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3. Consider a simplified version of the Solow model discussed in class without technological innovations and no depreciation of the capital stock (i.e. g = 6 = 0). The production function is Cobb-Douglas: Y = Ko Ll-. Given these assumptions and the fact that labor grows at the constant rate n, the rate of change of capital in levels is given by: K = sK"' (L(0) ent ) 1-a where L (0) denotes the initial endowment of labor. Show that the solution to the above first-order nonlinear differential equation is given by: K (t) = K(0) - - -1(0)1-+-1(0)1-a n(1-a)t 1-a n where K (0) denotes the initial capital stock (Hint: Define an auxiliary variable z as z (t) = K (t) -").(3) Consider an exchange economy with two consumers and two goods. Suppose that consumers share the same utility function u (71, 12) = 2172 and have the following endowment bundles: e' = (el, 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. (c) 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).Chapter 2 Question Type 1 (Only one of the following models will be given) For a simple linear regression model: Write complete model Derive least squares estimators of regression coefficients Find expected value of least squares estimator of each regression coefficient and explain whether it is unbiased or reasonable Find variance of least squares estimator of each regression coefficient Write estimated error variance formula and prove MSE is an unbiased estimator for error varianceConsider the following utility function of 2 goods, x and y: U[x,y]|= [{x 101'2 + {yli'i ]; x,yal] The prices of good x and y is 10 and 20 respectively. The income is denoted by m. a] Draw the indifference curves for the utility function and use arrows to explain in which direction utility increases or decreases. [3 points] b} Find the consumption bundle or bundles that maximizes utility for the consumer. [4 poirrts] c] Find the Engel curve for both goods. [4 points] Suppose the utility function changes and now it is: mosh-101*: + {yrlure ; was: dj Draw the indifference curves for the utility function and use arrows to explain in which direction utility increases or decreases. [3 points] e] Find the consumption bundle or bundles that maximizes utility for the consumer. [4 points] f} Find the Engel curve for both goods. [4 points] 2. Consider a firm that produces output using only labor input, E, and capital input, K, using the production technology f(E, K) = E"KB. The firm is a price taker in input markets, where w is the per unit price of labor and r is the per unit price of capital. The firms sells its output in a competitive product market at price p. c. What two conditions hold if the firm chooses the (E,K) bundle that produces output level y at minimum long run cost? d. Use the conditions in (c) to find the firm's conditional (i.e., output constant) demand functions for labor and capital. What is the wage elasticity of conditional labor demand? What is the output elasticity of conditional labor demand? e. Find the firm's long run total cost and marginal cost functions, given the conditional input demand functions that you found in (d). f. Find the firm's profit-maximizing output level, given the long run total cost function that you found in (e). What is the wage elasticity of output supply? g. Use your answers to (d) and (f) to find the firm's unconditional (i.e., not holding output constant) demand functions for labor and capital. What is the wage elasticity of unconditional labor demand? answer a-g