * k(t)
Since the savings function in (6) is general, the difference equation may feature complicated dynamics resulting in multiple equilibria (see the slide above). To make some tight predictions, we need to put more structure on utility and production function (next).
Restrictions on utility and production function Utility function is CRRA: U t (c 1 (t), c 2 (t + 1)) = c 1(t) 1 θ 1 θ + β c 2(t + 1) 1 θ, 1 θ where θ > 0, and β (0, 1). Production function is Cobb-Douglas: Y (t) = K(t) α L(t) 1 α and so Thus, the Euler equation is c 2 (t + 1) c 1 (t) f(k) = k α. = (βr(t + 1)) 1/θ.
The Euler equation in terms of savings is (verify) so that s(t) θ βr(t + 1) 1 θ = (w(t) s(t)) θ, s(t) = w(t) ψ(t + 1), where ψ(t + 1) [ 1 + β 1/θ R(t + 1) (1 θ)/θ] > 1. Note that 0 < s w = s(t) w(t) = 1 ψ(t + 1) < 1.
Savings and the real interest rate s R = s(t) R(t + 1) = 1 θ (βr(t + 1)) 1/θ }{{ θ }}{{} >0 0 s(t) ψ(t + 1) } {{ } >0. Note that s R < 0 if θ > 1: the income effect (positive for c 1 (t) since consumer is a net saver) dominates the substitution effect (negative for c 1 (t) since the cost of the time t-consumption is higher); s R > 0 if θ < 1: the substitution effect dominates, and individuals reduce their consumption when young and increase s(t); s R = 0 if θ = 1 (log-preferences): income and substitution effects exactly cancel out.
or k(t + 1) = s(t) 1 + n = w(t) (1 + n)ψ(t + 1), k(t + 1) = In the steady state k = f(k(t)) k(t)f (k(t)) (1 + n) [ 1 + β 1/θ f (k(t + 1)) (1 θ)/θ]. f(k ) k f (k ) (1 + n) [ 1 + β 1/θ f (k ) (1 θ)/θ].
Proposition In the OLG model with two-period lived households, Cobb-Douglas technology, and CRRA preferences, there exists a unique steady-state equilibrium with the capital-labor ratio k, and for any θ > 0, this equilibrium is globally stable for all k(0) > 0.
Equilibrium dynamics in the canonical overlapping Introduction to Modern Economic Growth generations model k(t+1) 45 k* 0 k(0) k* k (0) k(t)
The canonical OLG model Utility function: U t (c 1 (t), c 2 (t + 1)) = log c 1 (t) + β log c 2 (t + 1). Production function is f(k) = k α. At the optimum, and so c 1 (t) = w(t) 1 + β, The savings rate is s(t) = w(t) c 1 (t) = β 1+β β 1 + β w(t). constant (recall the Solow model).
The canonical OLG model The dynamics of capital accumulation is k(t + 1) = s(t) 1 + n βw(t) = (1 + β)(1 + n) β(1 α)k(t)α = (1 + n)(1 + β).
In the unique steady state, [ k = β(1 α) (1 + n)(1 + β) ] 1 1 α.
Overaccumulation and Pareto optimality of competitive equilibrium in the OLG model Suppose the social planner attaches weight ξ t on generation t s utility and maximizes max ξ t U t (c 1 (t), c 2 (t + 1)) = t=0 ξ t [u(c 1 (t)) + βu(c 2 (t + 1))] s.t. the resource constraint F (K(t), L(t)) = K(t + 1) }{{} aggr. inv. at t + L(t)c 1 (t) + L(t 1)c 2 (t), }{{} aggr. cons. at t which, in per capita terms, is t=0 K(t + 1) L(t + 1) f(k(t)) = + c 1 (t) + c 2(t) L(t + 1) L(t) 1 + n = (1 + n)k(t + 1) + c 1 (t) + c 2(t) 1 + n.
The F.O.C is: u (c 1 (t)) = β f (k(t + 1)) u (c }{{} 2 (t + 1)). =R(t+1) The social planner prefers to allocate consumption (over time) of a given individual in exactly the same way as the individual himself would do. Allocations of resources across generations can be different since the weights can be different. Is competitive equilibrium Pareto optimal?
The resource constraint at the steady-state equilibrium implies f(k ) (1 + n)k = c 1 + (1 + n) 1 c 2 c. Maximum consumption per capita at the steady state is obtained when c k = f (k ) (1 + n) = 0. The golden-rule capital labor ratio is defined from At k gold, f (k gold ) = R = 1 + r = 1 + n. r = n.
Suppose that k > k gold, then, by concavity of f( ), c k < 0 reducing savings can increase total consumption. In this case, the economy is said to be dynamically inefficient: it overaccumulates capital. It happens when r < n.
