9.2 Cost Minimization
Utility maximization and cost minimization are, in many ways, two sides of the same coin. Assuming monotonicity, which ensures that the solution lies along the constraint, both are constrained optimization problems of the form \(\begin{aligned} \max_{x_1,x_2}\ \ \ & f(x_1,x_2) \\ \text{s.t.}\ \ \ & g(x_1,x_2) = 0 \end{aligned}\) In this kind of constrained optimization problem we call the function $f(x_1,x_2)$ the objective function and the equation $g(x_1,x_2) = 0$ the constraint.
Utility maximization with a budget constraint
For what we’ve seen thus far, the objective function has been the “utility function” whose output is measured in utils, and the constraint has been the budget constraint: that is, \(\begin{aligned} \text{Objective function: }f(x_1,x_2) &= u(x_1,x_2)\\ \text{Constraint (when set equal to zero): }g(x_1,x_2) &= m - p_1x_1 - p_2x_2 \end{aligned}\) Visually, we can picture the budget constraint and a number of indifference curves (i.e., the level sets of the objective function), and see that our objective is to get to the highest indifference curve (i.e., the highest level set of the objective function) while staying on the constraint:
We wrote the optimal bundle for this problem as $(x_1^\star, x_2^\star)$. This bundle was a function of the market prices $p_1$ and $p_2$, as well as the exogenously given income of $m$. We called the solution functions for this problem demand functions, though technically we should have called them the ordinary or Marshallian demand functions: \(\text{Ordinary demand functions: }\begin{cases}x_1^\star(p_1,p_2,m)\\x_2^\star(p_1,p_2,m)\end{cases}\)
Cost minimization with a utility constraint
What if we were to flip this script, though? What if we were to say that we wanted to find the cheapest way of affording some utility level $U$? Then our problem would become \(\begin{aligned} \min_{x_1,x_2}\ \ \ & p_1x_1 + p_2x_2 \\ \text{s.t.}\ \ \ & u(x_1,x_2) = U \end{aligned}\) Now, using the same $f()$ and $g()$ framing as above, we have \(\begin{aligned} \text{Objective function: }f(x_1,x_2) &= p_1x_1 + p_2x_2\\ \text{Constraint (when set equal to zero): }g(x_1,x_2) &= U - u(x_1,x_2) \end{aligned}\) Visually, we now think of the target indifference curve as the constraint, with the level curves of the objective function being “iso-cost” lines with slope $-p_1/p_2$; now the objective is to find the least-cost way of affording the level of utility shown by the indifference curve:
Let’s write the optimal bundle for this problem as $(x_1^c, x_2^c)$. This bundle is also a function of the market prices $p_1$ and $p_2$, but instead of the income $m$ being the exogenous factor that determines the constraint, now the constraint is determined by the exogenously given utility of $U$. The solution functions for this problem are called the compensated or Hicksian demand functions: \(\text{Compensated demand functions: }\begin{cases}x_1^c(p_1,p_2,U)\\x_2^c(p_1,p_2,U)\end{cases}\) As with utility maximization, there are two possible kinds of solutions: when the Lagrange method finds an interior solution characterized by a tangency between the level sets of the objective function and the constraint, and when Lagrange fails because the solution occurs at a corner or kink. Let’s see how the math works in each of these cases.
When Lagrange works
Mathematically, when we solved the utility maximization subject to a budget constraint problem using the Lagrange method, we said that the optimal solution was characterized by two equations: \(\begin{aligned} \text{Tangency condition: } & MRS(x_1^\star,x_2^\star) = {p_1 \over p_2}\\ \text{Constraint condition: } & p_1x_1^\star + p_2x_2^\star = m \end{aligned}\) If we were to solve for this point using the Lagrange method, we would end up with the optimal bundle $(x_1^c,x_2^c)$ being characterized by two similar conditions: \(\begin{aligned} \text{Tangency condition: } & MRS(x_1^\star,x_2^\star) = {p_1 \over p_2}\\ \text{Constraint condition: } & u(x_1,x_2) = U \end{aligned}\) Because the constraint is not a budget constraint, but rather a utility constraint – that is, we’re constraining ourselves to be along the indifference curve representing bundles, the optimal bundle is the one along the utility constraint where the tangency condition holds.
