Multiply both sides of (3) by (1 + ) and the equality is obviously preserved: (14 p)M; = m(Y,r+ pu)pe(1+ p) (4) Using the expressions above for M;_ ; and p; 1, equation (4) can be rewritten as: M, =m(Y,r+ u)pria (5) Equation (5) is telling us that the money market is also in equilibrium at # + 1. Economically, what's going on is the following. Since GDP and nominal interest rates are constant, people want to hold constant real money balances. Since the money supply is growing, prices must be growing too in order to keep money balances constant. 1.1.3 A Growing Economy Now suppose that the economy is in a steady-state-with-growth, with growing at a constant rate g and a constant real interest rate r. The money supply grows at a constant rate u. Let's try to find the inflation rate in this economy. Start from (1), take derivative with respect to time and divide on both sides by (1), to reach finally the following: H=ng+m = T=p-1g (5) where is the elasticity of money demand with respect to GDP. So, indeed, inflation is constant. Equation (5) tells us that, other things being equal, a growing economy will have lower inflation. Why is this? A growing economy means a growing number of transactions and therefore a growing demand for real money balances. This means that the economy can absorb growing quantities of money without resulting in inflation. Why does show up in the formula? measures how much the demand for money increases when the economy grows. The higher this number, the faster the money supply can grow without leading to inflation Exercise 3: Price Level and Inflation Consider an economy in a steady-state with growing money supply, growing output and a constant real interest rate. Money supply grows at a constant rate # = 0.05, such that Mit = (1 + #) Mi. Output grows at a constant rate g = 0.02, such that yi+1 = (1 + g)yt. 2 Econ 4721-Money and Banking HW 2 Recall the equilibrium condition for money: Mi = mi(y, i)p. where M' is the money supply, m" (y, i) is real money demand and p, is the price level of the economy (think of this as the price index). Recall from our discussion in class, that we can think of the nominal interest rate as: it = n + 7 where it is the inflation rate. (For the following questions, you are advised to follow closely your class notes) a) (10 points) Assume that real money demand takes the following form: my (yt, it) = you Find the implied inflation rate as a function of the growth rate of money, the growth rate of output and the elasticity of money demand with respect to output (as we did in class, let's call this term n).b) (10 points) Repeat part a) but assume instead that the money demand is of the following form: my (yt, it) = yi(1 + it)- c) (10 points) Repeat part a) but assume instead that the money demand is of the following form: my (ye, it) = 2it where a = 0.5 (notice that this is just the Baumol-Tobin demand function that we derived in class).1.1.1 An Economy in Steady State with a Constant Money Supply Imagine first that the economy is in a steady state where Y and r are constant and the central bank holds the money supply MS constant as well. We'll conjecture that in this economy the price level will be constant as well, and then verify that this is consistent with an equilibrium in the money market. If indeed the price level is constant, then the nominal interest is equal to the real interest rate. Therefore, solving for p in (1) we get: MS = 2 P= (Y, r) @ which indeed is constant. Equation (2) tells us that an economy where the money supply is higher will, other things being equal, have higher prices. People want a certain level of real money balances given by mP, so the price level will be such that % corresponds to these desired real money balances. 1.1.2 An Economy in Steady State with a Growing Money Supply Maintain the assumption that and r are constant but now assume that the money supply grows at a constant rate y, i.e. M, | = (1 + p)M;j. In this economy the price level will also grow at rate p. Let's check that this is consistent with equilibrium in the money market. If p;1 = (14 p)py, then inflation . is ,_1=l";,;'l=|u And therefore the nominal interest rate is ipi=r+mua=r+pu If the money market is in equilibrium in period t, then: Mi = m'(Y,r+p)p; 3 After that, the money supply will again be constant, except that the level will be higher. Therefore we i mi[Yr)* are going to be back in a constant-money-supply steady state, where p' = The effect on prices is therefore: pr - M p M (6) In other words, prices jump immediately to their new level, and the size of the jump is proportional to the size of the increase in M*. Note that one condition for this reasoning to be correct is that prices must be flexible, reacting immediately to changes in the supply of money. However, we can also think on the possibility that prices might be \"sticky\" and react slowly to changes in the money supply. This will be a source of monetary non-neutrality, i.e. of interaction between money and the real economy. 1.1.5 A Change in the Rate of Growth of the Money Supply Now let's do a slightly more subtle exercise. Suppose we start at a steady state with the money supply growing at rate u and, therefore, an inflation rate of u. At time t there is a sudden, unexpected increase in the rate of growth of the money supply, form u to pj. After this, the rate of growth of the money supply is expected to remain at y' forever. What's going to happen? A naive guess would be to say that inflation would simply increase from u to pi'. This guess is not wrong, but it's incomplete. If the inflation rate changes from y to |u", then the nominal interest rate rises fromi = r+ ptoi =r+ ,uj. Using (1), this implies that real money balances must fall: M= ' =m'(Y,r+u) 7) P Higher nominal interest rates increase the opportunity cost of holding money, so people want to hold less of it. But the level of M* does not change at time t: it simply starts growing at a different rate. What makes the money market clear? The price level must rise! 1.2 The Velocity of Money The \"velocity\" of money refers to the number of times a unit of money is used per period. There is a general (very famous) formula that captures the relationship between price level and velocity!: M vV = p Y (8) S~ S i S Money Supply Velocity Price LevelReal GDP Nominal GDF i M . . Using (1) to replace T in (8) and re-arranging we get: () Equation (9) says that any theory of money demand, summarized by a function m*(Y, i), is also a theory of velocity. Once we have a function for m?(Y, i), we can simply plug it into (9) to obtain velocity as a function of Y and i. Our theory of money demand implies that velocity is an increasing function of the nominal interest rate (since m? is decreasing in i, which implies V' is increasing in 7). Economically, what this is saying is that if interest rates are higher, people will hold less money, so in order to carry out the same amount of real transactions, each dollar will have to change hands more times. Figure 1 below shows how the velocity of money has evolved over time. Notice that the velocity of M1 is higher than the velocity of M2. Recall from the definitions of M1 and M2 that M2 includes more things than M1