kindly solve these for me
1. (22 points) A price-taking firm has this production function: q = L1/3 /1/3 Assume that: w =1 r=1 K = 1 a) (6 points) Calculate the Average Product of Labor when L = 8. What information does this calculated value tell you? b) (4 points) Show that the total cost function is TC = q' + 1 c) (6 points) Show that the short-run supply function is q = ( 1/2 d) (6 points) Find these functions: Average Variable Cost and Average Fixed Cost. At what value of q is Average Cost minimized?(3) Consider an exchange economy with two consumers and two goods. Suppose that consumers share the same utility function u (r,, 12) = 12, and have the following endowment bundles: el = (e], e;) = (2, 1) and e? = (e], e?) = (2,3). (a) Draw the Edgeworth box and draw the indifference curves that pass through the endowment point. (b) Characterize the contract curve. (e) Characterize the core. d) Characterize the core if consumer I has all the endowments (i.e., e = (4.4) and e' = (0.0)). e) Redo (b) to (d) with initial endowments el = (3, 1) and e? = (3,3)Question 5 (20 points): Weighted Least Squares vs Ordinary Least Squares Suppose that Assume that Eje. = 0 and that Var(e,) - Ir, Jo-, ie., we violate the constant variance assumption in linear model (a) Represent the above model using matrix notation. What is Var(e)?.3. Consider the model Y, = Bo + BIX; +; where ; ~ indNV(0, o?) for observations i = 1, . .., n. Suppose X, > 0 and o? = 02 X,. Let Q(3) = Clo. {Y, -(Bo + B1X,)} denote the weighted error sum of squares. Define Y = (Y1, . . ., Yn)': n x 1 vector of response variables X: n x 2 design matrix with I's in the first column and (X1, ..., X,,)' in the second column. B = (80, 31 )': 2 x 1 vector of regression coefficients V = diag{ X1, . ... X,}: n x n diagonal matrix with X1, .... X,, along the diagonal (a) Derive the distribution of Y including the analytical forms of the mean vector / and variance-covariance matrix E. (b) Write the weighted error sum of squares Q ( 3) in matrix terms. (c) Let 3 = (30. 3, )' denote the weighted least squares estimates of B. Derive & by minimizing Q(B). (d) Derive the mean vector and the variance-covariance matrix of 3. (e) Despite the weighted variances, derive the ordinary least squares estimates of 3 by minimizeConsider an individual making choices over two goods, x and y with prices px and py, with income I , and the utility function u(xy) = xyl/2. You already know that this yields the demand functions x* = 21 3px and y* = 1 3py (no need to calculate). (a) Find the indirect utility function, expenditure function and the compensated (Hicksian) demands for x and y. Show your work. (b) Use your expenditure function to find the compensating variation for a price increase from $2 to $3 for good x, given income | = 60, and py = 1. (c) Use your answer from part (a) to determine which of these options is best for this consumer: (i) px = 1, py = 3, 1 = 100; (11) px = 2, py = 2, 1 = 90;(iii) px = 3, py = 1, 1 = 120; (iv) px = 2, py = 1, I = 110
