1.4 — Utility Maximization — Appendix
Solving the Constrained Optimization Problem with Calculus
Example
A consumer has a utility function of
and faces prices of
The problem can be expressed as
There are a few strategies you can use to solve this problem using calculus.
Substitution Method: take the budget constraint and solve it for one good in terms of the other good and income. Let’s solve for
Now we can plug this in for
Then we take the derivative with respect to
Now that we know
So we’ve found the optimal bundle is 5 of
Lagrangian Method: Recall the Lagrangian adds the objective function and the Lagrange multiplier
Solving for the First Order Conditions (setting all partial derivatives to 0):
Taking the first two equations, and rearranging each to equal
Setting them equal to one another, and solving for
So the consumer will buy twice the amount of
Since we now know
So the optimum consumption bundle (
You may be asking: so what in the world is
Mathematically, it can be proved that
Economically, it is known as the “shadow price,” or sometimes the “marginal utility of wealth.” It tells us how much your utility would increase if you were able to increase your constraint (income) by one unit (e.g. $1). Alternatively, we can think of it as the marginal benefit of relaxing the constraint (having $1 more to spend), or the marginal cost of strengthening the constraint (having $1 less).
In our example,
Recall that since utility is ordinal and not cardinal, the number of “utils” your utility were to change by is meaningless.
Let’s lastly confirm that this is the optimum using the definition
General Case for Goods & Marshallian Demand Functions
In the case of
That is, as usual, choosing the consumption bundle that maximizes utility, subject to the budget constraint.
Using the Lagrangian method, the Lagrangian is:
The first order conditions are a series of
Solving for each individual
Each equation here is a demand function for that respective good, as a function of all market prices