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1 THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 35 URL: Economics 31 Topics in Macroeconomics 3 Fall 21 Aims of this course Clear, critical thinking Analytical/mathematical methods Application of theory to facts Lecture 1: Introduction to OLG models 1 Why OG models? Realism: people do have finite lives (and imperfect altruism?). The framework enables us to study issues that are related to the fact that people grow old and die. E.g. public pension systems, age-consumption profiles. Mathematical simplicity. We can do dynamic macroeconomics in a completely rigorous way without knowing anything about functional analysis. 1

2 2 Describing the environment Time: t =..., 2, 1,, 1, 2,... We call t < the past, t = the present and t > the future. (M & W say t = 1 is the present.) Population: Each generation t consists of infinitely many small individuals of mass N(t). Generation t is born in period t. Members of generation t are young in period t and old in period t + 1. In subsequent periods, they are dead. Generation t overlaps with generations t 1 and t + 1. Resources: There is exactly one good in each period. In this sense, there are infinitely many goods. Let Y (t) denote the economy s total endowment of the time t good. We will be making various assumptions about the feasibility of transforming good t into good s, s t. impossible. Initally we will assume that it is Consumption allocations: Let c h t (s) represent the consumption of the time s good by individual h from generation t. Thus the consumption profile of an individual h from generation t is [c h t (t), c h t (t + 1)]. Meanwhile, a time t consumption allocation consists of two mappings, c h t (t) : [, N(t)] R (for the young) and c h t 1(t) : [, N(t 1)] R (for the old). Finally, a consumption allocation is a sequence of time t consumption allocations for t =, 1,.... Feasible allocations: An allocation is said to be feasible if it can be achieved given the total resources and the (for now, trivial) technology. We write C(t) = N(t) c h t (t) dh + N(t 1) c h t 1(t) dh, and we say that an allocation is feasible if C(t) Y (t) for all t =, 1,.... 2

3 Warning: Feasible allocations are often confused (by students at all levels) with sequences of affordable consumption profiles. Don t make this mistake! An allocation is said to be efficient if it is feasible and there is no alternative feasible allocation with more total consumption of some good and no less of any other good. At this stage, this means C(t) = Y (t) for all t =, 1,.... Warning: Don t confuse efficiency with Pareto optimality! Notice that we cannot yet talk about Pareto optimality, since we have not introduced preferences. Preferences: A preference ordering ( ) on the set of consumption profiles [c h t (t), c h t (t + 1)] is a relation that is symmetric (x x), transitive (if x y and y z, then x z) and complete (either x y or y x or both). If x y but not y x, then we write x y. Preferences are designed to describe actual and hypothetical behaviour. What we mean when we say that x y is that an individual with preferences described by would not choose y if x is available. When we (later) talk about welfare, we will (implicitly) make the additional (strong) assumption that an individual who weakly prefers x to y is no worse off having x than y. (Some people think that this last statement is a matter of definition. It seems to me that, to the extent that it is true, it is rather a matter of fact.) Utility functions: As Herman Wold ( ) showed in the 194s, certain (not all) preference orderings can be represented by a utility function. What this means is just that for a given preference ordering there is a real-valued function u such that u(x) u(y) just in case x y. Warning: For a given preference ordering, if there is a utility function that represents it, then there are infinitely many others that represent the same preferences. None have pride of place; they are equivalent from the point of 3

4 view of describing behaviour (and, if the additional assumption is made, interests). In this context, we write u h (ĉh t t (t), ĉ h t (t + 1) ) u h h t ( c t (t), c h t (t + 1) ) if the profile with a circumflex (hat) is (weakly) preferred by individual h of generation t to the profile with a tilde. Note that different individuals are allowed to have different preferences. On the other hand, we do not allow individuals to care about anything except their own consumption profile. This is a selfcenteredness assumption. Pareto optimality: We say that an allocation A is Pareto superior to an allocation B if there is no person who strictly prefers her allocation under B to that under A and there is (at least) one person who strictly prefers her allocation under A to that under B. An allocation is said to be Pareto optimal if it is feasible and there is no feasible allocation that is Pareto superior to it. Example 1 Let the population be given by N(t) = 1 for all t =, 1,.... Suppose resources are given by Y (t) = 4 for all t =, 1,.... There is no technology for transforming one good into another. Meanwhile, preferences are common to all and represented by u h t (x, y) = u(x, y) = x y. Then the following allocation is feasible. c h t (t) = 3 and c h t 1(t) = 1 for all h [, 1] and all t. Notice that this involves specifying how much members of generation 1 get to consume as old. Is this allocation Pareto optimal? Well, it is certainly feasible. It is also efficient. But there is another feasible allocation that is Pareto superior to it, namely the following. Set c h t (t) = c h t 1(t) = 2 for all relevant h and all t =, 1,.... Notice that the initial old (the ones born in period 1) now get 2 instead of 1 as old. So they are no worse off. Meanwhile, all subsequent generations get a utility of 2 2 = 4 rather than 3 1 = 3. So they are all get a strictly preferred profile. We conclude that the original allocation is not Pareto optimal. 4

