Section 3

Preferences, MRS, and Individual Rationality

Preferences as utility functions

Each agent $i \in \{A, B\}$ has a utility function $u^i : \mathbb{R}^2_+ \to \mathbb{R}$ that assigns a real number to every bundle of the two goods. A higher number means a preferred bundle. Throughout this module we represent preferences with the Cobb-Douglas form:

u^i(x_1, x_2) = x_1^{\alpha_i} \, x_2^{1 - \alpha_i}

The exponent $\alpha_i \in (0, 1)$ measures agent $i$ 's relative taste for good 1. When $\alpha_i$ is close to 1, an extra unit of good 1 contributes a lot to utility and an extra unit of good 2 contributes little — the agent tilts consumption toward good 1. The complementary weight $1 - \alpha_i$ plays the symmetric role for good 2. Agents A and B may differ in $\alpha$ , and that difference is the engine of every mutually beneficial trade in the model.

Indifference curves

An indifference curve is the locus of all bundles that give agent $i$ exactly the same utility level $\bar{u}$ — formally, the set $\{(x_1, x_2) \in \mathbb{R}^2_+ : u^i(x_1, x_2) = \bar{u}\}$ . The agent is indifferent between any two points on the same curve; a point on a higher curve is strictly preferred. For Cobb-Douglas preferences these curves are smooth and strictly convex toward the origin, reflecting the idea that variety is weakly preferred to extremes. Solving $u^i(x_1, x_2) = \bar{u}$ for $x_2$ as a function of $x_1$ gives the indifference curve explicitly:

x_2 = \left(\frac{\bar{u}}{x_1^{\alpha_i}}\right)^{1/(1-\alpha_i)}

As $x_1$ rises (and $x_2$ falls along the curve to keep utility constant), the curve becomes flatter — the agent demands less and less of good 2 in compensation for each additional unit of good 1. That diminishing slope is exactly what convexity means, and it plays a central role in characterising efficient allocations.

Marginal rate of substitution

The marginal rate of substitution (MRS) is the absolute value of the slope of the indifference curve at a given bundle — how many units of good 2 the agent is willing to give up in exchange for one additional unit of good 1 while remaining on the same indifference curve. We derive it from the ratio of marginal utilities.

Step 1
$\frac{\partial u^i}{\partial x_1} = \alpha_i \, x_1^{\alpha_i - 1} x_2^{1-\alpha_i}$
Marginal utility of good 1.
Step 2
$\frac{\partial u^i}{\partial x_2} = (1-\alpha_i) \, x_1^{\alpha_i} x_2^{-\alpha_i}$
Marginal utility of good 2.
Step 3
$\mathrm{MRS}^i = \frac{\partial u^i / \partial x_1}{\partial u^i / \partial x_2} = \frac{\alpha_i}{1-\alpha_i} \cdot \frac{x_2}{x_1}$
Ratio of marginal utilities. The MRS rises in $x_2/x_1$ : the more relatively scarce good 1 is, the more good 2 the agent will surrender for it.

Notice that the MRS depends only on the ratio $x_2/x_1$ , not on the level of utility or the scale of the bundle. At a bundle where the agent holds a lot of good 2 and little good 1, $x_2/x_1$ is large, so the MRS is large — the agent values good 1 highly at the margin. As good 1 becomes more plentiful relative to good 2, the MRS falls. This is the algebraic content of convex indifference curves.

Individual rationality

Every agent $i$ arrives at the market with an initial endowment $\omega^i = (\omega^i_1, \omega^i_2)$ . Participation in trade is voluntary: an agent will reject any allocation that leaves them worse off than simply consuming their endowment. An allocation $x^i$ is individually rational for agent $i$ if:

u^i(x^i) \ge u^i(\omega^i)

Any agent who is worse off at $x^i$ than at their own endowment can simply walk away from trade and consume $\omega^i$ . So the only allocations that can arise under voluntary exchange are individually rational ones. In the Edgeworth box, the set of feasible allocations satisfying this for both agents is the lens — the area bounded by agent A's indifference curve through $\omega$ and agent B's indifference curve through $\omega$ . The endowment point itself lies on the boundary of the lens; every interior point of the lens is Pareto-superior to it. Trade, if it happens, must land inside or on the boundary of this lens.