PRACTICE PROBLEMS 1. where X is consumption of widgets and Y is consumption of all other goods.

Transcription

1 ECONOMICS S-1010 HARVARD UNIVERSITY PRACTICE PROBLEMS 1 1. Consider a consumer who derives utility from two sources, consumption of widgets and consumption of all other goods. Her income is $1500, and her utility function is given by: U = X 2 Y where X is consumption of widgets and Y is consumption of all other goods. a) Suppose the price of widgets is $4 and the price of all other goods is $10. Use the condition for utility maximization to find the quantities of each good she will choose each month. Explain the economic reasoning for the answer you found. b) Find the equation for the consumer s demand curve for widgets when the price of all other goods is $10 and her income is $1500. c) Suppose now that the price of widgets increases to $5. What quantities of each good will she now choose? Describe this change in terms of income and substitution effects. (Be as precise as you can.) How much would the consumer be willing to pay to avoid this price increase? 2. Consider a firm that produces output according to the following production function: q = 2K 1/2/ L 1/2 where q = output, K = capital, L = labor. The firm s total cost is determined by TC = rk + wl, where r = rental cost of capital, and w = the wage rate for labor. Assume that w = $1 and r = $4. a) Find the firm s total cost function in terms of q. What is the total cost of producing 10 units of output? What is the marginal cost? b) In the short run, the firm s capital is fixed 16 units and it can sell as much output as it wants at a price of $2 per unit. How much output should the firm produce? How much labor should it employ? How much profit will it earn? c) Assume there are 50 identical firms, each producing according to the production function given above. The market is perfectly competitive, and market demand is given by: Q D = P. Find the market quantity and price in the long run equilibrium. How much output will an individual firm produce and how much capital and labor will it employ?

2 d) Suppose now demand increases to: Q D = P. Find the market quantity and price in the short run. How much output will an individual firm produce and how much capital and labor will it employ? Will a typical firm make a profit? How much? What will happen in the long run? Be as specific as possible. 3. Consider a consumer whose preferences for X and Y can be described by the utility function: U = (X + 20)(Y + 20). a) Suppose the price of X is $10, the price of Y is $2, and the consumer s weekly income is $120. Use the condition for utility maximization to find the quantities of each good she will consume. b) Suppose the price of X falls to $2 per unit. By how much does the consumer adjust her consumption of X and Y due to the substitution effect? By how much does she adjust her consumption of X and Y due to the income effect? c) Find the equation for this consumer s demand curve for X when the price of Y is $2 and her income is $120. Find the equation for her income expansion path when the price of X is $5 and the price of Y is $2. Also, find the consumer s Engel curve for X given these prices. 4. Consider a firm that produces output according to the following production function: q = K 2/3 L 1/3 Where q is output, K is capital in machine-hours and r is the hourly wage. Initially, the wage rate is $4 per labor-hour and the rental rate on capital is $64 per machine-hour. a) Find the firm s total cost function in terms of q. What will it cost to produce 100 units of output? What is the marginal cost? b) Suppose the firm s capital stock is fixed in the short run at 8 machine- hours. Find the firm s short-run total cost function in terms of q. What will it cost to produce 100 units of output? What is the firm s short-run marginal cost function? c) Suppose the firm can sell as much of its output as it wants at a constant price of $32 per unit. Find the firm s profit maximizing level of output in the short-run. How much profit will it earn in the short-run? How much profit will it earn in the long-run?

3 5. Cantabridgia has two residents, Smith and Jones. Both work for the only employer in town, Hartech, which produces education according to the following function: E = 2Y Where E is the amount of education produced (measured by the number of credits awarded) and Y is the hours per week of labor employed in education production. Smith and Jones can work up to 40 hours (per week) and their preferences for the consumption of education (E) and leisure (L) can be represented by the following utility functions: U S = E 1/3 L 2/3 U J = E 1/4 L 3/4 The wage rate is $10 per hour. Note that despite the small number of actors in this economy, all behave as price-takers. a) Find the total cost function for Hartech. How much does it cost to produce 10 units of education? b) What must the price of education be in a general competitive equilibrium in Cantabridgia? How much labor (in hours per week) will Smith and Jones supply? How much education will be produced/consumed? c) Now suppose the government of Cantabridgia requires everyone to have a minimum of 30 credits worth of education. Is this policy efficient? How do you know?

