Question

1. Suppose a price-taking consumer chooses goods 1 and 2 to maximize her utility given her wealth. Her budget constraint could be written as p1x1 + p2x2 = w, where (p1,p2) are the prices of the goods, (x1,x2) denote quantities of goods 1 and 2 she chooses to consume, and w is her wealth. Assume her preferences are such that demand functions exist for this consumer: xi(p1,p2,w),i = 1,2. Prove these demand functions must be homogeneous of degree zero.

Answer 1

As per above question we are given that consumer has option of purchasing two goods ( x1 and x2) at their respective given prices (p1 and p2). The consumer has to choose what quantity of x1 and x2 to be purchased at p1 and p2 given total wealth w to maximize the utility level which is determined by tangency ofBudget constraint ( p1.x1 + p2.x2 = w) and Indifference curve to maximie the utiity with the given wealth.

ll (marshallian) demand functions are homogenous of degree zero in prices and income. This is a well known result.

So if we multiply all prices and income by the same amount, demand doesn't change. This is very intuitive. Imagine the government would just add a zero to all prices and to your paycheck. There is no reason for you to change your demand as nothing really got cheaper nor did your income increase in real terms.