Question

For each of the following, graphically decompose the total effect of the price change into the substitution and income effects on a well-annotated graph. Be sure to clearly label each of the three effects.

2. Paula’s preferences for goods x and y are given by the utility function, U = x 1/2 y 1/2 . Her income is $160, and the price of good y is always $20. Suppose the price of x starts at $20 and then decreases to $5.

A. Calculate the income, substitution, and total effects of the price decrease on both goods. Show your work for full credit and round to 2 decimal places where appropriate.

B. Use a clearly labeled graph to illustrate the total, income, and substitution effects of the price decrease on Paula’s consumption of goods x and y.

Answer 1

The total effect of a price change on a consumer's choice can be divided into the substitution effect and the income effect, graphically represented by changes in the consumer's budget line and levels of utility in response to the price change. For Paula, with her utility function and given income, the decrease in the price of good x results in a new consumption bundle that can be analyzed by drawing a compensated budget line parallel to the new budget line to separate the substitution effect from the income effect.

Step-by-step explanation:

To understand the question, it is essential to first identify what the total effect, substitution effect, and income effect mean in the context of consumer choice theory in economics. When the price of a good changes, these three effects describe how a consumer's purchasing decisions might alter.

The substitution effect occurs when a consumer responds to a price change by purchasing more of the cheaper good and less of the more expensive good, keeping the level of utility the same. The income effect happens when a price change increases a consumer's purchasing power, allowing them to reach higher levels of utility. The total effect is the combination of both the substitution and income effects. To graphically identify these effects, a dashed line is drawn, indicating a hypothetical situation where the consumer's income adjusts to keep utility constant amid price changes.

Let's take Paula's situation with her preferences defined by the utility function U = x1/2y1/2 and her initial income of $160. With prices initially set at $20 for both goods x and y, and then the price of x decreases to $5, we can illustrate these effects on a graph.

The initial budget constraint is drawn given the income and prices of the two goods. When the price of x decreases, the budget line rotates outwards, showcasing a new set of combinations of goods x and y that Paula can now afford. To isolate the substitution effect, we draw a hypothetical ('compensated') budget line, parallel to the new budget line but tangent to the original indifference curve at a point representing a different combination of goods but the same level of utility.

The movement from the original consumption bundle to the point on this hypothetical budget line represents the substitution effect, while the movement from the hypothetical point to the new consumption bundle on the actual new budget line represents the income effect. The combination of these two movements shows the total effect of the price change on Paula's consumption of goods x and y.