Integrability of Demand and Consumer Surplus

Observed consumer demand must be consistent with rational utility maximization. The theory of integrability provides the answer: when is this satisfied? Consumer surplus is a key tool for welfare analysis of price and policy changes. Accurate welfare measurement is critical for evaluating tax reforms, tariffs, and subsidies.

The Slutsky Matrix and Its Properties

Slutsky equation: Marshallian demand $x_l(p,m)$ is related to Hicksian demand $h_l(p,u)$:

$ \frac{\partial x_l}{\partial p_k} = \frac{\partial h_l}{\partial p_k} - x_k \cdot \frac{\partial x_l}{\partial m} $

Decryption of the three components: the left side is the observed change in Marshallian demand as $p_k$ rises. The first term is the "substitution effect" (the Hicksian change at fixed utility, only the price effect). The second is the "income effect": an increase in $p_k$ reduces real income, which changes demand for $x_l$ through $\partial x_l/\partial m$.

Slutsky matrix $S = \left(\frac{\partial h_l}{\partial p_k}\right)$: Symmetry: $S_{lk} = S_{kl}$ (cross-substitution effects are symmetric—a consequence of Shepard's lemma for the expenditure function). Negative semi-definiteness: $x^\top S x \leq 0$ for all $x$ (eigenvalues $\leq 0$—the substitution effect of own-good is negative). $p^\top S = 0$ (zero-degree homogeneity: scaling all prices doesn’t change demand).

Integrability theorem: Functions $x_l(p,m)$ are compatible with rational maximization if and only if the Slutsky matrix is symmetric, negative semi-definite, and $x_l$ are homogeneous of degree zero in $(p, m)$. Corollary: with market demand data, one can test rationality.

Consumer Surplus

Marshallian surplus $\Delta CS = \int_{p_1}^{p_0} x(p,m) , dp$ when the price drops $p_0 \to p_1$. Geometrically, it is the area under the demand curve between the two prices. The problem: the Marshallian surplus is only an exact welfare measure if the income effect is zero.

Hicksian compensated measures:

Compensating variation $CV = e(p^1, u^0) - m$: how much income must be changed after price changes to return to the initial welfare $u^0$. With a price decrease (a gain): $CV > 0$—you can withdraw $CV$ from income and the consumer remains at the previous welfare level.

Equivalent variation $EV = m - e(p^0, u^1)$: how much income would need to change at old prices to reach the new welfare level $u^1$. $EV$ equals the consumer’s willingness to pay for the price change.

Relation for a normal good: $EV \leq \Delta CS \leq CV$ when price decreases. For quasilinear utility $u = x_1 + v(x_2)$: zero income effect $\Rightarrow EV = \Delta CS = CV$—Marshallian surplus is exact.

Revealed Preferences

WARP (Samuelson, 1938): If at $(p^0, m^0)$ the consumer chose $x^0$ and $x^1$ was available ($p^0 \cdot x^1 \leq m^0$), then at $(p^1, m^1)$ with $p^1 \cdot x^0 \leq m^1$, $x^1$ must not be chosen ($x^0$ was already "rejected" in favor of $x^1$?—a logical contradiction). WARP is a weak requirement of transitivity.

GARP (Varian, 1982): A stronger cyclical condition. $GARP \iff$ Slutsky matrix is symmetric if continuity holds. Varian's test—a polynomial-time algorithm $O(n^3)$ for $n$ observations. Used in labour supply studies: tests whether household consumption data are consistent with GARP.

Nonlinear search of revealed preferences: Violations of GARP $\rightarrow$ irrationality. But: data may be noisy, choices discrete. Afriat (1973): it is always possible to find a “nearly rational” utility function explaining the observations with minimal violations.

Numerical Example

Linear demand for gasoline: $x = 100 - 5p$, $p_0 = 10$ (base price), $p_1 = 8$ (after subsidy). $\Delta CS$ (Marshallian) $= \int_8^{10} (100-5p) dp = [100p - 2.5p^2]_8^{10} = (1000 - 250) - (800 - 160) = 750 - 640 = 110$ rubles.

With quasilinear utility: $EV = CV = \Delta CS = 110$. For a normal good with income effect $\partial x/\partial m = 0.5:$ $CV = 115,$ $EV = 105,$ $\Delta CS = 110$ (in between).

Tax instead of subsidy: $p_1 = 12$. $\Delta CS = \int_{10}^{12} (100-5p) dp = -(100 \cdot 2 - 5 \cdot (144-100)/2) = -(200 - 110) = -90$. Tax revenue $= (12-10)\cdot x(12) = 2\cdot 40 = 80$. Deadweight loss $= DWL = 90 - 80 = 10$.

Assignment: Consumer with $u(x, y) = x^2 y$, $m = 120$. Price of $y$: $q_x = 2$, $q_y = 3 \rightarrow q_y' = 6$. (1) Find Marshallian $\Delta CS$. (2) Compute $CV$ and $EV$. (3) Compare all three measures—does the income effect exist? (4) Check WARP for two observations: at $(q_x=2, q_y=3, m=120)$ consumes $(30,20)$; at $(q_x=4, q_y=2, m=120)$ consumes $(20,20)$. Is WARP violated?