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In: Economics

Consider a consumer with the Utility function: U = C^1/5 O^ 4/5 and facing a budget...

Consider a consumer with the Utility function: U = C^1/5 O^ 4/5 and facing a budget constraint: M ≥ PcC +PoO

Note: For this utility function MUC = (1/5)C^-4/5 O^ 4/5 and MUo = (4/5)C^1/5 O^ -1/5 Where C denotes the consumption of corn, and O denotes the consumption of other goods.

A) For corn, characterize the income elasticity of demand, the price elasticity of demand, the cross price elasticity of demand and explain what each represents. (You do not need to calculate each elasticity, just tell me if it is positive or negative.) Is corn a normal good?

B) Determine the indirect utility function for the consumer and describe what the indirect utility function represents. Suppose, as in C), PC = 1 and PO = 1. What is the value of the indirect utility function? Suppose PC increases to 2; calculate the new value for the indirect utility function. Explain why the indirect utility function increased or decreased

Solutions

Expert Solution

The utility function is U= C1/5O4/5

MUc= 1/5C-4/5O4/5

MUo= 4/5C1/5O-1/5

The budget conraint: M ≥PcC +PoO

From the optimality condition:

MUc/MUo = Pc/ Po

1/5C-4/5O4/5/ 4/5C1/5O-1/5 = Pc/Po

O/4C= Pc/Po

O= 4c Pc/P0

Putting this value of O in the budget constraint the constraint becomes:

M= Pc*C + Po( 4c Pc/P0)

M= PcC +4PcC

M= 5PcC

C= M/ 5Pc.......(i)

This is the demand curve for C.

in a similar way,

O= 4M/ 5Po........(ii)

This is the demand curve for 0

The income elasticity of demand for C is given by:

now, differentiating the demand curve of C with respect to M

The income elasticity becomes:

(1/5Pc)* M/C

=

The income elasticity of corn is postitve, thus corn is a normal good.

The price elasticity of demand is given By:

Differentiating the demand equation with respect to Pc.

The price elasticity of demand becomes:

The income elasticity of corn is negative which implies that with the increase in the price of corn, the demand would decrease.

The cross price elasticity would be 0, as the demand for corn depends only on its own price and income, and not on the price pf O.

Rahul Sunny answered 2 years ago
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