thank you so much.
Consider an Overlapping generations economy in which N, young individuals are born each period. Individuals are endowed with y = 15 units of the consumption good when young and nothing when old. The utility function of one typical agent is a typical time-separable CRRA: u(c], (2) = 4 , + AY.. Ac (0, 1] is the time discount factor and a > 0 is the inverse of the elasticity of intertemporal substitution. Population of the future generations are determined by Nitz = n.N, for all t 2 1, No = 100. 1. What is the equation for the feasible set of this economy? 2. Portray the feasible set on a graph. 3. Let's solve the Planner's Problem. (i) State the Planner's problem as a constrained maximization problem. (ii) Write down the Lagrangean for this problem. (iii) What are the FOCs? (iv) Assuming a stationary equilibrium, find the optimal allocations as a function of A, a and n only. 4. How does consumption when young respond to changes in 8? What about ? And n? 5. Now, 8 = 0.5 and o = 2 and n = 2. Substitute and find c] and cz.6. With arbitrarily drawn indifference curves, illustrate the stationary combination of c and ez that maximizes the utility of future generations. 7. Now look at a monetary equilibrium. Write down equations that represent the con- straints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint. 8. Suppose the initial old are endowed with a total of M = 400 units of fiat money. What condition represents the clearing of the money market in an arbitrary period 7 Use this condition to find the real rate of return of fiat money. 9. Let's Find the competitive equilibrium allocation: (i) Write down the future genera- tions problem for this economy (ii) State the definition of a competitive equilibrium. (iii) Write down the Lagrangean for the future's generation problem. (iv) What are the FOCs? (v) Find the stationary equilibrium allocation as a function of a and # only . 10. How does the return of money affect consumption when young, when old, and real money holdings (v,m.)? 11. What is the value of money in period t (":)? What is the price of the consumption good p? 12. Suppose n increases. What would happen to the rate of return of fiat money and real money holdings of any generation ? What would happen to the value of a unit of fiat money in the initial period and the utility of the initial old? Explain your answers. Hint: Answer these questions in the order asked.] 13. Suppose instead that the initial old were endowed with a total of 200 units of fiat money. What would happen to the price level and consumption allocations? Are the initial old better off with more units of fiat money? Explain. 14. What is the utility function when a = 17 (HINT: Take limits and use the L'Hopital's rule)
