Consider an economy in which people wish to hold money balances worth a total of 5,000,000 goods. They are indifferent between money issued by the central bank and money issued by private banks (so long as both offer the same rate of return). In the initial period, the central bank issues $1,000,000 and uses the proceeds to purchase capital. The central bank owns a stock of capital equal to its stock of money and uses the return to pay interest on its money. Assume that  x = 1.2 and a dollar always buys two goods. Intermediation, including the payment of interest on money, is costless.
a. What rate of interest ρ must the central bank offer to induce people to accept its money? Does this satisfy the central bank’s budget constraint?
b. What is the real value of the total amount of money issued by private banks?
c. Is there an equilibrium in which a dollar always purchases three goods? In this case, what is the real value of money issued by private banks?
d. Argue that the people are indifferent between the equilibrium in which a dollar is worth three goods and the equilibrium in which a dollar is worth two goods.
e. Suppose the central bank pays no interest on its money but maintains a constant stock of capital, using the net return from the capital it owns to buy up and burn a fraction of its money. Find z, the rate of change of the nominal central bank money stock. Check that the government budget constraint is met. (You should no longer assume that in all periods.)