Economics:

1. According to the postulate of Leontieff, the value ry of goods shipped from the ith sector of the economy to the jth sector is proportional to the activity level r, of the latter: my = a;joy. Also, the activity level of the ith sector is reckoned as the sum of (the values of) the output, Ti, consumed within that sector, the goods, caj; j =1, ... ,n, shipped to other sectors, and the goods, yi, consumed in final demand. Imagine a closed economy of three sectors which is characterised by the following activity levels and trade flows: 100 C11 10 30 10 V/1 50 = 200 C21 C22 ) 50 20 92 100 100 133 10 20 20 50 Construct the complete input-output table including a row for the value added to each sector by factor services, and confirm that the various ac- counting identities have been observed in the construction of the table. Calculate the matrix A = [a;] of input-output coefficients. Use the method of Gaussian elimination and the method of back-substitution to solve the equation (I - A)r = y to find the vector = = [:1, 12, x3]' of the activity levels in the three sectors when the levels of final demand are given by y = [y1, y2, y3]' = [60, 120, 60]'.a. (6 points) Bob enjoys cookies (x) according to the utility function U(x) 20x-tx-, where t is a parameter that reflects how hungry he is. Cookies are costless in Bob's world and so there is no income constraint. Using the envelope theorem, calculate how Bob's maximum utility from eating cookies varies with t. b. (8 points) Are the following utility functions quasi-concave? Show why. D) U(X, Y) = In(X) + In(Y) ii) U(X, Y) = min(X, Y) (Hint: You can use a diagram or sample values here)2. Utility Maximization (15 points) A consumer faces income constraints and has CES preferences of the following form: U(x, y) = x + yo a. (8 points) Find the consumer's demand for x as a function of prices and income. b. (4 points) Are these preferences homothetic? Explain why or why not.c. (3 points) Calculate the consumer's income elasticity of demand. 3. Slutsky Decomposition (21 points) The utility function for an individual is given by U(X, Y) = X-75y-25. Prices for the two goods are Pr and P respectively, and income is I. The uncompensated demand functions for the two goods ' and ) are: 31 X ( Px. Py, ] ) = AP and Y(Px, Py.! ) = AP a. (6 points) Derive the compensated demand function for good X, X"(Px, PY,D).b. (3 points) For a small increase in the price of X , what is the total change in the quantity demanded of X' ? Express your answer in terms of price(s) and income. c. (6 points) Using the Slutsky decomposition, calculate the substitution and income effects for the price change in part (d). Again, express your answer in terms of price(s) and income.d. (6 points) Imagine that I $1000, px-py $1. What is the maximum income tax the government could impose if they didn't want to reduce consumer utility by more than half? 4. Uncertainty (15 points) a. (5 points) Show graphically that if an individual has diminishing marginal utility of wealth, she will prefer certain income to a fair gamble. Be sure to mark all important points on the graph clearly.b. (10 points) If the individual has utility over wealth (W) = -e-AW , show that the premium that this person would pay to avoid a fair gamble of h is independent of initial wealth