Take Home Exam #2 - Answer Key. ECON 500 Summer 2004.

Transcription

1 Take Home Exam # - Answer Key. ECO 500 Summer 004. ) While standing in line at your favourite movie theatre, you hear someone behind you say: like popcorn, but m not buying any because it isn t worth the high price. Letting x = (buckets of popcorn) and x = (dollars spent on everything else), graphically illustrate the solution to this individual s utility maximization problem. (8 points) x X x 0

2 ) Consider a consumer with U ( X ) = x + x. Suppose the price of commodity one is p = 4, the price of commodity two is p =, and the income of the consumer is = 80. a. Rank the following bundles in order of preference, from least preferred to most preferred : A = (4,0), B = (5,3), and C = (6,6). (6 points) U ( A) = =, U ( B) = = 8, and U ( C) = = 0. This implies A C B. b. Which of the bundles from part (a) are affordable? Explain. (6 points) Given p = 4 and p =, the expenditure necessary to purchase a bundle X = ( x, x ) is E X = 4x + x. A particular bundle X is affordable if and only if E X = 80. oting that E A = 6 < 80, E B = 03 > 80, and E C = 70 < 80, we have that A = (4,0) and C = (6,6) are affordable, while B = (5,3) is not. 3) Consider a utility maximizing individual with monotonic preferences who allocates income between two commodities. Her optimal consumption of 3 commodity one is given by the function x ( p, p =. 4 p a. s commodity one a normal good or an inferior good? Explain. (6 points) 3 As is increased, x ( p, p = increases. Since consumption of 4 p commodity one increases as income is increased, commodity one is a normal good. b. s commodity two an ordinary good or a Giffen good? Explain. (6 points) A consumer with monotonic preferences will spend all income when maximizing utility (that is, x and x must satisfy p x + px = ). As a 3 result, since x ( p, p =, it follows that x ( p, p =. 4 p 4 p From here, it is clear that x ( p, p = decreases as p is 4 p increased, implying that commodity two is an ordinary good.

3 4) oel and his brother Liam enjoy going to the pub and consuming x =(cigarettes) and x =(alcohol). Liam s utility is given by U L ( X ) = 4xx, while oel s utility is given by U ( X ) = [ 544 ( 0 x ) ( x ) ]. As a result, Liam s marginal utility for cigarettes is MU L ( X ) = 4x and for alcohol is MU L ( X ) = 4x. Finally, oel s marginal utility for cigarettes is MU ( X ) = ( 0 x ) and for alcohol is MU ( X ) = ( x ). a. Determine Liam s Marginal Rate of Substitution (of cigarettes for alcohol), denoting this function by MRS,. (4 points) MRS, MU 4x x L = = =. MU L 4x x b. Does Liam have convex preferences? Clearly explain. (4 points) Yes. As x is increased and x is decreased, x MRS, = becomes x smaller, indicating that Liam's Marginal Rate of Substitution is "Diminishing." Therefore, his preferences are "convex." c. Derive Liam s Demand function for cigarettes and Demand function for alcohol, clearly showing all work. (8 points) The two conditions that Liam's optimal consumption bundle must x p simultaneously satisfy are: MRS, = = and p x + px = x p Solving this system of two equations with two unknowns, we have x ( p, p = and x ( p, p =. p p. d. Are oel s preferences complete? Clearly explain. (4 points) [ ] Yes. The function ( X ) 544 ( 0 x ) ( x ) U = can be evaluated at any consumption bundle X = ( x, x ) with x 0 and x 0. After doing so for any two bundles Y = ( y, y ) and Z = ( z, z ), either U ( Y ) > U ( Z) (in which case Y Z ), U ( Y ) < U ( Z) (in which case Y Z ), or U ( Y ) = U ( Z) (in which case Y ~ Z ). Since "any two bundles can be compared," the preferences are complete.

