Using Cagan model(appendix of chapter of mankiw book), derive the inflation rate. Just assume that we are in the perfect foresight world. If the money supply grwth rate is given by some constant, what is the inflation rate? Answer

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1.Using Cagan model(appendix of chapter 5 of MANKIW BOOK), derive the inflation rate. Just assume that we are in the perfect foresight world. If the money supply grwth rate is given by some constant , what is the inflation rate?
Answer.
if the amount of real money balances required depends on the cost of holding the money, the price level depends on both. current money supply and future money supply. This application will be developed Cagan model to show more clearly how this relationship works.To simplify the math as much as possible, we put the money demand function it is linear in the natural logarithms of all variables. Money demand function .
mt - pt = - g (pt + 1 - pt)
where (mt) is the log of the quantity of money at time (t), (pt) is the log of the price level at time (t), and gis a parameter that governs the sensitivity of money demand to the rate of inflation. By the property of logarithms, (mt – pt) is the log of real money balances, and (pt+1 – pt) is the inflation rate between period t and period (t + 1). This equation states that if inflation goes up by 1 percentage point, real money balances fall by (g) percent. We estimate the perfect inflation forecast by introducing real inflation through this function. We find from this function (g) and the resulting function is important in determining the future forecast of this inflation. Finally, let’s relax the assumption of perfect foresight. then we should write the money demand function as
mt − pt = −g(Ept+1 − pt), this function is derived from the above formula.
2. According to classical economic theory, money is neutral: the money supplydoes not affect real variables. Therefore, classical theory allows us to studyhow real variables are determined without any reference to the money supply. The equilibrium in the money market then determines the price leveland, as a result, all other nominal variables. This theoretical separation ofreal and nominal variables is called the classical dichotomy. Just explain classical dichotomy in more details.
Answer.
In macroeconomics, the classical dichotomy is the idea, attributed to classical and pre-Keynesian economics, that real and nominal variables can be analyzed separately. To be precise, an economy exhibits the classical dichotomy if real variables such as output and real interest rates can be completely analyzed without considering what is happening to their nominal counterparts, the money value of output and the interest rate. In particular, this means that real GDP and other real variables can be determined without knowing the level of the nominal money supply or the rate of inflation. An economy exhibits the classical dichotomy if money is neutral, affecting only the price level, not real variables.[citation needed] As such, if the classical dichotomy holds, money only affects absolute rather than the relative prices between goods.The classical dichotomy was integral to the thinking of some pre-Keynesian economists ("money as a veil") as a long-run proposition and is found today in new classical theories of macroeconomics. In new classical macroeconomics there is a short-run Phillips curve which can shift vertically according to the rational expectations being reviewed continuously. In the strict sense, money is not neutral in the short-run, that is, classical dichotomy does not hold, since agents tend to respond to changes in prices and in the quantity of money through changing their supply decisions. However, money should be neutral in the long run, and the classical dichotomy should be restored in the long-run, since there was no relationship between prices and real macroeconomic performance at the data level. This view has serious economic policy consequences. In the long-run, owing to the dichotomy, money is not assumed to be an effective instrument in controlling macroeconomic performance, while in the short-run there is a trade-off between prices and output (or unemployment), but, owing to rational expectations, government cannot exploit it in order to build a systematic countercyclical economic policy.Keynesians and monetarists reject the classical dichotomy, because they argue that prices are sticky. That is, they think prices fail to adjust in the short run, so that an increase in the money supply raises aggregate demand and thus alters real macroeconomic variables. Post-Keynesians reject the classic dichotomy as well, for different reasons, emphasizing the role of banks in creating money, as in monetary circuit theory. Neutrality of money is an important idea in classical economics and is related to the classical dichotomy. It implies that the central bank does not affect the real economy (e.g., the number of jobs, the size of real GDP, the amount of real investment) by creating money.
3. Suppose that consumption depends on the levelof real money balances (on the grounds that realmoney balances are part of wealth). Show that if real money balances depend on the nominal interest rate, then an increase in the rate ofmoney growth affects consumption, investment, and the real interest rate. Does the nominal interest rate adjust more than one-for-one or less than one-for-one to expected inflation? This