Prove Hicksian Demand Function is Continuous

Here is a detailed discussion on ordinary and compensated demand function.

1. Ordinary Demand Function :

A consumer's ordinary demand function (called a Marshallian demand function) shows the quantity of a commodity that he will demand as a function of market prices and his fixed income. Demand functions can be derived from the utility-maximising behaviour of the consumer (i.e., maximisation of u = f(x₁, x₂), subject to m̅ = p₁x₁ + p₂x₂.

The first-order conditions for maximisation consists of the following three equations in the three unknowns: x₁, x₂ and λ:

The demand functions are derived by solving this system for the unknowns. The solutions for x₁ and x₂ are in terms of the parameters p₁, p₂ and m̅. In general, the quantity of x₁ or x₂ that the consumer buys depends upon the prices of all commodities and his income.

Let us take a difference type of utility function which appears in a multiplicative form:

u = x₁x₂.

The budget constraint remains the same:

m̅ – p₁x₁ – p₂x₂ = 0.

Now we form the expression:

However, the quantities demanded by the consumer which can be derived from these demand functions are the same as those derived directly from the utility function.

Properties:

Ordinary demand functions have two important properties.

These are the following:

1. The demand functions are single-valued:

The demand function of commodity is a single valued function of prices and income. At a particular price and fixed income, a certain quantity will be demanded — neither one unit more nor one unit less. This property follows from the strict quasi-concavity of the utility function.

The ordinal utility function is u = f(x₁, x₂), where (x₁ and x₂) are the quantities of the two commodities which the consumer purchases. It is assumed that f(x₁, x₂) is continuous, has continuous first-and second-order partial derivatives, and is a regular strictly quasi- concave function.

A strictly quasi-concave function is one for which:

2f₁₂ f₁ f₂ – f₁₁ f₂ ² – f₂₂ f₁ ² > 0.

A single maximum and, therefore, a single commodity combination corresponds to a given set of prices and fixed income. If the utility function were quasi-concave and not strictly quasi-concave, the indifference curves would possess straight line portions, and maxima would not need to be unique.

In this case, more than one value of the quantity demanded may correspond to a given price and the demand relationship is called a demand correspondence than a demand function. See Fig. 7.1 where the three points E, F and G are equilibrium points. Although equilibrium exists it is not unique.

In other words, Fig. 7.1 shows a situation of multiple equilibria which arises when demand functions are not single valued.

2. The demand functions are homogeneous of degree zero in prices and income:

This means that if all prices and income change in the same proportion the quantities demanded remain unchanged. Alternatively stated, the consumer will not behave as if he were richer (or poorer) in terms of real income if his money income and prices rise in the same proportion. To prove this let us assume that all prices and money income change in the same proportion and in the same direction.

Now the budget constraint becomes:

αm̅ – αp₁x₁ – αp₂x₂ = 0

where a is the proportionality factor.

The Lagrange function now becomes:

L = f (x₁, x₂) + λ (αm̅ – αp₁x₁ – αp₂x₂)

And the first-order conditions for maximisation are:

f₁ – λαp₁ = 0

f₂ – λαp₂ = 0 …. (1)

αm – αp₁x₁ – αp₂x₂ = 0

The last equation of (1) is the partial derivative of L with respect to the Lagrange multiplier and can be expressed as

a(m̅ – p₁x₁ – p₂x₂) = 0

Since a ≠ 0

m̅ – p₁x₁ – p₂x₂ = 0

Eliminating a from the first two equations of 1 by moving the second term to the right hand side and dividing the first equation by the second we get-

f₁/f₂ = p₁/p₂

which is the original equilibrium condition of the consumer. Therefore the demand function for the price-income set (αp₁, αp₂, αm) is derived from the same equations as the price- income set (p₁, p₂, m). It is also easy to prove the validity of the second-order conditions. This proves the homogeneity property of the demand function. Such functions are homogeneous of degree zero in prices and income.

If all absolute prices and consumer's money income rise proportionately his real income (purchasing power) will remain unchanged. So there is no reason why the consumer should alter his equilibrium purchase of any of the commodities.

Interpretation of the Result:

This property implies that the consumer will not behave as if he were richer (poorer) in terms of real income if his income and market prices rise in the same proportion. A rise in money income is no doubt beneficial for the consumer, ceteris paribus, but its benefits are offset if prices rise proportionately. If such proportionate changes leave his behaviour unaltered, there is an absence of money illusion.

A rise in money income is desirable for the consumer cet. par, but its benefits are illusory if prices change proportionately. If such proportionate changes leave his behaviour unaltered, there is an absence of money illusion. If prices rise due to an increase in money supply but money income remains constant, the quantity theory of money holds.

While some production functions are homogeneous by assumption, all demand functions are homogeneous by nature (at least if we ignore money illusion).

In terms of the generalized demand function involving n number of commodities we have:

X(αP₁, αp₂, … αp_n, am) = X(p₁, p₂, … p_n, m)

for all p₁, p₂, … p_n and m. Here X = x₁ + x₂ + … + x_n.

Since α⁰ = 1 (where 0 is the degree of homogeneity) this equation states that demand is homogeneous of degree zero in prices and income. Since each individual demand function is homogeneous of degree zero, the sum of these individual demand functions, aggregate demand, is also homogeneous of degree zero.

2. Compensated Demand Function:

Let us now consider a situation in which the government taxes or subsidizes a consumer in such a way as to leave his total utility unchanged after a price change. This can be done by making a lump- sum payment that will give the consumer the minimum income necessary to achieve his initial utility level.

The consumer's compensated demand functions give the quantities of the two goods that he will buy as functions of p₁ and p₂ under these unchanged utility conditions. These functions are arrived at by minimising the consumer's expenditure subject to the constraint that his utility remains fixed at the level u̅.

Using the multiplicative, rather than implicit, utility function, we form the Lagrangian:

These functions are also homogeneous of degree zero in prices, but not in income because total utility instead of money income appears in the Lagrangian (L').

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Source: https://www.microeconomicsnotes.com/demand/demand-function/properties-of-demand-function-microeconomics/13701

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