Pareto improvement Suppose at time T k > k gold. Suppose the next period capital is reduced by k, and this capital is kept forever at k k. At time T, f(k ) = c + (1 + n)k, and so f(k ) = (1 + n)(k k) + (c + c(t )), implying that c(t ) = (1 + n) k > 0. After time T, c (t) = [ f (k k) (1 + n) ] k > 0. }{{} <0 since f (k )<1+n The increase in consumption for each generation can be allocated equally during the 2 periods of their lives, increasing the utility of all generations.
Proposition In the baseline OLG economy, the competitive equilibrium is not necessarily Pareto optimal. More specifically, when r < n, the economy is dynamically inefficient. In this case, it is optimal to reduce the capital stock starting from the competitive steady state and increase the consumption level of all generations.
Why inefficiency? Pecuniary externalities are important: individuals from generation t face wages determined by the savings (capital stock) decisions of those from generation t 1; an individual from generation t 1 receives a rate of return on her savings determined by the savings decisions of others of generation t 1. Dynamic inefficiency arises from overaccumulation that results from the need of the current generation to save for old age. The more they save, the lower is the return, the more they are encouraged to save. Thus, if there was some way available to provide for the consumption when old, overaccumulation problem might be ameliorated.
The role of social security in capital accumulation 1 Fully funded system: the young make contributions to the social security system, and the contributions are paid back to them in their old age. 2 Pay-as-you-go (unfunded) system: transfers from the young go directly to the current old. Discourages aggregate savings; in case of dynamic inefficiency, may lead to a Pareto improvement.
Fully funded social security At date t, the government raises d(t) from the young (e.g., through compulsory contributions to social security accounts); invests into the capital stock; workers receive R(t + 1)d(t) when old.
Fully funded social security The individual maximization problem becomes max u(c 1(t)) + βu(c 2 (t + 1)) c 1 (t),c 2 (t+1),s(t) s.t. c 1 (t) + s(t) + d(t) w(t) c 2 (t + 1) R(t + 1)(s(t) + d(t)), where d(t) is chosen by the government. Clearing of the capital marker requires S(t) + D(t) = K(t + 1). Dividing through by L(t), this implies s(t) + d(t) = (1 + n)k(t + 1). From the second constraint at equality s(t) + d(t) = c 2(t+1) R(t+1). Plug this result into the first constraint to obtain c 1 (t) + c 2 (t + 1)/R(t + 1) w(t) the same constraint as without d(t).
Proposition Consider a fully funded social security system whereby the government collects d(t) from young at date t. 1. (non-binding constraint) Suppose that s(t) 0 for all t. If, given the feasible sequence {d(t)} t=0, the utility-maximizing sequence of savings {s(t)} t=0 is such that s(t) > 0 for all t, then the set of competitive equilibria without the social security is the same as the set of competitive equilibria with social security. 2. (no constraint) Without the constraint s(t) 0, given any feasible sequence {d(t)} t=0, the set of competitive equilibria with social security is identical to the set of competitive equilibria without social security.
Unfunded social security The government collects d(t) from the current young at t, and distributes to the current old. Note that per capita benefit for the old is D(t) L(t 1) = D(t) L(t) = d(t)(1 + n) = b(t). L(t) L(t 1) (For each person from generation t 1 there are 1 + n persons from generation t, each of them contributes d(t) to social security.) The individual maximizes max u(c 1(t)) + βu(c 2 (t + 1)) c 1 (t),c 2 (t+1),s(t) s.t. c 1 (t) + s(t) + d(t) w(t) c 2 (t + 1) R(t + 1)s(t) + (1 + n)d(t + 1)
The net return on social security payments is n rather than R(t + 1) 1; only s(t) out of s(t) + d(t) contribute towards capital accumulation. This discourages aggregate savings. Pareto improvement: Suppose individuals of generation t may choose d(t); it s given to the old as consumption, and they receive 1 + n dollars for every dollar invested when they become old. There will be no investment into physical capital until r(t + 1) n. The unfunded social security would increase r(t) enough so that the economy is no longer in the dynamic inefficiency region.
OLG with Impure Altruism: warm glow preferences Parents derive utility from (the warm glow of) their bequest, rather than from the offspring s utility or consumption. The economy consists of a continuum of individuals of measure 1. Each individual lives for 2 periods: childhood (born to life, no consumption in the first period) and adulthood (works, supplying 1 unit of labor inelastically; begets an offspring). Population is constant at 1.
