ECONOMICS 723 Models with Overlapping Generations 5 October 2005 Marc-André Letendre Department of Economics McMaster University c Marc-André Letendre (2005).
Models with Overlapping Generations Page i Contents 1 Endowment Economy: An Overview of the Model 1 1.1 Agent s Optimization Problem.......................... 2 1.2 Feasible, Efficient and Optimal Consumption Allocation............ 5 1.3 Competitive Equilibrium............................. 6 1.4 Extension I: Population Growth......................... 8 1.5 Extension II: Heterogeneity within a Cohort (2-Countries).......... 9 2 Production Economy: an Overview of the Model 12 2.1 Households.................................... 12 2.1.1 The Savings Decision........................... 12 2.1.2 Supply of Inputs............................. 15 2.2 Firms....................................... 16 2.3 Competitive Equilibrium and Transition Equation............... 17 2.4 Growth, Transition Period and Steady State.................. 18 2.5 Computing Time Paths.............................. 19 2.6 Comparative Dynamics.............................. 22 2.7 Growth in Population and Technology..................... 24 2.7.1 Growth in Population.......................... 24 2.7.2 Growth in Technology.......................... 25
Models with Overlapping Generations Page ii 2.8 (Un)Importance of Initial Conditions...................... 27 3 Fiscal Policy in the Diamond Model 27 3.1 Government Budget Constraint and National Income Identity........ 27 3.2 Fiscal Policy with a Zero Deficit......................... 28 3.2.1 Household Savings Decision with Taxes and Transfers......... 28 3.2.2 The Transition Equation......................... 29 3.2.3 Taxing the Young............................. 29 3.2.4 Taxing the Old.............................. 30 3.2.5 Pay-as-you-go Social Security System.................. 31
Models with Overlapping Generations Page iii Acknowledgements This set of notes borrows from McCandless with Wallace (1991) Introduction to dynamic macroeconomic theory: an overlapping generations approach, Harvard University Press; Auerbach and Kotlikoff (1998) Macroeconomics: an integrated approach, MIT Press; and lecture notes by Gregor Smith (1999). For more on OLG models, consult these sources and Romer (2001, 1996).
Models with Overlapping Generations Page 1 Introduction Most of the discussion in the section of the course on two-period economies focused on models where all agents are identical (representative agents models). As we have seen in the section on open-economy models, allowing for heterogeneity across agents in that framework is relatively easy. Obviously economic agents can differ in many respects. A fundamental characteristic of overlapping generations (OLG) models is that there are agents from different age groups (or cohorts) alive at the same time. Just like in the section on two-period economies, we assume that an individual lives for two periods. Note however that the economy itself goes on forever. Since a new cohort of agents is born each period, this implies that in any given period there are young and old agents in the economy. We will see that many (but not all) results from the section on two-period economies carry over to the OLG framework. OLG models are appealing for a number of reasons. First, they are tractable and relatively easy to work with even though they can be used to analyze complex issues. Second, they allow us to study behaviour of economic agents over their life-cycle. Third, many policy issues (like transfer payments) make more sense when the population is heterogeneous. Fourth, they provide examples of competitive equilibria which are not Pareto optimal. Our study of OLG models starts with an endowment economy framework that is related in a number of ways to the two-period economy setting we studied in section II of the course. Then we consider a model with production related to the Diamond OLG model. We are going to use it to study growth and fiscal policy issues. While we will encounter a version of the model with money and will look at fiscal policies, the discussion of macro policy will be quite limited compare to what you will have a chance to see in other graduate courses like ECON 728 and ECON 741. 1 Endowment Economy: An Overview of the Model We work in discrete time and the time index t can take values from to (the economy never ends). By convention, we normally look at the economy starting in period t = 1. The history of the economy (periods to 0) is taken as given and determines initial conditions.
Models with Overlapping Generations Page 2 Individuals live for two periods only. Each period t, a new generation/cohort is born. This generation is called generation t and has N t members. At this point we assume that all members of a generation are identical. 1 We will relax this assumption in section 1.5. A member of generation t lives in period t (young) and t + 1 (old). Therefore, in period t, there are N t young agents and N t 1 old agents (who were born in period t 1). Figure 1.1 below explains the overlapping generations pattern FIGURE 1.1 Time periods... 0 1 2 3 4.... old 0 young-0 old-0 1 young-1 old-1 Generations 2 young-2 old-2 3 young-3 old-3. young We assume there is only one good each period. In period t, we talk of the period-t good (or the time-t good). We normalize to price of that good to unity. Therefore, we measure all real variables in the model in units of this unique good (e.g. bushels of corn). For now there is no storage technology allowing agents to carry goods from period t into period t + 1. Since we work with an endowment economy, there is no firms producing goods. Rather, each individual is endowed with a non-negative quantity of goods in both periods of life. 1.1 Agent s Optimization Problem An person born in period t has preferences defined over his consumption in young age (denoted c 1t ) and consumption in old age (denoted c 2t+1 ). Notice that consumption has two subscripts. The first one indicates the age of the person (1=young age and 2=old age) and the second one indicates the period in the economy. The preferences of a person born in period t are represented by the utility function U(c 1t, c 2t+1 ). 1 That is, all members of a generation have the same preferences and same endowments.
