Constrained Maximization with Lagrangian
This is the basic, prototypical problem in economics and often in life: A consumer has a utility function over two goods, which relates quantities it consumes of each good to a level of well-being of each basket. Even though the consumers likes both goods, they cannot consume unlimited amounts. His consumption choice must be within his budget constraint, which is the set of all consumption baskets the consumer can afford given the prices of each good and his income.
Consider a world with two goods, X and Y . Given prices for each good {Px, Py} and an income level I. Intuitively, the budget constraint is plotted here, with the slope being - Px/Py.
The utility function captures how much the agent likes baskets of goods (X; Y) . It is a function U(X; Y) that relates the quantities consumed of each good to a number. The larger the number, the happier the agent (e.g. your customer) is. An indifference curve is a set of points that give a consumer the same amount of utility.
The indifference curve is this case decreasing in the X - Y diagram. This means that we must compensate the consumer by increasing X if we are decreasing Y . Indifference curve typically is also convex. This means that the agent has a taste for variety. For large values of X, a decrease of one unit of Y must be compensated by increasing X by large amounts (while the opposite is true for X small). In generally this is also adheres to the diminishing marginal utility of something.
Mathematical solution to solving constrained maximization
λ equals the “shadow price” of the budget constraint, i.e. it expresses the quantity of utils that could be obtained with the next dollar of consumption. Note that this expression only holds when x = x∗ and y = y∗. If x and y were not at their optimal values, then the total derivative of L with respect to I would also include additional cross-partial terms. These cross-partials are zero at x = x∗ and y = y∗.
The right-hand side shows a classic Lagrangian setup.
Rearranging (1) and (2), you have Ux/Uy = Px/Py. This means that the psychic trade-off is equal to the monetary trade-off between the two goods.
Interestingly, we also have Ux/Px = Uy/Py = λ.
What’s the meaning of λ?
What does the “shadow price” mean? It’s essentially the “utility value” of relaxing the budget constraint by one unit (e.g., one dollar). Note that this shadow price is not uniquely defined since it corresponds to the marginal utility of income in “utils”–an ordinal value. Thus, the shadow price is defined only up to a monotonic transformation.
We could also have determined that dL/dI = λ without calculations by applying the envelope theorem. Note that the envelope theorem for constrained problems says that dU∗ dI = ∂L ∂I = λ. At the utility maximizing solution to this problem, x∗ and y∗ are already optimized and so an infinitesimal change in I does not alter these choices. Hence, the effect of I on U depends only on its direct effect on the budget constraint and does not depend on its indirect effect (due to re-optimization) on the choices of x and y.
This “envelope” result is only true in a small neighborhood around the solution to the original problem.
The Expenditure Function and Duality
So far, we’ve analyzed problems where income was held constant and prices changed. This gave us the Indirect Utility Function. Now, we want to analyze problems where utility is held constant and expenditures change. This gives us the Expenditure Function. These two problems are closely related—in fact, they are ‘duals.’ Most economic problems have a dual problem, which means an inverse problem. For example, the dual of choosing output in order to maximize profits is minimizing costs at a given output level; cost minimization is the dual of profit maximization. Similarly, the dual of maximizing utility subject to a budget constraint is the problem of minimizing expenditures subject to a utility constraint. Minimizing costs subject to a minimum utility constraint is the dual of maximizing utility subject to a (maximum) budget constraint.