Demand Functions

Definition and motivation

A demand function gives the quantity of a good that a consumer chooses as a function of prices and income (or utility). In more formal terms, a consumer with utility function $u(x_1, x_2, \dots, x_n)$, facing prices $p = (p_1,\dots,p_n)$ and income $m$, solves a constrained optimization problem and selects a bundle $x^*(p,m)$. The Marshallian (uncompensated) demand is

Example: If $u(x_1,x_2)=x_1^{\frac12}x_2^{\frac12}$ (Cobb–Douglas), prices $p_1=2$, $p_2=1$, income $m=100$, then standard Cobb–Douglas algebra gives $x_1^M=\alpha m/p_1$ and $x_2^M=(1-\alpha)m/p_2$ where $\alpha=1/2$. Numerically, $x_1^M=0.5\cdot100/2=25$, $x_2^M=0.5\cdot100/1=50$.

Marshallian vs Hicksian: two ways to get demand

• •Marshallian (uncompensated) demand $x^M(p,m)$ comes directly from utility maximization under a budget constraint. In "Utility Theory" we used Lagrangian methods to solve problems like this. Use Roy's identity (below) to obtain demands from indirect utility functions.

• •Hicksian (compensated) demand $h(p,u)$ comes from expenditure minimization: given target utility level $u_0$, the consumer minimizes expenditure to reach that utility:

The Hicksian demand isolates substitution effects because income is adjusted to keep utility fixed; the Marshallian contains both substitution and income effects. A numeric illustration: with quasilinear utility $u(x_1,x_2)=v(x_1)+x_2$ and budget $m$, the Hicksian demand for $x_1$ at target utility $u_0$ solves minimizing $p_1 x_1 + p_2 x_2$ s.t. $v(x_1)+x_2 \ge u_0$. Eliminating $x_2$ gives the same first-order condition as the Marshallian when income shifts appropriately; we will show explicit numbers later.

Inverse demand and market demand

Inverse demand expresses price as a function of quantity. For a single-good quasilinear problem where utility is $u(q)+y$ and $y$ is money numeraire, the first-order condition is $u'(q)=p$ so the inverse demand is

Numeric example: if $u(q)=10\sqrt{q}$ then $u'(q)=5/\sqrt{q}$, so $p(q)=5/\sqrt{q}$. At $q=4$, $p(4)=5/2=2.5$.

Market demand is the horizontal sum of individuals' Marshallian demands. If two consumers have demands $x^M_1(p,m_1)$ and $x^M_2(p,m_2)$, the market demand is $X(p)=x^M_1(p,m_1)+x^M_2(p,m_2)$; for inverse market demand, invert that function where possible.

Why care? Practically every empirical demand estimation, welfare calculation, tax incidence, and monopoly pricing analysis starts from these demand functions. Understanding their derivation from utility ensures you interpret income and substitution effects correctly.

Mechanic: Lagrangian optimization and Roy's identity

Start from the utility maximization problem. In "Utility Theory" we solved similar problems using the Lagrangian. For two goods, the Lagrangian is

First-order conditions (assuming interior solution):

Solving yields Marshallian demand $x^M(p,m)$. A shortcut: compute the indirect utility function $v(p,m)=\max\{u(x)\;:\;p\cdot x\le m\}$ and use Roy's identity:

Numeric worked mini-example (Cobb–Douglas)

Let $u(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}$ with $\alpha=1/2$, $p_1,p_2$ arbitrary, and income $m$. Solve via standard steps (or Roy's identity). The Marshallian demands are:

Numerical example: $\alpha=1/2$, $p_1=2$, $p_2=1$, $m=100$ gives $x_1^M=25$, $x_2^M=50$, as above.

An alternative: Quasilinear case

If utility is quasilinear in money, $u(x,y)=v(x)+y$ where $y$ is the numeraire and price of $y$ is 1, the consumer maximizes $v(x)+y$ s.t. $p_x x+y\le m$. Eliminating $y$ gives maximize $v(x)+m-p_x x$, so choose $x$ to satisfy

This directly yields inverse demand $p_x(x)=v'(x)$. Numeric example: $v(x)=10\sqrt{x}$, then $v'(x)=5/\sqrt{x}$. If price is $p_x=2.5$, solve $5/\sqrt{x}=2.5$ so $\sqrt{x}=2$ and $x=4$.

Roy's identity numeric check

For Cobb–Douglas $u=x_1^{1/2}x_2^{1/2}$, the indirect utility is $v(p,m)=\left(\frac{m}{2\sqrt{p_1p_2}}\right)$ up to a constant factor (work through substitution). Taking derivatives and applying Roy's identity yields the same $x_1^M, x_2^M$ above. A concrete verification: with $p_1=2,p_2=1,m=100$, compute $v(2,1,100)$ numerically and then compute $-\partial v/\partial p_1\,/\,(\partial v/\partial m)$ to recover $25$.

