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Question: Consider an OLG economy where each generation lives for 2 periods. Preferences are given by:  U^h...

Show transcribed image text Consider an OLG economy where each generation lives for 2 periods. Preferences are given by: U^h_t[c^h_t(t), c^h_t(t + 1)] = c^h_t (t) (c^h_t(t + 1))^beta Suppose that there exists two segmented markets and 2 types of agents: rich and poor. In particular suppose that N/2 individuals of each generation are rich and receive an endowment omega^R, h(t) = [omega_t^R, h (t), 0] and N/2 individuals of each generation are poor and receive an endowment omega^R, h(t) = [omega_t^P, h (t), 0], with omega_t^R, h (t) > omega_t^P, h (t). The population is constant. Assume that there exists a capital technology that pays a return of x each period per unit invested (linear technology). Moreover, suppose that the minimum level of capital investment is k*, with omega_t^R, h (t) > k* > omega_t^P, h (t). Suppose that in the economy there is a government that issues money and that the quantity of money M is fixed. In conclusion, suppose that there is no possibility of financial intermediation (i.e., there are no bank institutions that collect deposits). 1. Write down the budget constraints of rich and poor individuals in the first and second period (recall to consider the restriction on the capital accumulation). Moreover, suppose that there exists a market for debt, where individuals can buy and sell l^h(t), as in the models we have seen in class. 2. Write down the inter temporal budget constraint (IBC). Explain. 3. Use the IBC to discuss the portfolio saving decision of the rich individuals. On what does it depend? Use the no arbitrage condition to show it. Provide the economic intuition. 4. Solve the problem of the agent h (both rich and poor), i.e. maximise the utility function subject to the budget constraints. Write down the FOC and find the consumption and saving of each agent (both rich and poor) in every period. 5. Show that the rich save only in capital if x > 1.

Consider an OLG economy where each generation lives for 2 periods. Preferences are given by: U^h_t[c^h_t(t), c^h_t(t + 1)] = c^h_t (t) (c^h_t(t + 1))^beta Suppose that there exists two segmented markets and 2 types of agents: rich and poor. In particular suppose that N/2 individuals of each generation are rich and receive an endowment omega^R, h(t) = [omega_t^R, h (t), 0] and N/2 individuals of each generation are poor and receive an endowment omega^R, h(t) = [omega_t^P, h (t), 0], with omega_t^R, h (t) > omega_t^P, h (t). The population is constant. Assume that there exists a capital technology that pays a return of x each period per unit invested (linear technology). Moreover, suppose that the minimum level of capital investment is k*, with omega_t^R, h (t) > k* > omega_t^P, h (t). Suppose that in the economy there is a government that issues money and that the quantity of money M is fixed. In conclusion, suppose that there is no possibility of financial intermediation (i.e., there are no bank institutions that collect deposits). 1. Write down the budget constraints of rich and poor individuals in the first and second period (recall to consider the restriction on the capital accumulation). Moreover, suppose that there exists a market for debt, where individuals can buy and sell l^h(t), as in the models we have seen in class. 2. Write down the inter temporal budget constraint (IBC). Explain. 3. Use the IBC to discuss the portfolio saving decision of the rich individuals. On what does it depend? Use the no arbitrage condition to show it. Provide the economic intuition. 4. Solve the problem of the agent h (both rich and poor), i.e. maximise the utility function subject to the budget constraints. Write down the FOC and find the consumption and saving of each agent (both rich and poor) in every period. 5. Show that the rich save only in capital if x > 1.