Expenditure Minimization and Hicksian Demand

1. The Duality of Consumer Choice: Expenditure Minimization

The Expenditure Minimization Problem (EMP) flips the utility maximization problem on its head.

• Objective: Minimize total expenditure, $E=p_1{x}_1+p_2{x}_2$.
• Constraint: Achieve a fixed, target level of utility, $u(x_1,x_2)=\bar{u}$.

Formally, the problem is:
${\min }_{x_1,x_2}E=p_1{x}_1+p_2{x}_2$
$\text{s.t.}u(x_1,x_2)=\bar{u}$

Notice the beautiful symmetry with the Utility Maximization Problem (UMP). What was the objective in the UMP (utility) is now the constraint, and what was the constraint (the budget) is now the objective (expenditure). This relationship is known as duality.

The solution to this problem yields Hicksian demand functions, named after economist John Hicks. They are denoted as $h_1(p_1,p_2,\bar{u})$ and $h_2(p_1,p_2,\bar{u})$. Crucially, they depend on prices and a target utility level, not income. They are also called compensated demand functions because they describe how consumption would change if, after a price change, the consumer were given just enough income (i.e., "compensated") to remain on their original indifference curve.

For a clearer understanding of this concept, you can refer to the following resource.

Graphical Intuition

The graphical representation of the EMP is distinct from that of the UMP. In the UMP, the budget line was fixed, and we sought the highest attainable indifference curve. Here, the indifference curve is fixed ($\bar{u}$), and we seek the lowest possible budget line that just touches it.

The tangency condition—that the slope of the indifference curve (MRS) equals the slope of the budget line (price ratio)—remains the same. The difference is the problem we are solving to find that tangency point.

2. Deriving Hicksian Demand: The Lagrangian Method

Given your physics background, you'll find that the mathematical process for solving the EMP is very familiar. We will again use the method of Lagrange multipliers.

The Lagrangian function for the EMP is:
$\mathcal{L}(x_1,x_2,\lambda )=(p_1{x}_1+p_2{x}_2)+\lambda (\bar{u}-u(x_1,x_2))$

Note the structure: $\mathcal{L}=\text{Objective Function}+\lambda \times (\text{Constraint})$.

The following video provides a clear, step-by-step walkthrough of solving the EMP for a specific utility function.

The core steps from the video are:

1. Set up the Lagrangian: $\mathcal{L}=p_1{x}_1+p_2{x}_2+\lambda (\bar{u}-x_1{x}_2^2)$
2. Find the First-Order Conditions (FOCs):
   • $\frac{\partial \mathcal{L}}{\partial x_1}=p_1-\lambda x_2^2=0$
   • $\frac{\partial \mathcal{L}}{\partial x_2}=p_2-2\lambda x_1{x}_2=0$
   • $\frac{\partial \mathcal{L}}{\partial \lambda }=\bar{u}-x_1{x}_2^2=0$
3. Solve for the Tangency Condition: By combining the first two FOCs to eliminate $\lambda$, we get a relationship between $x_1$ and $x_2$, which is equivalent to $\text{MRS}=p_1/p_2$.
4. Substitute into the Constraint: This relationship is then substituted into the utility constraint ($x_1{x}_2^2=\bar{u}$) to solve for each good individually as a function of prices and the utility level $\bar{u}$.

The final result is the Hicksian demand functions, which depend on $(p_1,p_2,\bar{u})$.

3. The General Cobb-Douglas Case

In the previous lesson, we saw that for a Cobb-Douglas utility function $U=x_1^{\alpha }{x}_2^{\beta }$, a consumer spends a constant fraction of income on each good. Let's see what the Hicksian demand looks like for this same utility function. This will highlight the difference between Marshallian and Hicksian demand very clearly.

The following video derives the Hicksian demand for a general Cobb-Douglas utility function. It reinforces the Lagrangian method and provides a general solution.

As derived in the video, the Hicksian demand functions for $U=x_1^{\alpha }{x}_2^{\beta }$ are:

$h_1(p_1,p_2,\bar{u})={\bar{u}}^{\frac{1}{\alpha +\beta }}{\left(\frac{\alpha p_2}{\beta p_1}\right)}^{\frac{\beta }{\alpha +\beta }}$
$h_2(p_1,p_2,\bar{u})={\bar{u}}^{\frac{1}{\alpha +\beta }}{\left(\frac{\beta p_1}{\alpha p_2}\right)}^{\frac{\alpha }{\alpha +\beta }}$

Compare this to the Marshallian demands we found in the last lesson:
$x_1^{\ast }(p_1,p_2,I)=\left(\frac{\alpha }{\alpha +\beta }\right)\frac{I}{p_1}$

The difference is stark. Marshallian demand depends linearly on income $I$. Hicksian demand depends on the utility level $\bar{u}$ and a complex term involving the price ratio. This makes sense: Hicksian demand tells you how to combine goods to hit a utility target as cheaply as possible, which depends fundamentally on their relative prices.

Conclusion

Today we explored the dual problem of consumer choice, leading us to a new and important concept.

Key Takeaways:

• The Expenditure Minimization Problem (EMP) seeks the least costly consumption bundle to achieve a fixed utility level. It is the dual of the utility maximization problem.
• The solution to the EMP provides the Hicksian (compensated) demand functions, $h_i(\mathbf{p},u)$, which give the quantity demanded as a function of prices and a target utility level.
• The derivation involves the same Lagrangian method as before, but the objective function (expenditure) and the constraint (utility) are swapped.
• Graphically, Hicksian demand corresponds to points of tangency between budget lines and a single, fixed indifference curve.
• Unlike Marshallian demand, which depends on income, Hicksian demand depends on utility. This isolates the effect of price changes on consumption, holding well-being constant.

Preview for the Next Lesson:
We now have two types of demand functions: Marshallian $x^{\ast }(p,I)$ and Hicksian $h(p,u)$. Why do we need both? Because together, they allow us to perform one of the most important analyses in microeconomics: decomposing the total effect of a price change into two separate components. In the next lesson, we will introduce the Slutsky equation, which uses both Marshallian and Hicksian demand to separate the substitution effect from the income effect.