Quiz 5: Income and Substitution Effects

Business

a ) Let x denote 0.75-liter containers of water and y denote 2-liter containers of water. The two goods are perfect substitutes such that one unit of img gives as much utility as img If img and img are perfect substitutes such that one unit of img gives as much utility as img then the utility function can be written as: img Therefore, the utility function for the given scenario can be written as: img b ) From the utility function, calculate the marginal utilities of the two goods. img This is the case of perfect substitutes. Hence, the consumer will spend entire income on img if the price ratio img is less than or equal to the ratio of marginal utilities; otherwise, zero. That is, the consumer will buy img units of img if img Therefore, the demand function for img can be written as: img c) The demand curve for good img can be represented as the red curve in Figure 1. img d) There are two portions of the demand curve - the portion that intersects the horizontal axis and the potion that coincides the vertical axis. An increase in income will shift the former to the right. An increase in the price of good img will increase the length of the former and reduce the length of the latter. e) The compensated demand curve shows only the substitution effect and shows no income effect. In the case of perfect substitutes, there is no income effect. Therefore, the compensated demand curve is the same as the demand curve shown in Figure 1.

Complementary goods are the goods that are consumed together by a consumer. Consumption of a single commodity do not add to the utility and thus, when the price of good increases, the demand for its complement good falls. The income of the consumer is the total budget that the individual has to spend on all the available commodities. It is given that the person D likes to consume peanut butter and jelly sandwiches using 1 ounce of jelly and 2 ounces of peanut butter. Thus, the utility and demand function of the consumer is given as perfect complements. Now it is assumed that x stands for peanut butter and y stand for jelly. The utility function for the complementary goods is as follows: img The money income ( M ) of $3 and the price of peanut butter to be $0.05 and jelly to be $0.10. The budget constraint is as follows: img Since the two goods are used in fixed proportions, therefore img . Now, it is needed to calculate the demand function. Substitute the value of img in the equation (1) as follows: img Calculate the value of x as follows: img Hence, person D would buy peanut butter and img of jelly. b. When the price of jelly increases to $0.15 and it is needed to find the quantity the consumer would buy. As the price of jelly has increased to $0.15 and thus it is costlier than before for the consumer. The new budget constraint is as follows: img Substituting the value of img to get the new consumption bundle. The new consumption bundle for the consumer will be seen as done below: img Calculate the value of as follows: img Due to increase in price, person D consumes img of peanut butter and img of jelly. c. It is required to find the increasing amount of D's allowance in order to compensate for the rise in the price. The hike in the price of jelly has caused the consumption to fall. Thus, D must be compensated for the sacrificed units. Calculate the person D's allowance to compensation given a new income level and the change in the income level as follows: img Hence, person D 's allowance to compensate would be img . d. Represent the graph as follows: img All the above results are graphed in the above graph. e. It is required to graph the demand curve for the single commodity. The peanut butter and jelly are complements that are consumed together in Bread. Thus, bread consumption is the consumption of a single commodity. The units of bread consumed will stay fixed given the ounces of butter and jelly used. The demand curve shows the relationship between price and units of consumption pattern of an individual with a particular income level. Represent the graph as follows: img The units of bread consumed will stay fixed given the ounces of butter and jelly used. The demand curve shows the relationship between price and units of consumption pattern of an individual with a particular income level. f. Substitution effect refers to the consumption of units in such a way such that the original bundle is also affordable as income is substituted to the consumer. When the price rises to $0.15, the new income is given to be $3.75. Calculate the new bundle purchased as follows: img Calculate the value of x as follows: img Hence, the Substitution effect is img . There will be no change in the quantity of goods demanded at the new income level as they are used in fixed proportions and thus, no price hike can lead to a change unless income level is not substituted in such a way that the original bundle is affordable. Income Effect refers to the change in the purchasing power of the consumer due to the change in the price level of a commodity. The consumer has the same income level, thus a hike in price level leads to a fall in purchasing power. Calculate the new consumption bundle as follows: img Calculate the value of x as follows: img Calculate the income effect ( I.E.) as follows: img Hence, the income effect is img .

a) It is required to be shown that if a straight line through the origin cuts all indifference curves at the points of equal slope, then img is constant; that is, for any given set of prices, a given change in income always causes the same change in the quantity of good img bought/consumed. A straight line through the origin has a constant ratio of two goods at every point on the line. This means that if a straight line through the origin cuts all indifference cuts at the points of equal slope, then, MRS depends on the ratio img of the two goods. In this case, the following expression can be written: img where img is some constant For optimality, MRS should equal the price ratios. Hence, img Substitute the above expression in the budget equation. img img Find the partial derivative of the above equation with respect to img . img Hence, given a set of prices, the expression img is constant. Therefore, img is constant. (b) Find the partial derivative of the Equation (1) with respect to img . img Since img is negative. An increase in price of img causes a decrease in the quantity demanded of good img . Hence, Giffen's paradox cannot occur if a straight line through the origin cuts all indifference curves at the points of equal slope. That is, Giffen's paradox cannot occur in the case of a homothetic utility function.