4 ECONOMICS E-1010 FALL 2005 HARVARD UNIVERSITY MICRO PRACTICE PROBLEMS -- ANSWERS 1a) A rational consumer will choose the bundle that places her on the highest indifference curve given the market prices and her income. The condition for optimization is to set her marginal rate of substitution equal to the ratio of market prices. Thus: U = X 2 Y => MRS = 2Y/X = Px/Py = 4/10 => X = 5Y Substituting in the budget constraint: I = PxX + PyY; 1500 = 4(5Y) +10(Y) => Ya = 50; Xa = 250. b) 1500 = PxX + 10(PxX/20) => X = 1000/Px. c) When the price of widgets rises to $5, Px/Py = 5/10 = 1/2. The ratio of marginal utilities (MRS) hasn t changed, so MUx/MUy = 2Y/X. Setting this equal to the new price ratio, we find X = 4Y. Substituting in the new budget constraint: I = PxX + PyY; 1500 = 5(4Y) +10(Y) => Yb = 50; Xb = 200. Because the relative price of food has increased, she will consume less food and more all other goods. This is the substitution effect. But because the consumer is now relatively poorer (her purchasing power has decreased) she will consume less of both goods. This is the income effect. In this example, the two effects unambiguously decrease her consumption of widgets, but they are equal and opposite with respect to her consumption of all other goods. We can measure the effects as follows: we must find the bundle on the original indifference curve (i.e. holding income fixed) that is tangent to a budget line with the new slope. Therefore, we are looking for a bundle on the indifference curve that includes Y = 50; X = 250, for which MRS YX = 1/2: Ua = X 2 Y = (250 2 )50 = X= 4Y => 16Y 3 = => Y = 58.0; X = Sub effect: X decreases 17.9; Y increases 8. Inc effect: X decreases 32.1; Y decreases 8. The maximum amount this consumer would be willing to pay is the income reduction that leads to an optimum along the same lower indifference curve as the price increase. This is the equivalent variation measure of the loss of utility from the price increase. To calculate this amount, we must find the minimum income the consumer needs to purchase a bundle on the indifference curve corresponding to U = (her utility with 200 units of X and 50 units of Y), given the old prices. Under these circumstances, she would purchase the bundle for which Ub = X 2 Y =

5 X= 5Y => 25Y 3 = => Y = 43.1, X = The market price of this bundle is $ Therefore, Anne is willing to pay $ a) The firm s total cost function expresses its total cost as a function of the amount of output it produces. Total cost equals the total expenditure on capital plus the total expenditure on labor, or TC = rk + wl. To find total cost in terms of q, we must substitute expressions in terms of q for K and L. We use the cost minimization condition and the production function to do this. Cost minimization requires that the firm produce using a combination of inputs for which the ratios of the marginal products, or the marginal rate of technical substitution (MRTS), equals the ratio of the input prices. q = 2K 1/2 L 1/2 MRTS KL = MP L / MP K MP L = K 1/2 L -1/2 MP K = K -1/2 L 1/2 MRTS KL = K/L = w/r = 1/4 Thus, we know that at the optimum, L = 4K. Plugging L = 4K into production function, we find q = 2K 1/2 (4K) 1/2 = 4K => K = q/4; L = q Solving for the firm s total cost as a function of q, we find TC = 4(q/4) +1(q) = 2q. Thus the firm s total cost function is TC = 2q => MC = 2. Producing 10 units costs $20. b) With capital fixed in the short run at K = 16, the firm will maximize profit by choosing a quantity of labor to employ such that MR = MC (and MRP L = w). The short run production function is: q = 2(16) 1/2 L 1/2 = 8L 1/2 => L = q 2 /64 Solving for the firm s short run total cost as a function of q: TC = 4(16) +1(q 2 /64) = 64 + (1/64)q 2 => MC = q/32 Since the firm can sell as much output as it wants at a price of $2, P = MR =2. So to maximize profit, the firm will set output, such that MC = MR = 2. Thus: MC = q/32 = 2 => q* = 64. Also, MRP = P MP L = (2)K 1/2 L -1/2 = 8L -1/2 = w = 1 => L* = 64; q* = 64; K* = K = 16. c) To solve for the long run market equilibrium, we need to construct the long run market supply curve. Since each firm has the same MC LR = 2; the supply curve is horizontal at P PC = 2. Plugging this into the demand equation, we get: Q* = 4500; P* = $2. With 50 firms in the market, we find: q* = 90; K* = 22.5; L* = 90.

