Can someone please do a step by step solution to this! thank you! Suppose your u

Question

Can someone please do a step by step solution to this! thank you!

Suppose your utility function (i.e. level of satisfaction from consuming x and y) is given by U = a^1/3b^2/3 (Let "a" represent apple and "b" represent banana). Your total income is $500. The price of apple is $5 and the price of banana is $10. Write your budget constraint (BC). Write your Rationality condition (RC). Find the combination of consumption of bananas and apples that maximizes your satisfaction given your budget constraint. From your answer on (c), what is the utility (or level of satisfaction) that you get? Graph your results above

Explanation / Answer

A consumer consumes two commodities a and b and the utility function of the consumer is given by

U(a, b) = a1/3 b2/3

a)

The consumer can purchase the required amount of good a at a price $5 and the amount of good b at a price $10 . The consumer has exogenously determined income $500 to spend on both goods

First note that the consumer spends her entire income on purchasing both goods. Thus, her budget constraint is:

5a + 10b = 500

a + 2b = 100

b)

Consumer wishes to maximize her utility function given.

Rationality condition require consumer to be at an equilibrium where the marginal rate of substitution is equal to price ratio.

MRS = 5/10

MU(a)/ MU(b) = ½

(1/3)[b2/3/ a2/3] /(2/3) [a1/3/ b1/3] = ½

b/a = 2/2

a = b

c)

Substituting this in budget, we get

a + 2b = 100

b + 2b = 100

b= 100/3

a = b = 100/3

Hence, the required equilibrium quantity of apple is 100/3 and that of banana is 100/3.

d)

The level of satisfaction or utility is

U(a, b) = a1/3 b2/3

U(a, b) = (100/3)1/3 (100/3)2/3

U(a, b) = 100/3

So the utility derived at the equilibrium is 100/3

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