Let’s look at the example of a Cobb-Douglas utility function, $u(x_1,x_2) = x_1x_2$. The MRS for this utility function is \(MRS = {x_2 \over x_1}\) so the tangency condition is \(x_2 = {p_1 \over p_2} x_1\) For the utility maximization problem, we plug this into the budget constraint $p_1x_1 + p_2x_2 = m$, and get the optimal bundle \(x_1^\star(p_1,p_2,m) = {m \over 2p_1}\) \(x_2^\star(p_1,p_2,m) = {m \over 2p_2}\) For the cost minimization problem, we plug this in to the utility constraint to find the “compensated” bundle $(x_1^c,x_2^c)$: \(\begin{aligned} x_1x_2 &= U\\ x_1\left[{p_1 \over p_2} x_1\right] &= U\\ x_1^2 &= {p_2 \over p_1}U\\ x_1^c &= \sqrt{\frac{p_1}{p_2}U} \end{aligned}\) Plugging this back into the tangency condition gives us \(x_2^c = {p_1 \over p_2} x_1^c = \sqrt{\frac{p_1}{p_2}U}\) Visually, we can compare these two methods:
The left-hand graph shows the utility maximization problem with a budget constraint for income $m$; the right-hand graph shows the cost minimization problem with a utility constraint for utility $U$. In each case, the optimum is the intersection of the relevant constraint with the tangency condition.
When Lagrange Doesn’t Work
For the most part in this chapter, we’ll deal with the case in which the Lagrange method works. However, as with utility maximization subject to a budget constraint, the solution to a cost minimization problem may not be characterized by a point of tangency between an indifference curve and a budget line. Let’s think about one of our two standard “extreme” examples, perfect complements.
In the case of perfect complements, the cheapest way to afford a given level of utility will always be at the base of the “L” where the minimands are equal to one another, regardless of the prices. For example, if you think about the utility function \(u(x_1,x_2) = \min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\}\) if we set this equal to some utility level $U$, we would have the conditions \(\frac{x_1}{2} = \frac{x_2}{3} = U\) and therefore the cost-minimizing solution would be \(x_1^c = 2U\) \(x_2^c = 3U\) For example, if $U = 2$, we can see that the minimum cost would occur at the point (4,6), regardless of what prices are (try changing prices to see!):
In other words, we find the optimum visually by finding the intersection of the utility constraint with the “ridge condition” that the minimands are equal to one another.
Cost minimization and the IOC
In the Cobb-Douglas case, we said that the optimum occurred at the intersection of the tangency condition and the constraint; in the case of perfect complements, we said that it occurred at the intersection of the “ridge condition” and the constraint.
If you recall, the tangency condition gave the expression for the IOC when Lagrange works, and the ridge condition was the IOC for perfect complements. In fact, we can say in general that the cost-minimizing bundle occurs at the intersection of the utility constraint and the IOC.
Why is this the case? Think about what the IOC represents: it shows, for various levels of income, what the utility-maximizing bundle would be. That is, each point along the IOC represents some level of money; higher points along the IOC represent higher levels of money, and lower points represent lower levels of money.
Furthermore, each point along the IOC represents a point where there is no overlap between the budget set and the preferred set: for every point $X^\star$ along the IOC, all points which give you more utility than $X^\star$ must cost more than $X^\star$, and all points which cost less than $X^\star$ must give you less utility than $X^\star$.
When we’re solving a cost minimization problem, we’re trying to minimize the amount of money we spend to achieve a certain utility. In other words, we can think of ourselves as trying to find the lowest point along the IOC that still gives us a certain amount of utility. This is true regardless of whether the IOC represents the tangency condition or not.