Solutions for Chapter 5 Calculus In‐Text Exercises Solution for Calculus In‐Text Exercise 5.4: We start by finding Natasha’s marginal rate of substitution for films with concerts. Taking the .
0.25
derivative of her utility function with respect to F, we see that 0.75
taking the derivative of her utility function with respect to C, we see that Using the fact that , we have . Similarly, .
. . To identify which bundle Natasha purchases, we now solve two equations in two unknowns (just as in Worked‐Out Problem 5.2). The first equation tells us that the solution has to satisfy the tangency condition, . We have just learned that . The problem tells us that . Therefore, we can rewrite the tangency condition as follows: Tangency Condition: The second equation tells us that the solution must lie on the budget line, which requires . The problem gives us the following information: M = 200, PC = 30, and PF = 5. Therefore, we can rewrite the equation for the budget line as follows: Budget Line: 30
5
200 Now we solve the two equations for C and F. The tangency condition implies that F = 2C. Using this expression to substitute for F in the equation for the budget line, we obtain 30C + 5(2C) = 200, or equivalently 40C = 200. It follows that C = 5. Because F = 2C, we know that F = 10. Natasha purchases 5 concert tickets and 10 movie tickets. Solution for Calculus In‐Text Exercise 5.5: Once again we start by finding Natasha’s marginal rate of substitution. Taking the derivative of her utility function with respect to F, we see that 0.4
.
derivative of her utility function with respect to C, we see that that , we have . Similarly, taking the 0.6
.
. Using the fact . With a formula for Natasha’s marginal rate of substitution in hand, we identify which bundle she chooses at any concert ticket price, PC, by proceeding just as in the solution to Worked‐Out Problem 5.5. Once again, we solve two equations in two unknowns. The first equation tells us that the solution must satisfy the tangency condition, . We know that , and the problem tells us that PF = 5. Therefore, we can rewrite the tangency condition as follows: Tangency Condition: The second equation tells us that the solution must lie on the budget line, which requires PCC + PFF = M. The problem gives us the following information: M = 250 and PF = 5. Therefore, we can rewrite the equation for the budget line as follows: Budget Line: 5
250 Now we solve the two equations for C and F. The tangency condition implies that . Using that expression to substitute for F in the equation for the budget line, we obtain 5
250, or equivalently C = 150/PC, which is the formula for Natasha’s demand curve for film tickets. Substituting the last expression in turn into the equation for the budget line, we obtain 5
250, or equivalently F = 20. So, for example, when PC = 10, Natasha buys 15 concert tickets and 20 film tickets; when PC = 25, she buys 6 concert tickets and 20 film tickets. Figure (a) below shows Natasha’s price‐consumption curve, and figure (b) shows her demand curve for concert tickets. Natasha’s total spending on concert tickets is so the fraction of her income she spends on concert tickets is therefore does not depend on the price of concert tickets. 150, . That fraction Solution for Calculus In‐Text Exercise 5.6: Once again we start by finding Natasha’s marginal rate of substitution. Taking the derivative of her utility function with respect to F, we see that 0.7
derivative of her utility function with respect to C, we see that that , we have .
. Similarly, taking the 0.3
.
. Using the fact . With a formula for Natasha’s marginal rate of substitution in hand, we identify which bundle she chooses when she earns any level of income, M, by solving two equations in two unknowns. The first equation tells us that the solution must satisfy the tangency condition, . We know that , and the problem tells us that PF = 4 and PC = 12, so 1/3. Therefore, we can rewrite the tangency condition as follows: Tangency Condition: The second equation tells us that the solution must lie on the budget line, which requires PCC + PFF = M. The problem gives us the following information: PF = 4 and PC = 12. Therefore, we can rewrite the equation for the budget line as follows: Budget Line: 12
4
Now we solve the two equations for C and F. The tangency condition implies that 7 . Using that expression to substitute for F in the equation for the budget line, we obtain 12
4 7
, or equivalently C = M/40, which is the formula for Natasha’s Engel curve for concert tickets. Substituting the last expression into the equation for the budget line, we obtain 12 40
4
. Solving that equation for F, we obtain 7 ⁄40, which is the formula for Natasha’s Engel curve for film tickets. So, for example, when Natasha earns $40, she buys one concert ticket and 7 film tickets; when she earns $120, she buys 3 concert tickets and 21 film tickets. Figure (a) below shows Natasha’s income‐consumption curve, and figure (b) shows her Engel curves for concert and film tickets.