6 d) In the short run, each firm s capital is fixed (K* = 22.5), so the increase in demand will cause firms to move outward along the MC curve associated with this level of capital. q = 2(22.5) 1/2 L 1/2 = (9.49)L 1/2 => L = q 2 /90 TC = 4(22.5) +1(q 2 /90) = 90 + (1/90)q 2 => MC = q/45 The market short run supply curve is the horizontal sum of (50) individual MC curves: MC = q/45 => q = 45P; Qs = n(q) = 50(45P) => Qs = 2250P. Since market demand is Qd = P, we find: P** = $2.73; Q** = q** = ; K** 22.5; L** = A typical firm s profit, π = TR TC = P(q) (rk + wl) = 2.73(122.73) [(4) (1)167.36] => π = $ In the long run, firms will enter the market until the price is driven back to P = Ppc = $2. Given the new demand curve, Q D = P, we find Q*** = The industry will employ 6500 units of labor and 1625 units of capital to produce this output, but given the constant returns 3a) To make herself as well off as possible the consumer chooses the bundle that exhausts her income and for which her marginal rate of substitution equals the ratio of the market prices, if there is an interior solution; if there is no feasible bundle that meets both conditions, the consumer optimizes at a corner of her opportunity set. Starting with the optimization principle for an interior solution, we find the bundle along the budget constraint for which the marginal rate of substitution equals the price ratio. Budget Constraint: 10X + 2Y = 120 MRS YX =P X /P Y : (Y + 20)/(X + 20) = 10/2 Solving this pair of simultaneous equations, X = -2 and Y = 70. This is not a feasible bundle because the consumer cannot choose negative quantities of a good. Therefore, she must optimize at a corner solution. In this case, she spends his entire income on Y, so that X = 0 and Y = 60. With this bundle, her MRS YX is 4, which is less than the price ratio, as required at a corner solution that includes no X. b) We follow the same procedures to find the consumer s new optimal bundle when the price of X falls to $2 per unit. Budget Constraint: 2X + 2Y = 120 MRS YX =P X /P Y : (Y + 20)/(X + 20) = 2/2

