Answer.the following attachments.

1.Exchange Rate Overshooting

Use the foreign exchange and money market diagrams to answer the following questions about the relationship between the Indian rupee (INR) and the Chinese yuan (CNY). Let the exchange rate be defined as rupees per yuan EINR/CNY. Suppose there is a fall in the Indian nominal money supply. Make the usual assumptions: UIP holds, PPP holds in the long run, prices are sticky in the short run,

a.Assume first that the fall in money supply is temporary (so that the nominal money supply is put back at its original level in the long run). Illustrate the effects of this in a pair of graphs, one for the Indian money market and one for the foreign exchange market. Label the initial equilibrium as point A, the short-run equilibrium point B, and your long-run equilibrium point C.

b.Now assume instead that the fall in money supply is permanent. Illustrate this in a pair of graphs, one for the Indian money market and one for the foreign exchange market. Label the initial equilibrium as point A, the short-run equilibrium point B and your long-run equilibrium point C.

c.For the case you just analyzed above (permanent shock), plot a graph for each of the following variables over time showing the initial equilibrium, short run equilibrium, and the long run equilibrium: India's nominal money supply, India's interest rate, India's price level, India's real money supply, and the exchange rate EINR/CNY.

d.Does the theory of "exchange rate overshooting" apply to the case in part (a) above? How about to

the case in parts (b) and (c)? Explain the economic reason the two cases are different.

4) Consider two countries, Home and Foreign. Suppose the foreign country experienced a relatively slow output growth (1%), whereas the home had relatively robust output growth (6%). Suppose the Central Bank of foreign country allowed the money supply to grow by 2% each year, whereas the Central Bank of home country chose to maintain relatively high money growth of 12% per year. For the following questions, use the simple monetary model (where L is constant). [20 points]. a. What is the inflation rate in Home? In Foreign? b. What is the expected rate of depreciation in the home currency relative to the foreign currency? c. Suppose the Central Bank of home country increases the money growth rate from 12% to 15%. If nothing in the foreign country changes, what is the new inflation rate in the home country? d. Suppose the Central Bank of home country wants to maintain an exchange rate peg with the foreign currency. What money growth rate would the Central Bank of home country have to choose to keep the value of its currency fixed relative to the foreign currency?Part II (Proofs) Choose 3 of the following: a) Prove: Let A and B be invertible a xn matrices. Then AB is invertible, and (AB)' - Bad". b) Let F be the vector space of all functions mapping R into R. Show that the set S of all solutions in F of the differential equation /" + f = 0 is a subspace of F. c) Prove: A linear transformation 7 : V - F' is invertible if and only if it is one-to-one and onto " . Hint: Def. 1-1 v, * v, implies 7 (v, ).' for some vEV . d) Prove: Let I be an inner-product space, and let v and w be vectors in . Then Kv. w) =|~|wl. e) Prove: Let A be an axn matrix and let 4, 4,....A. be (possibly complex) scalars and V,V,:..V. be nonzero vectors in -space. Let C be the axe matrix having v, asjith column vector, and let D= . Then AC - CD if and only if 2. ...A. are 0 eigenvalues of A and v, is an cigenvector of A corresponding to A, for / = 1,2.. .. () Prove: Let A be an # xn matrix and let v, V,,...>. be eigenvectors of A corresponding to distiner eigenvalues A. A....A . respectively. the set fo, v....v. } is linearly independent and A is diagonalizable. Prove: If A is a Hermitian matrix, there exists a unitary matrix ( such that U" AU is a diagonal matrix and all eigenvalues of A are real. [ complete the proof started below] Proof. By Schur's lemma, there exists a unitary matrix ( such that U" AL is upper-triangular matrix. Because U is unitary, we have U"U = J . so U" = U" and because A is Hermitian, we also know Thus, we have which shows that the upper-triangular matrix CAU is also Hermitian. Because the conjugate transpose of an upper-triangular matrix is a lower-triangular matrix. we see that the entries above the diagonal in U" AU must all be zero; therefore. U AU - . Where is matrix. Thus, A is unitarily diagonalizable. Now we will show that each eigenvalues of A is a real number. [Finish this part of the proof] by Prove: The eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal. Proof: Let v and w be eigenvectors of a Hermitian matrix 4 corresponding to distinct eigenvalues 4, and 2. respectively. Using the fact that Am " and that the eigenvalues are real, so that by = 2. we have 2 (wv)- ... =(wv). [Fill in details]. Therefore, (2, - 2, )(wv)- 0. [Finish proof]. Since the eigenvalues are distinct, specifically