Intermediate Mathematical Methods for Economics — DU B.A. (Hons.) Economics Previous Year Question Papers

1. (a) (i) Define the rank of a matrix. (ii) Find the rank of the following matrix for various values of p and q: (b) The equation g(x-y) = 10 implicitly represents y as a function of x. Calculate the first and second derivatives of y with respect to x. (C) Let f(x, y) = 2 + 2x + 2y - x2 - y2 be defined on the domain D, a triangular region in the first quadrant bounded by the lines =0, y= 0 andy = 9-x. (i) Draw the domain D of f (x, y) (ii) Is the set closed, bounded, compact and convex? Give reasons for each. (iii) Find the absolute maximum and minimum values of the function f (x, y) over the domain D. 2. (a) If P is an n x n matrix whose column vectors are all of length 1 and are mutually orthogonal, then prove that P is an orthogonal matrix. (b) The function z = f(x, y) is homogeneous of degree k. (i) Check if the functions ox - and - - are homogeneous (ii) Find the degree of homogeneity for the function 8x. (c) Consider a competitive firm with the production function Q = f (K,L) = Lž + Kž where Q is output, L is labour, and K is capital. Suppose the price of the product is p, the wage rate is w, and the rental rate of capital is r. Find the profit-maximising labour and capital for the firm. Check the second-order sufficient conditions. 193-13) a - ( cambo expressed as a linear combination of the set {a1, 92}? (b) Demand functions for two commodities X and Y are: x= Px-Py and y = 10 + 2px -Py, where x and y are quantities and px and Py are prices of the two goods. (i) Calculate the partial elasticities of demand for the two commodities with respect to the price of the other commodity. (ii) Determine whether the two commodities are complementary goods. (c) Classify the stationary points of the function f(x, y, z, u) = 20y + 48z + 6u + (a) Let A = (G) such that xy ≥ 0} denote the union of the first and third quadrants in the xy-plane. (i) If u is a vector in 1 and 2 is any scalar, is du in 1? (il) Provide an example of two vectors u and v in A such that u + v is not in 1. Consider the following system of two equations: What conditions are required so that y and z can be expressed as functions of x near a point P = (xo, Yo, Zo) that satisfies the two equations? (c) Graph the following sets in the X-Y plane and comment whether they are convex. 5. (a) For the following system of linear equations, answer the questions that follow: If p and q lie between 0 and 1, how many solutions does this system have, and why? Ignoring the condition that p and q lie between 0 and 1, find values of p and q so that this system has no solution. (b) A utility function is given by the function U = f(x, y) = -. Calculate the elasticity of substitution of y for x. Let U be a function of many variables defined on a convex set S and let g be a function of a single variable defined on a set that contains the range of U. Suppose that U is quasi-concave and g is non-decreasing. Show that the function f defined on S given by f (x) = g (U(x)) is quasi-concave.