Question

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.

Answer 1

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= P_XA

Price of Commodity y in A= P_YA

Price of Commodity x in B= P_XB

Price of Commodity y in B= P_YB

Income of A= Income of B= m

P_XA> P_XB

P_YA<P_YB

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,

P_XA= $5, P_YA=$2

P_XB= $2, P_YB=$5

m= $20

Let us formulate the Budget line

For A= x_A P_XA+y_AP_YA=m

→ 5x_A+2y_A= 20 (Eq. 1)

For B= x_B P_XB+y_BP_YB=m

→ 2x_B+5y_B= 20 (Eq. 2)

Optimal Demand Function of A is given by the formula

x_A=m/2(P_XA)= 20/10= 2
x_A=2

y_A=m/2(P_YA)= 20/4= 5

y_A=5

Similarly Demand function of B
x_B=m/2(P_XB)= 20/4= 5
x_B=5

y_B=m/2(P_YB)= 20/10= 2

y_B=2

MRS= slope of the budget line= Marginal utility of x (MU_X)/ Marginal utility of Y (MU_Y)

For A

MU_XA= δu/δx_A=δx_Ay_A/δx_A
→ MU_XA= y_A

MU_YA= δu/δy_A=δx_Ay_A/δy_A
→ MU_YA= x_A

MRS_A=y_A/ x_A= 5/2

MRS_A=5/2

For B

MU_XB= δu/δx_B=δx_By_B/δx_B
→ MU_XB= y_B

MU_YA= δu/δy_B=δx_By_B/δy_B
→ MU_YB= x_B

MRS_B=y_B/ x_B= 2/5

MRS_B=2/5

Optimal Choices for

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

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

Ans: MRS_A is not equal to MRS_B 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.