Models with Overlapping Generations Page 3 Like in the two-period economy setting, we mostly work with the logarithmic utility function U(c 1t, c 2t+1 ) = ln c 1t + ln c 2t+1, 0 < < 1 (1) where is a discount factor indicating how the person value consumption in old age versus consumption in young age. A person born in period t (a member of generation t) is provided with an endowment stream {y 1t, y 2t+1 }. Since the agent owns her endowments, she can lend part of it to other agents when she is young and get the proceeds from the loan one period later. Obviously, an old person has no incentives to lend part of his endowment since he will not be alive next period when the loan repayment is made. Similarly, a young person is not interested in lending to an old since he will never have the loan paid back by the old who will not be alive next period. As a result, we won t observe intergenerational trade (i.e. trade among members of different generations). Therefore, we need only allow for intragenerational trade (i.e. trade among members of a given generation). Obviously, if all individuals born in a given period are identical, then there will be no trade in equilibrium just like there was no trade in equilibrium in the two-period endowment economy model when all agents were identical. In such a case, the equilibrium interest rate adjust to insure that a person s savings is exactly zero. The savings of an individual born in period t is and his consumption in old age is s t = y 1t c 1t (2) c 2t+1 = y 2t+1 + (1 + r t+1 )s t (3) where by convention r t+1 denotes the real interest rate linking periods t and t + 1. The sequence budget constraint (2)-(3) implies the present value budget constraint or intertemporal buget constraint for someone born in period t c 1t + c 2t+1 1 + r t+1 = y 1t + y 2t+1 1 + r t+1. (4) Therefore, the optimization problem solved by an individual born in period t is to choose c 1t and c 2t+1 to maximize U(c 1t, c 2t+1 ) subject to budget constraint (4). Figure 1.2 provides a graphical representation of the agents optimization problem. In the figure, we measure consumption in old age along the vertical axis while we measure consumption in young age along the horizontal axis. Intertemporal budget constraint (4) makes
Models with Overlapping Generations Page 4 clear that this constraint is linear and has a slope equal to (1 + r t+1 ). Accordingly, 1 + r t+1 represents the price of young age consumption in terms of old age consumption. To see this, think about how much old age consumption is given up in order to increase consumption in young age by one unit. The triangle OAB in Figure 1.2 represents the individual budget set for a given endowment stream (y 1t, y 2t+1 ). Consumption bundles that are outside of this budget set are not affordable at the current endowments and interest rate. The curves in Figure 1.2 are indifference curves. An indifference curve represents the combinations of consumption in young age and in old age that yield the same level of utility. Obviously, indifference curves are related to preferences and the utility function. The further from the origin an indifference curve is, the higher the level of utility. Accordingly, the individual prefers the consumption allocation given by point E over allocations given by points D or F because point E lies on a higher indifference curve. Actually, since point E is the point where the indifference curve for utility level u 2 is tangent to the intertemporal budget constraint, u 2 is the highest level of utility that can be achieved at the current interest rate, given endowments (y 1t, y 2t+1 ). While consumption allocations lying on the indifference curve for utility level u 3 are preferred to point E, these allocations are not affordable. Mathematically, when the individual preferences are represented by utility function (1) then the agent s optimization problem can be transformed into the following unconstrained maximization problem: The first-order condition is which implies Equations (4) and (7) imply c 2t+1 = max c 1t ln(c 1t ) + ln[y 2t+1 + (1 + r t+1 )(y 1t c 1t )]. (5) 1 (1 + r t+1 ) c 1t y 2t+1 + (1 + r t+1 )(y 1t c 1t ) = 0 (6) c 1t = 1 [ y 1t + y ] 2t+1 c 1t (r t+1, y 1t, y 2t+1 ). (7) 1 + 1 + r t+1 1 + [(1 + r t+1)y 1t + y 2t+1 ] c 2t+1 (r t+1, y 1t, y 2t+1 ). (8) Except for the slightly different notation, equations (7) and (8) are identical to the consumption functions derived in the section on two-period economies (see Smith page 20).
Models with Overlapping Generations Page 5 Finally, the savings function of an agent born in period t is given by s t (r t+1, y 1t, y 2t+1 ) = y 1t c 1t (r t+1, y 1t, y 2t+1 ) = 1 + y y 2t+1 1t (1 + )(1 + r t+1 ). (9) 1.2 Feasible, Efficient and Optimal Consumption Allocation Aggregate quantities are calculated by aggregating over all individuals (young and old) alive in a given period. For example, aggregate consumption of the time-t good is denoted C t and is calculated as C t = N t 1 c 2t + N t c 1t. (10) Similarly, aggregate endowment of the time-t good is denoted Y t and is calculated as Y t = N t 1 y 2t + N t y 1t. (11) A time-t consumption allocation is given by the period-t consumption of a representative young agent (c 1t ) and the period-t consumption of a representative old agent (c 2t ). In short, a time-t consumption allocation is the pair {c 1t, c 2t }. A consumption allocation is the sequence of time-t consumption allocations for all t 1. In short, a consumption allocation is the sequence {c 1t, c 2t } t=1. A feasible consumption allocation is a consumption allocation that can be achieved with the given total resources and the technology. More formally, a consumption allocation is feasible if the sequence of aggregate consumption {C t } t=1 satisfies C t Y t for all t 1. A feasible consumption allocation is efficient if there is no alternative feasible allocation with more total consumption of some good and no less of any other good. An efficient consumption allocation is a consumption allocation where C t = Y t for all t 1. Loosely speaking, a consumption allocation is efficient when there is no waste. Consumption allocation A is Pareto superior to consumption allocation B if (i) no one strictly prefers B to A, and (ii) at least one person strictly prefers A to B. A consumption allocation is Pareto optimal if it is feasible and if there does not exist a feasible consumption allocation that is Pareto superior to it.
