Question

2. You are a market analyst at Crusty Pies, the slogan for which is "We Are the Only Pie Company." But despite having a stranglehold on the apple pie (a) market, Crusty is concerned about the popularity of The Cake Mo- nopolist's bundt cake (b). You've been asked to determine how much consumption of apple pies will change if Crusty increases the price by one dollar. Here are the facts of the market: • The price of an apple pie, pa, is $4. . The price of a bundt cake, P, is $6. • There are 100 consumers of pies and cakes, each of whom has a dessert budget of $120. . You believe there are two kinds of consumers. Type I consumers view apple pie and bundt cake as substitutes and have the utility function u(a,b) = 3a + 46. Type 2 consumers view pie and cake as quite different goods and like both, fulfilling utility function u(a,b) = Vab. However, you do not know how many of each type of consumer there is for certain • Sales of apple pie last period, given all the above market features, were 2550 pies. Use the following procedure to determine how sales of apple pie will change if Pa is increased. (a) Find the optimal bundle (a,b) for each Type 1 consumer. (Hint: the tangency condition won't hold here - why not? If it helps, try drawing a graph of the budget set and indifference curves.) (b) Find the optimal bundle (a, b) for each Type 2 consumer. (c) Treating Pa as a variable, express a Type 2 consumer's optimal level of a in terms of Pa. Use this expression to find the price elasticity of demand at the optimal bundle and the current price of $4.

Answer 1

All the questions have been answered.

The usual methodology of Marshallian exercise by setting up a Lagrange Multiplier is followed.

The solution provided below only deals with the first order conditions as the second order condition of the Bordered Hessian Matrix to be positive is satisfied because of the diminishing marginal utility of both a and b (can be verified easily) in part b and c.

L with subscripts of lambda(λ), a and b denotes the first order partial derivative of L with respect to λ, a and b respectively.

Any 2 Given." Pa = $4; P = $6 No. of consumers - 100 Budget/Income of each consumer Hence we can write the of each consumer as = $120.- M. income/budget constraint M = 12040466 ²1 27 Type 1 consumers have to i uca,b) = 3a46 antil Type 2 consumers have in - uca, b) = Jab Cs want and - all we need to find the optimal bundle for Type 1 consumer If we go by the usual method of Marshallian analysis using a Lagrange multiplier, we get- na L = 3a + 4b + (120- 40-66) L E_120-40 - ab 20 7 First order Lot 3-4x=0 L ) X = 344 = 3 conditions. h: 4 - 6 = 2 X = 2/3 - 201 2020 2021 0210 But we can't have multiple is wrong here. Let us see such indifference curves. s. Something the graph of

We can see that the budget line I by many indifference curves in figure Indan indifference curve that just line. (figure 2) dol a dolu is intersected 1 we need to touch as the budad Indifference Indifference Curves curves - Income line Income line S 10 15 20 25 30 35 40 a 5 10 15 20 25 30 35 40 45 Figure 1 Figure 2

We can verify that the lined 3at16=80 and 3at46=90 touch the budget line 4at66 =120 at BC0, 20) and A (30,0) respectively. But the utility at point A (30, o) is go whereas the utility at point B (0,20) is 8o. Hence we can say that pt. A is the optimal bundle for Type 1 consumer -> Type 1 Consumer consumes az 30 and beo For the Type 2 consumer, we set up the Marshallianº Lag rangian multiplier - L = Tab + 1 ( 120 - 40 - 66) L = 120 - 40- 6b Fo r These are the first 2 4 ao poder conditions obtained by taking derivatives - 60=o of I wort a, b, a b) - This gives b - 23 = b = 2 a 120 - 40 -6 (22) = 0 120 - 8a=0 al =a = 1593 lb = 10

4 - A Budget / Income line 4ळ = 516 51015205303540 -

The utility in the above diagram is calculated using the point (15,10) in the utility function.
c) Now, we need to treat e as a variable we again set up the Lagrangian as follows- La Tab + x (120-pa- 66) LTE 120-pa-6b09: I La = 1/6 - Pa x=0 ht Alt Le La 2J b M GEO A Thus we get baha - - 66=pa a How To 120 - pat 66=0 oslu 2 120 - Pa- POOL OLC 7. = 28. a = 120 p a = 60 a= 60 / =) Now, we know that the price elasticity demand Ed is given by. of dp/

Here is - Quantity e demanded of a - Pa Price of . Eda-da ! مع عهدها - = da ala la - Ed = 60 Pa Using the Pa $4, optimal bundle, a 15 E = 60 4x15 Ed = 1 in absolute terms