Question 4 (15 p) Assume that national income, Y, is exogenously set at one fixed level (full employment) in both the short-run and in the long-run, and that the exchange rate is floating. Assume also that prices are sticky (the price level changes slowly over time toward its long-run equilibrium). The money supply has a zero growth rate. In each part of this question, you are asked to consider a change in money supply. That change is assumed to always be unanticipated, but once the change occurs, it is publicly known. Assume that the economy is initially in a state of long-run equilibrium. To help explain your answer to questions (a) and (b), use a separate graph in each that displays simultaneous equilibrium in the domestic money market and the foreign-exchange market (uncovered interest parity). (a) Show how a temporary increase in the money supply affects the domestic interest rate and exchange rate in the short-run. (b) Show how an permanent one-time increase in the money supply affects the domestic interest rate and exchange rate in the short-run and in the long-run. Explain what causes any curves to shift. (c) On a graph with the exchange rate on the vertical axis and time on the horizontal axis, show the time path of the exchange rate after an permanent one-time increase in the money supply.1. (a) Show that if A and B are similar n x n matrices then they have the same characteristics equation and hence the same eigenvalues. [8 Marks] (b) If A is a2 x 2 matrix find the characteristic polynomial and the characteristic equation of A. [6 Marks] (c) State the theorem of Cayley-Hamilton. [4 Marks] (d) Let A = -1 4 (i) If the characteristic polynomial of A is p(1) = 13 - 217 - 51 + 6 show that A satisfies its characteristic polynomial [7 Marks] (ii) Use Cayley-Hamilton's theorem to compute A-1 [10 Marks] 2. (a) Let V be a vector space over the scalar field F. Prove that if V, and V2 are subspaces of V thenV, n V2 is a subspace of V. [10 Marks] (b) Let C[0 , 2x] denote the vector space of all continuous real-valued functionsdefined on [0 , 2x]. Define an inner product on C[0 , 2x] by 2 x (f .g) = | f(x)g(x)dx ICUI Mous [uz' 0]5 3 6 / ned A f (x) = - 1 =Sin mx and g(x) = =cos nx form an orthonormal set. [15 Marks] (c) Let C' be a vector space and define an inner product onC by (x , y) = S. V pair x = (x1.x2), y = (y1. yz) in C?If x = (3, -i) and y = (2. 6i), show that x and y are orthogonal. [5 Marks] (d) Let ( ) : 12 x R2 - R be defined by(x , y) = xiyi + 3x2yz V pair x. y e R2. If x = (2,-3), find [| x]|. [5 Marks]The numbers in parentheses below the estimated coefficients are the standard errors. a) Test the hypothesis that higher deficits lead to higher interest rates. Use 5% level of significance. b) Test the hypothesis that higher inflation leads to higher interest rates, Use 5% level of significance. 6. The following regression results were obtained from 46 sample points for the period 2015; log Y = 4.30-1.34 log X, +0.17 log X2 se = (0.91) (0.32) (0.20) R2 = 0.27 Where I= Bread consumption, packs per year X1= real price per pack Xy= real disposable income per capita a) What is the elasticity of demand for bread with respect to price? Is it statistically significant? If so, is it statistically different from one? b) What is the income elasticity of demand for bread? Is it statistically significant? If not, what might be the reasons for it? 7. From the following sample of ten yearly observations a researcher wants to estimate the demand function for second-hand TV sets. No of TV sets(Y) 543 580 618 695 724 812 887 90 1186 1940 Price in Kwacha(X) 61 54 50 43 38 36 28 23 19 10 The researcher estimated the following demand function Y, = BoXe" or InY, = InBo + B, InX, +u; Preliminary analysis of the sample data produces the following sample information [(InY,)=67.25 _(InX,)=34.71 _(InX,)(InY,)=231.51 _(InX,)' =123.24 Ey? =-1.91 Exy= Ex =2.77 EM =1.33 0 =0.01 Where x, and y, are deviations of the InX and InY from their respective means. Using the above information please answer the following questions clearly showing all formulas and calculations. a) From the data above, estimate the given demand function using OLS method b) Interpret the estimated slope coefficient in (a). c) What is the price elasticity of demand for second hand TV sets? Interpret it d) What percentage of the variation in demand for second hand TV sets is not explained by the regression line e) Compute the standard errors of the estimated parameters. State two other factors that would affect the demand for second hand TV sets.1. Consider a perfectly competitive economy with two goods (I and y) and a single consumer. The consumer is endowed with s {> 0} units of thee which are supplied inelasticale to the labor market. Both goods (:1: and y} are produced by perfectly competitive rms using one unit of labor to produce one unit of good 3': or good 3;. The consumer chooses consumption of goods e and y to maximize his utility: U = olnm+lny, where lI] cm > {1:3 > U . 0 (16:1 5. ,33
