How to Derive Demand from Utility and an Endowment

A consumer with an endowment starts with a bundle of goods rather than cash income. To find demand, first calculate the endowment’s value at current prices, then maximize utility subject to that budget. The consumer’s chosen consumption bundle is not the same as the amount they buy: they may sell some of what they initially own.

1. Turn the endowment into a budget

Let $p=(p_1,\ldots,p_n)$ be the vector of goods prices, let $\omega=(\omega_1,\ldots,\omega_n)$ be the endowment, and let $x=(x_1,\ldots,x_n)$ be the consumption bundle chosen after trade. Assume $p_i>0$ and $\omega_i\geq 0$ for each good. In the standard model, the consumer can sell the endowment at market prices and spend the proceeds. Their wealth is therefore

$$m=p\cdot\omega=\sum_{i=1}^n p_i\omega_i.$$

The demand problem is

$$\max_{x\geq 0} u(x)\quad\text{subject to}\quad p\cdot x\leq p\cdot\omega.$$

If utility is locally nonsatiated, the budget constraint binds at an optimum: the consumer spends all available wealth. The notation $x(p,p\cdot\omega)$ emphasizes that demand depends on prices and on the endowment’s value at those prices.

2. Use the first-order conditions

For differentiable utility and an interior optimum, form the Lagrangian

$$\mathcal{L}(x,\lambda)=u(x)+\lambda\bigl(p\cdot\omega-p\cdot x\bigr),\qquad \lambda\geq 0.$$

The first-order condition for each good is

$$\frac{\partial u}{\partial x_i}=\lambda p_i.$$

Thus, at an interior solution, the marginal utility per dollar is equal across goods. Combine these conditions with the budget equality to solve for $x$. These conditions alone do not guarantee a global maximum. Concavity of utility is sufficient for global optimality here; second-order conditions can establish a local maximum, but a global comparison may still be needed. Feasibility also matters. If a solution would require negative consumption, check a boundary or corner solution instead.

3. Worked example: Cobb–Douglas utility

Suppose there are two goods, utility is

u(x_1,x_2)=x_1^{1/2}x_2^{1/2},$$ prices are $p_1=2$ and $p_2=1$, and the endowment is $\omega=(4,2)$. The endowment is worth

m=p_1\omega_1+p_2\omega_2=2(4)+1(2)=10.

For utility $u(x_1,x_2)=x_1^{1/2}x_2^{1/2}$, the consumer spends half of wealth on each good. Hence

x_1=\frac{m}{2p_1}=\frac{10}{4}=2.5,\qquad x_2=\frac{m}{2p_2}=\frac{10}{2}=5.

The chosen consumption bundle is $(2.5,5)$. Net demand, meaning consumption minus the initial endowment, is

x-\omega=(2.5-4,,5-2)=(-1.5,3).

The negative first entry means the consumer sells 1.5 units of good 1. The positive second entry means they buy 3 units of good 2. The trade balances at the stated prices: selling 1.5 units of good 1 earns $3, which pays for 3 units of good 2. The final bundle costs $2(2.5)+1(5)=10$, exactly the endowment’s value. More generally, for $u(x_1,x_2)=x_1^\theta x_2^{1-\theta}$ with $0<\theta<1$, the demands are

x_1=\frac{\theta(p\cdot\omega)}{p_1},\qquad x_2=\frac{(1-\theta)(p\cdot\omega)}{p_2}.

## 4. Consumption demand versus net demand These terms answer different questions: - **Consumption demand** is the final bundle $x$ the consumer wants. - **Net demand** is $x-\omega$. A positive component means a purchase; a negative component means a sale. Some texts call $x-\omega$ the excess demand of an individual consumer. Check the convention in your course, since “demand” sometimes refers to the final bundle and sometimes to trade relative to the endowment. A change in prices can affect demand in two ways: it changes relative prices, and it changes the market value $p\cdot\omega$ of the endowment. For instance, a higher price for a good the consumer owns raises their wealth, even as it makes that good more expensive to buy. Ignoring this wealth effect gives the wrong budget for an endowment economy. ## Common mistakes 1. **Using the endowment as the budget.** The budget is its value, $p\cdot\omega$, not the quantity of any one good. 2. **Reporting only purchases.** Solve for final consumption first, then subtract the endowment to get net trades. 3. **Treating endowment value as fixed when prices change.** With a price-taking consumer, wealth changes with prices unless there is separate cash income. 4. **Assuming every optimum is interior.** If utility has satiation, goods have restrictions, or a good has zero marginal utility at the relevant boundary, check whether the budget binds and whether a corner is optimal. The standard setup assumes the endowment can be traded at the same prices the consumer faces and that there are no transaction costs. With nontradeable goods or trading restrictions, the feasible set must be modified.