The Overlapping Generations Model (Romer chapter 2, Part B)

1. The OverlappingGenerations Model(Romer chapter 2, Part B) By Ole Hagen Jørgensen, OJ@cebr.dk 4/10 2006

2. Introduction I will teach three lectures: • One lecture: on the basic OLG model in Romer (2001), chapter 2, part B • One lecture: on a recently developed solution method for the OLG model: • One lecture: on the Real Business Cycle literature (RBC) • My own research actually applies RBC-techniques for solving OLG models! • There will be a presentation in PowerPoint on each subject - Therefore, you will receive 3 handouts (or download from Blackboard or www.cebr.dk/oj)

3. Intergenerational issues Motivation for life-cycle approach to economic dynamics • Life-cycle aspects of human behavior are important to study… • We model explicitly the different periods of life • Distribution of welfare over generations • How the choices of one generation can affect the succeeding generation • How different exogenous shocks to the economy affects different generations (demographic shocks, productivity shocks) • Intergenerational transfers • Purpose: If the market equilibrium allocates consumption unevenly across generations there may be a scope for redistribution. • How: Taxes and benefits • Case: Pensions, education, health

4. The model The OLG model will be presented according to the following outline: • Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources • Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State • Case study • Efficiency and welfare • Dynamic efficiency • Government policy

5. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

6. Basic assumptions Overall assumptions about the economy: • Time is discrete • There is one good, to be consumed or saved/invested • The economy “lives” on forever (no last generation) • Individuals have finite lifetime (finite horizon) • Infinite number of agents • Closed economy • Perfect competition • Absence of externalities • No government sector (could be included easily) • No uncertainty (perfect foresight) Of course the economy has more detailed characteristics – we turn to those when relevant…

7. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

8. Demographics • The life-cycle of generations • The lifetime is divided into two periods: young and old • When individuals are young they work • When individuals are old they are retired • One period therefore amounts to a half lifetime • Who are alive at the same time? (vertical box) • We want to inspect the behavior of one specific generation (horizontal box) • We trace generation “0” that is born at time t=0 and is old in t+1

9. Demographics • Assumptions on the demographic structure of the economy • Could be modeled in great detail • Different sexes • Survival probabilities • Different skills by different people • Very simple assumption in this model • Fixed growth rate of the population over generational periods: (Equivalent to the continuous time variant ) • where Lt is the number of individuals born at time t, • where n is the growth rate of the population

10. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

11. Household utility • Individuals derive utility only from consumption in their two periods of life • Two factors determine how individuals decide to divide consumption over time in a risk-free (certain/perfect foresight) environment • The consumption “smoothing” motive, captured by the term ρ • The consumption “fluctuation” motive, captured by the term θ We discuss each in turn…

12. Household utility • The consumption “smoothing” motive • Individuals generally like to smooth (evenly divide) their consumption over periods • The degree of impatience towards consuming today is captured by the discount rate, ρ • The discount rate of future consumption is generally 1/(1+ρ) so that household utility can be represented in present value terms as:

13. Household utility Consumption “fluctuation” motive • Uncertain environment • In an uncertain environment you might be risk averse, and might not be willing to shift consumption very freely over time. • Say, if you decide to smooth consumption 50/50 over your two periods, and if you are uncertain about how your consumption will vary your tend to stick to the safer level in each period. • If you expect the interest rate to increase in the next period, you would get a higher lifetime consumption if you shift some units of consumption. If you are risk averse you would rather stick to the safer levels of consumption – you then miss out on the extra consumption • θ measures the degree of consumption risk aversion • Certain environment • In this case there is no risk (perfect foresight) • You can still appreciate stable consumption levels in each period, so the parameter θ then measures the degree to which you like stable and consumption • Again, if you expect the interest rate to increase in the future period, and you prefer consumption not to fluctuate – you will then not take advantage of the potentially higher lifetime consumption. • θ measures the degree of consumption fluctuation aversion

14. Household utility • The utility function: • utility from consumption • features intertemporal consumption smoothing motive, ρ • features consumption fluctuation motive, θ • We divide by (1-θ) to ensure positive marginal utility in case θ>1 • Note: for ρ>0, second period utility is valued less than first period utility • We assume that ρ>-1: weight on second period consumption>0. • Also,

15. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

16. Life-cycle consumption • People live for two periods, as adults and as old, and they need to consume in each period • Adults (workers) • The adults work, consume, and save: where • Old (retirees) • The elderly are retired, and consume (they do not work, but live of their savings and interest earnings) • Intertemporal budget constraint (IBC)

17. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

18. Firms • Firms use two factors in production: labor and capital • Firms pay the wage rate, wt , to the labor, Lt, supplied by workers • Firms rent capital, Kt, from retirees at a rental price of rt • The production function is generally: • Due to CRS we can restate the capital in efficiency units, where: • The wage rate is the marginal product of labor in production: • Return to capital is defined by marginal product of capital in production (assume no capital depreciation, δ=0) • Total Returns:

19. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

20. Resources (economy-wide) • We have seen the consumption budget constraint for the household (IBC) • There is also a constraint on consumption for the economy as a whole: society cannot prioritize over more than is actually produced (closed economy) • In each period people save and consume (to save is to invest): • Resource constraint (RC): • In efficiency units:

21. Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources

22. The model The OLG model will be presented according to the following outline: • Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources • Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State • Case study • Efficiency and welfare • Dynamic efficiency • Government policy

23. Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State

24. Household utility maximization • Simple intertemporal utility maximization: C2t+1 Slope of Utility function (MRS) Slope of IBC = -(1+rt+1) C1t

25. Household utility maximization • Optimal consumption allocation depends on • The consumption “smoothing” motive, ρ • The consumption “fluctuation” motive, θ • The future interest rate, rt+1 • The intertemporal optimality condition (Euler equation) or • This equation is all we need derive through maximization!! The rest of the solution of the model is only based on simple math (insert/reduce)

26. Household utility maximization • To find first and second period consumption, C1t and C2t+1, we just insert the Euler equation into IBC: - reduce - The savings rate: • insert the optimal C*1t into The Euler equation to derive C*2t+1:

27. Household utility maximization • We need to derive how much people save, because this determines our consumption in the future. • As such, the intertemporal structure of this model evolves around savings (recall that today’s savings is equal tomorrows capital stock) • Savings can simply derived from first period consumption: or lets analyze the dynamics of the savings rate for alt. parameter assumptions…

28. Household utility maximization The dynamics of the savings rate • Substitution effect: relative price change  savings rate increases • Income effect: purchasing power  savings rate decreases • Special case, θ=1: • No consumption impatience, ρ=0: half of your lifetime income is: 1) consumed in period one, 2) and saved for period two C2 C’2 -(1+r’) C2 -(1+r) C1 C’1 C1

29. Household utility maximization Summary of households intertemporal choice The Euler equation: • rt+1↑: 2. period consumption becomes relatively more preferable • ρ↑: 1. period consumption becomes relatively more preferable • θ ↓: For a given change in (1+rt+1)/(1+ ρ) Consumption is shifted more freely over periods (larger increase in C2t+1/C1) • Inverse elasticity of intertemporal substitution is constant (CRRS): (your rate of marginal substitution, MRS, changes more when your consumption “fluctuation” aversion, θ, is low)

30. Household utility maximization Summary of the household’s intertemporal choice • The household maximization problem boiled down to deriving the savings rate, st(rt+1). this concludes the section on household utility maximization – we’ll move on to study the economy’s capital accumulation…

31. Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State

32. Capital accumulation and Steady State The equation of motion for the capital stock Next period’s capital stock is the current period’s investments! Thus: we know the level of next period’s capital stock in this period…

33. Capital accumulation and Steady State • We can now derive the dynamics of the economy (the way the capital stock evolves over time – thus, also all other variables) • Transform the expression for the motion of the capital stock into efficiency units to derive the Balanced Growth Path: (divide by ) • Insert the general expressions for rt+1and wt:

34. Capital accumulation and Steady State The general case • If we don’t know exactly how factor returns are determined relative to kt we obtain this expression for the evolvement of the economy: • The economy evolves over time, and households want to save/invest to generate the capital stock that will provide them with the highest possible utility. When they reach this capital stock they will keep their savings at the level that will re-generate this particular capital stock in all future periods. Equilibrium condition for the economy to be in its long run equilibrium: kt+1 = kt

