Infinite-Horizon and Overlapping Generations Models

1. Chapter 2 Infinite-Horizon and Overlapping Generations Models

2. Part A The Ramsay-Cass-Koopmans (Representative Agent) Model

3. 2.1 Assumptions

4. Firms Large number of profit maximizing firms Perfectly competitive on the output side Perfectly competitive on the input side Owned by households Identical, with constant-returns-to-scale Cobb-Douglass production is the easiest Use exogenous technology, A UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 4

5. Households Large, fixed number of H identical households Dynasties Households �live� forever Population, L, and household size both grow at rate n Everyone supplies 1 unit of labor No labor-leisure choice Households can own capital, which they rent to firms No depreciation (for simplicity � it can be harder to make plausible assumptions that allow a lot of cancellation in the math if we keep it in the model) UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 5

6. Households Utility U=?e-?tu[c(t)]L(t)H-1dt Period/instantaneous utility, u[c(t)] depends on current consumption only Discount at rate ? Typically, there needs to be a condition (like ? > n) so that utility doesn�t go to infinity as population size increases Multiplied by household size, L/H So it won�t be utility improving to beggar future generations to increase savings today Infinite time horizon UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 6

7. 2.2 The Behavior of Households and Firms

8. Firms Extensive form production function: Y=F(K,AL) Intensive form production function: y=f(k) Capital is paid its marginal product or rental rate r(t)=f�[k(t)] Labor is paid its marginal product. For a competitive firm, this is what is left over after paying capital W=f(k)-kf�(k), (this identity is quicker than calculus) Actual labor is paid w=AW UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 8

9. Household�s Budget Constraint The household takes r and w as given The household is not liquidity constrained In general, C = W, and ?C = ?W The size of the household, L/H doesn�t affect anything since both sides get multiplied by it The household discounts at the continuously compounded instantaneous rate of interest R(t)=?rdt The households own the initial capital stock, K(0)/H UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 9

10. Household�s Budget Constraint The model doesn�t make sense if capital can go to zero as time goes to infinity This is why we don�t use capital in a Fisher model: with finite time, why would you have capital at all in the second period? We require that the discounted value of the capital stock be non-negative in the limit lim(s ?8)e-R(s)k(s)?0 UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 10

11. Household�s Maximization Problem It�s useful to assume a CRRA utility function Consumption for an individual is C(t) Instantaneous utility is u(t)=C(t)(1-?)/(1-?) Using L�Hopital�s rule (that the limit of a fraction is the same as the limit of a second fraction comprised of the derivatives of the numerator and denominator of the first fraction) we can show that u(t)=ln[c(t)], when ? = 1 This is useful because it is simple, and because estimates of ? are near 1. UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 11

12. Household�s Maximization Problem We need to extract the effects of the intial value of technology Consumption per effective worker is c(t)=C(t)/A(t) So, u(t)=A(t)(1-?)c(t)(1-?)/(1-?) u(t)=[A(0)egt](1-?)c(t)(1-?)/(1-?) u(t)=[A(0)](1-?)egt(1-?)c(t)(1-?)/(1-?) Note that [A(0)](1-?) is a constant UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 12

13. Household�s Maximization Problem U=?e-?t{u[c(t)]}{L(t)H-1}dt Now we need to get the starting value for L out of the equation using L(t)=L(0)ent So: U=?e-?t{[A(0)](1-?)egt(1-?)c(t)(1-?)/(1-?)}{L(0)entH-1}dt Extract constants and gather exponents to get: U={[A(0)](1-?)L(0)H-1}?e-?t{egt(1-?)c(t)(1-?)/(1-?)}{ent}dt U={[A(0)](1-?)L(0)H-1}?e[-?+g(1-?)+n]t{c(t)(1-?)/(1-?)}dt UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 13

14. Household�s Maximization Problem It looks complex, but the utility function has a general form of: U=B?e-�t{c(t)(1-?)/(1-?)}dt Where B = [A(0)](1-?)L(0)H-1 is a constant capturing a weighted version of starting technology per household Where �=-[-?+g(1-?)+n] is a composite discount rate that needs to be positive (just like any discount rate in continuous time) This is asserted without proof on pg. 49, but justified on pg. 53 UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 14

15. Household�s Maximization Problem In the budget constraint Total income depends on the wage per worker, times technology, times population, divided by the number of households Total consumption clearly depends on population and households. It also depends on technology, although this might not be so obvious. These can be factored out once we account for starting capital UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 15

