Hi tutor. Kindly help me solve these
I1 Exercise 2. Let f: R" + R be a real-valued function on R", and Xo = E R a In point. (a) If f is differentiable at Xo, explain what it means to say that the function z = h(x), where h(x) = f(Xo) + DS(xo) (x -x) is the best linear function to approximate f at Xo- (b) Suppose f is differentiable at the point x X1, and Df (x, ) = [ 0 ... 0 ]. the O-matrix. Describe what may be happening to graph(f) at or near x = X1. (c) Suppose f is not differentiable at x = x2. At this point, all of the partial derivatives exist and are continuous as functions, except for one: In the ith coordinate direction, we found of (x2) = lim fxzte,h) - f(x2) =00 h Describe what may be happening to graph(f) at or near x =Consider an individual with preferences represented by the following utility function: U ( x 1, x 2) = x113x 223 The individual faces prices: p 1 = 0.75 ; p 2 = 1 And has income: M = 250 a. [5 points) What is the consumer's optimal bundle at the original set of prices and income? b. [5 points] What utility does the consumer achieve at the original set of prices and income? c. [5 points] The price of good 1 increases to 1.8. What bundle does the consumer choose at the new price of good 1 and original income? d. [5 points] What utility does the consumer achieve at the new price of good 1 and original income? e. [10 points] The price of good 1 increases to 1.8. How much does the consumer spend to achieve the original utility at this new (higher) price? [Write this answer in the space below.] f. [10 points] What is the consumer's compensating variation of this price change?Consider the following uni-variate linear regression model, Given n observations on 2, = (y. r,) with i = 1,2, , n, where a, 's are I-by-1 scalars. Suppose assump- tions A(0). A(1), A(2) and A(3) hold. We split the data sample in two with n, observations of s, in the first sub-sample and na observations of z, in the second sub-sample, note We are interested in estimating coefficient 8 = (0, b) and run linear regression (OLS) of y, on z, and a constant for each sub-sample. Thus we obtain estimate ) based on my observations in the first sub-sample. and estimate 32) based on no observations the second sub-sample. Now let's study the finite sample property of the following estimator of 3, 1. Show that S is an unbiased estimator for ? 2 Show that Aris a linear estimator in y where y is not Column after con sting of all n observations on , s Hint, the function4. (a) Let / be an interval in R and consider a function / / - R. (i) Define what it means to say that f is differentiable at c E I and define the derivative of f at c. (3 marks) (ii) Show that if f is differentiable at ce / then f is continuous at c. (6 marks) (b) Let f : R - R be defined by sin(1/)fore / 0 0 for 1 = 0. Show that the function f is not differentiable at be R. (6 marks) (c) Use the Mean-Value Theorem to prove that for all r. y E R. Use this result to conclude that the function cos : R - R is uniformly continuous. (5 marks)2. Suppose that X has mean ux and variance of , and suppose that Y has mean fly and variance of. Suppose the correlation between X and Y is p. Let U = aX + b and V = cy + d where a, b, c, d are constants. (a) Find Covariance(U. V ) in terms of a, b, c, d, ux. Ox, AY, of, and p. (Note: some of these parameters might not appear in the answer.) (b) Find Corr(U. V) in terms of a, b, c, d. ux. of, by, of, and p. (Note: some of these parameters might not appear in the answer.) (c) Suppose that W and I are two random variables with Var(W) = 4. Var(V) = 9, and Cov(W. V) = 2. Find Vor(-4WV + 3V - 2). (d) Suppose that X and Y are random variables with cov(X, Y) = 3. Find cov(2X - 1. 3 - 5Y)
