Indirect Utility and Expenditure Functions

1. The Indirect Utility Function: Utility as a Function of Prices and Income

The Marshallian demand function, $x^{\ast }(\mathbf{p},w)$, tells us the optimal bundle of goods a consumer will purchase given prices $\mathbf{p}$ and income $w$. A natural next question is: what is the level of utility the consumer achieves with this optimal bundle?

To find this, we simply substitute the Marshallian demands back into the original utility function. The resulting function is called the indirect utility function, denoted as $v(\mathbf{p},w)$:

$v(\mathbf{p},w)=u(x_1^{\ast }(\mathbf{p},w),x_2^{\ast }(\mathbf{p},w),\dots ,x_n^{\ast }(\mathbf{p},w))$

This function gives the maximum possible utility a consumer can attain for a given set of prices and income. The term "indirect" is used because it relates utility to prices and income, rather than directly to the quantities of goods consumed.

To see how this works in practice, let's watch a demonstration.

Properties and Interpretation of $v(\mathbf{p},w)$

The indirect utility function has several key properties that align with our economic intuition. You can find a formal list in the UCL lecture notes, which we will use throughout this lesson.

As the notes describe, the indirect utility function is:

• Non-increasing in prices ($\frac{\partial v}{\partial p_i}\le 0$): Higher prices can't make the consumer better off.
• Increasing in income ($\frac{\partial v}{\partial w}>0$): Higher income allows for more consumption, leading to higher utility.
• Homogeneous of degree zero in $(\mathbf{p},w)$: As we saw with Marshallian demands, doubling all prices and income leaves the budget set unchanged, so the maximum attainable utility also remains unchanged. This is the "no money illusion" principle again.
• Quasi-convex in prices: This is a more technical property, but it relates to the fact that consumers are better off with diversified prices rather than extreme ones.

Roy's Identity: From Utility back to Demand

Perhaps the most powerful feature of the indirect utility function is that it allows us to recover the Marshallian demand functions without re-solving the original optimization problem. This relationship is known as Roy's Identity:

$x_i^{\ast }(\mathbf{p},w)=-\frac{\partial v/\partial p_i}{\partial v/\partial w}$

This remarkable identity shows that the consumer's demand for good $i$ is the ratio of the marginal disutility of its price to the marginal utility of income. Your physics background gives you a strong appreciation for the power of the envelope theorem, which provides the formal basis for this result. A change in a parameter (like a price) affects the optimized value (utility) only through its direct effect on the constraint, not through the induced change in the choice variables.

The following video provides both a great intuitive argument and a practical demonstration of Roy's Identity.

For a formal derivation of Roy's Identity, you can refer back to the UCL notes (resource LINK) on page 9.

2. The Expenditure Function: The Dual Problem

So far, we have focused on the consumer's primal problem: maximizing utility for a given budget. Now, we turn to the dual problem: minimizing the expenditure needed to achieve a specific level of utility.

The dual problem is formulated as:
${\min }_{\mathbf{x}}\mathbf{p}\cdot \mathbf{x}\text{subject to}u(\mathbf{x})\ge \bar{u}$

The solution to this problem gives the Hicksian (or compensated) demand functions, $h_i(\mathbf{p},\bar{u})$. The value of the objective function at the optimum is the minimum expenditure required. This value, as a function of prices and the target utility level, is the expenditure function, $e(\mathbf{p},\bar{u})$:

$e(\mathbf{p},\bar{u})=\mathbf{p}\cdot \mathbf{h}(\mathbf{p},\bar{u})$

The expenditure function answers the question: "How much money do I need to be this happy?"

Properties and Interpretation of $e(\mathbf{p},\bar{u})$

Please refer back to the UCL lecture notes (LINK) and read the section titled "Expenditure function" on page 8.

As described there, the expenditure function is:

• Non-decreasing in prices ($\frac{\partial e}{\partial p_i}\ge 0$): To maintain the same utility level, expenditure must not decrease if a price goes up.
• Increasing in utility ($\frac{\partial e}{\partial u}>0$): Higher utility targets require more expenditure.
• Homogeneous of degree one in prices: If all prices double, you need to double your expenditure to buy the same bundle and achieve the same utility. $e(\lambda \mathbf{p},\bar{u})=\lambda e(\mathbf{p},\bar{u})$.
• Concave in prices: This property implies that as a price increases, the consumer substitutes away from that good, so expenditure increases at a decreasing rate.

Shephard's Lemma: From Expenditure to Demand

Analogous to Roy's Identity, there is a simple and direct way to recover the Hicksian demand functions from the expenditure function. This is known as Shephard's Lemma:

$h_i(\mathbf{p},\bar{u})=\frac{\partial e(\mathbf{p},\bar{u})}{\partial p_i}$

This result is even more direct than Roy's Identity. It states that the rate of change of minimum expenditure with respect to a price is exactly equal to the quantity of that good you need to buy. This is another consequence of the envelope theorem. The increase in cost from a small price rise is simply the quantity you're buying multiplied by that price rise.