16. Household�s Maximization Problem Starting capital per effective worker is capital per household times population, divided by households. Divide through by starting technology to get capital per worker So, starting capital per household can be expressed as: k(0)A(0)L(0)/H UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 16

17. Household�s Maximization Problem Discounted infinite lifetime consumption is ?{e-R(t)c(t)A(t)L(t)/H}dt ?{e-R(t)c(t)entA(0)egtL(0)/H}dt Discounted infinite lifetime income is: ?{e-R(t)w(t)A(t)L(t)/H}dt ?{e-R(t)w(t)entA(0)egtL(0)/H}dt Factoring out A(0)L(0)/H, we get constraint: ?{e-R(t)c(t)e(n+g)t}dt = k(0) + ?{e-R(t)w(t)e(n+g)t}dt UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 17

18. Household Behavior The maximization problem is now fairly simple The FOCs are then the budget constraint and: Be-�tc(t)-?=?e-R(t)e(n+g)t, for each dt Take logs to get: logB � �t �?ln[c(t)] = log? �R(t) + nt + gt Take the derivative with respect to time to get: � � �?[(dc/dt)/c(t)] = r(t) + n + g Substitute out � to get the Keynes-Ramsay rule: [(dc/dt)/c(t)] = [r(t)-?-?g]/? UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 18

19. Household Behavior The above is consumption per effective worker. To get consumption per capita we need: (dC/dt)/C(t)=(dA/dt)/A(t)+(dc/dt)/c(t) (dC/dt)/C(t)= g + [r(t)-?-?g]/? (dC/dt)/C(t)= [r(t)-?]/? Either one of these relationships can be thought of as the Euler equation for the problem UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 19

20. Household Behavior The Euler equation says that consumption will grow if: The rate of return on capital exceeds the discount rate The former is what you get for giving up consumption The latter is how willing you are to give up consumption If you are not willing to give up some consumption now, you won�t be able to grow. But, it is only sensible to give up some of your consumption if you can earn a reasonable rate of return on it. UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 20

21. 2.3 The Dynamics of the Economy UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 21

22. The Dynamics of c We are interested in describing when consumption is increasing, and when it is decreasing In order for [dc/dt]/c(t) to be equal to zero, the numerator of the RHS must be zero r(t)-?-?g=0 r(t)=f�[k(t)], so there is a single value of k for which this is true Call it k* UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 22

23. The Dynamics of c For k > k*, f�(k) , f�(k*) or r(t) < r*(t) This implies that for high value of k that c will fall By the same argument, for low values of k, c will be increasing UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 23

24. The Dynamics of k What will dk/dt look like? New investments in capital are f(k)-c. The amount of capital that needs to be purchased to supply new effective labor is (n+g)k So, dk/dt = f(k)-c-(n+g)k This is analogous to the distance between the sf(k) and (n+g+d)k lines in the Solow model It�s sort of parabolic UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 24

25. The Dynamics of k Below this curve, k is large relative to c, so k will be increasing Above this curve, c is large relative to k, so k will be decreasing UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 25

26. The Phase Diagram The intersection of the line and the curve is the steady state This type of intersection is called a saddlepoint A saddlepoint is an equilbrium that can only be reached along a single path There is a single path from the bottom-left to the top-right that leads to this steady state This is called the saddlepath UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 26

27. The Phase Diagram Is the steady state at the golden rule level of capital (the one where c is maximized)? The steady state is where r = ?+?g The golden rule is where r = n+g We already know that the composite discount rate must have -?+g(1-?)+n < 0 This can only hold where n+g < ?+?g This is where the rate of return is higher at the steady state, or the level of capital is lower UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 27

28. The Initial Value of c There are two rules for thinking about phase diagrams Paths that start at points in the same vertical or horizontal line can�t cross Each locus of points where a variable doesn�t grow can only be crossed going horizontally or vertically UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 28

29. The Saddle Path Because of the rules for movement in a phase diagram, there is only one point immediately before the steady state that will lead into the steady state By induction there is only one point immediately preceding that point that will connect ultimately to the steady state UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 29

30. Welfare By construction, the equilibrium of a Ramsay-Cass-Koopmans model is Pareto-optimal It satisfies the First Welfare Theorem Its markets are competitive and complete There are no externalities There are a finite number of agents (thus the fixed number of households) UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 30

31. 2.5 The Balanced Growth Path UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 31

32. Properties of the Balanced Growth Path It turns out that once the steady state is reached, the properties of the model are the same as the Solow growth model where the savings rate is fixed and sub-optimal Namely, that growth in per capita income depends on growth in technology UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 32

33. The Balanced Growth Path and the Golden-Rule Level of Capital Since the actual golden rule level of capital � where consumption is maximized � cannot be achieved, and Since the steady state is optimal The saddlepath is sometimes called a modified-golden-rule path UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 33

34. 2.6 The Effect of a Fall in the discount rate UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 34

35. Qualitative Effects This section focuses on the discount rate as an example of comparative dynamics. The discount rate only effects the dc/dt = 0 locus A fall must be balanced by a fall in f�(k), which can only be produced by a higher k UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 35

36. Qualitative Effects The new steady state is to the right of the initial one It will have a new saddlepath as well The old saddlepath is now explosive Capital cannot jump instantaneously, but consumption can So, consumption drops until the new saddlepath is reached UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 36

37. Rate of Adjustment and the Slope of the Saddlepath See the spreadsheet UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 37

38. The Speed of Adjustment See the spreadsheet UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 38

39. 2.7 The Effects of government purchases UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 39

40. Adding Government to the Model Start out by eliminating just about everything that is important about government Government does nothing for utility, or budget constraints Those are relaxed in later chapters Government does is place a drag on the economy (it isn�t perfectly efficient) We model this by subtracting G(t) from dk/dt This shifts its graph downward UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 40

41. The Effects of Permanent Changes In Government Purchases A permanent increase/decrease in G changes the position of the steady-state and the saddlepath The old saddlepath is now explosive Capital cannot change instantaneously Consumption can change instantaneously , and it changes by the negative of the change in G to move the economy to the new saddlepath UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 41

42. The Effects of Temporary Changes In Government Purchases A temporary increase/decrease in G cannot take use to the new saddlepath and then back to the old one Since the change is temporary, the �new� saddlepath is never there Instead consumption jumps instantaneously to a point connected to an explosive path that intersects the saddlepath at the time when the temporary purchase cease UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 42

43. Empirical Application: Wars and Real Interest Rates Wars induce temporary changes in government spending The temporary transit on an explosive path corresponds to higher real interest rates Barro �87 offers some evidence in support of a correlation between temporary military spending and real rates UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 43

44. Part B: The Diamond Model UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 44

45. 2.8 Assumptions UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 45

46. Assumptions UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 46 Note that the richness of behavior caused by having an infinite number of finitely-lived individuals � instead of one infinitely-lived individual � makes OLG models contain RA models as a special case.

47. Assumptions The model is discrete Agents live for 2 periods New agents enter the model each period It is the sense in which the number of optimizers is infinite that gives the Diamond overlapping generations model its qualitative differences from the Ramsay-Cass-Koopmans representative agent model UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 47

48. Assumptions Lt young agents are �born� in each period The rate of population growth is still n Agents supply 1 unit of labor when young, and 0 when old Because agents have finite lives, no limit needs to be placed on the discount rate of the model Depreciation rate is 100% UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 48

49. Household Behavior An agent�s income in period 1 is their per capita wage times aggregate technology, wtAt Savings earn the exogenous rate rt=f�(kt) This equals the marginal product of capital The per capita wage is wt Consumption of an old agent is: C2t+1=(1+rt+1)(wtAt-C1t) The lifetime budget constraint is then: C1t+1(1+rt+1)+C2t+1=(1+rt+1)wtAt UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 49

50. Household Behavior With CRRA utility, the Lagrangian is: L=(C1t)(1-?)/(1-?) + [1/(1+?)](C2t+1)(1-?)/(1-?) + ?{(1+rt+1)wtAt-C1t+1(1+rt+1) - C2t+1} Maximization of the Lagrangian yields the Euler equation: C2t+1(1+rt+1)(-1/?)=C1t(1+?)(-1/?) Since: C1t-? - ?(1+rt) = 0 And: [1/(1+?)]C2t+1-? � ? = 0 UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 50

