Question

Why is it that the MU per dollar of each good is equal for optimal consumption? Explain in detail, and use equations and graphs to answer the question.

Answer 1

The utility preference would be , for x and y be two goods with usual preferences (convex to origin), and the budget constraint would be . The Lagrangian function to maximize the utility would be or . The FOCs are as below.

or or or or .

or or or or .

or or or .

Comparing the first two FOCs, we have , as the necessary condition for utility maximization, meaning that, at the otimum consumption, the condition must be satisfied. This is the mathematical formulation to the assertion that at the optimum level, the marginal utility per unit dollar of both goods must be equal.

The graph is as below.

As is shown in the graph, the BL is the budget constraint , and three different utility curves are given as , and , for c<a<b. The budget line represents the set of affordable bundle which would exhaust the income. The points above the BL are unaffordable, while the points below the BL are affordable, but would not exhaust the income.

The utility curve U=c possess affordable bundles, but one may still afford bundles by increasing c. The utility curve U=b possess unaffordable bundles bundles, and one must decrease the utility (b) to get to afford a bundle. Both c and b would tend to a, in which the utility exists which is the maximum utility that possess an affordable bundle, at the tangent of the utility curve and the budget line.

The slope of a particular utility curve would be as, for , we have or or or or . The slope of the budget line is . At the tangent, the slope of the utility curve and budget line are equal, and hence, the optimal bundle would be where or , which is the formulation found above. This is the reason that the MU per dollar of each good is same at the optimum. The reason can be asserted as since or should be satisfied, meaning that the relative price of good x with respect to y must be equal to the relative marginal utility of x with respect to y, at the optimum. On the left of (x*,y*) in the U=a, the price of x for a unit price of y is smaller than the marginal utility of x for a unit marginal utility of y, and the consumption of good x must be increased so that both be equal. On the right of (x*,y*) in the U=a, the price of x for a unit price of y is larger than the marginal utility of x for a unit marginal utility of y, and the consumption of good x must be decreased so that both be equal.