Question

Suppose an individual consumes two goods – X and Y, which are substitutes such that the consumer is willing to trade 2 units of X for one unit of Y (that is, two units of X give the same utility as one unit of Y).

(a) If Px = $5 and Py = $6, and the income of the individual is $600, then what is the consumption bundle that will maximize the utility of the individual? Show the calculation/intuition behind your answer. Also draw the corresponding indifference curves and budget lines that demonstrate your answer.

(b) What is the new optimal bundle if price of good X falls to $2 per unit? Indicate this equilibrium on a new graph (with the corresponding indifference curve and budget line).

(c) For the answer in part (b) please show the corresponding substitution and income effects o n a graph. Also discuss the associated calculation/intuition.

(d) Assume the conditions in (a) with one difference – the individual now considers the two goods to be perfect complements (instead of substitutes), that is, the consumer use the goods X and Y in a 1:1 ratio (akin to the right shoe-left shoe example). What is the new optimal bundle of the consumer? Explain your answer using both a graph, and mathematical/economic intuition.

Answer 1

The optimal consumption bundle for a consumer treating goods X and Y as substitutes would involve spending all their income on the cheaper per unit utility good. When the price of good X drops, the substitution and income effects lead to purchasing more of good X due to its increased attractiveness and the consumer's greater purchasing power. If the goods are perfect complements, the consumer divides their income to buy equal quantities of both goods.

Step-by-step explanation:

To determine the consumption bundle that will maximize utility for an individual considering goods X and Y as substitutes with prices Px = $5 and Py = $6 and an income of $600, we need to compare the utility gained per dollar spent on each good. Because the consumer is willing to trade 2 units of X for 1 unit of Y, the consumer will gain more utility per dollar by spending entirely on X if the price of X is less than twice the price of Y. With Px = $5 and Py = $6, the individual would spend all $600 on good X, thereby buying 120 units of X (since $600/$5 per unit of X = 120 units of X).

When the price of good X falls to $2 per unit, the new optimal bundle will still involve buying only good X, now with even more units being able to be purchased due to the price drop. With $600 of income and Px now at $2, the individual can buy 300 units of X ($600/$2 per unit of X).

For part (b), the substitution and income effects explain why more of good X is purchased when its price drops. The substitution effect encourages buying more of the cheaper good (X) and less of the relatively more expensive good (Y). As for the income effect, because the purchasing power has increased, the individual can now afford to buy more of both goods but will still prefer X entirely due to the constant rate of substitution.

If goods X and Y are perfect complements and used in a 1:1 ratio, then the consumer will spend their income equally on X and Y to match the consumption one for one. For prices Px = $5 and Py = $6, and an income of $600, we divide the income equally to spend $300 on each good. This results in buying 60 units of X ($300/$5) and 50 units of Y ($300/$6), ensuring no units of X are left without a corresponding unit of Y and vice versa.