Models with Overlapping Generations Page 6 1.3 Competitive Equilibrium In most cases, competitive equilibria are not Pareto optimal in OLG models. Therefore, we have to solve for the competitive equilibrium explicitly instead of using the second welfare theorem and a central planner problem. We first define a competitive equilibrium. A competitive equilibrium is a consumption allocation and a price system such that (i) the quantities that are relevant for a particular person maximize that person s utility subject to the relevant budget constraint, taking prices as given. (ii) the quantities clear all markets at all dates. Note that in any given period, there are only two markets. A market for the consumption good and a market for private borrowing and lending. Recall that there is no intergenerational trade in the model (see subsection 1.1). Therefore, only the young agents are involved in the borrowing and lending market. Obviously, for the borrowing and lending market to clear in period t the sum of the savings of all young agents alive in period t must be zero. When solving for a competitive equilibrium, the key is to use the competitive equilibrium condition S t (r t+1 ) = 0. (12) where the function S t (r t+1 ) represents total savings of all young people alive in period t. In an abuse of notation, we write the total savings of all young people alive in period t S t (r t+1 ) when this savings function also depends on current and next period endowments. When all individuals who are born in period t are identical and have log utility, then we simply have S t (r t+1 ) = N t s t (r t+1, y 1t, y 2t+1 ) = 1 + N N t y 2t+1 ty 1t (1 + )(1 + r t+1 ) (13) where the last equality comes from using (9). Equilibrium condition (12) can be used to solve for the equilibrium interest rate. Then the consumption allocation is simply found by substituting the equilibrium interest rate in the consumption functions ((7) and (8) for the log utility case). The competitive equilibrium condition (12) is derived using the two conditions appearing in the definition of a competitive equilibrium. We start with the market clearing condition for good t N t c 1t + N t 1 c 2t = N t y 1t + N t 1 y 2t. (14)
Models with Overlapping Generations Page 7 Then summing the budget constraint of the old agents alive in period t we have (see equation(3)) N t 1 c 2t = N t 1 y 2t + (1 + r t )N t 1 s t 1 (15) Clearing of the period t 1 borrowing and lending market implies that N t 1 s t 1 = 0 which in turn implies N t 1 c 2t = N t 1 y 2t. (16) Subtracting (16) from (14) implies N t c 1t = N t y 1t N t (y 1t c 1t ) = 0. (17) Note that utility maximization implies the consumption function c 1t (r t+1, y 1t, y 2t+1 ) for a young agent alive in period t. To ease notation, we drop the endowments from the list of arguments of the consumption and savings functions. Substituting the consumption function in (17) implies N t (y 1t c 1t (r t+1 )) = 0 N t s t (r t+1 ) = 0 S t (r t+1 ) = 0 (18) where s t (r t+1 ) denotes the saving function of a young agent alive in period t and S t (r t+1 ) the aggregate saving function of the young agents alive in period t. The derivation of the competitive equilibrium condition S t (r t+1 ) = 0 highlights the fact that this condition takes into account of market clearing on the goods market and on the financial market as well as utility maximization. We complete this section by solving for the competitive equilibrium consumption allocation and interest rate for the case where all members of a generation are identical and have log utility. The first step is to perform utility maximization and derive the consumption and savings functions. This yields equations (7), (8) and (9). Then we impose the equilibrium condition (12) which implies the equilibrium interest rate 1 + N N t y 2t+1 ty 1t (1 + )(1 + r t+1 ) = 0 r t+1 = y 2t+1 1. (19) y 1t To find consumption of a person born in period t in equilibrium, plug the equilibrium interest rate (19) into consumption functions (7) and (8) to get c 1t = y 1t and c 2t+1 = y 2t+1. As expected, because all members of a generation are identical, we end up with an autarkic competitive equilibrium. Example 1.3.1 below works out a numerical example.
Models with Overlapping Generations Page 8 Example 1.3.1 Suppose that in each period, 100 identical individuals are born (e.g. N t = 100 for t = 0, 1, 2,...). The utility of an individual born in period t is given by U(c 1t, c 2t+1 ) = ln c 1t + 0.95 ln c 2t+1. (20) Finally, an individual born in period t has the following endowment stream {y 1t, y 2t+1 } = {1, 1.25}. Performing the utility maximization for an agent born in period t yields the consumption functions (7) and (8). Using the numerical values in the current example we have c 1t = 1 [ 1 + 1.25 ] = 0.5128 + 0.6410 c 1t (r t+1 ). (21) 1.95 1 + r t+1 1 + r t+1 The individual savings function is then and the aggregate savings function is s t (r t+1 ) = y 1t c 1t (r t+1 ) = 0.4872 0.6410 1 + r t+1 (22) S t (r t+1 ) = 100 s t (r t+1 ) = 48.72 64.10 1 + r t+1. (23) Imposing the equilibrium condition S t (r t+1 ) = 0 yields the equilibrium real interest rate Using this result in the consumption functions yields r t+1 = 64.10 1 = 0.32. (24) 48.72 c 1t = 1, c 2t+1 = 1.25. (25) Since all periods are identical in this example, the competitive equilibrium is the sequence of interest rate {r t+1 } t=1 = {0.3157} t=1 and the consumption allocation {c 1t, c 2t } t=1 = {1, 1.25}. This equilibrium consumption allocation is not surprising. In an economy where nobody trades, agents always consume their endowments. 1.4 Extension I: Population Growth Allowing for population growth is straightforward. Instead of having a constant cohort size over time (i.e. N t constant for all t) we have N t = (1 + η)n t 1. (26)
