15.4 The Relationship between Money and Utility: Indirect Utility and Expenditure Functions
We’ve established solutions to the utility-maximizing bundle as a function of income, and the cost-minimizing bundle as a function of target utility. By plugging these bundles back into their respective objective functions, we can now ask two related questions:
- If she has a certain amount of money, what’s the maximum amount of utility she can achieve?
- If she wants to achieve a certain amount of utility, what’s the least amount of money she would need to achieve it?
The answer to the first question is the indirect utility function; the answer to the second is the expenditure function. As we’ll see, these are actually inverses of one another!
Indirect Utility Functions
The indirect utility function is a function of prices and income that describes the utility from the utility-maximizing bundle given those prices and income. That is, if a consumer has ordinary (“Marshallian”) demand functions \(x_1^\star(p_1,p_2,m)\) \(x_2^\star(p_1,p_2,m)\) then the indirect utility function $V(p_1,p_2,m)$ may be found by plugging those functions back into the utility function: \(V(p_1,p_2,m) = u(x_1^\star(p_1,p_2,m),x_2^\star(p_1,p_2,m))\) In other words, this asks: if a consumer with income $m$ faces prices $p_1$ and $p_2$, what is the maximum amount of utility they could achieve?
Indirect utility functions are often used when analyzing situations in which a consumer is choosing their income in a given situation: for example, choosing how much of their current income to save. In that case, the choice variable is the amount of money to spend in each period; so the utility they get will depend on how happy having an additional dollar to spend in each period would make them.
Expenditure Functions
The expenditure function is a function of prices and a utility level $U$ that describes the cost of the cost-minimizing way of achieving that utility given prices. That is, if a consumer has compensated (“Hicksian”) demand functions \(x_1^c(p_1,p_2,U)\) \(x_2^c(p_1,p_2,U)\) then the expenditure function $E(p_1,p_2,U)$ is the cost of the cost-minimizing bundle: \(E(p_1,p_2,U) = p_1 x_1^c(p_1,p_2,U)+ p_2 x_2^c(p_1,p_2,U)\) As we’ll see in the next section, expenditure functions are particularly useful in measuring changes in utility in terms of dollars. That is, if we’re interested in seeing how big of a deal some change in utility is, we can analyze the implied change in expenditure for a given set of prices.
Example: Cobb-Douglas
Suppose someone has Cobb-Douglas preferences of the form \(u(x_1,x_2) = (x_1 \times x_2)^{1 \over 2}\) We’ve shown many times that the ordinary demand functions for this utility function are \(x_1^\star(p_1,p_2,m) = \frac{m}{2p_1}\) \(x_2^\star(p_1,p_2,m) = \frac{m}{2p_2}\) To find the indirect utility function, we plug these optimized values back into the utility function: \(\begin{aligned} V(p_1,p_2,m) &= u(x_1^\star(p_1,p_2,m),x_2^\star(p_1,p_2,m))\\ &= (x_1^\star(p_1,p_2,m) \times x_2^\star(p_1,p_2,m))^{1 \over 2}\\ &= \left(\frac{m}{2p_1} \times \frac{m}{2p_2}\right)^{1 \over 2}\\ &= \tfrac{1}{2}p_1^{-{1 \over 2}}p_2^{-{1 \over 2}}m \end{aligned}\) Conversely, using the cost-minimization techniques we just developed, the compensated demand functions for this utility function are \(x_1^c(p_1,p_2,U) = p_1^{-{1 \over 2}}p_2^{1 \over 2}U\) \(x_2^c(p_1,p_2,U) = p_1^{1 \over 2}p_2^{-{1 \over 2}}U\) To find the expenditure function, we evaluate how much it would cost to buy the bundle $(x_1^c,x_2^c)$ at prices $p_1$ and $p_2$: \(\begin{aligned} E(p_1,p_2,U) &= p_1x_1^c(p_1,p_2,U) + p_2x_2^c(p_1,p_2,U)\\ &= p_1 \times p_1^{-{1 \over 2}}p_2^{1 \over 2}U + p_2 \times p_1^{1 \over 2}p_2^{-{1 \over 2}}U\\ &= 2p_1^{1 \over 2}p_2^{1 \over 2}U \end{aligned}\)
Relationship between indirect utility and expenditure functions
Let’s fix prices at $p_1 = p_2 = 1$, to focus on the relationship between money and utility. With these prices, the indirect utility function says that with money $m$, this consumer could achieve \(U = \tfrac{1}{2}m\) while the expenditure function says that to afford utility $U$, the consumer would need an income \(m = 2U\) In other words, these functions are just inverses of one another! Put another way, they each describe the relationship between utility and money. We can see why by looking at this graph:
[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/welfare/indirect_utility_and_expenditure ]
The top graph shows the budget-line/indifference-curve diagram we’re so familiar with by now. If this is a utility maximization problem, the constraint is the budget line; if it’s representing a cost-minimization problem, the constraint is the indifference curve.
The bottom graph shows the relationship between money and utility, as described by the indirect utility and expenditure functions. The indirect utility function expresses $U$ as a function of $m$; the expenditure function expresses $m$ as a function of $U$. But they both describe the same relationship.
Notice the effect of a price change: an increase in the price of either good means you can’t afford as much happiness for any level of income, so the indirect utility function shifts down. Thought of another way, every level of utility gets more expensive, so the expenditure function shifts to the right.
The Lagrange Multiplier as the derivative of the Indirect Utility and Expenditure Functions
Finally, let’s note the role the Lagrange multiplier plays in understanding indirect utility and expenditure functions.
Recall that in general, the Lagrange multiplier for a utility maximization problem measures the “bang for the buck” – both in the sense of the utility of the last dollar spent on each good, and in the sense of the additional utility the consumer would get from having another dollar. For example, with the utility function above, the Lagrangian is \(\mathcal{L}(x_1,x_2,\lambda) = (x_1 \times x_2)^{1 \over 2} + \lambda(m - p_1x_1 - p_2x_2)\) The first-order condition with respect to $x_1$ is \({\partial \mathcal{L} \over \partial x_1} = {x_2^{1 \over 2} \over 2 x_1^{1 \over 2}} - \lambda p_1 = 0 \Rightarrow \lambda = {x_2^{1 \over 2} \over 2p_1 x_1^{1 \over 2}}\)
If we plug in the optimal values of $x_1 = m/2p_1$ and $x_2 = m/2p_2$, we get \(\lambda = {(m/2p_2)^{1 \over 2} \over 2p_1 (m/2p_1)^{1 \over 2}} = \tfrac{1}{2}p_1^{-{1 \over 2}}p_2^{-{1 \over 2}}\) Note that this is just the derivative of the indirect utility function with respect to $m$! \(V(p_1,m_2,m) = \tfrac{1}{2}p_1^{-{1 \over 2}}p_2^{-{1 \over 2}}m \Rightarrow {\partial V(p_1,p_2,m) \over \partial m} = \tfrac{1}{2}p_1^{-{1 \over 2}}p_2^{-{1 \over 2}}\)
I’ll leave it for a homework problem to illustrate that the Lagrange multiplier for the cost-minimization problem yields the additional money you’d need to have to afford one more “util”.