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Question 4 Consider three consumers, each with a different utility function for two goods: apples

`(x_(1))`

and oranges (

`x_(2)`

). The price of apples is 82 , and the price of oranges is

`$3`

. Each consumer has an income of

`$120`

. Consumer 1: Cobb-Douglas utility function

`U_(1)(x_(1),x_(2))=x_(1)^(0.5)x_(2)^(0.5)`

Consumer 2: Linear utility function

`U_(2)(x_(1),x_(2))=4x_(1)+3x_(2)`

Consumer 3: Leontief utility function

`U_(3)(x_(1),x_(2))=min(x_(1),2x_(2))`

A) Draw the indifference curve for each consumer corresponding to a utility level of

`U=50`

. B) For each consumer, calculate the marginal utility of apples (

`{(:)/(M)U _(()()x_(1)))`

and the marginal utility of oranges

`(()/(MU) _(()()x_(2)))`

. C) For each consumer, compute the Marginal Rate of Substitution (MRS) between apples and oranges. Recall that

`MRS=(()/(MU) _(()()s_(1)))/(()/(MU) _(()()s_(2)))`

. D) The consumers are considering the following consumption bundles: Bundle 1: 15 apples and 10 oranges (for consumer 1) Bundle 2: 20 apples and 20 oranges (for consumer 2) Bundle 3: 10 apples and 5 oranges (for consumer 3) For each consumer, check whether these bundles are optimal by comparing the MRS to the price ratio

`(P_(s_(1)))/(P_(s_(2)))=(2)/(3)`

. In other words, discuss whether

`MRS=(P_(s_(1)))/(P_(2_(2)))`

for each bundle E) Which of the bundles are optimal and why? Identify any bundles that are not optimal and explain the direction in which the consumer would adjust their consumption to reach the optimal point. (Footnote:) For this problem: Bundle 1 is not optimal for Consumer 1 . Bundle 2 is optimal for Consumer 2 . Bundle 3 is not optimal for Consumer 3.