7 Solving this pair of simultaneous equations, X = 30 and Y = 30. This is a feasible interior solution. As a result of the price change, consumption of X has increased by 30 and Y has decreased by 30. To identify the substitution effect, we must find the bundle on the original indifference curve (i.e. holding real income fixed) that is tangent to a budget line with the new slope. Therefore, we are looking for a bundle on the indifference curve that includes X = 0, Y = 60, for which MRS YX = 1: Indifference Curve: (X + 20)(Y + 20) = (0 + 20)( ) = 1600 MRS YX =P X /P Y : (Y + 20)/(X + 20) = 2/2 Solving this pair of simultaneous equations, X = 20 and Y = 20. The movement along the original indifference curve from X = 0, Y = 60 to X = 20, Y = 20 is the substitution effect of the fall in the price of X. Stated differently, the substitution effect implies an increase in consumption of X of 20 and a reduction in consumption of Y of 40. The movement from X = 20, Y = 20 to X = 30, Y = 30 is the income effect; that is, the income effect of the fall in the price of X is an increase in consumption of X of 10 and an increase in consumption of Y of 10. c) We find the consumer s demand curve for X by solving the set of equations for the optimization principle for X in terms of Px, Pv, and I, eliminating Y from the pair of equations: MRS YX = (Y+20)/(X+20) = P X /P Y => 2Y = P X X + 20P X 40 P X X + P Y Y = I Using the first equation to substitute for Y in the second: P X X + P X X + 20P X 40 = I If the consumer s income is $120, her demand curve is: X = 80/P X 10. The income expansion path shows the consumer s optimal bundle at every possible level of income, given fixed prices in the market. It is the set of bundles for which her marginal rate of substitution equals the ratio of market prices. Therefore, the equation of the consumer s income expansion path is MRS YX = (Y+20)/(X+20) = P X /P Y => Y = (5/2)X The consumer s Engel curve shows the amount of X she consumes at each level of income, given the fixed market prices. To find the equation of the consumer s Engel curve given these prices, we must solve for X as a function of income (I). We do this by stating the consumer s optimization conditions with I as a variable and solve them simultaneously for X as a function of I by eliminating Y: 5X+2Y = I Y = (5/2)X + 30 => X = (I 60)/10.

8 4a) The firm s total cost function expresses its total cost as a function of the amount of output it produces. Total cost equals the total expenditure on capital plus the total expenditure on labor, or TC = rk + wl. To find total cost in terms of q, we must substitute expressions in terms of q for K and L. We use the cost minimization condition and the production function to do this. Cost minimization requires that the firm produce using a combination of inputs for which the ratios of the marginal products, or the marginal rate of technical substitution (MRTS), equals the ratio of the input prices. q = K 2/3 L 1/3 MRTS KL = MP L / MP K MP L = 1/3K 2/3 L -2/3 MP K = 2/3K -1/3 L 1/3 MRTS KL = K/2L = w/r = 4/64 Thus, we know that at the optimum, L = 8K. Plugging L = 8K into production function, we find: q = K 2/3 (8K) 1/3 = 2K => K = q/2; L = 4q Solving for the firm s total cost as a function of q, we find TC = 64(q/2) +4(4q) = 48q. Thus the firm s total cost function is TC = 48q. Producing 100 units costs $4800. Marginal cost MC = 48. b) In the short run, each firm s capital is fixed (K = 8), so the increase in demand will cause firms to move outward along the MC curve associated with this level of capital. q = K 2/3 L 1/3 = (8) 2/3 L 1/3 => q = 4L 1/3 => L = q 3 /64 Solving for the firm s short-run total cost as a function of q, we find TC: = 64(8) + 4(q 3 /64) TCsr = q 3 /16. Producing 100 units costs $ Marginal cost MCsr = 3/16q 2. c) The firm s short-run profit function is π = TR TC = Pq (512 + q 3 /16) = 32q 512 q 3 /16. To maximize profits, set the first derivative = 0 (or MR = MC) dπ/dq = 32 3/8q 3 = 0 => q* = π = 32q 512 q 3 /16 => π = [Note: This is less than fixed costs.] In the long-run, the firm will exit the market.

9 5a) TC = 5E. To produce 10 units of education costs $50. b) MRS LE S = L S /(2E S ) = P E /P L MRS LE J = L J /(3E J ) = P E /P L The price of education must equal the marginal cost of producing it if the firm is to be maximizing profit. Each credit of education requires 0.5 unit of labor. Therefore, P E = 0.5P L. => P E = = P L L S + P E E S 400 = P L L J + P E Ej L S = 26.67; L J = 30; Ej = 20; Es = 26.67; Yj =10; Ys = In answer to the question, Smith works hours, Jones works 10 hours and units of education are produced. c) Inefficient