# Hicksian Compensated Demand

The Hicksian demand is the solution to the cost minimization problem in which the consumer chooses a bundle of goods to minimize the expenditure subject to a utility-level constraint. The Hicksian demand is also called the compensated demand.

## Intuition

In this minimization problem, we give the consumer a required level of utility and a set of prices, and ask the consumer to minimize the amount of money she needs to spend to achieve that level of utility. Another way of saying it is we are asking the consumer what the cheapest bundle of goods to purchase in order to achieve a certain level of welfare.

This is different from the Marshallian demand in which the consumer is given a certain amount of money and a set of prices, and she is trying to maximize her utility given the amount of money in her pocket. We are asking the consumer how happy she could be given some amount of money.

## Mathematical representation

The mathematical formulation of the cost minimization problem is as follows:

${\displaystyle min\sum p_{i}x_{i}\quad s.t.\ u(x_{1},x_{2},...,x_{n})\geq {\bar {u}}}$

where ${\displaystyle p_{i}}$ are a set of prices, ${\displaystyle x_{i}}$ are the quantity of goods and the choice variables, and ${\displaystyle {\bar {u}}}$ is the required level of utility.

The problem can be solved by the Lagrange method.

## Connection to Marshallian Demand

If the utility achieved in the Marshallian problem given prices (often called the value-function) and some income is used as the required level of utility in the cost-minimization problem, then the expenditure required to achieve the level of utility will be the same as the income given in the Marshallian problem. Mathematically,

${\displaystyle V(p_{i},E(p_{i},{\bar {u}}))={\bar {u}},or\ E(p_{i},V(p_{i},m))=m}$

where E is the expenditure function ${\displaystyle E(p_{i},u)}$ and V is the value function ${\displaystyle V(p_{i},m)}$

Moreover, the optimal choices of goods are the same in the two problems, under the above condition. Denote the compensated demand ${\displaystyle h(p_{i},u)}$, then

${\displaystyle h(p_{i},V(p_{i},m))=x(p_{i},m),or\ x(p_{i},E(p_{i},u))=h(p_{i},u)}$