51. Household Behavior Rearrange the Euler equation and budget constraint to get a solution for C1t: C2t+1(1+rt+1)(-1/?)=C1t(1+?) (-1/?) C1t(1+rt+1)+C2t+1=(1+rt+1)wtAt C2t+1=(1+rt+1)wtAt-C1t(1+rt+1) {wtAt-C1t}(1+rt+1)[(?-1)/?]=C1t(1+?)(-1/?) (1+?)(1/?){wtAt-C1t}(1+rt+1)[(?-1)/?]=C1t C1t={(1+?)(1/?)/{(1+?)(1/?)+(1+rt+1)[(?-1)/?]}}wtAt UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 51

52. Household Behavior So, first period consumption is a not a simple function Let C1t=[1-s(rt+1)]wtAt, where [1-s(rt+1)] = {(1+?)(1/?)/{(1+?)(1/?)+(1+rt+1)[(?-1)/?]}} The saving rate depends on ? Increasing in r for ? < 1 Decreasing in r for ? > 1 UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 52

53. 2.10 the dynamics of the economy UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 53

54. The Equation of Motion of k For aggregate capital: Kt+1 = s(rt+1)LtAtwt rt+1 is paid at t+1 based on decisions made at t Divide by next period�s effective labor: Kt+1/Lt+1At+1 = s(rt+1)LtAtwt/Lt+1At+1 But Lt/Lt+1=1/(1+n) and At/At+1 = 1/(1+g) kt+1 = s(rt+1)wt/(1+n)(1+g) Or, in terms of capital only kt+1 = s[f�(kt+1][f(kt)-ktf�(kt)]/(1+n)(1+g) UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 54

55. The Evolution of k Because the equation for k includes multiple functions of k, it can be highly non-linear This leads to a lot of interesting theoretical behavior if there are multiple steady-states For example, if the gross investment curves to create multiple steady-states, they each have to be evaluated as sinks, sources or saddlepoints. In this case, it is possible to have more than one stable steady-state, and to then rank them as better or worse � so poverty traps are possible. UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 55

56. Logarithmic Utility and Cobb-Douglas Production In this case, the dynamics of k are simple, and very much like the Solow model UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 56

57. The Speed of Convergence An important difference with this model is that convergence to the steady state is much faster See the spreadsheet UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 57

58. The General Case The phase diagram relating kt+1 to kt need not be monotonic This means there can be multiple steady states Some sources Some sinks Self-fulfilling prophecies are possible See the spreadsheet UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 58

59. 2.11 The Possibility of Dynamic Inefficiency Even though they are competitive, OLG models need not be Pareto efficient The capital stock can exceed the golden rule level of capital � this occurs over time so it is known as dynamic inefficiency. Recall that the Solow and Ramsay-Cass-Koopmans models end up with capital below the golden rule level � so it is typically dynamically efficient This means that everyone can be made better off by reducing capital and consuming instead UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 59

60. 2.11 The Possibility of Dynamic Inefficiency In particular, since there are an infinite number of generations, each larger than the last, it is possible to: Transfer from the current young to the current old to make them better off Transfer from the future young to the future old (who are the current young) to make them better off too Repeat infinitely UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 60

61. 2.11 The Possibility of Dynamic Inefficiency Dynamic inefficiency is really about overinvestment in capital The economy would be more efficient if capital was cut because consumption would go up instead of down Alternatively, the rate of return on capital is too low UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 61

62. Empirical Application: Are Modern Economies Dynamically Inefficient? Na�ve Approach Measure the real rate of return and the real rate of growth (or the nominal rates) If the real rate of return is lower, then the economy is dynamically inefficient The problem is what asset to use Safe ones suggest dynamic inefficiency Risky ones suggest dynamic efficiency Either way, the na�ve approach confirms dynamic inefficiency UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 62

63. Empirical Application: Are Modern Economies Dynamically Inefficient? Better Approach: You can�t be overinvesting if some of your income from capital is not being reinvested in more capital The difference must be going to consumption Evidence suggests that capital income exceeds investment, and therefore economies are not dynamically inefficient This means that we do not have to use OLG models to capture this feature of reality. But, we still might use them for other reasons. UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 63

64. 2.12 Government In the Diamond Model The result is similar to the Ramsay-Cass-Koopmans model The deadweight loss of government shifts the capital locus (however graphed) towards lower capital growth UNO, ECON 6204, Summer 2008, Dr. Tufte Notes drawn from Chapter 2 of Romer's "Advanced Macroeconomics" 64