Question #63256

Q3.A consumer has a Cobb-Douglas utility function (x1,x2)= 0.5inx1 + 0.5inx2. His budget constraint is . Set a lagregian function; hence derive the optimal demand function for goods X1 and X2.
Q4. If the demand function face by the consumer for good X is given by
X=20+
Where X = Quantity demanded, M = income and P = Price of product X.
Assume his original income is Kshs. 6400 per month and price of good X has increased from Kshs. 20 per unit to Kshs. 40 per unit. Calculate the magnitude of total effect (TE), substitution effect (SE) and income effect (IE) resulting from this change in price

Expert's answer

Q3. Lagregian functon: µ(X1, X2, ʎ) = 0.5InX1 + 0.5InX2 + µ (M - X1 - X2)

dµ/dX1 = 0.5/X1 - ʎ = 0,

dµ/dX1 = 0.5/X2 - ʎ = 0,

dµ/X – P1X1 – P2X2 = M,

X1 = X1/0.5,

X2 = 0.5X1,

Substituting these into the budget constraints:

X1(m) = 0.5m/7, substituting this back into X2 = X1/0.5 and X2 = 0.5X1 then gives us X2(m) = M/7 and X2(m) = m/7

The consumpton of each of the two goods is therefore a constant fraction of income – which implies the 2 goods are normal and borderline between luxuries and necessites.

Q4. If The demand functon face by the consumer for good X is given by X = 25 + MP^(-1)/10, then:

X(P,M) = X(20, 6400) = 25 + (6400 * 1/20)/10 = 57 units per day

X(P,M) = X(40, 6400) = 25 + (6400 * 1/40)/10 = 47 units per day

Total change = reduced 10 units of good X per day

∆M = X1∆P1 =57(40 - 20) = 1140

M’ = M + ∆M

6400 + 1140 = 7540

X(P’1M’) = X(40, 7540) = 25 + (7540 * 1/40)/10 = 43.85 - Substitution effect.

∆Xs1 = X(P’1M’) – X(P1M)

Income efect:

X(P’1M) = X(40,6400) = 47,

X(P1M) = X(40,7540) = 43.85

Thus: Total effect ∆Xn1 = X1(40,6400) – X1(40,7540) = 47 – 43.85 = 3.15

dµ/dX1 = 0.5/X1 - ʎ = 0,

dµ/dX1 = 0.5/X2 - ʎ = 0,

dµ/X – P1X1 – P2X2 = M,

X1 = X1/0.5,

X2 = 0.5X1,

Substituting these into the budget constraints:

X1(m) = 0.5m/7, substituting this back into X2 = X1/0.5 and X2 = 0.5X1 then gives us X2(m) = M/7 and X2(m) = m/7

The consumpton of each of the two goods is therefore a constant fraction of income – which implies the 2 goods are normal and borderline between luxuries and necessites.

Q4. If The demand functon face by the consumer for good X is given by X = 25 + MP^(-1)/10, then:

X(P,M) = X(20, 6400) = 25 + (6400 * 1/20)/10 = 57 units per day

X(P,M) = X(40, 6400) = 25 + (6400 * 1/40)/10 = 47 units per day

Total change = reduced 10 units of good X per day

∆M = X1∆P1 =57(40 - 20) = 1140

M’ = M + ∆M

6400 + 1140 = 7540

X(P’1M’) = X(40, 7540) = 25 + (7540 * 1/40)/10 = 43.85 - Substitution effect.

∆Xs1 = X(P’1M’) – X(P1M)

Income efect:

X(P’1M) = X(40,6400) = 47,

X(P1M) = X(40,7540) = 43.85

Thus: Total effect ∆Xn1 = X1(40,6400) – X1(40,7540) = 47 – 43.85 = 3.15

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