The utility of x can be written as u(x). It is continuous and strictly increasing on Rn+. There are six properties of the indirect utility function.

Such properties are explained along with proof as follows:

Property 1:

Continuous on Rn++ * R+:

Proof:

The utility of x, u(x) is a continuous function. There might be the change in the income and price. The change in price is always observed with change in income. But first change in price is also possible. It means ΔP = ΔY, now ΔY = ΔM. Therefore consumer achieves the same utility which he/she was getting before the change in income and price. It is also called as compensation variation. Change in income is equivalent to the change in consumer’s budget constraint. Utility level remains unaffected and it remains continuous over the period of time. It is elaborated in more detail in the following property.

Property 2:

Homogenous of Degree Zero in (p, y):

Proof:

The proof of the property one and two of indirect utility function is given in a simple form.

Suppose v (tp, ty) = [max u(x) subject to t.p.x ≤ ty) which is clearly equivalent to max u(x) subject to p.x ≤ y]. This is because we may divide both sides of the constraints by t, where t > o and it is without affecting the set of bundles satisfying it. Consequently, v (tp, ty) = [max u(x) subject to p.x ≤ y] = U (p, y). It is simple proof of above property. It means utility of x is subject to price and income.

Property 3:

It is Strictly Increasing in y:

Such property is easy to prove. Therefore it is not explained here but the proof can be explained in the following property.

Property 4:

Indirect Utility Function is Decreasing in p:

Proof:

Property 3 and 4 is explained simultaneously as follows. Any consumer is always like to increase his/her utility. The consumer’s budget constraint can never cause the maximum level of achievable utility to decrease. With the given level of income and price consumer is always achieve highest utility on indifference curve. We assume that the utility is strictly positive and differentiable, where (p, y) » 0 and that u (0) is differentiate with (∂u/x) for all x » 0.

Homogeneity of the indirect utility function can be defined in terms of prices and income. Here u (.) is strictly increasing in this utility function. The utility constraint must stick to the optimum level. Therefore utility function is equivalent to

We can add Lagrangian for equation (24) then it becomes

For (p, y » 0), let us assume that x* = x (p, y). Now we need to solve it for equation (24). We have to add assumption x* » 0. Now we may apply again Lagrange’s theorem to conclude that there is a λ*ԑR such that

Both pi and δu(x*)/ δxi are positive in the above equation. We now apply Envelope theorem to establish that v (p, y) is strictly increasing in y. The partial derivative of the minimum value function v (p, y) with respect to y is equal to the partial derivative of the Langarangian. It is with respect to y evaluated at (x*, λ*),

Thus v (p, y) is strictly increasing with income. It is because v is continuous and increasing.

The elementary proof of equation does not rely on any additional hypothesis. If p0 ≥ p1 and equation can be solved when p = p0. Then this is x0 ≥ 0, (p0⎯ p1), x0 ≥ 0, hence p1.x0 ≤ p0 .x0 ≤ y. The x0 is feasible set when p = p1. The conclusion of the above property is that v (p1, y) ≥ u(x0) = v (p0 y). It is desirable conclusion from previous property.

Property 5:

Quasi Convex in (p, y):

In order to prove this property, we need to assume that a consumer would prefer one of any two extreme budget sets. The point is to show that v (p, y) is quasi-convex in the vector of prices and income (p, y). This proof is to concentrating on the budget sets.

Suppose β1 which is the budget set available to consumer then the budget sets are available when prices and income are (p1y1) (p2y2) and (ptyt) respectively. The available prices and income denoted as pta” tp1 + (1-t)p2 and yt = y + (1-t)y2. Here, we have taken three probabilities.

The choice is made by consumer when he/she faces the budget constraint βt. It is a choice made when he/she face either β1 or budget set β2. At each level of utility, he/she can achieve utility βt budget set. It is simple to understand but such level of utility which should have achieved either when facing β1 or β2 budget set. The maximum level of utility is achieved over βt budget set and it could be no longer than at least of the following.

The maximum level of utility he/she can achieve at β1 or at β2 budget set. For simplicity, the maximum level of utility achieved at βt. It can be no longer and larger of these two budget sets. Suppose the statement is correct then, we know that u(pt, yt) ≤ max [max [v[p1, y1), v(p2,y2)] ∀t ϵ [0,1]. It is equivalent to the statement that u (p, y) which is quasi-convex in (p, y).We want to show that if xϵβt then xϵβ1 or xϵβ2 for all t ϵ [0, 1], It is easy to understand. Suppose we choose either extreme value for t, then βt budget set co-insides with either β1 or β2 budget set. The relationship holds with little effect. It remains to show that they hold for all tϵ (0, 1).

Some Eϵ βt such that:

It is because tϵ (0, 1). First, we multiply by t and the second equation by (1-t).

We preserve the inequality to obtain following equation:

Adding above two equations, we obtain

Or

Above line says that x is not equal to ât. It is contradicting with our original assumption. We can conclude that if xϵβ1 then xϵβ1 or xϵβ2 for all tϵ (0, 1). From previous points, it can be desired that v (p, y) is quasi-convex in (p, y).

Property 6:

Roy’s Identity:

It explains that, consumer’s Marshallian demand for good is simply the ratio of the partial derivatives of indirect utility. It is with respect to pi and y after a change of sign. We have assumed that x* = x (p, y) is strictly positive solution. If we apply the envelope theorem to evaluate ∂U (p, y)/∂pi, then it gives following equation

According to above equation λ* = ∂u(p,y)/∂y > 0, hence equation (29) can be interpreted as

It is a desired function and proof of the above property.