Question

George is making a consumption choice over apples (a) and bananas (b). George’s preferences are represented by u (a, b) =(2a^2)b where a > 0 and b > 0. Denote by f = (a, b) a fruit bundle.

1. In terms of preference relations, how does George rank the three bundles f1 = (3, 1), f2 = (1, 3), and f3 = (2, 2) relative to each other?

2. State George’s marginal rate of substitution (MRS) function, MRS(a, b) (where MRS is the absolute value of the slope of the indifference curve in a diagram with a on the horizontal axis and b on the vertical).

3. State the equation for the indifference curve that has f1 on it (that is, all the fruit bundles such that George is indifferent between f1 and the bundle in question). State the equation such that b is a function of a. If George is given only one apple, how many bananas does he require to be indifferent relative to f1?

4. George has an income of m = 12 and the price of apples and bananas are pa = 2 and pb = 1, respectively

(a) State George’s budget line. What is the slope of the budget line?

(b) Determine George’s optimal fruit bundle choice f ∗ = (a ∗ , b∗ ).

5. Now, instead of the utility function above, let George’s preferences over fruit bundles f = (a, b) where a > 0 and b > 0 be represented by the utility function u (a, b) = 4 ln(a) + 2 ln(b). Evaluate the statement (true or false): “Since George’s utility function has changed, his fruit bundle preferences are also different.” Provide a clear and concise explanation of your answer

Answer #1

.

u(f1) = 2*3^{2}*1 = 18

u(f2) = 2*1^{2} *3 = 6

U(f3) = 2*2^{2}*2 ^{=}16

so f1>f3>f2

b) MRS =mua/mub = 2*2a*b / 2a^{2} = 2b/a

c) MRS (F1) = 2*1/ 3 = 2/3

U (F1) = 18

if george has 1 apple U = 2*1^{*}b = 2b

now to be indifferent to U(f1)

2b = 18

b = 9

so , he needs 9 bananas

4.a) b..l

pa.a +p.b.b = m

2a + b = 12

b) for optimal

mrs = pa/pb

2b/a = 2/1

2b = 2a

b = a

so putting in b.l

2a + a= 12

3a=12 ,

**a = 4 = b**

**5.** U can be wriiten

as U =a^{4}b^{2}

Taking ln on both sides and we will get the utility fn as given .

Preferences have not been changed , as it still represents cob douglas utility function.

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