Question

In: Economics

Amy's utility function is U(x1 , x2) = x1x2, where x1 and x2 are Amy's consumption...

Amy's utility function is U(x1 , x2) = x1x2, where x1 and x2 are Amy's consumption of banana and apple, respectively. The price of apples is $1, the price of bananas is $2, and his income is $40.


(a) Find out the Amy's optimal consumption bundle. (Note that Amy's utility function is Cobb-Douglas.)
(b) If the price of apples now increases to $6 and the price of bananas stays constant, what would Amy's income have to be in order to just be able to afford his old bundle?
(c) Based on (b), measure the changes in Amy;s consumption of apple due to the substitution effect and the income effect.

**IMPORTANT** I don't need explanation for (a), (b), but I want a detailed explanation of (c). (Please don't copy solutions from previous questions asked on this website)

Solutions

Expert Solution

Ans: . Utility function, U = x1 * x2

Marginal Utility of x1, MU1 = dU/dx1 = x2

Marginal Utility of x2, MU2 = dU/dx2 = x1

=> Marginal Rate of Substitution, MRS = dx2/dx1 = MU1/MU2 = x2/x1

At utility maximizing level,

MRS = p1/p2

=> x2/x1 = p1/p2

=> x2 = x1*p1/p2 ---> Eq1

Substituting Eq1 in budget constraint, x1*p1 + x2*p2 = M, we get,

x1*p1 + x1*p1*p2/p2 = M

=> x1 = M/2p1 and thus, x2 = M/2p2 --> Eq2

a) At p1 = $2 and p2 = $1 and M = $40

Fron Eq 2

x1 = 40/2*2 = 10 units of bananas

and x2 = 40/2*1 = 20 units of apples

b) If new p2 = $6, p1 = $2 and M = $40

From Eq2, we get,

x2 = 40/6*2 = 3.33 units and x1 = 40/2*2 = 10 units

So, compensating income required to be paid to Amy so that she is able to afford the old bundle of (10,20) is,

M' = 10*2 + 20*6 = $140

c) For substitution effect,

p1 = $2, p2 = $6 and M' = $140

From Eq2,

x2 = 140/2*6 = 11.667 units

So, decrease in demand for apples due to substitution effect = Original demand for apples - 11.667 = 20 - 11.667 = 8.333 units

For Income effect,

p1 = $2, p2 = $6 and M = $40

Fron Eq2,

x2 = 40/2*6 = 3.333 units

Decrease in demand for apples due to income effect = Demand from substitution effect - 3.333 = 11.667 - 3.333 = 8.333 units

Thus, total decrease in demand for apples = Income Effect + Substitution effect = 8.333 + 8.333 = 16.666 units

Thus, new demand for apples = 20 - 16.666 = 3.333 units

Please like .... Thank you...


Related Solutions

1. Amy's utility function is U(x1 , x2) = x1x2, where x1 and x2 are Amy's...
1. Amy's utility function is U(x1 , x2) = x1x2, where x1 and x2 are Amy's consumption of banana and apple, respectively. The price of apples is $1, the price of bananas is $2, and his income is $40. (a) Find out the Amy's optimal consumption bundle. (Note that Amy's utility function is Cobb-Douglas.) (b) If the price of apples now increases to $6 and the price of bananas stays constant, what would Amy's income have to be in order...
Charlie’s utility function is U(x1, x2) = x1x2, where x1 and x2 are the Charlie’s consumption...
Charlie’s utility function is U(x1, x2) = x1x2, where x1 and x2 are the Charlie’s consumption of banana and apple, respectively. The price of apples is $1, the price of bananas is $2, and his income is $40. (a) Find out the Charlie’s optimal consumption bundle. (Note that Charlie’s utility function is Cobb-Douglas.) (b) If the price of apples now increases to $6 and the price of bananas stays constant, what would Charlie’s income have to be in order to...
Bilal’s utility function is U(x1; x2) = x1x2 (assume x1 and x2 are normal goods). The...
Bilal’s utility function is U(x1; x2) = x1x2 (assume x1 and x2 are normal goods). The price of good 1 is P1, the price of good 2 is P2, and his income is $m a day. The price of good 1 suddenly falls. (a)Represent, using a clearly labelled diagram, the hicks substitution effect, the income effect and the total effect on the demand of good 1. (b) On a separate diagram, represent using a clearly labelled diagram, the slutsky substitution...
Amy has utility function u (x1, x2) = min { 2*(x1)^2*x2, x1*(x2)^2 }. Derive Amy's demand...
Amy has utility function u (x1, x2) = min { 2*(x1)^2*x2, x1*(x2)^2 }. Derive Amy's demand function for x1 and x2. For what values (if any) of m, p1, and p2 are the goods gross complements or gross substitutes of each other?
Chapters 8. Slutsky Equation 1. Consider the utility function u(x1, x2) = x1x2. Suppose that the...
Chapters 8. Slutsky Equation 1. Consider the utility function u(x1, x2) = x1x2. Suppose that the prices are given 1 for each good and the income is 10. (1) Find out the optimal choice(s). Figure out the utility level at this optimal choice. (2) Suppose now that the price of good 2 increases from 1 to 2. Figure out the value of the bundle that you obtained in (1) under the new prices. Draw the budget set associated with this...
(a) Calculate the marginal utility of x1 and x2 for the following utility function u (x1;...
(a) Calculate the marginal utility of x1 and x2 for the following utility function u (x1; x2) = x 1 x 2 (b) What must be true of and for the consumer to have a positive marginal utility for each good? (c) Does the utility function above exhibit a diminishing marginal rate of substitution? Assume that and satisfy the conditions from Part b. (Hint: A utility function exhibits a diminishing marginal rate of substitution if the derivative of the marginal...
Your preference is represented by the utility function: u(x1, x2) = x10.5x20.5 where x1 is potato...
Your preference is represented by the utility function: u(x1, x2) = x10.5x20.5 where x1 is potato chips (in bags) and x2 is chocolate bars. The price of a bag of potato chips is $5 and the price of a chocolate bar is $10. (a)You have no income, but you received a gift from your uncle. The gift is 9 bags of potato chips and 1 chocolate bar. What is your utility from consuming the gift? Assume that you cannot exchange...
Bridgit’s utility function is U(x1, x2)= x1 + ln x2 x1 - stamps x2 - beer...
Bridgit’s utility function is U(x1, x2)= x1 + ln x2 x1 - stamps x2 - beer Bridgit’s budget p1 x1 + p2 x2 = m p1 – price of stamps p2 – price of beer m – Bridgit’s budget a) What is Bridgit’s demand for beer and stamps? b) Is it true that Bridgit would spend every dollar in additional income on stamps? c) What happens to demand when Bridgit’s income changes (i.e. find the income elasticity)? d) What happens...
Chizzy’s has a utility function U(x1,x2) = x1x2. He originally faces the prices ($1,$1) and has...
Chizzy’s has a utility function U(x1,x2) = x1x2. He originally faces the prices ($1,$1) and has income $50. If the price of good 1 falls $0.5. What is the change in consumer surplus, compensating and equivalent variation?
Sophie's utility function is given as U = 2(x1)^(1/2) + 8(x2) where x1 and x2 represent...
Sophie's utility function is given as U = 2(x1)^(1/2) + 8(x2) where x1 and x2 represent the two goods Sophie consumes. Sophie's income is $3400 and the prices are given as x1 = $2 and x2 = $160 a) derive & represent in two separate diagrams the demand for x1 and x2 ( for any income and prices) b) If Sophie's income is increased to $6600, what is the income effect on x1? Is x2 a normal good? Justify your...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT