Explain why the following is true: If preferences are homothetic, then both good

Question

Explain why the following is true: If preferences are homothetic, then both goods are normal.

Explain why the following is false: If both goods are normal, then preferences are homothetic.

True or False: For U(X, Y) and the income offer curve is either a vertical line or horizontal line not out of the origin, then the Engels curve for one of the goods will be vertical line not out of the origin.

Explain why the following is true: If the Engels curve is upward sloping then the demand curve is downward sloping.

The utility equals the square of x times the square of y and he spends $347 on good y.

a. How much does he spend on good x if its price is $113?

A person has homothetic preferences. He only buys brandy and cigars. He is a profligate old codger and spends all of his income on these two goods. He spends 76% of his income on brandy.

a. If his income increases by $200 how much money does he spend on cigars?

Explanation / Answer

1) The statement is true because homothetic preferences means those preferences which are monotonic transformation of homogenous preferences. Now, the homogenous preferences have this property that these preferences are linear in prices and income. It means, demand for the given commodity increases (decreases) in the same proportion to increase (decrease) in income. If a commodity's demand increase/decrease with the rise/fall in consumer's income then the goods are said to be normal goods.

2) If both goods are normal, then preferences are homothetic. This statement is not true because homothetic preferences have this fundamental property that the income offer curve of such preferences is a straight line passing through origin which means that the as income of the consumer increase/decreases, the expenditure on both goods increase/decrease in the same proportion. However, this is not true even if both the goods are normal goods. For some goods, like luxury cars, expenditure rise more than proportionaltely to rise in consumer's income.

3) For U(X, Y) and the income offer curve is either a vertical line or horizontal line not out of the origin, then the Engels curve for one of the goods will be vertical line not out of the origin.

This is true. Engel curve represents the relationship of a commodity with income of consumer. Engel curve is drawn in income-quantity space. A vertical engel curve means that a commodity's demand is not at all affected by income of the consumer. If this is to be shown in an income offer curve, with all other commoditites that the consumer buys, then the income offer curve can not pass through origin, it will be either horizontal or vertical.

4) It is true that if the Engels curve is upward sloping then the demand curve must slope downwards because an upward sloping engel curve shows positive income effect. A positive income effect means the good is a normal good and for such goods, demand always slopes downwards.

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