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Econ 4110/6110: Homework 1
Math Primer
1. Compute the slope between the following pairs of points where the X value is written
first and the Y value second as (X 1 , Y 1 ) and (X 2 , Y 2 ):
a. (1, 1) and (5, 5)
b. (2, 3) and (4, 9)
a. Slope = 1
b. Slope = 3
2. Compute the derivative of the following functions:
a. Y(X) = 25
b. Y(X) = 15 + 5X
c. Y(X) = 2X^3 – 10X^2 + 5X + 12
d. TR(Q) = 100Q – 10Q^2
e. TC(Q) = 25 + 5Q + .5Q^2
a. Y'(X) = 0
b. Y'(X) = 5
c. Y'(X) = 6X^2 – 20X + 5
d. TR'(Q) = 100 – 20Q
e. TC'(Q) = 5 + Q
Chapter 2
1. Suppose the demand curve for a product is given by Q = 300 – 2P + 4I, where I is average income measured in thousands of dollars. The supply curve is Q = 3P – 50.
a. If I = 25, find the market clearing price and quantity for the product. Given I = 25, the demand curve becomes Q = 300 2P + 4(25), or Q = 400 2P. Setting demand equal to supply we can solve for P and then Q: 400 2P = 3P 50 P = 90 Q = 220. b. If I = 50, find the market clearing price and quantity for the product. Given I = 50, the demand curve becomes Q = 300 2P + 4(50), or Q = 500 2P. Setting demand equal to supply we can solve for P and then Q: 500 2P = 3P 50 P = 110 Q = 280. c. Draw a graph to illustrate your answers. It is easier to draw the demand and supply curves if you first solve for the inverse demand and supply functions, i.e., solve the functions for P. Demand in part (a) is P = 200 0.5Q and supply is P = 16.67 + 0.333Q. These are shown on the graph as Da and S. Equilibrium price and quantity are found at the intersection of these demand and supply curves. When the income level increases in part (b), the demand curve shifts up and to the right. Inverse demand is P = 250 0.5Q and is labeled Db. The intersection of the new demand curve and original supply curve is the new equilibrium point.
4. A vegetable fiber is traded in a competitive world market, and the world price is $9 per pound. Unlimited quantities are available for import into the United States at this price. The U.S. domestic supply and demand for various price levels are shown as follows: U.S. Supply U.S. Demand Price (Million Lbs.) (Million Lbs.) 3 2 34 6 4 28 9 6 22 12 8 16 15 10 10 18 12 4 a. What is the equation for demand? What is the equation for supply? Price Quantity 90 110 200 250 220 280 450 500 16.67 Da Db S
$5000. She also observes that as the gas-mileage index rises by one unit, the price of the car increases by $2500. a. Illustrate the various combinations of style (S) and gas mileage (G) that Brenda could select with her $25,000 budget. Place gas mileage on the horizontal axis. For every $5000 she spends on style the index rises by one so the most she can achieve is a car with a style index of 5. For every $2500 she spends on gas mileage, the index rises by one so the most she can achieve is a car with a gas-mileage index of 10. The slope of her budget line is therefore 1/2 as shown by the dashed line in the diagram for part (b). b. Suppose Brenda’s preferences are such that she always receives three times as much satisfaction from an extra unit of styling as she does from gas mileage. What type of car will Brenda choose? If Brenda always receives three times as much satisfaction from an extra unit of styling as she does from an extra unit of gas mileage, then she is willing to trade one unit of styling for three units of gas mileage and still maintain the same level of satisfaction. Her indifference curves are straight lines with slopes of 1/3. Two are shown in the graph as solid lines. Since her MRS is a constant 1/3 and the slope of her budget line is 1/2, Brenda will choose all styling. You can also compute the marginal utility per dollar for styling and gas mileage and note that the MU/P for styling is always greater, so there is a corner solution. Two indifference curves are shown on the graph as solid lines. The higher one starts with styling of 5 on the vertical axis. Moving down the indifference curve, Brenda gives up one unit of styling for every 3 additional units of gas mileage, so this indifference curve intersects the gas mileage axis at 15. The other indifference curve goes from 3.33 units of styling to 10 of gas mileage. Brenda reaches the highest indifference curve when she chooses all styling and no gas mileage. c. Suppose that Brenda’s marginal rate of substitution (of gas mileage for styling) is equal to S/(4G). What value of each index would she like to have in her car? To find the optimal value of each index, set MRS equal to the price ratio of 1/ and cross multiply to get S = 2G. Now substitute into the budget constraint, 5000S + 2500G = 25,000, to get G = 2 and S = 4. d. Suppose that Brenda’s marginal rate of substitution (of gas mileage for styling) is equal to (3S)/G. What value of each index would she like to have in her car? Now set her new MRS equal to the price ratio of 1/2 and cross multiply to get G = 6S. Now substitute into the budget constraint, 5000S + 2500G = 25,000, to get G = 7.5 and S = 1.25. Styling 10 Gas Mileage
5 15 Optimal bundle