Models with Overlapping Generations Page 9 The net population growth rate is given by η. To calculate the gross growth rate, divide the population in period t by the population in period t 1 N t + N t 1 N t 1 + N t 2 = (1 + η)2 N t 2 + (1 + η)n t 2 (1 + η)n t 2 + N t 2 = (1 + η)[(1 + η) + 1] [(1 + η) + 1] = 1 + η. (27) Population growth at rate η implies that aggregate endowment (Y t ) and aggregate consumption (C t ) also grow at rate η. For example, the gross growth rate of Y is Y t Y t 1 = N t 1 y 2t + N t y 1t N t 2 y 2t + N t 1 y 1t = (1 + η) = 1 + η (28) N t 2 y 2t 1 + N t 1 y 1t 1 N t 2 y 2t 1 + N t 1 y 1t 1 where the last equality comes from the fact that all members of all generations receive the same endowment stream which implies y 1t = y 1t 1 and y 2t = y 2t 1. Little else is changed. We still impose the equilibrium condition (12) to solve for the equilibrium interest rate. The sequence of equilibrium interest rate and the consumption allocation is unchanged by the addition of population growth. Exercise: Use the environment described in example 1.3.1 with the addition of population growth. Assume N 0 = 1000 and that η = 0.1. Show that in the competitive equilibrium of this economy with population growth, the sequence of interest rate is {r t+1 } t=1 = {0.3157} t=1 and the consumption allocation is {c 1t, c 2t } t=1 = {1, 1.25} (as in example 4.1). 1.5 Extension II: Heterogeneity within a Cohort (2-Countries) As seen in previous sections, when all members of a cohort are identical, there is no trade in equilibrium. However, allowing for heterogeneity within a cohort can create incentives for trade. In this section we see how the model can be extended to deal with heterogeneity within cohorts. We assume there are two types of agents born in each period. To facilitate the exposition (and to make an obvious parallel to what we did in the section on two-period economies earlier in the course), think of two countries in free trade (same interest rate in both countries). In the home country, N t agents are born in period t. These agents have preferences given by the utility function U(c 1t, c 2t+1 ) and receive the endowment stream {y 1t, y 2t+1 }. In the foreign country Nt agents are born in period t. These agents have preferences given by the utility function u (c 1t, c 2t+1) and receive the endowment stream {y1t, y2t+1}. Note that the two countries can differ in three respects. First, there can be a different number of agents
Models with Overlapping Generations Page 10 born at home and abroad (i.e. N t does not necessarily equal Nt ). Second, the preferences can be different across countries (for example, there could be differences in the discount factor at home and abroad). Third, the endowment streams can differ across countries. 2 Let us consider the case where all residents of a country are identical and that cross-country differences come from differences in the discount factor ( in the home country and in the foreign country) and in the endowment stream. We assume everyone has log utility. The total savings of all young people alive in period t in the home country is given by S t (r t+1 ) = N t s t (r t+1, y 1t, y 2t+1 ) = 1 + N N t y 2t+1 ty 1t (1 + )(1 + r t+1 ) (29) while in the foreign country we have S t (r t+1 ) = N t s t (r t+1, y 1t, y 2t+1) = 1 + N t y1t Nt y2t+1 (1 + )(1 + r t+1 ). (30) In equilibrium, the total savings of all young people in both countries should be equal to zero (when one country is a net lender, the other country must be a net borrower) so the equilibrium condition used to solve for the equilibrium interest rate is now S t (r t+1 ) + S t (r t+1 ) = 0 or [ 1 + N ty 1t which implies ] [ N t y 2t+1 + (1 + )(1 + r t+1 ) 1 + N t y1t 1 + r t+1 = N t y 2t+1 + N t y 2t+1 (1+) (1+ ) 1+ N ty 1t + 1+ N t y 1t Nt y2t+1 ] (1 + )(1 + r t+1 ) = 0 (31) The world interest rate can then be plugged in consumption functions (like (7) and (8)) for the home country residents and foreign country residents to find out the consumption allocation. The following example shows a case where endowment streams and country size differ. Example 1.5.1 Consider two countries where individuals differ in their endowment streams. The characteristics of the two countries are summarized in the following table 2 Note that we talk of heterogeneity within a cohort when the agents in the two countries have different endowment streams or have different utility functions (or both). (32) If the only difference between the two groups is that N t N t, then the model reduces to the model discussed in previous sections.
Models with Overlapping Generations Page 11 Home country Foreign country N t = 60, t 0 N t = 40, t 0 {y 1t, y 2t+1 } = {1, 1.25}, t 0 {y 1t, y 2t+1} = {1, 1}, t 0 U(c 1t, c 2t+1 ) = ln c 1t + 0.95 ln c 2t+1, t 0 U(c 1t, c 2t+1) = ln c 1t + 0.95 ln c 2t+1, t 0 From (7)-(9) we can easily calculate the consumption and savings functions for an agent born in period t in the home country c 1t = 0.5128 + 0.6410 1 + r t+1, c 2t+1 = 0.4872(1 + r t+1 ) + 0.6090, s t (r t+1 ) = 0.4872 0.6410 1 + r t+1 (33) and for an agent born in period t in the foreign country c 1t = 0.5128 + 0.5128 1 + r t+1, c 2t+1 = 0.4872(1 + r t+1 ) + 0.4872, s t (r t+1 ) = 0.4872 0.5128 1 + r t+1. The aggregate savings function in each countries are (34) S t (r t+1 ) = 29.23 38.46 1 + r t+1, S t (r t+1 ) = 19.49 20.51 1 + r t+1. (35) We solve for the equilibrium interest rate by imposing the condition that savings of all young agents equal zero S t (r t+1 ) + S t (r t+1 ) = 0 48.72 58.97 1 + r t+1 = 0 r t+1 = 0.21. (36) Using the value of the equilibrium interest rate in equations (33)-(34) we find the solutions c 1t = 1.04, c 2t+1 = 1.20, s t (r t+1 ) = 0.04 (37) c 1t = 0.94, c 2t+1 = 1.08, s t (r t+1 ) = 0.06. (38) Note that contrary to example 1.3.3, there is intragenerational trade in equilibrium. The heterogeneity of agents across countries creates opportunity for trade. We can easily verify whether free trade is superior to autarky by comparing the welfare of home and foreign agents in free trade and in autarky. In free trade, the utility of a domestic agent is ln 1.04 + 0.95 ln 1.20 = 0.2124 and the utility of a foreign agent is ln 0.94 + 0.95 ln 1.08 = 0.0112. In autarky, all agents simply consume their endowments so the utility of a domestic agent is ln 1 + 0.95 ln 1.25 = 0.2120 and the utility of a foreign agent is ln 1 + 0.95 ln 1 = 0. Clearly, free trade is Pareto superior to autarky.
