**A** **consumer’s indirect utility function** may be expressed as : **v (p, m)**

(**Where p stands for vector of prices of commodities **

**and m stands for his/her budget under constraint)**

Actually, it reflects the consumer’s maximum attainable utility and his/her preferences and the market conditions.

Here, the derivatives so far used are more fundamentally related to Slutsky equation.

Now, the **above function is termed “indirect” because consumers usually think about their preferences in terms of what they consume rather than the price**.

## Properties & Characteristics :

Suppose, the **consumer’s utility function** i.e. **u (x) is continuous and represents non-satiated** (that is more is always preferred to less) preference relation, then **Hicksian demand function correspondence h (p, u)** satisfies the following properties:

**A)** Homogeneity of zero in price and income. If price and income are all multiplied by a given constant (say a) the same bundle of consumption represents a maximum.

**B)** No excess demand.

**C)** The indirect utility function is the inverse of the expenditure function when prices are kept constant i.e. for every price vector p and utility u, v (p,e (p,u) ) = u.

This is basically known as **Roy’s identity.** Actually, Roy’s identity is a major result in microeconomics and related to the application in consumer choice and the theory of the firm.

It also relates to Marshallian demand function to the derivatives of the indirect utility function.

**D)** The indirect utility function is closely related to the utility maximization problem (UMP).

## QUASILINEAR UTILTY as per GORMAN POLAR:

When the utility function of agent” i “has the form:

**(x, m) = (x) + m**

The indirect utility function has the form:

**(p, m) = (p) +m.**

Actually, Marshallian demand maximizes utility subject to consumer’s budget; this is a function of price and income.

Substituting Marshallian demand in the utility function, we obtain indirect utility as a function of prices and income.

The duality between Marshallian and Hicksian demand is that all information about the consumer captured by u(x) is also captured by v(p,m).