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

(a) Formulate the linear programming problem and compute the number of pants and jackets which maximises profits. (b) Find the shadow prices of the resources. Alternate Question for Pwd Candidates Consider the linear programming problem of Max c,x, + c,x, + c,x, subject to 2, 0, 1 a 2x2 + a 3*3 ≤ b, with non-negativity constraints Xy, X2, X3 ≥ 0. (a) Formulate the dual of the problem with 91, 92, and 93 denoting the dual prices associated with the 3 (b) If (x,* , x,* , x,*) is feasible in the primal problem and (9,* *, 92*, 93*) is feasible in the dual problem, and that cx,* +C_*,* + C3x3* = b,9,* + b,92* + b,93*, prove that (x, * *, x2*, x,*) solves the primal and (9,* , 92* *, 93*) solves the dual. (5+5) 3. Solve the following difference equations. In each case, discuss the stability of the equation. 4. (a) The total income of an economy (in ? billion per year) is given by where t is measured in years (0≤t≤ 5) The income share of the top 10% is s(t) = 0.5 - 0.04t. Find the total income of the bottom 90% over the 5-year period. (b) Find the general solution of the differential equation: 5. Consider the given problem: Max x' + 2y2 - x, subject to x2'+ y' ≤ 1 (a) Write down the lagrangean function and the first order conditions. (b) What is complementary slackness condition? (c) Find all pairs (x, y) that satisfy all the necessary conditions. (3+1+6) Max x2 + 2y2 - x, subject to x? + y2 ≤ 1 6. Consider a producer who generates electricity by burning a fuel. The demand for electricity varies between peak periods and off-peak periods. The peak period demand for power is given by and the off-peak period demand is given by The variable cost is 20 per unit (paid in both markets) and the capacity cost is 10 per unit, which is only paid once and is used in both periods. (a) Write down the Lagrangian and Kuhn-Tucker conditions for this problem. (b) Find the optimal outputs and capacity for this problem. (c) How much of the capacity is paid for by each market? (3+5+2) "b - 00+ = 'd i a tale te pane dlte te lge fole lete petele de (a) Write the dual problem and solve it. (b) Find the solution of the primal problem. (6+4) Alternate question for PwD candidates (a) Write the dual of the following problem: (b) What are the solutions to this problem and the dual? (c) What is the change in the criterion function of the primal problem if the constraints change to : (a) An oil company is planning to extract oil from a field, starting today at t = 0, where t is measured in years. g(t) gives the rate of flow of oil measured in barrels per year. The company sells its oil at a price per barrel given by p(t) = 1 + 1/(1 + +) Find the total revenue from this extraction project over t e [0,6]. (b) Let f' (x) = kx? + 4x where k is a constant. Find 9. Solve the given problem: subject to the constraints: 10. (a) Determine the area of the region bounded by y=2x2+10, y= 4x+16. x=0, x=1 (b) Draw the phase diagram associated with the following differential equation :