Models with Overlapping Generations Page 12 2 Production Economy: an Overview of the Model Endowment and production economies share several characteristics, so here we outline only the difference between the two structures. First, individuals are endowed with time rather than units of goods. Second, there is still only one good but it is storable. So it can be used for consumption and investment. Third, the model includes firms who employ workers and capital to produce. The model we work with in section 2 is closely related to the Diamond model. 2.1 Households 2.1.1 The Savings Decision As usual, the preferences of an agent born in period t are represented by the utility function U(c 1t, c 2t+1 ). We continue working with the logarithmic utility function U(c 1t, c 2t+1 ) = ln c 1t + ln c 2t+1, 0 < < 1 (1) Each person has the following time endowment: 1 unit in young age and 0 unit in old age. As you can see from the utility function above, we simplify the analysis by leaving leisure out of the utility function. Accordingly, a young person always supply his/her entire unit of time inelastically to the labour market. We assume that people do not work in old age. Therefore, the young people in the economy are the workers while the old people are the retirees. Deciding how much to consume in young age and in old age involves a savings decision. Recall that individuals do not work in their second period of life. Therefore, they must save in their first period of life to finance consumption in old age. The real labour income earned in young age is simply equal to real wage rate (w) since a young person works exactly one unit of time. The savings made in young age will be carried over to old age and invested in physical capital at the very beginning of old age. Since investment in physical capital provides a riskless positive rate of return (r), and there are no other investment instruments, all of an individual s savings end up being invested in physical capital. We assume that capital does not depreciate. Therefore, at the end of the old age period, the individual gets
Models with Overlapping Generations Page 13 back his/her entire investment in physical capital plus the return on that investment. As a result, an individual born in period t faces the following sequence of budget constraints when making consumption and savings decisions: s t + c 1t = w t (39) c 2t+1 = s t (1 + r t+1 ) (40) The timing on the real interest rate r in the latter equation reflects the fact that savings from period t are invested in physical capital only at the very beginning of period t + 1. Note that individuals are price takers. Accordingly, they take w and r as given when making consumption and savings decisions. The utility maximization problem solved by someone born in period t can be represented graphically. With this graphical analysis in mind, we derive the intertemporal budget constraint faced by an individual born in period t. We derive the intertemporal budget constraint by combining budget constraints (39) and (40) is such a way that we eliminate s t c 2t+1 = w t (1 + r t+1 ) (1 + r t+1 )c 1t. (41) The intertemporal budget constraint is also referred to as the present-value budget constraint because it can also be written as c 1t + c 2t+1 1 + r t+1 = w t (42) where the left-hand side represents the present-value of consumption whereas the right-hand side represents the present value of labour income. In Figure 2.1, we measure consumption in old age along the vertical axis while we measure consumption in young age along the horizontal axis. Intertemporal budget constraint (41) makes clear that this constraint is linear and has a slope equal to (1 + r t+1 ). Accordingly, 1 + r t+1 represents the price of young age consumption in terms of old age consumption. To see this, think about how much old age consumption is given up in order to increase consumption in young age by one unit. The triangle OAB in Figure 2.1 represents the individual budget set for a given wage rate and real interest rate. Consumption bundles that are outside of this budget set are not affordable at the current wage rate and interest rate. The curves in Figure 2.1 are indifference curves. An indifference curve represents the combinations of consumption in young age and in old age
Models with Overlapping Generations Page 14 that yield the same level of utility. Obviously, indifference curves are related to preferences and the utility function. The further from the origin an indifference curve is, the higher the level of utility. Accordingly, the individual prefers the consumption allocation given by point E over allocations given by points D or F because point E lies on a higher indifference curve. Actually, since point E is the point where the indifference curve for utility level u 2 is tangent to the intertemporal budget constraint, u 2 is the highest level of utility that can be achieved at the given wage rate and interest rate. While consumption allocations lying on the indifference curve for utility level u 3 are preferred to point E, these allocations are not affordable. Formally, the problem solved by a person born in period t is to choose c 1t, c 2t+1 and s t to maximize (1) subject to (39) and (40) [or (41), or (42)] taking w t and r t+1 as given. While there are a few different ways to solve this maximization problem in order to get savings and consumption functions, perhaps the easier way to proceed is to use equations (39) and (40) to substitute out c 1t and c 2t+1 from the objective function (1). Proceeding that way leaves us with the optimization problem max s t U(s t ) = ln(w t s t ) + ln(s t (1 + r t+1 )). (43) The first-order condition corresponding to problem (43) is found by setting equal to zero the partial derivative of the function U(s t ) with respect to s t (the only remaining choice variable). This first-order condition is which yields U(s t ) s t = 1 w t s t ( 1) + s t = s t (1 + r t+1 ) (1 + r t+1) = 0 (44) 1 + w t s t (w t ). (45) where s t (w t ) denotes the savings function of a young person alive in period t. The consumption function in young age and in old age (still for someone born in period t) is found by substituting the above savings function in budget constraints (39) and (40) c 2t+1 = c 1t = 1 1 + w t c 1t (w t ) (46) 1 + w t(1 + r t+1 ) c 2t+1 (w t, r t+1 ). (47) Notice that the consumption function in young age does not depend on the interest rate at all. You can think of the consumption function in young age as generally having the
