slove each question step by step in paper hand wrritenn problem 1-10 show akk the work and be oragnize thank you

Course: ECO10002 Course: ECO10002 Instructor: Marian Melnyk Instructor: Marian Melnyk Problem Set 10 Problem 4 Due date: 12/13/2023 (11:59 PM) Let Y = F(K, L) = 144K /21 / be the output of a production process when K > 0 units of capital and L > 0 units of labour are used as inputs. Problem 1 (a) Compute F/(100, 64). What is the economic interpretation of this partial derivative? Find the elasticities of the functions given by the following formulas: (b) If capital input increases from K = 100 to K = 101 and labour input is fixed at 64, what is the approxi- (a) 3x-3 (b) -100x 100 (c) Vx mate increase in output? What is the exact increase? ELASTICITY Problem 5 If f is differentiable at x and f(x) # 0, the elasticity off w.r.t. x is Find the degree of homogeneity (if the function is homogeneous) for El, f (x) = f(x) (7.7.3) (a) f(x1, x2) = 17'x3+ 3xx72 ( b ) F ( K, L ) = 100 V X3 L (c) f (x,y) = xayB + xBy Problem 2 Problem 6 Let f(x,y) = >y-y. A firm produces Q = K1/2L /4 units of its output good when it uses K units capital and L units of labor. The (a) Compute f(2, 3), f(3,2), f(a, 2a), and f(x th, y) - f(x,y). firm sells its output at the price of $16 per unit of output; it pays $4 per unit of capital and $2 per unit of (b) Compute f (2x, 2y) - 8f(x, y). labor. (c) Compute f(tx, ty) - 13 f(x, y). (a) Is Q homogeneous in K and L? (b) Find the firm's profit. Problem 3 (c) Find the values of K and L that maximize profit. Let Y = F(K,L) = 100(;K-1/2 + ;L-1/2)-2 denote the production function specifying the number of units Problem 7 of output that can be produced using positive inputs of K units of capital and L of labour. (a) Compute F (1, 1), F(4,9), and F(16, 25). A function f of two variables is given for all x and y by (b) Show that F(2K, 2L) = 2F(K, L), and give an economic interpretation. f(x, y) = 5x2 -2xy+ 2y2 - 4x- 10y+5 (c) Show in general that F(IK,IL) = tF(K, L) for all t > 0. (a) Find the first- and second-order partial derivatives of f. (b) Find the critical points of f, and classify them as maxima or minima, or as saddle points.+ EIO Problem 8 A firm uses K units of capital and L units of labour to produce Q units of a commodity, where Q : 5K + 31. + 2K1. Each unit of capital costs 20 euros; each unit of labour costs 4 euros. (a) Find the values of K and L that maximize the number of units produced subject to the constraint that total expenses of the two factors of production is 140 euros. Use the insertion method. Problem 9 A rm produces two commodities. A and B. The inverse demand functions are pa = 900 2x 2y and pg 2 I400 2x 4y respectively, where the rm produces and sells x units of commodity A and y units of commodity B. Its costs are given by CA :7000+100.i+x2 and CB 210000+6y2 (a) Show that the rm's total prot is given by f5(x.y)* 3x2 10y2 4x3; l 800x 1 1400y 17000 (b) Find the values of .r and y that maximize prots. (c) Suppose the rm is required to produce a total of exactly 60 units. Find the values of x and y that maximize prots. Problem 10 (a) Compute the rst- and second-order partial derivatives of the function dened by x. y) = x3 + 2x}: 7 5x i y2 (b) Find the critical points and classify them