Optimization Methods for Economic Analysis — DU B.A (Prog.) Economics as Major Previous Year Question Papers

1. (a) Naveen derives satisfaction from coffee (X) and croissants (Y). His utility is: The price of coffee is Px = 40 per cup and the price of croissants is Py = 60 per piece. Naveen's monthly income is, 1 = 2000. i) Using the Lagrange multiplier method, determine Naveen's optimal consumption bundle (X, Y). (5) ii) Is the second order condition satisfied? (4) (b) Consider the time-dependent production function: Y = A(t)K(t) L(t)", where A(t), K(t) and L(t) represent the level of technology, capital stock and labour at time t. i) Find the partial derivatives ö2 and 8k (4) ii) Determine the total derivative dY/dt. (5) 2. a) (i) The relationship between utility and consumption is given implicitly by Use the Implicit-Function rule to find ox and sy (5) (i) Using the inverse-function rule, find ax for the following function: b) The supply function of an agricultural good is: Qs = a + bp? + T0.5 • a, b> 0 where Qs is the quantity supplied, P is the market price and T is temperature. (i) Determine the elasticity of supply with respect to temperature and elasticity of supply with respect to price. Also interpret the sign of each elasticity in economic terms. (5) (ii) Using derivatives, show the effect of an increase in average temperature T on the elasticities obtained in part (i). (4) id 3. a) A firm faces cost C(Q) = 03 - 15Q? + 63Q + 50 and inverse demand R P(Q) = 90 - 3Q. (i) Find the profit-maximizing output and the maximum profit. (5) (i) Also, verify the second-order condition(s) for a maximum. (4) (b) Consider an open economy where equilibrium in the goods market and money market is determined by the following equations: 1+x=S + M and M. = M where 1 Imports) is Investment expenditure, X = X, is export, S = g(Y, r) is savings, M = h(Y) is M. = M(Y, r) is the quantity demanded of money, and M s= M is money supply. Y denotes national income and r denotes the rate of interest. All functions are continuously differentiable. And the following properties hold: Here X and M are exogenous variables, while Y and r are endogenous variables. (i) Verify whether the conditions of the Implicit Function Theorem are satisfied for this system. If so, write the equilibrium identities expressing Y and r as functions of exports and imports (X, and M.). (5) (i) Interpret the economic meanings of the derivatives 9y and h'(r). (4) 4) (a) Consider a firm producing goods in two different markets. The profit function of the firm is100, + 802 - 201 2-22-9,92. (i) Find the profit maximizing output in both the markets. Also find the maximum profit of the firm. (5) (ii) Check the second order conditions for profit maximisation. (4) (b) (i) Determine whether q = 3u,? - 2u,14, + 44, 4 + 54, + 44, - 2u,uz is either positive definite or negative definite? (5) (ii) Find: lim (4) 5(a) In the market for wheat, the demand and supply functions are given by: 9D= A +BP QS = C + DP where A, B, C and Dare parameters. A,D> 0 and B,C<0.