The Cagan Model. Lecture 15 by John Kennes March 25

Transcription

1 The Cagan Model Lecture 15 by John Kennes March 25

2 The Cagan Model Let M denote a country s money supply and P its price level. Higher expected inflation lowers the demand for real balances M/P by raising the opportunity cost of holding money. We write Cagan s model in log-linear form m d t p t E t p t1 p t (1) where m log M, p log P, and is the semielasticity of demand for real balances with respect to expected inflation. In eq. (1), m t d denotes (the log of) nominal balances held at the end of period t. Assume rational expectations, Muth (1961).

3 The LM curve? Cagan s equation (1) is a simplified version of the LM curve. M t d P t LY t, i t1 (2) Cagan argued that during a hyperinflation, expected inflation swamps all other influences on money demand. Thus we ignore changes in real output and the real interest rate.

4 The Fisher equation Note that under perfect foresight the real interest rate links the nominal interest rate to inflation through the Fisher parity equation 1 i t1 1 r t1 P t1 P t (3) The Fisher equation implies that in equilibrium, the gross real rates of return on real and on nominal bonds must be the same. Thus the nominal interest rate and expected inflation will move lockstep if the the real interest rate is constant. This explains Cagan s simplified model as a version of the LM curve.

5 Solving the Model Assume that the supply of money m is set exogenously. In equilibrium, demand equals supply: m t d m t Equation (1) therefore becomes the monetary equilibrium condition m t p t E t p t1 p t (4) Equation (4) is a first-order stochastic difference equation explaining price level dynamics in terms of the money supply, which is an exogenous forcing variable here. (Difference equations in appendix C). We will solve the model from first principals.

6 The non-stochastic case Equation (4) becomes m t p t p t1 p t (5) Start by rewriting equation (5) as p t 1 1 m t 1 p t1 (6) So that this periods price level depends on the foreseen future price level..lead eq (6) by one period to obtain p t1 1 1 m t1 1 p t2 then use this expression to eliminate p t1 in equation (6). p t 1 1 m t 1 m t1 1 2 p t2

7 Repeated Iteration Repeating to eliminate p t2, p t3,... p t 1 1 lim T st 1 1 s t m s T p tt (7) Tentatively, assume that the second term on the right hand side of (7) is zero: lim T 1 T p tt 0 (8) This limit is zero unless the absolute value of the log price level grows at an exponential rate 1 / or above. Rules out speculative bubbles. Convergence of the first term requires the log money supply does not grow at an exponential rate of 1 / or above

8 Equilibrium Price Level Condition (8) implies that the equilibrium price level is p t 1 1 st 1 Notice that the coefficients on the money-supply terms in equation (9) is Thus the price level depends on a weighted average of future money suppies, with weight decline geometrically as the future unfolds. s t m s (9) 2...

9 Money Neutrality What happens if we double the money supply? The fact that the weights on future money supplies sum to one implies money is fully neutral This is version of long-run money neutrality. However, money is not neutral in the short run in this model.

10 Guessing a Solution Consider some cases for (9) that are so simple that we can guess the solution. For example, if the money supply is expected to remain at m forever. In this case, it is logical to think that inflation should be zero too, p t1 p t 0. But in this case eq (5) implies the price level is constant at p m, which is the solution (9) also implies.

11 Constant Money Growth A second case, is money supply growing at a constant rate per period, m t m t (If the log of a variable is growing at a constant rate, then the level of the variable is growing at per year.) In this case, it makes sense to guess that the price level is growing at rate,. Substitute this guess into the Cagan equation (5) yields p t m t (10) This, too, is the answer equation (9) implies. Needs computation.

12 The Computation Term by term multiplication of the product yields p t 1 1 m t st s t m t s t st m t 1 s t 1 1 m t

13 More General Processes Consider the effects of an unanticipated announcement that the money supply is going to rise sharply and permanently on a future date T. Specifically, suppose m t m m t T t T Given this money-supply path, eq (9) gives the path of the price level as p t m 1 T t m m m t T t T Illustration (in class)

14 The Assumption of No Speculative Bubbles If we allow speculative bubbles, there might be solution to equation (5) of the form p t 1 1 st 1 s t m s b 0 1 where b 0 is the initial deviation of p 0 from its fundamental value. That is b 0 p s0 1 s m s When we introduce maximizing agents, as we do in the next section, we will ask whether there is a basis for believing equation (8). For now, we simply assume that b 0 0 and that the equilibrium price level is always given by equation (9) t (1

15 The Stochastic Cagan Model Because the Cagan equation is linear, extending it to a stochastic environment is simple. When the future money suply is uncertain, the no-bubbles solution to the Cagan model (solution to equation 4) is p t 1 1 st 1 s t E t m s (12) as you can check by substitution. The only difference is the replacement of perfectly forseen money supplies by their expected values.

16 An Example Suppose, for example, that the money supply process is governed by m t m t 1 t (13) where t is a serially correlated white noise money supply shock such that E t t1 0. Substitute the money supply process into equation (12) and note that E t ts 0 for s t. Theresult p t m t 1 st 1 s t m t m t 1 (14) The limiting case 1 (money shocks are expected to be permanent), the solution reduces to p t m t in analogy

17 with the nonstochastic case

18 Continuous Time In continuous time, the Cagan money demand function is m t p t p t (15) where dlog P/dt P /P is the anticipated inflation rate in continuous time. Using methods of first order differential equations, one finds the general solution is given by p t 1 t exp s t/nms ds b o expt/n (16) which has a strong resemblence to (11). Speculative bubbles are ruled out by setting b o to zero. Confirm long run money neutrality

19 Limit of discrete time (1) Consider the discrete time Cagan model. Let the time interval between successive dates be of arbitrary lenght h. Then the perfect foresight Cagan equation becomes m t p t h p t1 p t (17) We can divide by h because a given price level increase lowers the real rate of return of money in inverse proportion to the time interval. Taking the limit as h 0 yields equation (15).

20 Limit of discrete time (2) Solving equation (17) forward as before (h1 results in equation 11) we get p t 1 1 /h b 0 st 1 /h /h /h 1 /h where the preceding sum is over s t, t h, t 2h, etc. Rewriting this expression t/h s t/h m s p t 1 h b 0 1 h st 1 h t/h s t/h ms h and take the limit as h 0 gives the continuous time model

21 A Simple Example Suppose that the money supply grows at a constant rate m. Assuming no speculative bubbles and by applying integration by parts to equation (16) yields p t 1 t exp s t/nms ds m t t exp s t/n m s ds m t (18) If we dont impose no bubbles, p t m t b 0 e t/n (19) where b 0 p 0 m 0 is the gap between the initial price level and its fundamental, no-bubbles value.

22 Seignorage Seignorage represents the real revenue a government aquires by using newly issued money to buy goods and nonmoney assets. Most hyerinflations stem from the government s need for seignorage revenue. A government s seignorage revenue in period t is Seignorage M t M t 1 P t (20) For next class. Derive the formula for maximum seignorage. Try not to use the book. max 1 (23)