Models with Overlapping Generations Page 15 following format: consumption is equal to a fraction of the present value of lifetime income (see for example equation (7)). Here, the present value of lifetime resources is simply w t (which does not depend on r) since there is no labour income earned in old age. For this reason, changes in the interest rate have no wealth effects on consumption. 3 The fraction of lifetime resources here is constant because the utility function we are employing is such that the income and substitution effects of interest rate changes perfectly cancel out. However, This is not the case for all utility functions. Exercise: Derive the savings function of an individual who has a lifetime utility function of the type U(c 1t, c 2t+1 ) = c1 1/σ 1t 1 1/σ + c1 1/σ 2t+1 1 1/σ, σ > 0 where σ is the elasticity of intertemporal substitution. 2.1.2 Supply of Inputs Let L t denote the labour input used by firms in period t. What is the labour supply in period t? Well, we know that (1) only young people work, (2) each young person is endowed with one unit of time, (3) individuals supply their entire time endowment to the job market. Since there are N t young people in period t, and that they all work one unit of time, labour supply in period t is simply N t. Accordingly, the labour supply curve is vertical and we have L t = N t since there are no frictions in the model preventing the labour market from clearing. As we will see in the next section, production depends on another input, physical capital. Let K t denote the capital stock installed in the economy and available for production at the beginning of period t. As mentioned above, the savings of a young person in period t are carried over to old age to be invested in physical capital. Therefore, new capital in the amount N t s t (w t ) is formed at the beginning of period t + 1. Since the old people alive in period t completely reverse their investment in capital at the end of period t to finance their consumption 4 in that period, we have K t+1 = N t s t (w t ). Therefore, the capital stock in period t comes entirely from the savings of all young people in period t. 3 This will no longer be true once we allow for taxes to be paid in old age. 4 Since old age is the last period of life, an old person consumes everything he/she owns. For an old person, there is no point giving up consumption (and therefore reducing utility) to save since the savings cannot be used to finance consumption in the period following old age.
Models with Overlapping Generations Page 16 2.2 Firms We assume there is a large number of identical firms (that is, all firm use the same technology) acting as perfect competitors. We assume that the production technology is represented by a production function F (K, L) which has constant returns to scale (CRS), is increasing in both inputs, is concave and satisfies the Inada conditions lim F 1(K, L) =, K 0 lim F 1(K, L) = 0, K lim F 2 (K, L) =, L 0 lim F 2 (K, L) = 0 (48) L where F i (K, L) denotes the partial derivative of the production function with respect to its i th argument. In cases where we have identical firms and a CRS production function, the number of firms is indeterminate. For convenience, we analyze the model as if there were a single firm. Aggregate output in period t is denoted Y t function and is given by a Cobb-Douglas production Y t = A t K α t L 1 α t, 0 < α < 1 (49) where A t represents the level of technology in the economy in period t. We measure profits in units of the consumption/investment good in the economy. Therefore, total revenue in period t equals Y t. Since the firm must pay workers a wage rate w and must pay a rental rate (or rate of return) on the capital invested by households, its period t profits are given by Π t = Y t w t L t r t K t. (50) The firms optimization problem is to maximize profits given the technological constraint represented by equation (49). Because all markets are competitive, firms take factor prices (w and r) as given. Therefore, the firm s problem can be written max Π t = A t Kt α L 1 α t w t L t r t K t. (51) K t, L t The first-order conditions corresponding to the above problem are Π t = αa t Kt α 1 K t L 1 α t r t = 0 (52) Π t L t = (1 α)a t K α t L α t w t = 0. (53) Condition (52) shows that the rental rate/real interest rate is equal to the marginal product of capital while equation condition (53) shows that the wage rate is equal to the marginal
Models with Overlapping Generations Page 17 product of labour. Plugging the factor prices implied by conditions (52) and (53) in the profit function shows that profits are zero in equilibrium. At various stage of our analysis of the life-cycle model, it will prove convenient to write conditions (52) and (53) in a few different ways. Here are different versions of (52) and (53) r t = α Y t K t, r t = αa t k α 1 t (54) w t = (1 α) Y t L t, w t = (1 α)a t k α t (55) where k t denotes the capital-labour ratio, that is k t = K t /L t. Equations (54) and (55) make clear that factor prices depend on the size of the capital stock relative to the labour supply. The larger the capital stock is relative to the labour supply (i.e. the larger k), the smaller is the return on capital and the larger is the wage rate. Looking at the signs of the first and second derivatives of the marginal product of capital and labour, we find that w is a concave function of k whereas r is a convex function of k. Let s summarize the interactions between households and firms before studying the equilibrium of the model. Young people supply the labour input needed by firms. The savings of the young people alive in a given period will be invested to form the capital stock in the following period. Firms hire workers (young people) and rent capital to produce output. With their output, they pay wages to workers and a return to investors. Young people take their wage income and allocate a share 1 to immediate consumption and a share to savings. 1+ 1+ Old people get back their initial investment in capital plus the return on that investment and consume all of that. 2.3 Competitive Equilibrium and Transition Equation In the current environment, a competitive equilibrium is a price system (w, r) and an allocation (c 1, c 2, K) such that (1) individuals maximize utility subject to their budget constraints (taking prices as given), (2) firms maximize profits given prices and technology, and (3) all markets clear. As we know by now, the capital stock K t+1 is equal to the savings of all young people in