15. Jane receives utility from days spent traveling on vacation domestically (D) and days spent traveling on vacation in a foreign country (F), as given by the utility function U(D,F) = 10DF. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4000. a. Illustrate the indifference curve associated with a utility of 800 and the indifference curve associated with a utility of 1200. The indifference curve with a utility of 800 has the equation 10DF = 800, or D = 80/F. To plot it, find combinations of D and F that satisfy the equation (such as D = 8 and F = 10). Draw a smooth curve through the points to plot the indifference curve, which is the lower of the two on the graph to the right. The indifference curve with a utility of 1200 has the equation 10DF = 1200, or D = 120/ F. Find combinations of D and F that satisfy this equation and plot the indifference curve, which is the upper curve on the graph. b. Graph Jane’s budget line on the same graph. If Jane spends all of her budget on domestic travel she can afford 40 days. If she spends all of her budget on foreign travel she can afford 10 days. Her budget line is 100D + 400F = 4000, or D = 40 4F. This straight line is plotted in the graph above. c. Can Jane afford any of the bundles that give her a utility of 800? What about a utility of 1200? Jane can afford some of the bundles that give her a utility of 800 because part of the U = 800 indifference curve lies below the budget line. She cannot afford any of the bundles that give her a utility of 1200 as this indifference curve lies entirely above the budget line. d. Find Jane’s utility maximizing choice of days spent traveling domestically and days spent in a foreign country. 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
a. Suppose you are given the following information about the choices that Bill makes over a three-week period:
x 1 x 2 P 1 P 2 I
Week 1 10 20 2 1 40
Week 2 7 19 3 1 40
Week 3 8 31 3 1 55
Did Bill’s utility increase or decrease between week 1 and week 2? Between week 1 and week 3? Explain using a graph to support your answer. Bill’s utility fell between weeks 1 and 2 because he consumed less of both goods in week 2. Between weeks 1 and 2 the price of good 1 rose and his income remained constant. The budget line pivoted inward and he moved from U 1 to a lower indifference curve, U 2 , as shown in the diagram. Between week 1 and week 3 his utility rose. The increase in income more than compensated him for the rise in the price of good 1. Since the price of good 1 rose by $1, he would need an extra $10 to afford the same bundle of goods he chose in week 1. This can be found by multiplying week 1 quantities times week 2 prices. However, his income went up by $15, so his budget line shifted out beyond his week 1 bundle. Therefore, his original bundle lies within his new budget set as shown in the diagram, and his new week 3 bundle is on the higher indifference curve U 3. b. Now consider the following information about the choices that Mary makes:
x 1 x 2 P 1 P 2 I
Week 1 10 20 2 1 40
Week 2 6 14 2 2 40
Week 3 20 10 2 2 60
Did Mary’s utility increase or decrease between week 1 and week 3? Does Mary consider both goods to be normal goods? Explain. Mary’s utility went up. To afford the week 1 bundle at the new prices, she would need an extra $20, which is exactly what happened to her income. However, since she could have chosen the original bundle at the new prices and income but did not, she must have found a bundle that left her slightly better off. In the graph to the right, the week 1 bundle is at the point where the week 1 budget line is tangent to indifference curve U 1 , which is also the intersection of the week 1 and week 3 budget lines. The week 3 bundle is somewhere on the week 3 budget line that lies above the week 1 indifference week 1 bundle week 3 bundle Good 1 Good 2 week 2 bundle U 1 U 2 U 3 week 1 bundle week 3 bundle Good 1 Good 2 U 1 U 3
curve. This bundle will be on a higher indifference curve, U 3 in the graph, and hence Mary’s utility increased. A good is normal if more is chosen when income increases. Good 1 is normal because Mary consumed more of it when her income increased (and prices remained constant) between weeks 2 and 3. Good 2 is not normal, however, because when Mary’s income increased from week 2 to week 3 (holding prices the same), she consumed less of good 2. Thus good 2 in an inferior good for Mary. c. Finally, examine the following information about Jane’s choices:
x 1 x 2 P 1 P 2 I
Week 1 12 24 2 1 48 Week 2 16 32 1 1 48 Week 3 12 24 1 1 36 Draw a budget line-indifference curve graph that illustrates Jane’s three chosen bundles. What can you say about Jane’s preferences in this case? Identify the income and substitution effects that result from a change in the price of good X 1****. In week 2, the price of good 1 drops, Jane’s budget line pivots outward and she consumes more of both goods. In week 3 the prices remain at the new levels, but Jane’s income is reduced. This leads to a parallel leftward shift of her budget line and causes Jane to consume less of both goods. Notice that Jane always consumes the two goods in a fixed 1:2 ratio. This means that Jane views the two goods as perfect complements, and her indifference curves are L-shaped. Intuitively if the two goods are complements, there is no reason to substitute one for the other during a price change, because they have to be consumed in a set ratio. Thus the substitution effect is zero. When the price ratio changes and utility is kept at the same level (as happens between weeks 1 and 3), Jane chooses the same bundle (12, 24), so the substitution effect is zero. The income effect can be deduced from the changes between weeks 1 and 2 and also between weeks 2 and 3. Between weeks 2 and 3 the only change is the $12 drop in income. This causes Jane to buy 4 fewer units of good 1 and 8 less units of good 2. Because prices did not change, this is purely an income effect. Between weeks 1 and 2, the price of good 1 decreased by $1 and income remained the same. Since Jane bought 12 units of good 1 in week 1, the drop in price increased her purchasing power by ($1)(12) = $12. As a result of this $12 increase in real income, Jane bought 4 more units of good 1 and 8 more of good 2. We know there is no substitution effect, so these changes are due solely to the income effect, which is the same (but in the opposite direction) as we observed between weeks 1 and 2.
