HHelp me to solve the questions as attached below.
3. Consider an economy with A consumers and B rms. Each consumer is endowed with one unit of time and one piece of capital. A representative consumer's utility function is u(:r, r) = ins: + 21n r, where .r is consumption of goods and r = 1 l is leisure time and I. the time working. Each rm hires consumers to work and rents capital from consumers to produce goods according to technology: y = KaLl'a', where a 6 (0, 1) is a. parameter, K is the amount of capital rented by the rm and L is the amount of labor hired by the rm. All consumers and rms take goods price p, wage rate to and rental rate q as given. Normalize p = 1. (a) [10 points] Set up a representative consumer's utility-memorization problem. Derive the Marshallian demands for .1: and r. (b) [10 points] Set up a representative rm's prot-maximization problem. List the rst-order conditions regarding the choices of K and L. (c) [10 points] List the marketclearing conditions for markets of capital, labor and goods respectively. (d) [20 points] Use the results from (a)(c) to calculate equilibrium rental rate 4} and the Marshallian demand for r. 3. Consider the problem on multiple linear regression in which you have p predictor variables 22(1), . .. ,scu'). The model is given by Y = 50 + 5193(1) + - ' - + pmm + 6 where 5 ~ N(0,02). Suppose that we have 7; random samples of Y, i.e. Y; for i = 1, . . . ,n that satisfy sz=o+61$n+~-+p:rp)+6i with 5,- N N (0, 0'2). Perform the following: (a) Formulate the model above in terms of matrices. Clearly write down the model in matrix form by dening the matrices Y, X , , 6. You can express your answer in terms of realizations of K, 61-. (b) Assuming that 02 is known, write down the minimization problem (in matrix form) that needs to be solved to nd an estimate of i3. (c) Argue that the calculations to nd El are identical to what we did in lecture for simple linear regression. (d) Consequently, write down the least squares estimate of [:3 in matrix form. Also write down its distri bution. A special endowment policy pays a sum assured of E20,000 to a life who is currently aged exactly 57 after three years or at the end of the year of earlier death. Annual reversionary bonuses are declared at the end of each policy year, and an additional terminal bonus is payable at maturity only. Policies may be surrendered only at the end of each policy year. On surrender, the policyholder receives a return of premiums with interest calculated at the rate of 3% per annum. A level premium of $8,000 is paid at the start of each year. The premium basis is as follows: Interest: 7% per annum Mortality: AM92 Select Surrender rates: 15% of all policies in force at the end of year 1 5% of all policies in force at the end of year 2 Reversionary bonuses: 6% per annum compound Terminal bonus: 10% of all other benefits payable at maturity Expenses: Initial E500 Renewal E30 at start of year 2 E35 at start of year 3 Termination f100 per termination (death, surrender or maturity) Reserves: Net premium reserves, using AM92 Ultimate mortality and 4% per annum interest (i) Calculate the profit signature for this policy according to the premium basis. [14] (ii) By accumulating the elements of the profit signature to the maturity date, explain briefly whether you think the company expects to declare the bonus rates it has assumed in its premium basis, assuming all the other assumptions in the basis are realistic. [2] [Total 16]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]