3. Duality: Tying It All Together

The primal and dual problems are not independent; they are two sides of the same coin. The indirect utility function and the expenditure function are intimately related: they are mathematical inverses of each other.

This gives rise to a set of fundamental identities:

1. $e(\mathbf{p},v(\mathbf{p},w))=w$: The minimum expenditure needed to achieve the maximum utility you can get from income $w$ is simply $w$.
2. $v(\mathbf{p},e(\mathbf{p},\bar{u}))=\bar{u}$: The maximum utility you can get from the minimum expenditure needed to achieve utility $\bar{u}$ is simply $\bar{u}$.
3. $h_i(\mathbf{p},v(\mathbf{p},w))=x_i^{\ast }(\mathbf{p},w)$: The Hicksian demand for the utility level $v(\mathbf{p},w)$ is the same as the Marshallian demand at income $w$.
4. $x_i^{\ast }(\mathbf{p},e(\mathbf{p},\bar{u}))=h_i(\mathbf{p},\bar{u})$: The Marshallian demand at an income level equal to $e(\mathbf{p},\bar{u})$ is the same as the Hicksian demand for utility $\bar{u}$.

These relationships provide a complete and elegant summary of consumer theory.

The diagram below provides a powerful visual summary of all the connections we have discussed in this module.

4. A Worked Example

Let's work through an example to see how these concepts connect. We will use Example 1 from the UCL lecture notes (LINK, page 12).

Consider the direct utility function $u(q_1,q_2)=(q_1-a_1)(q_2-a_2)$.

1. Find Marshallian Demands: Solving the utility maximization problem (which involves setting the MRS equal to the price ratio and substituting into the budget constraint) yields the Marshallian demands:
   $f_1(y,p_1,p_2)=\frac{y+p_1{a}_1-p_2{a}_2}{2p_1}$
   $f_2(y,p_1,p_2)=\frac{y-p_1{a}_1+p_2{a}_2}{2p_2}$

2. Construct Indirect Utility Function: Substitute these demands back into $u(q_1,q_2)$:
   $v(y,p_1,p_2)=(f_1-a_1)(f_2-a_2)=\left(\frac{y-p_1{a}_1-p_2{a}_2}{2p_1}\right)\left(\frac{y-p_1{a}_1-p_2{a}_2}{2p_2}\right)$
   $v(y,p_1,p_2)=\frac{(y-p_1{a}_1-p_2{a}_2{)}^2}{4p_1{p}_2}$

3. Construct Expenditure Function: We can find $e(u,\mathbf{p})$ by inverting $v(y,\mathbf{p})$. We set $v=u$ and solve for $y$, which is our expenditure $e$.
   $u=\frac{(e-p_1{a}_1-p_2{a}_2{)}^2}{4p_1{p}_2}$
   $\sqrt{4u{p}_1{p}_2}=e-p_1{a}_1-p_2{a}_2$
   $e(u,p_1,p_2)=2\sqrt{u{p}_1{p}_2}+p_1{a}_1+p_2{a}_2$

4. Find Hicksian Demands using Shephard's Lemma: Now we can easily find the Hicksian demands by differentiating the expenditure function.
   $h_1(u,p_1,p_2)=\frac{\partial e}{\partial p_1}=2\sqrt{u}\left(\frac{1}{2\sqrt{p_1}}\sqrt{p_2}\right)+a_1=a_1+\sqrt{u\frac{p_2}{p_1}}$
   $h_2(u,p_1,p_2)=\frac{\partial e}{\partial p_2}=a_2+\sqrt{u\frac{p_1}{p_2}}$

This example beautifully illustrates the entire workflow, from a utility function to all the associated demand and value functions, highlighting the power of duality and the associated lemmas.

Conclusion

This lesson concludes our deep dive into the theory of consumer choice. We have moved beyond simple supply and demand curves to understand the rigorous microfoundations based on constrained optimization.

Key Takeaways:

• The indirect utility function $v(\mathbf{p},w)$ is the value function of the utility maximization problem. It tells you the maximum utility you can achieve given market conditions.
• The expenditure function $e(\mathbf{p},u)$ is the value function of the expenditure minimization problem. It tells you the minimum cost to achieve a certain level of happiness.
• These two functions are duals and mathematical inverses of each other, containing all essential information about a consumer's preferences.
• Roy's Identity and Shephard's Lemma are powerful results derived from the envelope theorem that allow us to recover Marshallian and Hicksian demand functions directly from these value functions through differentiation.

Preview for the Next Module:
We now transition from the consumer to the firm. In the next module, Producer Theory and Firm Supply, you will find that the mathematical structure of the firm's problem is strikingly similar to the consumer's. The firm's cost minimization problem, for instance, is a direct analogue of the consumer's expenditure minimization problem. The tools of constrained optimization and the concept of duality will be just as central to our analysis of the firm's behavior.