4 e. Are oel s preferences monotonic? Clearly explain. (4 points) o. MU ( X ) ( 0 x ) ( X ) ( x ) = is negative for x 0 and MU = is negative for x >. Over these ranges, increased consumption leads to a lower level of utility, indicating that oel is actually worse off. > f. Suppose a pack of twenty cigarettes costs $5 (so that the per unit price of cigarettes is p =. 5) and that each drink costs $4 (that is, each unit of alcohol costs p = 4 ). f oel goes to the pub with $75, how many units of x and how many units of x will he consume? Clearly explain. (6 points) From part (e) it follows that X = (0,) is "oel's overall best bundle," in that U ( X ) > U ( X ) for all X X. As a result, if X = (0,) is affordable, then oel will maximize his utility by purchasing X = (0,). At the given prices, X = (0,) is affordable (since () ( 4) = < ). Therefore, oel will optimally consume 0 cigarettes and drinks. 5) Richard has utility of ( X ) min{ x x } x = and p + p U =,, and as a result optimally consumes x =. Suppose that initially Richard has income of p + p = = 00, and faces prices of p 3 and p =, but then the price of commodity one decreases to p =. a. Explain how Richard s consumption of commodity one changes as a result of this decrease in price. (6 points) nitial 00 nitially Richard consumes x = = 0. After the price change he 3 + Final 00 consumes x = = 5. Thus, his consumption of commodity one + increases by 5 0 = 5 units.

5 b. Decompose the change in consumption of commodity one into that resulting from the substitution effect and that resulting from the income effect. Clearly explain. (6 points) nitial nitial nitially Richard consumes x = 0 and x = 40, resulting in utility of U ( X ) = min{ (0),40} = 40. Regardless of the price ratio, in order to achieve U ( X ) = 40 as inexpensively as possible, Richard will want to consume x = 0 and x = 40. That is, his "decomposition" level of Decomposition commodity one is x = 0. As a result, the change in consumption of commodity one can be decomposed as follows: Final Decomposition ( ncomeeffe ct) = x x = 5 0 = 5 and Decomposition nitial Substituti oneffect) = x x = ( =

6 6) Suppose that at "initial prices and income" of ( p, p, a consumer maximizes utility by choosing the bundle X illustrated below. For each of the following scenarios, identify the area in ( x, x ) -space where the "new optimal bundle" could be. Clearly explain your answer. a. good one is normal and good two is a complement to good one; the price of commodity one increases, with income and the price of commodity two held fixed. (8 points) x X x Since commodity one is ormal, it must also be Ordinary. Thus, an increase in p will result in a decrease in x. Further, since commodity two is a complement to commodity one, an increase in p will also result in a decrease in x. As p increases (with p and fixed), the budget line will rotate inward, pivoting around the vertical intercept. Together these insights imply that the new optimal consumption bundle must be somewhere in the "Bright Green"-shaded area above.

7 b. good one is "normal"; the price of each commodity increases by 0%, with income held fixed. (8 points) x X x n terms of impact on the set of affordable bundles, a simultaneous increase in both prices by 0% is equivalent to decrease in income of ( ) (with prices held fixed). Therefore, the new budget line will have the same slope as the initial budget line, but the value of each intercept will be 0 as large as the value of the corresponding intercept on the initial ( ) budget line. Since commodity one is normal, the change must lead to decrease consumption of commodity one. Consumption of commodity two may increase, decrease, or stay the same. Therefore, the new optimal bundle will lie somewhere along the "Plum"-shaded line segment above.

8 0 7) Suppose demand is given by D( p) =. + p a. Graphically identify the change in Consumers' Surplus as a result of an increase in price from p = 7 to p =. (4 points) 7 As a result of this increase in price, Consumers' Surplus decreases by an amount equal to the area of the "Red"-shaded region illustrated above. b. oting that the demand curve becomes flatter as we move to lower price, higher quantity pairs, place a numerical bound on the change in Consumers' Surplus as a result of an increase in price from p = 7 to p =. (6 points) 0 0 Begin by noting that D ( 7) = = 5 and D ( ) = = 0. oting that 8 the demand curve becomes flatter as we move to lower price, higher quantity pairs, the following bounds can be placed on the change in Consumers' Surplus: ( 0 0)( 7) < CS < ( 0 0)( 7) + ( 5 0)( 7) or more simply 40 < CS < 50.