35. Capital accumulation and Steady State The Steady State capital stock can be determined as illustrated: • Given well-behaved preferences, and given Cobb-Douglas technology • We will discuss the more general case later!... kt+1 45o kt k*

36. Capital accumulation and Steady State • Existence • Determined by Inada conditions, where the slope of kt+1 is approaching 0 for lim. kt ∞, and approaching ∞ for lim. kt 0. This is related to the decreasing marginal product of capital through the production function and the requirement that kt+1=kt • Uniqueness • Also determined by the Inada conditions. Hence, the slope is falling for k getting larger and larger – therefore the motion of capital can only cross the 45-degree line once. • Stability • Determined through inspection of the phase diagram:

37. Capital accumulation and Steady State Convergence to Steady State – two cases • Initial over-accumulation: A too high capital stock is caused by too much savings by households (too high for utility to be maximized over intertemporal consumption allocation). If utility could be higher by changing the consumption allocation then the current level of capital is not sustainable and is not compatible with household utility maximization. Consequently, people will start saving less, spending more in the current period – total savings fall – investments fall – the capital stock decreases until savings has reached its optimal level compatible with household preferences and utility maximizing… • Initial under-accumulation: Opposite: People will save more to maximize utility – the capital stock rises… kt+1 45o kt kB k* kA

38. Capital accumulation and Steady State • To summarize: If we do not know the relationship between the factor returns, rt+1 and wt and the level of capital, kt, then the savings can shift up and down. As such, the path of the capital stock can also shift up and down. • This is determined through the expression for the path of the economy: the equation of motion of the capital stock Different versions of the relationship between the capital stock and factor returns can be illustrated graphically…

39. Capital accumulation and Steady State

40. Capital accumulation and Steady State Summary of the section on Capital accumulation: We have determined the path of the economy in two ways: • Analytically - through the expression for the dynamic evolution of capital, kt+1(kt) • Graphically - through the steady state condition for capital, kt+1=kt • Recall key relationships when the capital stock has been derived, all other variables in the model can be determined!

41. Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State

42. The model The OLG model will be presented according to the following outline: • Description of the economy • Basic assumptions • Demographics • Household utility • Life-cycle consumption • Firms • Resources • Dynamics of the economy • Household utility maximization • Capital accumulation and Steady State • Case study • Efficiency and welfare • Dynamic efficiency • Government policy

43. Case study The Diamond OLG model Three assumptions: • Logarithmic utility: • Cobb-Douglas technology: • No capital depreciation:

44. Case study Recall the general expression for the law of motion of capital (and the economy): For assumed log utility insert the savings rate (for θ=1): For assumed Cobb-Douglas technology insert the wage rate:

45. Case study • If we should draw curve for this fundamental difference equation then note that and we have again: • Consequently we can derive a Steady State… kt+1 45o kt k*

46. Case study Comments on existence, uniqueness, and stability • Existence: Is the slope positive and decreasing in k? • Uniqueness: Since the expression for kt+1 is unchanged for increasing values for kt, the Inada conditions ensure that for low k’s the curve is very steep and for high k’s the curve flattens • Stability: The function is based on well-behaved preferences and Cobb-Douglas technology so the analysis of the phase-diagram before also applies here: thus stability

47. Case study Steady State • Remove subscripts: • This value for the Steady State capital stock can be calibrated and a numerical estimate can be derived • Hence, all other variables can also be derived numerically (since they all ultimately depend on the capital stock) • One could then make experiments with the model: • Change parameter values and trace the effects on variables • How would workers consumption change? • How would retirees consumption change? lets do some sensitivity analyses…

48. Case study Experiments Change parameter values and trace the effects on variables • How would workers’ consumption change? • How would retirees’ consumption change?

49. Case study Sensitivity analyses • What if you get less impatient with your consumption (ρ) • What if the growth rate of the population increases (n )

50. Case study Sensitivity analysis: A fall in ρ: • Why: People want to consume less today and more tomorrow – this increases savings – increases the long run capital stock per effective worker – increases wages – decreases return to capital! kt+1 45o kt k’* k*