Elle preferences over apples (A) and bananas (B) are represented by the Cobb-Dou

Question

Elle preferences over apples (A) and bananas (B) are represented by the Cobb-Douglas utility function;

u(A; B) = AB.

(a) Write down the optimization problem. What is the objective function? What are the choice variables?

What is the constraint?

(b) What are the optimality conditions? Explain.

(c) Derive her individual demand curves for apples and bananas as functions of the prices and income.

(d) What is her optimal bundle if her income is $100, and bananas are $1 a pound and apples are $2 a pound.

(e) What is her new optimal bundle if price of apples go down by 1 dollar? What is the substitution e§ect of this price change? What is the income e§ect of this price change?

Explanation / Answer

u(A; B) = AB

a) Optimizaton Problem is to maximize the utility of Elle at the purchase of units of apples and bananas such that her utility gets maximize.

Objective function: To maximize u(A; B) = AB

Choice variables are units of apples and bananas that are A and B.

Constraint is budget function that is I = PaA + PbB where I = income; Pa is price of apple, Pb is price of banana.

b) To find optimality, it is required that the ratio of marginal utiliy derived from consumption of each product should be equal to ratio of their respective price. That is at optimal condition, MUA/MUB = Pa/Pb.

c) From utility function lets first find out Marginal utility of apple and banana:

MUA = dU/dA = B

and MUB = dU/dB = A

Now, MUA/MUB =B/A = Pa/Pb.

B = APa/Pb .............. eq i

Putting this value into budget constraint,

I = PaA + Pb(APa/Pb)

2APa = I

A = I/2Pa This is individual demand curve/function for apples as functions of the prices and income.

and from above equation and eq i,

B = (I/2Pa)Pa/Pb = I/2Pb This is individual demand curve/function for bananas as functions of the prices and income.

d) Budget constraint : I = PaA + PbB

Putting given values into above

100 = 2A + 1B

B = 100/2A ... eq ii

and

MUA/MUB =B/A = Pa/Pb = 2/1 = 2

B = 2A ...... eq iii

from eq ii and iii

100/2A = 2A

A = 5 units.

hence B = 2*5 = 10 units.

Optimal bundle (5 apples and 10 bananas).

e) New Budget constraint : 100 = 1A + 1B

B = 100/A ... eq iv

and

MUA/MUB =B/A = Pa/Pb = 1/1 = 1

B = A ...... eq v

from eq iv and v

100/A = A

A = 10 units.

hence B = 10 units.

Optimal bundle (10 apples and 10 bananas).

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