Money and banking econ class

Reference - Modeling Monetary Economics (2016, Fourth Edition, Cambridge University Press) by Bruce Champ, Scott Freeman and Joseph Haslag.

Problem 2 Consider an overlapping generations economy where people live for two periods - young and old. Population and fiat money stock are constant with M = 1000. The banking sector is perfectly competitive. Banks can borrow from consumers and invest in capital. One unit of capital produces x = 1.5 units of output in the next period. There are no reserve requirements (i.e. y = 0). There is a fixed cost $ = 10 of making a deposit at the bank. This means, if r denotes the gross interest paid by banks, s; units of the consumption good deposited lead to a gross return of rs; -o. There are three types of consumers, each with different endowment when young and none when old. There are 100 consumers of each type. Assume consumers of each type want to save half of their total endowment. They can save in deposits or fiat money but cannot directly invest in capital. The following table shows endowments of the types. Endowment Type 1 10 Type 2 30 Type 3 50 (a) Let s* denote the value of desired savings such that a consumer is indifferent between saving in deposits or fiat money. What is the value of s*? (b) What is the deposits-to-currency ratio in the economy? (c) What is the value of the money multiplier and M1? (d) What is the value of real GDP? (e) Suppose now that a increases to 2. Do parts (a) to (d) with this new value. (f) Based on your answers to the previous parts, what is the sign of the correlation be- tween changes in total money stock and changes in output? Is this consistent with the observations discussed in class? (g) Can you find a value of a such that the resulting value of MI is the same as in (c) but that of GDP is different from the one in (d)? (h) Can you find all values of a that satisfy the above