Question

Sophie's given utility function is U(V, C) = VC + V where V = # of visits per month and C = # of minutes per month (hundreds). V = $1 and C = $4 and M = $44.

a) Calculate the equation that represents the consumers demand for number of visits (V).

b) Say that V increases to $16. Calculate the new equilibrium. Compute Sophie's substitution and income effect and represent it within a diagram.

Answer 1

Sophie's given utility function is;

U(V,C) = VC + V

Price of V, P_V = $1
Price of C, P_C = $4
Income, M = $44

The budget constraint;

P_V V + P_C C = M
V + 4C = 44

Marginal rate of substitution;

MRS = MUv / MUc
= d/dV (VC+V) / d/dC (VC+V)
MRS = C + 1 / V

a) The optimal condition will be when;

MRS = P_V / P_C
C + 1 / V = P_V / P_C
V = P_C(C+1) / P_V

Putting this in budget constraint;

P_V V + P_C C = M
P_V (P_C(C+1) / P_V) + P_C C = M
P_C(C+1) + P_C C = M
P_C C + P_C + P_C C = M
P_C + 2P_C C = M
C = (M-P_C) / 2P_C

V = P_C [((M-P_C) / 2P_C)+ 1] / P_V
V = M - P_C + 2P_C / 2P_V
V = (M+P_C) / 2P_V

The demand function for C = (M-P_C) / 2P_C
The demand function for V = (M+P_C) / 2P_V

b) When,

Price of V, P_V = $1
Price of C, P_C = $4
Income, M = $44

The budget constraint;

V + 4C = 44

The optimal condition will be when;

MRS = P_V / P_C
C + 1 / V = P_V / P_C
C + 1 / V = 1/4
4(C+1) = V
4C + 4 = V

Putting this in budget constraint;

V + 4C = 44
4C + 4 + 4C = 44
8C = 40
C = 5

V = 20 + 4
V = 24

When price of V increases to; P'_V = $16

The optimal condition will be ;

MRS = P'_V / P_C
C + 1 / V = P'_V / P_C
C + 1 / V = 16/4
C+1 = 4V
C = 4V - 1

Putting this in budget constraint;

16V + 4C = 44
16V + 4(4V-1) = 44
16V + 16V - 4 = 44
32V = 48
V' = 1.5

C = 4(1.5) - 1
C' = 5

Therefore, the total effect of increase in price of V is;

Good V; T.E = 1.5-24 = 22.5
Good C; T.E = 5-5 = 0

The total effect is divided into 2 parts; Substitution effect and income Effect

Substitution effect

At point A, the old bundles are ; (24,5)

U = VC +V
U(24,5) = 24*5 + 24
U = 144

As price of V increases, the substitution effect will be that consumer will consume more of C and less V, the point will move from A to B. At point B the utility will be same because it lies on same IC;

When price of V increases to; P'_V = $16, the new optimal bundle is Point C (1.5,23)

Substitution effect will be;

S.E_V = 6 - 24
S.E_V = - 18

S.E_C = 23 - 5
S.E_C = 18

Incoem effect

As price of V increases, the income effect will be that consumer's income will decrease and he/she will consume less of both the goods, the movement from point B to C is income effect.

When price of V increases to; P'_V = $16, the optimal bundle will be at Point C (1.5,5)

Income effect will be;

I.E_V = 1.5 - 6
I.E_V = - 4.5

I.E_C = 5 - 23
I.E_C = - 18