Hicksian Demand

Micro

This is also a question in Consumer Theory

Normally, from the Consumer Theory, we are studying the Marshallian Demand question, i.e., given prices and income, what is the optimal consumption bundle that maximizes the utility.

So this is also why we say this is a dual (对偶) perspective, because in the Marshallian Demand, we are maximizing utility given budget constraint, while in the Hicksian Demand, we are minimizing expenditure given utility constraint. Formally, it is:

$$\underset{h\in {\mathbb{R}}_{+}^L}{\min }\sum _{\ell =1}^L{p}_{\ell }{h}_{\ell }\text{s.t.}u(h)\ge U$$

Here $h$ means the consumption bundle in the Hicksian Demand context. (We use $x$ to denote consumption bundle in the Marshallian Demand context).

Notice that $h(p_1,\dots ,p_L)=x(p_1,\dots ,p_L,e(p,u))$, where $e(p,u)$ is the expenditure function that tells us the minimum expenditure needed to achieve utility level $u$ at prices $p$.

A relationship between $h$ and $x$:

$$\frac{\partial e(p,u)}{\partial p_{\ell }}=h_{\ell }(p,u)=x_{\ell }(p,e(p,u))$$

The intuition behind this equation is that, the change in expenditure with respect to price change is equal to the quantity demanded of that good in the Hicksian Demand context, which is also equal to the quantity demanded of that good in the Marshallian Demand context when income is set to the minimum expenditure needed to achieve utility level $u$.

The proof method is so called Envelope Theorem. But in this page, we mainly care about what we could use these equations to derive the Slutsky Equation.

Slutsky Equation

Two key words: Income Effect and Substitution Effect

Recall that $h(p_1,\dots ,p_L)=x(p_1,\dots ,p_L,e(p,u))$, and it is then obvious that $w=e(p,u)$.

We do the following: