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Consider an individual's saving problem in Blanchard's "perpetual youth" model (standard notation): max () =0 0 = Z 0 (ln )(+) s.t. 0 = ( + ) + where 0 is given, lim 0 (+) 0. a) Briefly interpret the objective function, the constraints, and the parameters. b) Derive first the first-order conditions and the transversality condition, next the Keynes-Ramsey rule. c) Derive the full solution to the problem, i.e., find the consumption function. Hint: combine the Keynes-Ramsey rule with strict equality in the intertemporal budget constraint. d) How will a rise in the interest rate level affect current consumption and saving? Comment in terms of the Slutsky effects. IV.3 Short questions a) State with your own words what the No-Ponzi-Game condition says. b) The No-Ponzi-Game condition belongs to problems with an infinite horizon. What is the analogue condition for a problem with finite horizon? c) In the consumption/saving problem of a household, is the household's transversality condition a constraint in the maximization problem or does it express a property of the solution to the problem complete

d) Write down the perfect foresight transversality condition of a household with infinite horizon. In fact there are three ways of writing it. Indicate all three. e) State with your own words what the transversality condition says in each of the three versions. IV.4 Demography and the rate of return The Blanchard OLG model for a closed economy is described by the two differential equations = ( ) ( + + ) 0 0 given, (1) = h 0 ( ) i ( + ) (2) and the condition that for any fixed pair ( 0) where 0 0 and 0 lim 0 (0 (())+) = 0 (3) Notation: () and () where and are aggregate capital and aggregate consumption, respectively, is population, and is the technology level, all at time () is a production function on intensive form, satisfying (0) = 0 0 0, 00 0 and the Inada conditions. Finally, is financial wealth of an individual born at time and still alive at time The remaining symbols stand for parameters and we assume all these are strictly positive. Furthermore, 0. a) Briefly interpret the three above equations, including the parameters. b) Draw a phase diagram and illustrate the path the economy follows, given some arbitrary positive initial value of . Can the divergent paths be excluded? Why or why not? In the last more than one hundred years the industrialized countries have experienced a gradual decline in the three demographic parameters and Indeed, has gone down, thereby increasing life expectancy, 1 Also has gone down, hence has gone even more down than . The question is what effect on the long-run interest rate, we should expect? Below you are asked to give a "rough answer" based on stepwise curve shifting (comparative analysis) in the phase diagram. In this context it is convenient to consider and as the basic parameters and + as a derived one. So in (1) and (2) substitute + 25 c) How does a lower affect the position of the = 0 locus? Illustrate by a new phase diagram. Comment. d) Given how does a lower affect the position of the = 0 locus? Illustrate in the phase diagram. Comment. e) Given how does a lower = + affect the position of the = 0 locus? Illustrate in the phase diagram. Comment. Hint for an explanation: Sign the effect of the lower on the proportion of young people in the population, then on the ratio next on the ratio [( + )] = (+) (standard notation), and finally on ( ). f) What is your overall conclusion as to the sign of the effect of the demographic change on ? g) The method of analysis has a limitation which explains the proviso hinted at by the expression "rough answer" above. What is this limitation?

Productivity speed up The basic model for this problem is the same as in Problem IV.4. Assume the economy has been in steady state until time 0 Then an unanticipated shift in to a higher positive level occurs. Hereafter everybody rightly expects to remain at this new level forever. a) What happens to the real interest rate on impact? Comment. b) Illustrate by a phase diagram the evolution of the economy for 0. There might be different possibilities to consider. Comment. c) What happens to the real interest rate in the long run? Comment. d) Compare two closed economies, and that can be described by this model and have the same production function, the same and the same initial conditions, 0 0, and 0, and the same The only difference is that country for some reason has a higher health level and therefore lower than country (and lower since is the same) "Country will in the long run experience a higher level of labor productivity, than country ". True or false? Why?

a) Make a list of motives for individual saving. Are some of these motives more in focus in an OLG framework than in a Ramsey framework? b) In standard long-run models with perfect competition (like Blanchard's OLG model with exponential retirement or the Ramsey model), the real rate of interest, , and the real rental rate, , for physical capital (i.e., a price on the market for capital services) may or may not coincide for all . Give a necessary and sufficient condition that they coincide. IV.7 Short questions. a) What is the golden rule capital intensity? b) A steady-state capital intensity can be in the "dynamically efficient" region or in the "dynamically inefficient" region. What is meant by "dynamically efficient" and "dynamically inefficient"? Give a simple characterization of the two regions. c) Compare some long-run properties of the Blanchard OLG model with the corresponding long-run properties of the Ramsey model. Hint: For example, think of the long-run interest rate and/or the possibility of dynamic inefficiency. d) The First Welfare Theorem states that, given certain conditions, any competitive equilibrium (Walrasian equilibrium) is Pareto optimal. Give a list of circumstances that each tend to obstruct the needed conditions and thus make the conclusion untrue. IV.8 Short questions. Consider the Blanchard OLG model with Harrodneutral technical progress at rate a) Can a path below the saddle path in ( ) space be precluded as an equilibrium path with perfect foresight in the Blanchard OLG model? Why or why not? b) Can a path above the saddle path in ( ) space be precluded as an equilibrium path with perfect foresight in the Blanchard OLG model? Why or why not?

a) "If and only if the production function is Cobb-Douglas, does the Blanchard OLG model predict that the share of labor income in national income is constant in the long run." True or false? Give a reason for your answer. b) Are predictions based on the Blanchard OLG model (with exogenous Harrod-neutral technical progress) consistent with Kaldor's stylized facts? Why or why not? c) Suppose we want a concise economic theory giving the long-run level of the average rate of return in the economy as an explicit or implicit function of only a few parameters and/or exogenous variables. Does the Blanchard OLG model give us such a theory? Why or why not? d) Briefly, assess the theory of the long-run rate of return implied by the Blanchard OLG model compared with that of the Ramsey model. That is, mention what you regard as strengths and weaknesses of the Blanchard theory. IV.10 Short questions a) What does Barro's dynasty model conclude about the hypothesis of Ricardian equivalence? b) What does Blanchard's OLG model conclude about the hypothesis of Ricardian equivalence? c) What is the basic reason that the two models lead to different conclusions in this regard? IV.11 Short questions a) "Considering the different Slutsky effects, the consumption function of the individual in the Blanchard OLG model (with logarithmic instantaneous utility) is such that a higher tax on interest income lowers current consumption." True or false? Why? b) "When the real interest rate remains above the GDP growth rate of the economy, then the NPG condition for the government is a necessary and sufficient condition for fiscal sustainability." True or false?