Exercises: Demand theory
1. Assume that a representative consumer has the following utility function of her daily consumption of bananas,
?(? = kilos of bananas) = 5? - ?2?
(a) Plot this function in a diagram for different values of ? in the interval 0 = ? = 5? Does the consumer become satiated with bananas, and if so at which quantity?
(b) Plot the marginal utility function and find is slope for different value of ? in the appropriate interval along the ?- axis. Does the utility function exhibit increasing or decreasing marginal utility?
(c) If the price, in kronor, of one kilo of bananas, is ?? = 2? How many kilos does the consumer buy each day? What’s her total utility at this price and what is her marginal utility?
(d) Derive the consumer’s direct and indirect demand curves for bananas.
2. Assume that our consumer can choose between bananas and oranges for her daily intake of fruit; and define ? = kilos of oranges per day.We change the utility function to the Cobb-Douglas type,
?(???) = 5?2?2?
(a) Find the marginal utility functions for ? and ?? Do these functions exhibit increasing or decreasing marginal utility?
(b) Derive an expression for the marginal rate of substitution betweenbananas and oranges Plot this function in a diagram for ?(???) = 10 (solve for ? as a function of ? when you put 5?2?2 = 10?)
(c) Repeat ?) for ?(???) = 5? and ?(???) = 15? Plot in the same diagram as before.
(d) Assume that the consumer budget is ? = 6? the market price of bananas, ?? = 2? and the market price of organes, ?? = 3? Find an expression for the consumer’s budget line and plot it in the diagram. From the diagram, make a best “guess” of the optimal consumption quantity and utility level, at this income and prices.
(a) Solve the consumer’s utility maximization problem with the Lagrangemethod and derive the ordinary (Marshallian) demand functions for oranges and bananas, ?(???????)? Put in the given values of ?? ??? and ??? to find the optimal quantities and the utility level at the optimum. Also find an expression for the Lagrange multiplier and its value at the optimum. Give and interpretation of its economic meaning.
(b) Assume now that ? increases to ?0 = 7? Find the new optimal values of ?? ?? ? and ?? By how much does the optimal utility change? How does this compare with the values of the Lagrange multiplier at both optima?
(a) Find general expression for the income elasticities, ??? and ???, as well as the actual values around the initial optimum.
(b) Find general expression for the own price elasticities, ??? and ???, as well as the actual values around the initial optimum.
(c) Find general expression for the cross-price elasticity, ???, as well as the actual value around the initial optimum.
5. Let’s return to question 2 and the utility function over bananas and organges,
?(???) = 5?2?2?
(a) Assume that the price of bananas increases to ?1? = 3? ceteris paribus. Find the new optimum solution, i.e. the new quantity demanded and the utility at the optimum.
(b) Formulate and solve the consumer’s ???? ???????, i.e. to minimize expenditure for a given utility level. Use the optimum utility in 2?) as the utility restriction. Verify that the minimum expenditure to fulfill the utility constraint equal the income (budget) used as a constraint in the ?????? ???????? Verify that the utility constraint holds as an equality.
(c) Repeat 5?) but use the optimal utility level at the initial prices as the restriction.
(d) Find the necessary income compensation so that the consumer’s utility is equal at the new prices, as with the old price and old income (”Hicks’-income compensation”).
(e) Find the necessary income compensation so that the consumer canafford to buy the initial optimal quantities at the new prices (”Slutsky’s income compensation”). Compare the compensations in 5? and ?; why do they differ?
(a) Derive, from the dual problem, a general expression for the incomecompensated (Hicksian) demand function for bananas, given the utility level ?0? and prices of oranges, ??? (??(??;?0??0?)) derived in 5?)?
(b) Draw the income-compensated demand function in a regular demanddiagram (with ?? on the vertical axis), together with the ordinary
(Marshallian) demand function for bananas, (??(?0?????0?))?
(c) Calculate the slopes of the functions ( and around the initial equilibrium point (at the intersection of the two curves). What do the slopes of these curves show?
(d) Verify that the Slutsky equation holds, i.e., that,
When the price of a good changes, ceteris paribus, the consumer surplus will change. In certain situation, such as in tax reforms, the income changes as well (changes in the income tax rate of in transfers). In this case it is relevant to compute the change in the consumer’s surplus along the income compensated demand curve.
(a) Calculate the change in the income compensated consumer’s surplus(????)? as well as the ordinary (uncompensated) consumer’s surplus (????)? when ?? decreases from ?0? = 3? to ?1? = 2. Use the utility after the price changes in the compensated demand curve. Compare these surpluses and explain the difference.
(b) Now use the utility before the price change in the income compensated demand curve. Derive the change in the consumer’s surplus. Comment on the difference from the answer in 7?)? Illustrate both answers in a diagram. What are the changes in the consumer’s surpluses for the two different income compensated demand curves called? How are they related? Explain.