(a) Derive Marshallian demand curve for an inferior good in a two-commodity framework by using income and substitution effects…

(a)

Let commodities be x and y, prices p_x and p_y, income M. The Marshallian demand for x, x(p_x,p_y,M), is obtained from:

max U(x,y) subject to p_x x + p_y y = M.

The Lagrange function is L = U(x,y) + λ(M − p_x x − p_y y). The first-order conditions are:

U_x = λ p_x, U_y = λ p_y, p_x x + p_y y = M.

Dividing the first two gives the tangency condition U_x/U_y = p_x/p_y, which together with the budget constraint solves for x = x(p_x,p_y,M). This is the Marshallian demand curve.

To decompose a change in p_x, define the Hicksian demand h_x(p_x,p_y,u) from the expenditure minimisation problem:

min p_x x + p_y y subject to U(x,y) = u.

The Slutsky identity, holding p_y and M constant, is:

∂x/∂p_x = ∂h_x/∂p_x − x ∂x/∂M.

Here ∂h_x/∂p_x is the substitution effect. For a convex indifference map, it is always negative. For an inferior good, ∂x/∂M < 0, so the income effect −x ∂x/∂M is positive. Thus:

∂x/∂p_x = (negative substitution effect) + (positive income effect).

If the substitution effect dominates, ∂x/∂p_x < 0, so the Marshallian demand curve slopes downward. If the income effect dominates, ∂x/∂p_x > 0, giving an upward-sloping demand curve. Such a good is a Giffen good. Therefore, an inferior good's demand curve is not always negatively sloped. It is negatively sloped when the substitution effect exceeds the inferior income effect; it becomes positively sloped in the Giffen case when the income effect dominates.

(b)

Let total market output be Q = q₁ + q₂. Duopolist II has 1/3 share of the whole market, so:

q₂/Q = 1/3 ⇒ 3q₂ = q₁ + q₂ ⇒ q₂ = q₁/2.

Substitute q₂ = q₁/2 into Duopolist I’s demand:

p₁ = 200 − 4q₁ − 2q₂ = 200 − 4q₁ − 2(q₁/2) = 200 − 5q₁.

Total revenue of I is:

R₁ = p₁ q₁ = 200q₁ − 5q₁².

So marginal revenue is:

MR₁ = dR₁/dq₁ = 200 − 10q₁.

Cost is c₁ = 5q₁², so marginal cost is:

MC₁ = dc₁/dq₁ = 10q₁.

Profit maximisation requires MR₁ = MC₁:

200 − 10q₁ = 10q₁ ⇒ 200 = 20q₁ ⇒ q₁ = 10 units.

Then:

q₂ = q₁/2 = 5 units.

Price:

p₁ = 200 − 5(10) = 150 monetary units per unit.

Profit:

π₁ = p₁ q₁ − c₁ = 150×10 − 5(10²) = 1500 − 500 = 1000 monetary units.

Second-order condition: d²π₁/dq₁² = −20 < 0, so this is a maximum.

Final: q₁ = 10 units, q₂ = 5 units, p₁ = 150 monetary units per unit, π₁ = 1000 monetary units.

(c)

The Scitovsky Paradox is the possibility that the Kaldor-Hicks compensation criterion can rank two allocations cyclically: allocation B may be declared superior to A, while A may also be declared superior to B. It shows that potential-compensation tests are not always consistent when income distribution and prices change.

Under the Kaldor test, a move from A to B is desirable if the gainers in B can compensate the losers from A and still be better off. Under the Hicks test, B is desirable if the losers from the move A→B cannot bribe the gainers to forgo the move (equivalently, cannot bribe them to return to A). The Kaldor-Hicks criterion uses such potential compensation, not actual compensation.

The paradox arises because the identity of gainers and losers, and the amount needed for compensation, depends on the prevailing distribution and relative prices. Thus it is possible that:

• gainers at B can compensate losers from A, so B passes the Kaldor test over A;
• gainers at A can compensate losers from B, so A also passes the Kaldor test over B.

Both moves are then potential Pareto improvements, although they are opposite. This makes the compensation test non-transitive and potentially contradictory.

Scitovsky’s double criterion avoids the paradox: B is socially preferable to A only if B is Kaldor-superior to A and A is not Kaldor-superior to B. That is, the reverse compensation test must fail. The Scitovsky Paradox therefore exposes a fundamental weakness of the Kaldor-Hicks test: welfare ranking by potential compensation alone is not always consistent unless checked against the reverse move. It also shows that the Kaldor-Hicks criterion is not a complete social welfare ordering; it requires supplementary distributional judgments or the Scitovsky double test to avoid contradictory rankings.