econmics 4589

(d) [10 pts.] How much extra income must Barb be given in order to compensate her for the increase in the price of good x in part (c)? Explain. Exercise #3. Consider the following statements and say whether they are true or false and why. To get credit you should provide a clear justification for your answers. (a) [4 pts.] If two goods are perfect complements and the price of one of them increases, the quantity demanded of both goods decreases. (b) [4 pts.] A non-transitive preference relation > can be represented by some utility function. (e) [4 pts.] Consider two goods x and y. If preferences are strictly convex, the absolute value of the marginal rate of substitution between x and y is decreasing along an indifference curve as x increases. (d) [4 pts.] The following Cobb-Douglas utility functions represent two different preference relations: U1 (x.y) = 0.3 log(x) + 0.6 log(y) U2 (x,y) = 0.6 log(x) + 1.2 log(y)- (e) [4 pts.] If a consumer is making an optimal choice between two goods x and y, then, independently of his preferences, the following condition must always hold: Py Px = MRS (x.y) . (f) [4 pts.] If the following condition holds Px = MRS (x.y) Py then a consumer must be making the optimal choice between x and y, independently of his preferences. (g) [4 pts.] A cigar is a luxury good for a consumer that has Cobb-Douglas preferences over cigars and food. (h) [4 pts.] Consider two goods x and y, with prices px and Py, respectively. A 0.07 percent value tax on these two goods does not affect the relative price of x in terms of y. (i) [4 pts.] The marginal rate of substitution measures the rate at which the market is willing to substitute one good for the other. (i) [4 pts.] An indifference curve represents the collection of all the bundles that a consumer can buy.Exercise #1. A firm produces output using the technology y = 7 1.000 where capital, K, is measured in machine-hours, labor, L, is measured in person-hours, and y denotes the yearly output. The hourly wage rate un = 10, and the hourly rental rate of capital is Wk = 20. (a) Show that this technology displays increasing returns to scale. (b) Compute the marginal products of labor and capital. (c) Suppose that at the end of 1999 the firm has signed a contract to rent K = 1,000 machine hours over the course of the year 2000. Derive the firm's short run cost function in the year 2000. (d) What is the firm's short run marginal cost function? What is the firm's short run average cost function? At which point do these two curves intersect? (e) On a diagram plot the firm's short run average and marginal cost curves. Exercise #2. The aggregate labor supply in the state of Bahnanas is Ls = 2,000ur, where w is the hourly wage, measured in Bahnanas $ and L, denotes the number of person- hours supplied by the workers of Bahananas in a year. The aggregate labor demand in the state of Bahananas is La = 12,000 - 2,000w. where La denotes the number of person-hours demanded by the firms of Bahnanas in a given year. (a) Compute the equilibrium hourly wage and the amount of hours worked in a given year. (b) The government of Bahnanas introduces a minimum wage law requiring firms to pay an hourly wage not lower than $4. Compute the equilibrium number of hours worked by the workers of Bahnanas. (c) On a diagram that has the wage rate on the y-axis and the number of hours on the x-axis, plot the labor demand and supply functions and the equilibrium prices and quantities that you found in points (a) and (b). (d) Compute the deadweight loss induced by this policy, and show it on the diagram of point (c). Exercise #3. Short questions. (a) Compute the price and income elasticities of the following demand function Im Id (p.m) = 2p (b) A firm produces output, denoted by y, using the following production function y = 10VL, where L represents the labor input, measured in person-hours. The unit price of output is $10. Let w denote the hourly wage rate. Compute and plot on a diagram the inverse labor demand curve for the firm. (c) Explain how it can be possible that as tax rates decrease, government tax revenues increase (i.e. the Laffer curve).5. Consider the indirect utility function given by (a) What are the demand functions (b) What is the expenditure function? (c) What is the direct utility function? 6. "Consider the utility function: w(x, ) = min[2n + r r + 2r) (a) Draw the indiference curve for u(x,, 4;) = 20. Shade the area where u(x, , 1,) 2 20 (b) For what values of " will the unique optimum be s: = 0 (c) For what values of " will the unique optimum be as = 0 (d) If neither ss and so is equal to zero, and the optimum is unique, what must be the value of 7. Assume that there is a consumer with weakly monotonic, convex preferences and who is a utility maximizer. For each of the following pairs of bundles, specify if bundle 1 is & 3, or uncomparable to bundle 2. (a) Suppose you have no data: 1. Bundle 1: r =3, r, =3, Bundle 2: n =6,n =25 2. Bundle 1: 21 =3,27 =3, Bundle 2: n = 25,21 =2.5 (b) Suppose that you observe that when ph = 1, pr = 1, m = 10 the consumer chooses 1. Bundle 1: 3, =4, x, = 1, Bundle 2: x1 =3, x, =6 2 Bundle 1 x1 = 6, r = 4, Bundle 2 x =1, x =8 (c) Suppose that we have two observations When p, = 1, p. = 1, m = 10 the consumer chooses s = 2,n = 8. When pi = 1, pr = 3, m = 15 the consumer chooses x, = 15,r, =0 1. Bundle 1: x, =5, ry = 2, Bundle 2: x, =.0,17 =2.5 2 Bundle 1: x, =5, r, = 2, Bundle 2: * =65,n =0