Mechanic: expenditure minimization and compensated demand

The Hicksian demand isolates substitution effects by holding utility fixed. The consumer solves

and $h(p,u_0)$ denotes the minimizer. The function $e(p,u)$ is the expenditure function; Shephard's lemma states

Numeric example (Cobb–Douglas)

For $u(x_1,x_2)=x_1^{1/2}x_2^{1/2}$ and target utility $u_0$, we can solve for $h(p,u_0)$ analytically. The expenditure function is

Check numerically: let $p_1=4$, $p_2=1$, and $u_0=10$. Then $e=2\cdot10\cdot\sqrt{4\cdot1}=20\cdot2=40$. By Shephard's lemma,

Plug in numbers: $h_1=10\sqrt{1/4}=10\cdot\tfrac12=5$. Similarly $h_2=10\sqrt{4/1}=20$. Check utility: $5^{1/2}\cdot20^{1/2}=\sqrt{100}=10=u_0$ and expenditure $4\cdot5+1\cdot20=20+20=40$.

From Hicksian to Marshallian: income makes the difference

Marshallian demand can be obtained by plugging the indirect utility's $u^*=v(p,m)$ into Hicksian demand:

This equation simply says: the Marshallian demand at $(p,m)$ is the Hicksian demand that achieves the utility level the consumer actually attains at $(p,m)$.

Slutsky equation (decompose price effects)

For any good $i$ and price $p_j$, the total derivative of Marshallian demand splits into substitution (compensated) and income effects:

Numeric example: use Cobb–Douglas with $\alpha=1/2$, $x_1^M=\alpha m/p_1$. Compute derivatives: $\partial x_1^M/\partial p_1=-\alpha m/p_1^2$. The income effect term is $-(\partial x_1^M/\partial m)x_1^M = -(\alpha/p_1)(\alpha m/p_1)= -\alpha^2 m/p_1^2$. So the compensated substitution effect equals

Plug numbers: $\alpha=1/2$, $m=100$, $p_1=2$ gives $\partial x_1^M/\partial p_1 = -0.5\cdot100/4 = -12.5$, income derivative $\partial x_1^M/\partial m = 0.5/2 = 0.25$, so income effect magnitude is $0.25\cdot25=6.25$, and thus compensated effect is $-12.5+6.25=-6.25$. This shows the substitution effect is negative (as expected) and smaller in magnitude than the total effect when income effects go in the opposite direction.

Interpretation

• •Hicksian demand and Shephard's lemma let you compute how much expenditure must change when prices change to preserve utility, a building block for compensated price indices and welfare measures (compensating/equivalent variation).

• •The Slutsky decomposition is central to distinguishing consumer surplus changes into substitution vs income channels; numerical examples show how magnitudes arise from functional forms.

Inverse demand and pricing

In many applied problems you know desired quantity and want to know the price that would induce it. For quasilinear utilities inverse demand is immediate: $p(q)=v'(q)$. Example: $v(q)=10\sqrt{q}$ implies $p(q)=5/\sqrt{q}$. This form is widely used in monopoly pricing and public goods provision where marginal willingness to pay drives optimal provision rules.

Market aggregation

If there are $N$ consumers with Marshallian demands $x^M_{i}(p,m_i)$, market demand is

Numeric illustration: two consumers with identical Cobb–Douglas demands $x^M=\alpha m/p$ but different incomes $m_1=100,m_2=50$, $\alpha=1/2$, $p=2$ gives $x^M_1=25$, $x^M_2=12.5$, and market demand $X=37.5$. Inverse market demand may not be analytically invertible but can be constructed numerically for aggregate analysis.

Empirical demand estimation and structural interpretation

Estimating demand curves often produces Marshallian demand (observational: prices and incomes vary). To interpret estimated elasticities correctly for welfare or tax incidence you must convert to Hicksian demand using Slutsky if you want substitution-only effects. For instance, consumer surplus approximations using Marshallian demand ignore income effects and can be biased for large price changes.

Welfare analysis: compensating and equivalent variation

Hicksian demand and the expenditure function allow exact welfare measures. Compensating variation (CV) holds prices at new prices and asks how much income change would restore original utility; mathematically CV solves

where $u_0$ is original utility and $p'$ new prices. Numerical computation uses Hicksian demands via Shephard's lemma.

Downstream topics that use these ideas

• •Tax incidence and optimal taxation: requires decomposition of price changes into income and substitution effects and uses Hicksian demands for deadweight loss.

• •Monopolistic pricing and Ramsey pricing: need inverse demand functions derived from consumers' marginal willingness to pay.

• •Empirical demand estimation and identification: linking reduced-form elasticities to underlying preferences uses Roy's identity and the Slutsky matrix.

• •General equilibrium welfare decomposition: aggregate Hicksian demands enter social welfare functions and compensating equivalence calculations.

Concrete example tying pieces together

Consider a small subsidy that lowers $p_1$ from 2 to 1.5 for the Cobb–Douglas consumer with $\alpha=1/2$, $m=100$. Marshallian demand jumps from $x_1^M=25$ to $x_1^M=0.5\cdot100/1.5\approx33.33$. Using Slutsky we can attribute how much of this increase is substitution vs income effect numerically (substitution effect ~6.25 in earlier calculation; income effect ~1.08 here), and compute compensating variation using the expenditure function to find the budget change that would offset the subsidy.

In summary, demand functions are the operational translation of preference and budget constraints into quantities and prices; mastering both Marshallian and Hicksian perspectives enables precise comparative statics, welfare measurement, and aggregation.