Exercise 3
UTILITY MAXIMIZATION

I. Objectives

• To obtain the optimal bundle at the point of tangency between the indifference curve and the budget constraint
• To analyze how a change in the price of a good affects consumption
• To distinguish between the substitution effect and the income effect
• To derive a negatively-sloped demand curve

II. Data

• A consumer buys two goods, coconuts and muffins. Prices of each are denoted by P₁ and P₂; the consumer's income by M. Denote the quantity of coconuts by X, that of muffins by Y.

• Assume the following: P₂ = $1, M = $100. The budget constraint may then be written as:
  P₁X + Y = 100

• Using the calculator below:
  1. Select a value for the price of coconuts.
  2. Click on Gimme Maximum Utility! to obtain the optimal quantity of each good.
  
  
• Technical note: For those curious about the underlying computations, the utility function used is: u = X^0.25Y^0.75, and the corresponding optimal amounts are:
  X = 0.25(M/P₁) and Y = 0.75(M/P₂).

III. Questions

1. Select a value for P₁. Sketch the budget constraint. Sketch the indifference curve and indicate the optimal bundle on the graph. [Obtain the optimal values from the calculator above.]

2. Increase P₁ by 10%. The quantity demanded of coconuts will [ increase / decrease ] to ________. Sketch the new budget constraint and indifference curve. Indicate the new bundle on the graph. Has the consumer's utility increased? Explain.

   [Note a peculiar thing here: The quantity demanded of muffins has not changed. This result is unusual, and it is due to the choice of the utility function in the exercise. In a typical case, the quantity of muffins too would have changed as a result of the increase in the price of coconuts.]

3. Consider the case where coconuts become cheaper. Since a decrease in P₁ leads to [ more / less ] consumption of coconuts, we may conclude that the substitution effect is [ greater / less ] than the income effect. Explain.

4. Consider the effect of a decrease in P₁ on the quantity demanded of muffins. According to the substitution effect, the consumer will buy [ more / fewer ] muffins, while the income effect leads the consumer to buy [ more / fewer ] muffins. [Since the quantity of muffins has remained unchanged in this exercise (see table above), we may conclude that the two effects have exactly offset each other. As I said earlier, highly unusual.]

5. Change P₁ a few more times. Obtain the corresponding quantity of coconuts in each case. Sketch a demand curve for coconuts.