Question

In: Economics

In a two goods (x and y) world, two districts (A and B) are identical, except...

In a two goods (x and y) world, two districts (A and B) are identical, except the prices of good x (Px) and good y (Py) are higher and lower in district A, respectively. Suppose two identical individuals (i.e. same preferences and income) live in the two districts separately and their optimal choices are interior solutions. Evaluate the following statement: ‘The MRS at the optimal choices of two individuals are the same’. True, false, or uncertain? Explain your answer intuitively and graphically.

Solutions

Expert Solution

Given,

Two commodities: x & y

Two Individuals of both districts denoted by= A (for district A) & B (for district B)

Price of Commodity x in A= PXA

Price of Commodity y in A= PYA

Price of Commodity x in B= PXB

Price of Commodity y in B= PYB

Income of A= Income of B= m

PXA> PXB

PYA<PYB

Since, it is given that there will be interior solution let us assume that utility of both A & B is as follows:
u=xy

To solve this question let us take any random values that fit the above criteria

Let,

PXA= $5, PYA=$2

PXB= $2, PYB=$5

m= $20

Let us formulate the Budget line

For A= xA PXA+yAPYA=m

5xA+2yA= 20 (Eq. 1)

For B= xB PXB+yBPYB=m

2xB+5yB= 20 (Eq. 2)

Optimal Demand Function of A is given by the formula

xA=m/2(PXA)= 20/10= 2
xA=2

yA=m/2(PYA)= 20/4= 5

yA=5

Similarly Demand function of B
xB=m/2(PXB)= 20/4= 5
xB=5

yB=m/2(PYB)= 20/10= 2

yB=2

MRS= slope of the budget line= Marginal utility of x (MUX)/ Marginal utility of Y (MUY)

For A

MUXA= δu/δxA=δxAyA/δxA
→ MUXA= yA

MUYA= δu/δyA=δxAyA/δyA
→ MUYA= xA

MRSA=yA/ xA= 5/2

MRSA=5/2

For B

MUXB= δu/δxB=δxByB/δxB
→ MUXB= yB

MUYA= δu/δyB=δxByB/δyB
→ MUYB= xB

MRSB=yB/ xB= 2/5

MRSB=2/5

Optimal Choices for

A(x,y)=(2,5)

B(x,y)=(5,2)

Ans: MRSA is not equal to MRSB at the optimal choices.


The price of the commodity ultimately decides the slope of the budget line which determines the Marginal Rate of Substitution (when all other factors are the same)

When the price differs, the optimal consumption changes and therefore MRS would not be the same for both districts.


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