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Solutions for HW #3 Economics 301 Chapter 4 Problems 7. (a) P = $3 Q = 7000 Revenue = P*Q = $21,000 (b) E Q, P Q P 3 3 1000 .429 P Q 7 7000 (c) Yes, since the price elasticity of demand is below one in absolute value, they can increase revenue by increasing price (since quantity will fall by less than price rises). (d) Since substitution opportunities are greater, demand for the bridge will become more elastic. 10. First we need to determine the inverse slope of the demand curve (which will be constant since we are told that the demand curve is linear). inverse slope Q Q2 Q1 P P2 P1 P1 $400 Q1 300 P2 $600 Q2 280 Thus, Q Q2 Q1 280 300 20 1 P P2 P1 600 400 200 10 When P1 $400 , Q1 300 E Q, P Q P 1 400 .133 P Q 10 300 When P1 $600 , Q1 280 E 11. Q, P Q P 1 600 .214 P Q 10 280 P = 100 – Q If P = $30 Q = 70 Revenue = P*Q = $2100 E Q, P Q P 30 1 * .429 P Q 70 Since demand is inelastic at the current price, they can increase revenue by increasing price. 13. (a) Q = 300 P = 100 cents / $1.00 (Be careful here; even though the elasticity itself is a unit-less measure, the demand curve is based on a price measured in cents, so you need to maintain this as the unit of measurement when calculating the elasticity). Revenue = P*Q = 30,000 cents (or $300) (b) E Q, P Q P 100 15 * 5 P Q 300 (c) Since demand is elastic, he can increase revenue by reducing price. (d) Maximum revenue occurs where P 1 1 15 Q, P Q Q 15P E Substituting this into the equation for the demand curve Q = 1800 – 15P and solving yields P * 60 cents 16. (a) Complements cross-price elasticity < 0 (b) Complements cross-price elasticity < 0 (c) Substitutes cross-price elasticity > 0 Chapter 5 Problem 4. The maximum membership fee you would be willing to pay is equal to the consumer surplus that you receive from being able to rent all the movies you want at a price of $4 per rental. Rearranging the demand curve to express quantity as a function of price yields Q = 10 - (1/2)P Thus, at a price of $4, you will rent 8 videos. Since the vertical intercept for this demand curve is 20, the consumer surplus received from renting 8 videos at a price of $4 is CS = ½*(20 – 4)*8 = $64 Additional Problems 1. Since the budget line shifts inward, consumption of both goods will decline if both are normal. 2. False. By definition, an increase in price increases consumption of a Giffen good. Thus, demand curves for Giffen goods will be upward sloping. 3. True. If X is normal good, then the income effect of a price increase will reduce demand for good X. Since the substitution effect is always away from the good whose price has risen, the total effect of the price increase will reduce demand for good X. 4. False. By definition, an increase in income will reduce consumption of an inferior good. Thus, Engel curves for inferior goods will be downward sloping. 5. True. By definition, an increase in income will increase consumption of a normal good. Thus, Engel curves for normal goods will be upward sloping. 6. Please consult your class notes for the correct diagram. 7. Please consult your class notes for the correct diagram. 8. True. Giffen goods are the subset of inferior goods for which the income effect is larger than the substitution effect. 9. False. An inferior good for whom the substitution effect is larger than the income effect will not be Giffen. 10. First, we must find the optimal consumption bundles at each of the three prices. When the price of X is $2, we obtain the tangency / optimality condition Y 2 X 1 which can be rewritten as 2X = Y. Substituting this into the budget constraint 2X + Y = 36, and solving, we obtain an optimal bundle of X = 9, Y = 18. When the price of X is $4, we obtain the tangency / optimality condition Y 4 X 1 which can be rewritten as 4X = Y. Substituting this into the budget constraint 4X + Y = 36 and solving, we obtain an optimal bundle of X = 4.5, Y = 18. When the price of X is $6, we obtain the tangency / optimality condition Y 6 X 1 which can be rewritten as 6X = Y. Substituting this into the budget constraint 6X + Y = 36 and solving, we obtain an optimal bundle of X = 3, Y = 18. This leads to the following table, which can be used to plot a demand curve. 11. Price of X Quantity of X $2 $4 $6 9 units 4.5 units 3 units Solving for the vertical intercepts (choke prices) for each demand curve, we obtain Kubik: P = 50 Black: P = 100 Thus, for P greater than or equal to 100, market demand is zero. For P between 50 and 100, market demand is equal to Professor Black’s individual demand curve, and for P below 50, market demand is equal to the (horizontal) sum of both professor’s individual demand curves. Thus, market demand is given by 0 for Q 100 P for 125 (3 / 2) P for P 100 50 P 100 P 50 12. Since the income elasticity of demand for the product is negative, the projected increase in consumer incomes will decrease the demand for the product. Thus, you should order less PVC pipe. 13. This is just a thinly disguised version of the ``shipping the good apples out'' paradox that we discussed in class. Here the per-unit cost (which is independent of which good is purchased) is the cost of the babysitter. Adding this constant cost to the price of both goods reduces the relative price of the more expensive good for the Andersons, but not for the Smiths. Since both couples are sufficiently wealthy that their expenditure on movies and plays is a negligible fraction of their total expenditures, there will be no income effect and the total effect of the price differential will be determined by the substitution effect (which always causes consumers to substitute away from the good whose relative price has risen). As a result, all other things equal, the Andersons will go to relatively more plays and fewer movies than the Smiths. 14. To illustrate why this is so, consider the case where the price of good X rises while the price of good Y remains the same. Y D A U2 U0 U1 C E B X In the above diagram, budget line AB represents the prices faced by the consumer in the base year, while budget line AC represents the prices faced by the consumer in the following year (after the price of good X has risen). Recall the CPI cost of living adjustment gives consumers an income transfer that allows them to purchase their original (pre-inflation) bundle at the new (higher) prices (represented by the budget line DE in the diagram). This over compensates consumers in the sense that it allows them to achieve a higher level of utility than prior to the price increase (U2 vs. U0).