Models with Overlapping Generations Page 18 period t. Therefore, we have the equilibrium condition on the capital market K t+1 = N t s t (w t ) K t+1 = 1 + N tw t. (56) Using equation (53) to substitute out the wage rate we get K t+1 = 1 + N tw t = 1 + (1 α)a tkt α L 1 α t (57) where the last equality uses the fact that N t = L t. Now, dividing both sides by L t+1 yields the transition equation (1 α) k t+1 = A t kt α. (58) (1 + )L t+1 /L t Unless otherwise indicated, we assume that there is no growth in technology nor in population (that is, L t+1 /L t = 1 and A t = A for all t). In such a case, the transition equation becomes k t+1 = (1 α) 1 + Akα t. (59) 2.4 Growth, Transition Period and Steady State As we will see shortly, the model without growth in technology and in population eventually reaches a steady state where all variables are constant over time. By convention we denote the steady-state value of a variable using. capital-labour ratio is denoted k. For example, the steady-state value of the The key variable to focus on to determine whether the economy has reached its steady state is the capital-labour ratio. If in period t it is the case that k t k, then the economy is not in steady state. When the economy is not in steady state, then it is going through a transition period. The adjustments taking place during a transition period are referred to as transitional dynamics. Obviously, we need to solve for k if we want to be able to check whether the economy is in steady state or not. This is simple to do. Using the fact that k t1 = k t = k in steady state, replace both k t and k t+1 by k in transition equation (59) k = [ ] 1 1 + (1 α)ak α k 1 α = (1 α)a. (60) 1 + To find out the convergence properties of the model we use a transition path diagram (see Figure 2.2). The transition path diagram measures k t+1 on the vertical axis and k t on the
Models with Overlapping Generations Page 19 horizontal axis. It includes a 45 o line and a transition line corresponding to the transition equation (59). Since k t1 = k t = k in steady state, the value of k on the graph is found at the intersection of the 45 degree line and of the transition line. The first and second derivatives of the right hand side of the transition equation with respect to k t indicate that the transition line is increasing and concave. Since the first partial derivative goes to zero as k (and vice versa), we know that the slope of the transition line is very steep for small values of k t and almost flat for very large values of k t. Therefore, we know that the transition line cuts the 45 o line only once for positive values of k t and that it cut it from above. Therefore, the steady state is unique and stable. That is, for any positive k 0, the time path of k will always converge to k. Figure 2.2 shows an example where the capital stock is initially small such that k 0 < k. An immediate implication of the fact that the economy with constant population and constant technology eventually converges to a steady state is that there cannot be growth in the long-run in this model. If the economy has too much capital for its number of workers, the economy shrinks in the transition period to attain its steady state. If the economy has too little capital for its number of workers, the economy grows in the transition period to attain its steady state (so there could be economic growth in the short-run). 2.5 Computing Time Paths The previous section focussed exclusively on the capital labour ratio. In this section, we show how to calculate the time path of all variables appearing in the model. The first step is to calculate the time path of the capital-labour ratio. This is accomplished by iterating on the transition equation starting from some given initial condition k 0 k 1 = 1 + (1 α)akα 0 k 2 = 1 + (1 α)akα 1 k 3 = 1 + (1 α)akα 2 and so forth. With the time path of k on hands, we can easily calculate the time path of the wage rate and interest rate using (54) and (55). r 0 = αak α 1 0, w 0 = (1 α)ak α 0
Models with Overlapping Generations Page 20 and so forth. r 1 = αak α 1 1, w 1 = (1 α)ak α 1 r 2 = αak α 1 2, w 2 = (1 α)ak α 2 r 3 = αak α 1 3, w 3 = (1 α)ak α 3 Since L t = N we have that K t = Nk t and Y t = ANk α t. Therefore K 0 = Nk 0, Y 1 = ANk α 0 K 1 = Nk 1, Y 1 = ANk α 1 K 2 = Nk 2, Y 2 = ANk α 2 and so forth. K 3 = Nk 3, Y 3 = ANk α 3 Using the factor prices calculated above, we calculate individual savings and consumption using (45), (46) and (47) s 1 = 1 + w 1, c 11 = 1 1 + w 1, c 21 = 1 + w 0(1 + r 1 ) and so forth. s 2 = s 3 = 1 + w 2, c 12 = 1 1 + w 2, c 22 = 1 + w 1(1 + r 2 ) 1 + w 3, c 13 = 1 1 + w 3, c 23 = 1 + w 2(1 + r 3 ) Aggregate consumption is calculated by summing the consumption of all young and old people. Since there are N young people and N old people alive in any given period, aggregate consumption in period t is given by C t = Nc 1t + Nc 2t (61) Obviously, using the time paths of consumption in young age and in old age calculated above we find C 1 = Nc 11 + Nc 21 C 2 = Nc 12 + Nc 22 C 3 = Nc 13 + Nc 23
Models with Overlapping Generations Page 21 and so forth. Finally, the definitions of national savings (S) and national investment (I) are S t = Y t C t, I t = K t+1 K t (62) where the formula for national investment is consistent with our assumption of zero capital depreciation. Using the latter definitions, we calculate S 1 = Y 1 C 1, I 1 = K 2 K 1 and so forth. S 2 = Y 2 C 2, I 2 = K 3 K 2 S 3 = Y 3 C 3, I 3 = K 4 K 3 The steady-state values of all variables can be computed following the steps above but starting with the steady-state value of the capital-labour ratio (see (60)) rather that some arbitrary initial condition k 0. The MS-Excel file timepaths.xls shows the time paths calculated for the case k 0 = 1, N = 100, = 0.9, α = 0.3 and A = 20. In that numerical example, it is clear that the capital-labour ratio is smaller than its steady state value. In other words, the capital stock is very small. Therefore, it is not surprising to see that savings by young people in period 0 is larger than the dissaving of the old people. Equation (55) shows that w depends positively on k, so an increase in k between period 0 and 1 increases the wage rate (w 1 > w 0 ). Since the wage rate determines savings of the young people, the fact that young people in period 1 have a larger labour income means that they will invest more in capital than the previous generation which means an even larger capital stock in period 2. This story repeats itself up to a point where the economy reaches its steady state. We know that such a state will arise because as capital accumulation proceeds, diminishing marginal returns to capital imply smaller and smaller increases in the wage rate over time. At some point, the wage rate stops growing completely. At that point, each generation saves as much as the previous one and the capital stock (and capital-labour ratio) does not change anymore. Since all other variables depend on k, when k becomes constant, that is also the case for all the other variables. Remember that the real interest rate depend inversely on the ratio K/L. Since L is constant and there is capital accumulation in the transition, then the real interest rate falls to its steady- state value over time. Also, capital accumulation means that there is output growth in the transition period. The growth in the wage rate in the transition implies that
