Deriving the Utility Maximization Condition in Consumer Choice Theory

One of the most common conditions in consumer choice theory is

$$\frac{MU_X}{P_X}=\frac{MU_Y}{P_Y}.$$

This condition means that utility is maximized when the marginal utility per unit of money spent is equal across goods. I understood the intuition, but when I tried to derive it directly, I got stuck on how to differentiate the utility expression. So I rewrote the problem using the budget constraint and total utility as the integral of marginal utility.

Assume that a consumer only consumes two goods, $X$ and $Y$. Their prices are $P_X$ and $P_Y$, and the consumed quantities are $Q_X$ and $Q_Y$. The consumer's income is fixed at $I_0$. The budget constraint is

$$P_XQ_X+P_YQ_Y=I_0.$$

Solving for $Q_Y$ gives

$$Q_Y=\frac{I_0-P_XQ_X}{P_Y}.$$

Since the budget is fixed, consuming more of $X$ requires consuming less of $Y$. Differentiating this with respect to $Q_X$ gives

$$\frac{dQ_Y}{dQ_X}=-\frac{P_X}{P_Y}.$$

This is the slope of the budget line. It represents how much $Y$ must be given up in order to consume one more unit of $X$.

Now express total utility as the accumulated marginal utility. Since marginal utility measures how much total utility increases when consumption increases by a small amount, total utility can be written as

$$U(Q_X,Q_Y) = \int_0^{Q_X} MU_X(q)\,dq + \int_0^{Q_Y} MU_Y(q)\,dq.$$

The important point is that, on the budget line, $Q_Y$ is not an independent variable. It is determined by $Q_X$. So total utility can be treated as a function of $Q_X$ alone:

$$U(Q_X) = \int_0^{Q_X} MU_X(q)\,dq + \int_0^{Q_Y(Q_X)} MU_Y(q)\,dq.$$

Now differentiate this expression with respect to $Q_X$.

The first term is straightforward. By the fundamental theorem of calculus,

$$\frac{d}{dQ_X} \int_0^{Q_X} MU_X(q)\,dq = MU_X(Q_X).$$

The second term requires a little more care. The integration variable is $q$, but the upper limit $Q_Y$ is a function of $Q_X$. Therefore, we need to apply the chain rule.

In general,

$$\frac{d}{dx} \int_0^{g(x)} f(t)\,dt = f(g(x))g'(x).$$

Applying this to the second term gives

$$\frac{d}{dQ_X} \int_0^{Q_Y(Q_X)} MU_Y(q)\,dq = MU_Y(Q_Y)\frac{dQ_Y}{dQ_X}.$$

From the budget constraint, we already have

$$\frac{dQ_Y}{dQ_X}=-\frac{P_X}{P_Y}.$$

Therefore,

$$\frac{d}{dQ_X} \int_0^{Q_Y(Q_X)} MU_Y(q)\,dq = MU_Y(Q_Y)\left(-\frac{P_X}{P_Y}\right).$$

So the derivative of total utility with respect to $Q_X$ is

$$\frac{dU}{dQ_X} = MU_X(Q_X) + MU_Y(Q_Y)\left(-\frac{P_X}{P_Y}\right).$$

Simplifying,

$$\frac{dU}{dQ_X} = MU_X(Q_X) - MU_Y(Q_Y)\frac{P_X}{P_Y}.$$

At an interior optimum, slightly increasing or decreasing $Q_X$ should no longer increase total utility. Therefore,

$$\frac{dU}{dQ_X}=0.$$

So

$$MU_X(Q_X) - MU_Y(Q_Y)\frac{P_X}{P_Y} =0.$$

Rearranging gives

$$MU_X(Q_X) = MU_Y(Q_Y)\frac{P_X}{P_Y}.$$

Dividing both sides by $P_X$ gives

$$\frac{MU_X(Q_X)}{P_X} = \frac{MU_Y(Q_Y)}{P_Y}.$$

Therefore, the utility maximization condition in consumer choice theory is

$$\boxed{ \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} }.$$

The meaning of this condition is simple. The consumer allocates a limited budget across goods. The additional utility gained from spending one more unit of money on $X$ is $\frac{MU_X}{P_X}$, while the additional utility gained from spending one more unit of money on $Y$ is $\frac{MU_Y}{P_Y}$.

If

$$\frac{MU_X}{P_X}>\frac{MU_Y}{P_Y},$$

then spending money on $X$ is more efficient than spending money on $Y$. The consumer should reduce consumption of $Y$ and increase consumption of $X$.

If

$$\frac{MU_X}{P_X}<\frac{MU_Y}{P_Y},$$

then spending money on $Y$ is more efficient. The consumer should reduce consumption of $X$ and increase consumption of $Y$.

Eventually, the consumer has no reason to change the allocation when the two values become equal:

$$\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}.$$

In other words, utility maximization does not mean simply consuming more of the good with higher marginal utility. It means allocating the budget until the marginal utility per unit of money is equal across goods.