Question

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) Derive the Marshallian demand functions for C and O using the equilibrium conditions for an interior solution.

B) Graph and fully label the Demand Curve for corn if income is M=100. To do this, solve for demand when the price of corn is 10, 20, and 40, then approximate the demand curve using these three points.

Solutions

Expert Solution

(a) U = C^1/5 O^ 4/5

MUC = (1/5)C^-4/5 O^ 4/5

MUo = (4/5)C^1/5 O^ -1/5

At equilibrium condition (or utility maximization point), (MUc / MUo) = (Pc / Po)

=> [(1/5)C^-4/5 O^ 4/5 / (4/5)C^1/5 O^ -1/5] = (Pc / Po)

=> [(1/4) O^4/5 * O^1/5 / C^1/5 C^4/5] = (Pc / Po)

=>1/4 (O/C) = (Pc / Po)

=> O = 4C*(Pc / Po) ---------------(1)

---------

Budget constraint: M ≥ PcC +PoO

=> M =  PcC +PoO

Put eq(1)

=> M = Pc*C +Po*O

=> M = Pc*C + Po* [4C(Pc/Po)]

=> M = Pc*C + 4C*Pc

=> M = 5C*Pc

=> C = (M/5Pc) ---------------- Marshaliian demand function for C.

Put Marshallian demand function for C in eq (1)

O = 4C(Pc/Po)

> O = 4 * (M/5Pc) (Pc/Po)

=> O = (4/5) (M/Po) -------------------- Marshallian demand function of O

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(b) C = (M/5Pc)

Put M = 100

=> C = (100 / 5Pc)

=> C = (20 / Pc)

Price of corn (Pc) Demand for corn (C)
10 2
20 1
40 0.5

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