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Homework answers / question archive / Spring 2021, Tocoian PROBLEM SET 6 (2 problems, 4 points each → 8 points total) 1

Spring 2021, Tocoian PROBLEM SET 6 (2 problems, 4 points each → 8 points total) 1. Recall Abby from question 2 in Problem Set 5, whose utility function was ?(?, ?) = 10? 2 √?. (You can use the solutions from that problem set, if useful.) Suppose she has income ? = 500 and is facing prices ?? = 10 and ?? = 80 a) How much of each good does Abby consume at these initial prices? What if the price of good x increases to ??′ = 30? Draw an optimal choice x-y diagram showing the initial and final budget line and the chosen consumption bundles. b) If it is discovered that the reason for the increase in the price of good x was illegal collusion by the producers in that market, Abby might be entitled to compensation. Discuss and calculate the different ways this amount might be computed - Naïve method (assuming no substitution) - Compensating variation - Equivalent variation - Consumer surplus (you don’t have to find the numerical value for this case – but show clearly how you would do so) 2. Now consider Bobby, whose utility function is ?(?, ?) = √? + 2? (also from PS 5) Assuming the same initial parameters and price increase as for Abby (? = 500; ?? = 10 and ?? = 80; ??′ = 30), find: a) The quantities purchased in each of the two months b) Compensating variation c) Equivalent variation d) Draw an x-y graph that allows us to identify these variables (i.e. show the original and new consumption point, as well as the hypothetical bundles we call “B” and “D” in the lecture notes) As optional exercises (don’t turn in), you can redo problem 2, for other utility functions, such as: - ?(?, ?) = ? + 3? - ?(?, ?) = 6√? + 3? - ?(?, ?) = 2√min(?, 3?) - ?(?, ?) = √? + 3√? - ?(?, ?) = ? + 3√? - ?(?, ?) = ? + 3? 2 1 Spring 2021, Tocoian PROBLEM SET 5 There are 4 problems, each worth 3 points → 12 points total. 1. a) If x and y are both normal goods, indicate the likely sign of each term in the table below by entering a “+” or “−“ sign. Some of the cells are already filled out. substitutes Δ? ? ?? ↑ (Δ?? > 0) ?? ↓ (Δ?? < 0) − Δ? ? Δ? ∗ Δ? ∗ −? ( ) Δ?? Δ?? Δ? − − Δ? ∗ − Δ? ? Δ? ? Δ?? + + − −? ( Δ? ∗ Δ? ∗ ) Δ?? Δ? − Δ? ∗ complements Δ? ∗ Δ?? + Δ? ∗ − − b) Do the same below, this time if ? is an inferior good (ignore the Giffen goods case, for simplicity) substitutes Δ? ? Δ? ? Δ? ∗ Δ? ∗ −? ( ) Δ?? Δ?? Δ? Δ? ∗ Δ? ? Δ? ? Δ?? −? ( Δ? ∗ Δ? ∗ ) Δ?? Δ? Δ? ∗ complements Δ? ∗ Δ?? Δ? ∗ ?? ↑ (Δ?? > 0) ?? ↓ (Δ?? < 0) c) What if ? is an inferior good? substitutes? Δ? ? Δ? ? Δ? ∗ Δ? ∗ −? ( ) Δ?? Δ?? Δ? Δ? ∗ Δ? ? ?? ↑ (Δ?? > 0) ?? ↓ (Δ?? < 0) 1 Δ? ? Δ?? −? ( Δ? ∗ Δ? ∗ ) Δ?? Δ? Δ? ∗ complements? Δ? ∗ Δ?? Δ? ∗ Spring 2021, Tocoian 2. Consider a consumer (Abby), whose preferences are described by utility ?(?, ?) = 10? 2 √? a) Solve the utility maximization problem to find the ordinary demand functions and the indirect utility function. How much of the budget is spent on each good? b) Solve the expenditure minimization problem to find the compensated demand functions ? ? (?? , ?? , ?) and ? ? (?? , ?? , ?). c) Show that the elasticities-based Slutsky equation holds for good x, in both the own-price and the cross-price formulations: ??,?? = ?? ?,?? − ?? ??,? ??,?? = ?? ?,?? − ?? ??,? where ?? is the share of the budget spent on x, and ?? is the share spent on y. 3. Another consumer (Bobby) has utility ?(?, ?) = √? + 2? (you also saw this function in PS 4) a) Look up the ordinary (“Marshallian”) demand functions you found in Problem Set 4 (Q6), then solve the expenditure minimization problem to find the compensated (“Hicksian”) demand functions as well. b) At this part, you can assume that both goods are being consumed. Show that the own-price Slutsky equation holds for good y, by using the calculus-based formula ?? ∗ ??? ?? ? = ?? − ? ? ?? ∗ ?? ) 4. Suppose Cory has utility ? = √? + √?, and is facing prices ?? = 100, ?? = 300. a) Solve for the compensated demand functions ? ? (?? , ?? , ?) and ? ? (?? , ?? , ?). (Remember to check first for monotonicity and convexity.) b) Find the expenditure function ?(?? , ?? , ?) and check whether it increases (and by how much) if both prices increase by 40%. If only the price of good x increases by 40%, while the price of y remains unchanged, by what percentage does expenditure increase in order to remain at the same utility level? c) It turns out that initially the consumer was spending ¾ of the budget on good x and ¼ on good y (you should verify this by solving the utility maximization problem, but don’t need to turn that in). If we had assumed that the bundle consumed doesn’t change, by how much would we think the consumer’s income has to increase, in order to compensate them for the increase in price? 2

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