Macro & Micro economics

Question 1 a) Calculate the first-order partial derivatives for the following functions and evaluate them to the indicated values: 1. 3 2 3 2 z = f(x, y) = 3x - x + 3y - y ; rate fx(2,1) et fy(2,1). 2. z = f x y = x - y + x y - y x + y 3 2 2 3 ( , ) 3 2 2 ; rate fx(1,2), fx(3,4), fy(1,2) et fy(3,4). b) Calculate the second-order partial ¶ 2 z / ¶ x 2 , ¶ 2 z / ¶ y 2 , ¶ 2 z / ¶ x ¶ y and ¶ 2 z / ¶ y ¶ x the following functions and, for the function of the first sub-question, evaluate them to the indicated values: 1. z f (x, y) x x y 2xy 3 2 2 = = + + ; rate xx f (3,2), yy f (4,1), xy f (1,3), yx f (1,3). 2. z = f (x, y) = ln(2x + y). 3. 4 2 2 2 4 z = f (x, y) = 2x - 7x y + 5y - 3y . Question 2 The cost function of a company that jointly produces two outputs, x and y, is given by: ( , ) 4 5 3 525 2 2 CT = f x y = x + y - xy + . a) Calculate the first-order partial derivatives fx and fy. Interpret. b) Calculate the second order partial derivatives fxx, fyy, fxy and fyx. What information provides the sign of each of these derivatives? Question 3 Either the production function where L represents the labor input and K the capital input. Initially, the company uses L = 64 and K = 27 units of these inputs, then decides to increase the work of one unit and the capital of two units. How much will the output increase? Your answer must include an exact value and an approximate value Question 4 Are the following functions homogeneous? If yes, determine their degree of homogeneity using Euler's theorem: 1. 4 1 8 1 8 1 f (x, y) = 4x y - 2x . 2. xy x y g x y + ( , ) = . 3. 3 1 6 1 6 1 ( , , ) 6 - h x y z = x y z . 4. ( , ) 5 3 2 10 4 2 6 6 j x y = x y - x - y - . Question 5 An individual withdraws his satisfaction from the two goods he consumes x1 and x2. Its preferences are represented by the utility function ( , ) ( 2)( 1) u x1 x2 = x1 + x2 + and he faces the following budget constraint 4x1 +10x2 = 204 . a) Using the Lagrangian technique, find the quantities of goods that maximize the satisfaction of this individual, ( , ) * 2 * 1 x x b) What is the maximum utility of this individual ( * u ) Question 6 A company seeks to determine the highest level of production Q that it can achieve from its factors of production L, K and E, ie labor, capital and energy. However, the budget allocated to him for the purchase of his factors of production cannot exceed $ 60,000, which results in the following constraint: CT = 10L + 5K + 2E = 60,000$ Knowing that its technology is given by the production function: 1/ 2 Q = 2,5L 1/ 3 K 1/ 6 E . a) Find, using the Lagrangian technique, the quantities of factors that would allow it to maximize its output level (L *, K *, E *); b) What is the maximum output level (Q *)?