5 3 Competitive equilibrium The notion of competitive equilibrium is a concept designed for economies with private ownership, enforceable contracts and a large number of people so that no single one can affect prices. For heuristic purposes, we sometimes imagine that prices are set by a Walrasian auctioneer. Endowments: This is a specification of who owns what. The endowment profile of an individual h of generation t is written as ω h t = [ω h t (t), ω h t (t + 1)], where the notation should be obvious. We assume that no individual owns any claims to goods in periods when she is not alive. Note: the Greek letter ω is proncounced omega. The stress is on the first syllable. By definition of total resources, we have Y (t) = N(t) ω h t (t) dh + N(t 1) ω h t 1(t) dh. Trade: The first important point to note is that all trade is intragenerational and inter temporal. Thus trade is lending and borrowing among members of the same generation. We denote the lending (saving) of person h in generation t by l h (t). The old do not save. Denote the gross real interest rate the relative price of period t goods in terms of period t + 1 goods by r(t). The budget constraints of the young thus become c h t (t) ω h t (t) l h (t), (1) and those of the old become c h t (t + 1) ω h t (t + 1) + r(t)l h (t). (2) 5

6 We now suppose that r(t) >. Then Equations (1) and (2) together imply that c h t (t) + ch t (t + 1) r(t) ω h t (t) + ωh t (t + 1). (3) r(t) In a certain sense, the reverse is true. If (3) holds, then there is a number l h (t) such that (1) and (2) hold. Can you find what that number is? In any case, a profile c h t = [c h t (t), c h t (t + 1)] is said to be affordable at interest rate r(t) and endowment profile ω h t if (3) holds. Optimal choice: A consumption profile c h t is said to be optimal (for h) if there is no affordable profile strictly preferred to it (by h). Suppose the utility function representing h s preferences is differentiable. Then an optimal choice is characterized by r(t) = uh t / c h t (t) u h t / c h t (t + 1) Savings function: When the utility function is well behaved enough so that there is a unique optimal value of l h (t) for each specification of r(t) and ω h t, then we define the savings function as a function that delivers that optimal value. (Note: savings and lending are the same in this context, but they won t be later when there are other ways of saving than lending.) As a preliminary, we define a consumption function, this is the demand function for consumption when young. We write c h t (t) = χ h t ( (r(t), ω h t (t), ω h t (t + 1) ) and consequently s h t ( (r(t), ω h t (t), ω h t (t + 1) ) = ω h t (t) χ h t ( (r(t), ω h t (t), ω h t (t + 1) ) 6

7 Example: Cobb-Douglas. 1 Preferences are represented by ln c h t (t) + β ln c h t (t + 1). Then the savings function is given by s h t ( (r(t), ω h t (t), ω h t (t + 1) ) = βωh t (t) 1 + β ωh t (t + 1) r(t)(1 + β). Notice that if the endowment as old is zero, then savings is independent of the rate of return! Income and substitution effects cancel out. Definition: A competitive equilibrium is a sequence of prices and an allocation such that 1. The allocation is optimal for each individual. 2. The allocation clears markets at all dates. Market clearing: supply equals demand. In our context, there are two markets in each period: current goods and claims to future goods (lending). Thus our two market clearing conditions become 1. C(t) = Y (t) and 2. N(t) l h (t)dh =. The latter condition expresses the fact that there can be no intergenerational trade with this environment and this (primitive) asset market. Things would be different if either the technology were different or there existed an infinitely-lived asset. 1 The function that is now named after Cobb and Douglas was in fact first introduced to economics by Wicksell (1893). 7

8 References Wicksell, K. (1893). Über Wert, Kapital und Rente. Jena: G. Fischer. 8

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