Maximization problem of individual (i, t), reaching adulthood at t: s.t. max log(c i(t)) + β log(b i (t)) c i (t),b i (t) c i (t) + b i (t) y i (t) w(t) + R(t)b i (t 1), where w(t) = f(k(t)) k(t)f (k(t)), R(t) = f (k(t)), b i (t 1) is the bequest received by an individual from his parent; capital depreciates fully after its use. The total capital-labor ratio at t + 1 is obtained by aggregating the bequests of adults at t: k(t + 1) = 1 0 b i (t)di.
At the maximum, b i (t) = β 1 + β y i(t) = β 1 + β [w(t) + R(t)b i(t 1)]. Integrating over all agents in the economy, k(t + 1) = 1 In the steady-state 0 b i (t) = β [ w(t) + R(t) 1 + β 1 = β [w(t) + R(t)k(t)] 1 + β }{{} =f(k(t)) by Euler s theorem = β 1 + β f(k(t)). k = 0 ] b i (t 1) β 1 + β f(k ). (7)
The aggregate dynamics of k(t + 1) do not depend on the distribution of bequests or income across households. If the economy is at the steady-state capital-labor ratio k, individual bequest dynamics are given by b i (t) = β 1 + β [w + R b i (t 1)], where, for example, R = f (k β ). If 1+β R < 1, then, for any b i (0), b i (t) converges to a unique bequest (wealth): b = βw 1 + β(1 R ). By strict concavity, f (k )(0 k ) > f(0) f(k ) = f(k ). Thus, f (k ) = R < f(k ) k = 1+β β, where the last equality utilizes (7).
Consider the OLG economy with warm glow preferences. In this economy, there exists a unique competitive equilibrium; the aggregate capital-labor ratio converges monotonically to the unique steady-state capital-labor ratio k. The distribution of bequests and wealth ultimately converges to full equality, with each individual having a bequest (wealth) level of b, determined by w = f(k ) k f (k ) and R = f (k ).
OLG with perpetual youth Baseline OLG: individuals have finite lives and know when they die. Perpetual youth model: each individual is potentially infinitely-lived, but faces a probability ν (0, 1) that her life comes to an end at any date t (all realizations are independent). Let the pure time discount factor be β; normalize u(0) = 0 to be the utility after death. Expected utility at time 0 is given by U 0 (c(0), c(1),...) = u(c(0)) + β(1 ν)u(c(1)) + βν u(0) }{{} =0 + β 2 (1 ν) 2 u(c(2)) + β 2 (1 ν)ν u(0) +... }{{} =0 = (β(1 ν)) t u(c(t)). t=0
The expected lifetime of an individual is ν + 2(1 ν)ν + 3(1 ν) 2 ν + = j=0 j=0 jν(1 ν) j 1 j=0 = ν j(1 ν) j 1 d = ν dν (1 ν)j = ν d dν = ν d dν [ 1 1 (1 ν) ] = 1 ν <. (1 ν) j j=0 Expected lifetime is finite but those who lived up to a certain date have the same expectation of further life. Age has no effect on the future longevity and has no predictive power on how many more years individuals expect to live: individuals are perpetually young.
Individual i s flow budget constraint is a i (t + 1) + c i (t) = (1 + r(t))a i (t) + w(t) + z i (t), where z i (t) are the transfers to individual i. Competitive life insurance firms make annuity payments z i (a(t)) to individuals in return for capturing the assets when they die. The profits of an insurance company are: π(a(t), t) = (1 ν)z(a(t)) + νa(t). With free entry, π(a(t), t) = 0, and so z(a(t)) = ν 1 ν a(t).
The evolution of the total population in the economy is with L(0) = 1. L(t + 1) = (1 + n ν)l(t), The production side is the standard CRS in K and L; factor markets are competitive. It is not sufficient to specify aggregate consumption since the level of consumption is not the same for all individuals. Let c(t τ) be the consumption of a household born at τ t. The flow budget constraint is ( a(t + 1 τ) + c(t τ) = 1 + r(t) + ν ) a(t τ) + w(t). 1 ν
Definition: A competitive equilibrium consists of paths of capital stock, wage rates, and rental rates of capital, {K(t), w(t), R(t)} t=0, and paths of consumption for each generation, {c(t τ)} t=0,τ t, such that each individual maximizes utility, and the time path of factor prices, {w(t), R(t)} t=0, is such that given the time path of capital stock and labor, {K(t), L(t)} t=0, all markets clear. The consumption Euler equation for an individual of generation τ at time t is (show this): [ u (c(t τ)) = β(1 ν) 1 + r(t + 1) + ν ] u (c(t + 1 τ)) 1 ν = β [(1 + r(t + 1))(1 ν) + ν] u (c(t + 1 τ)) β [1 + r(t + 1)] u (c(t + 1 τ)), since (1 + r)(1 ν) 1 + r ν for small ν and r.