7. The director of a theater company in a small college town is considering changing the way he prices tickets. He has hired an economic consulting firm to estimate the demand for tickets. The firm has classified people who go the Good 1 Good 2 (^) week 1 and 3 bundle week 2 bundle
gp
5 P
Q
5 P Q 500 5 P
P 50
Q 250.
For the students: s
4 P
Q
4 P Q 200 4 P
P 25
Q 100.
These prices generate a larger total revenue than the $35 price. When price is $35, revenue is (35)(Qgp + Qs) = (35)(325 + 60) = $13,475. With the separate prices, revenue is PgpQgp + PsQs = (50)(250) + (25)(100) = $15,000, which is an increase of $1525, or 11.3%.
Chapter 5
1. Consider a lottery with three possible outcomes: $125 will be received with probability. $100 will be received with probability. $50 will be received with probability. a. What is the expected value of the lottery? The expected value, EV , of the lottery is equal to the sum of the returns weighted by their probabilities: EV = (0.2)($125) + (0.3)($100) + (0.5)($50) = $80. b. What is the variance of the outcomes? The variance, ^2 , is the sum of the squared deviations from the mean, $80, weighted by their probabilities: ^2 = (0.2)(125 - 80)^2 + (0.3)(100 - 80)^2 + (0.5)(50 - 80)^2 = $975.
c. What would a risk-neutral person pay to play the lottery? A risk-neutral person would pay the expected value of the lottery: $80.
6. Suppose that Natasha’s utility function is given by
u ( I ) 10 I , where I
represents annual income in thousands of dollars. a. Is Natasha risk loving, risk neutral, or risk averse? Explain. Natasha is risk averse. To show this, assume that she has $10,000 and is offered a gamble of a $1000 gain with 50 percent probability and a $1000 loss with 50 percent probability. Her utility of $10,000 is u (10) = √^10 (^10 )^ = 10. Her expected utility with the gamble is: EU = (0.5) √^10 (^11 )^ + (0.5) √^10 (^9 )^ = 9.987 < 10. She would avoid the gamble. If she were risk neutral, she would be indifferent between the $10,000 and the gamble, and if she were risk loving, she would prefer the gamble. You can also see that she is risk averse by noting that the square root function increases at a decreasing rate (the second derivative is negative), implying diminishing marginal utility. b. Suppose that Natasha is currently earning an income of $40,000 ( I = 40) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a .6 probability of earning $44,000 and a. probability of earning $33,000. Should she take the new job? The utility of her current salary is √^10 (^40 )^ = 20. The expected utility of the new job’s salary is EU = (0.6) √^10 (^44 )^ + (0.4) √^10 (^33 )^ = 19.85, which is less than 20. Therefore, she should not take the job. You can also determine that Natasha should reject the job by noting that the expected value of the new job is only $39,600, which is less than her current salary. Since she is risk averse, she should never accept a risky salary with a lower expected value than her current certain salary.
c. In (b), would Natasha be willing to buy insurance to protect against the variable
income associated with the new job? If so, how much would she be willing to pay
for that insurance? ( Hint : What is the risk premium?)
This question assumes that Natasha takes the new job (for some unexplained reason). Her expected salary is 0.6(44,000) + 0.4(33,000) = $39,600. The risk premium is the amount Natasha would be willing to pay so that she receives the expected salary for certain rather than the risky salary in her new job. In part (b) we determined that her new job has an expected utility of 19.85. We need to find the certain salary that gives Natasha the same utility of 19.85, so we want to find I such that u(I) = 19.85. Using her utility function, we want to