Models with Overlapping Generations Page 22 young age consumption grows. The effect on old age consumption depends on the wage rate and the interest rate (see (47)). As seen in the Excel file, the wage effect dominates and old age consumption is also growing in the transition. As a result, aggregate consumption is growing as well. As explained above, growth in the wage rate over time means national investment is positive since in any given period in the transition, the group of young people saves more than the old dissaves. However, because of diminishing marginal returns to capital, increases in the wage rate become smaller and smaller so national investment (I) actually falls in the transition. With zero capital depreciation, national investment is zero in steady state. The last column of the timepaths.xls shows the utility levels attained by the various generations. We see that in the transition period, any new generation is better off than the generation preceding it, which is not surprising given that both young age consumption and old age consumption grow in the transition. To understand the effect of growth on the utility maximization problem, refer to Figure 2.1. In the transition, the wage rate is increasing which pushes the intertemporal budget constraint up and to the right over time. The budget constraint does not shift up in a parallel fashion because the interest rate is falling. Therefore, the upward shift tends to be smaller than the rightward shift. 2.6 Comparative Dynamics The thought experiment in this sections are conducted as follows: (i) take the economy where there is no growth in population nor in technology and suppose that it is in steady state. (ii) describe a change in a parameter or a shock hitting the economy and study its implications for the capital-labour ratio. We start by looking at an increase in. Figure 2.3 provides a graphical representation of our first thought experiment. The economy is initially at point A, on the lower of the two transition lines represented on the graph. Then suppose that new generations of agents become more patient so that increases ( > say). How does that change affect the transition equation? To find that out, let s define h(k t ) = (1 α)akt α /(1 + ) and write transition equation (59) as k t+1 = h(k t ). You can show that for positive values of k, h(k t )/ k t > 0 which implies a counterclockwise rotation in the transition line in Figure 2.3. The capitallabour ratio k is no longer the steady-state capital labour ratio after the change in. So
Models with Overlapping Generations Page 23 the economy embarks on a transition path which creates some positive economic growth. As k, grows the economy converges to its new steady-state equilibrium denoted by point B. Once the economy has reached point B, economic growth stops and the economy is again in a steady state. However, this new steady state is characterized by a larger capital-labour ratio, k > k. What is the economics of this experiment? More patient individuals save more than impatient individuals. So, when rises, savings increase which pushes up k which pushes up w, which in turn pushes savings even higher. As k rises, diminishing marginal returns kick in and the increases in w become smaller and smaller, up to a point where k and w do not change over time anymore (a new steady-state is reached). In our second thought experiment, we look at the effect of a permanent increase in the level of technology (i.e. A > A). Clearly, an increase in A implies a counterclockwise rotation in the transition line in Figure 2.4. The capital-labour ratio k is no longer the steady-state capital labour ratio after the change in. So the economy embarks on a transition path which creates some positive economic growth. As k, grows the economy converges to its new steady-state equilibrium denoted by point B. Once the economy has reached point B, economic growth stops and the economy is again in a steady state. However, this new steady state is characterized by a larger capital-labour ratio, k > k. What is the economics of this experiment? The increase in A pushes the wage rate up (workers are more productive so their wage rate goes up), triggering an increase in savings which implies an increase in k, which pushes up w, which in turn pushes savings even higher. So the effect of an increase in A are similar to the effects of a change in. The third experiment is about a shock that hits the economy. Suppose the economy is in steady in period 0 and that a catastrophic event (the typical example is a war) destroys part of the capital stock at the beginning of period 1. The destruction of part of the capital stock does not imply a rotation of the transition line in Figure 2.5 since none of the parameters are affected. Rather, the economy jumps from point A to point B as a result of the destruction of the capital stock. As is evident from Figure 2.5, k 1 is not equal to the steady state capitallabour ratio. Therefore, the economy cannot stay at point B. There will occur a period of capital accumulation that will take place until the economy returns to point A. How does it work? For concreteness, suppose that K 0 = K which implies k 0 = k and that K 1 = K /2 which implies k 1 = k /2. Then, k 1 /k 0 = 0.5 and w 1 /w 2 = (k 1 /k 0 ) α > 0.5 because 0 < α < 1. Therefore, because of diminishing marginal returns, the drop in k is greater than the drop in w which implies that the drop in savings is less than the drop in k. This means that after the shock, the savings are large enough to push up the capital-labour ratio, which pushes