Provide solutions.to the following attachments.

Suppose that the business oyole in the United States is best described by R30 theory and that a new technology increases productivity. Show the effect of the new technology in the market for loaneble funds. Draw a demand for loanable funds curve. Label it DLFO. Draw a supply of loanable funds curve. Label it SLFD. Draw a point at the equilibrium quantity of loanable funds and real interest rate. Label it 1. Draw a curve that shows the effect of the increase in productivity. Label it. Draw a point at the new equilibrium quantity of loaneble funds and real interest rate. Label it 2. Real interest rate (percent per year) ,9 10 I i 1 I 0 I I i 1 i r I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Loanable funds (trillions of 2009 dollars) >>> Draw only the objects specied in the question. 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$n+~-+p:rp)+6i with 5,- N N (0, 0'2). Perform the following: (a) Formulate the model above in terms of matrices. Clearly write down the model in matrix form by dening the matrices Y, X , , 6. You can express your answer in terms of realizations of K, 61-. (b) Assuming that 02 is known, write down the minimization problem (in matrix form) that needs to be solved to nd an estimate of i3. (c) Argue that the calculations to nd El are identical to what we did in lecture for simple linear regression. (d) Consequently, write down the least squares estimate of [:3 in matrix form. Also write down its distri bution. 5. Consider a consumer who has preference towards specialization but that always prefers to consume positive of both goods rather than fully specialize in just one of them. Assume her preferences are represented by the following utility function: (POINTS: 18) x2+y2, ifx>0 and y>0 u(x,y)={0, ifx=0 or y=0 Note that this consumer preferences is not continuous (do not worry if you do not see that. we will work through the problem). We will show that the UMP and EMP may not generate the same outcome. Fixpx=py=1,I=$l. and u=1 (a) Argue that this agent will never choose zero quantities for x or y. in either problem (UMP or EMP), as long as I > O and u > 0. (Points: 3/18) (b) Explain why neither (1,0) nor (0,1) belong to 10(1). (Points: 3/18) 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