Question 1: (12 marks) Here we consider a model of liquidity and banking. Consider an OLG economy with an infinite horizon. Each period there are Mr yormg individuals born where the number of individuals born grows according to Mt : nNt_1 where n E l (i.e. the population is growing). Each individual lives for three periods, which we will call young, middle-age, and old. When young individuals are endowed with 1 unit of the consumption good. Individuals do not want to consume when yormg but want to consume both in middle-age and when they are old. There are two assets in the economy, at money and capital. There is a at money supply that is equal to M (i.e. z = 1). Money can be converted into consumption goods in any period by executing a trade with an individual with the consumption good. Capital has a two-period return X > \":12. Each unit of the consumption good can be used to create one unit of capital. Each unit of capital takes two periods to mature and cannot be converted into the consumption good prior to maturing. Individuals are unable to trade unmatured capital. (a). (2 marks) If a young individual uses their good to acquire money, how many units of money do they obtain when young? (b). (2 marks) How many units of the consumption good does the money provide in: (i) middle age?, and (ii) old age? (c). (1 marks) If a young individual uses their good to create capital, how many units of the consumption good does the capital provide in: (i) middle age?, and (ii) old age? ((1). (1 marks) If the individual wants to consume both in middle age and when old, what should they do? Suppose they want to consume the same amount in middle age and when old, do they need to hold more money or create more capital? (e). (1 marks) We now introduce a bank that is infmitely lived as is able to take deposits and issue IOU's that it connnits to. Suppose the bank maximizes prots, which it distributes to individuals in the form of a return on deposits. What one period return does a bank need to promise to get individuals to make deposits? What is the maximum one-period return the bank can promise on deposits? (f) (1 marks) Suppose the bank uses all deposits to create capital in period t. How does the bank pay the return to the period t depositors when they are in middle age in period I + I? (g) (2 marks) How does the bank solve the liquidity mismatch problem? Which of our assumptions guarantee it? (h) (2 marks) Is the output of the economy greater or less than the output of the economy without the bank? Explain carefully