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Behavior and Choice: EC 102

Andrew Monaco

Exercises 2.7 Practice Problems

Subsection 2.7.1 Preferences and Rationality

  1. Write about a situation where you formed preferences over several possible alternatives, and write out your preference ranking. These could be choices while shopping, or destinations for an activity or meal. What informed your preferences?
  2. What two requirements define rational decision making?
  3. What does mean for a preference ranking ...
    1. to be complete?
    2. to be transitive?
    3. What do we gain by requiring that preferences are complete?
    4. What do we gain by requiring that preferences are transitive?
  4. How realistic is the assumption of completeness (A1)? Explain your reasoning.
  5. How realistic is the assumption of transitivity (A2)? Explain your reasoning.

Subsection 2.7.2 Utility

  1. What is utility? What does it measure?
  2. How does the concept of utility relate to consumer preferences?
  3. What does mean for a preference ranking ...
    1. to satisfy monotonicity?
    2. to satisfy diminishing marginal utility?
    3. What do we gain by requiring that preferences satisfy monotonicity?
    4. What do we gain by requiring that preferences satisfy diminishing marginal utility?
  4. How realistic is the assumption of monotonicity (A3)? Explain your reasoning.
  5. How realistic is the assumption of diminishing marginal utility (A4)? Explain your reasoning.
  6. Consider your preferences for the following consumption bundles, where \((x, y)\) represents a number of slices of pizza (\(x\)) and soft drinks (\(y\)): \((0, 0); (1, 1); (2, 1); (2, 0); (1, 2); (3, 0)\text{.}\)
    1. Give your preference ranking across the bundles.
    2. Assign a utility value to each bundle.
    3. Describe how the utility value represents your preference ranking.
  7. Consider your preferences for the following living situations, where \((x, y)\) represents a number of human roommates (\(x\)) and number of dogs (\(y\)): \((0, 0); (0, 1); (1, 0); (2, 0); (0, 2); (0, 3); (1, 1)\text{.}\)
    1. Give your preference ranking across the bundles.
    2. Assign a utility value to each bundle.
    3. Describe how the utility value represents your preference ranking.
    1. Name and define the four assumptions which typically describe consumer preferences.
    2. Do we typically model consumers’ utility functions as increasing or decreasing? Explain why.
    3. Do we typically model consumers’ utility functions as concave or convex? Explain why.
  9. Consider the table below, which shows Andrew’s utility from having slices of pizza.
    slices of pizza (\(x\)) units of utility marginal utility
    0 0 –
    1 30
    2 50
    3 60
    4 65
    1. Fill in the far right column and calculate Andrew’s marginal utility for each additional slice of pizza.
    2. Does Andrew’s utility satisfy assumption A3? How can you see this in the utility column? How can you see this in the marginal utility column?
    3. Does Andrew’s utility satisfy assumption A4? How can you see this in the utility column? How can you see this in the marginal utility column?
    4. Draw Andrew’s utility function, with slices (\(x\)) on the x-axis, and utility on the y-axis.
    5. How would you describe the shape of this function? How does this shape relate to assumptions A3 and A4?
  10. On a graph with units of good \(x\) on the horizontal axis, and utility on the vertical axis, draw a utility function which satisfies assumptions A3 and A4. [Bonus: Give an example of a mathematical function which would give this shape!]
  11. Draw the following functions. Do they satisfy assumption A3? Assumption A4?
    1. Utility where the utility function is given by \(u(x) = \sqrt{x}\)
    2. Utility where the utility function is given by \(u(x) = -(x - 3)^2 + 5\)
    3. Utility where the marginal utility is given by \(MU_x = 10 - x\)
    4. Utility where the marginal utility is given by \(MU_x = \frac{1}{x}\)

Subsection 2.7.3 Budget sets

  1. A consumer considers the choice of consumption bundles \((x, y)\) which are made up of two goods: books (\(x\)) and pens (\(y\)).
    1. Write the general definition of the consumer’s budget constraint.
    2. Write the budget constraint when \(P_x = 20\text{,}\) \(P_y = 4\text{,}\) and \(I = 60\text{.}\)
    3. Draw the budget constraint when \(P_x = 20\text{,}\) \(P_y = 4\text{,}\) and \(I = 60\) in a clearly labeled (axes, intercepts) graph.
    4. Label at least 3 affordable bundles on your graph. Label at least 3 unaffordable bundles on your graph.
    5. What happens to the collection of affordable bundles if \(P_y\) increases from 4 to 6? Draw this change, and give a new equation for the budget constraint.
  2. Charlie, a consumer, wants to spend $30 on two goods: candy bars (\(x\)) and bottles of soda (\(y\)).
    1. If candy bars and soda each cost $1, what is Charlie’s budget constraint? Draw it and Charlie’s budget set.
    2. What is the slope of the budget constraint? How does it relate to the two prices?
    3. If the cost of a candy bar increases to $1.50, how does Charlie’s budget constraint shift? Illustrate the new budget constraint and new budget set on the same graph in a new color.
    4. What is the new slope of the budget constraint? Does it change relative to the original slope?
    5. Candy bars now cost $1.50, while bottles of soda cost $1. If Charlie find a crisp $10 bill on the ground, to add to his $30 to spend, where is his new budget constraint? Illustrate this and his new budget set.
    6. What is the new slope of the budget constraint? Does it change relative to the original slope?

Subsection 2.7.4 Consumer optimal choice & 2.6 Deriving demand

  1. What does the term \(\frac{MU_x}{P_x}\) represent? Explain both in words and with a numerical example.
  2. Give the two conditions which should be satisfied at a consumer’s optimal bundle of goods, \((x^*, y^*)\text{.}\) Interpret each condition in words.
  3. A consumer wants to choose an optimal consumption bundle of books (x) and pens (y).
    1. What two mathematical conditions should be satisfied at this optimal bundle?
    2. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(P_x = 5\text{,}\) \(P_y = 5\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    3. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) \(MU_y = \frac{1}{y}\text{,}\) and where \(P_x\) and \(P_y\) are parameters that have not been assigned numerical values. Notice that here, your solution will not be numerical.
  4. A consumer wants to choose an optimal consumption bundle of books (x) and pens (y).
    1. What two mathematical conditions should be satisfied at this optimal bundle?
    2. Find the optimal bundle \((x^*, y^*)\) when \(I = 20\text{,}\) \(P_x = 10\text{,}\) \(P_y = 2\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    3. Find the optimal bundle \((x^*, y^*)\) when \(I = 20\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) \(MU_y = \frac{1}{y}\text{,}\) and where \(P_x\) and \(P_y\) are parameters that have not been assigned numerical values. Notice that here, your solution will not be numerical.
    4. (Extra credit 2 points) From part c., what does \(x^*\) represent? What does \(y^*\) represent? Explain in words, and draw a carefully-labeled graph to support your explanation.
  5. A consumer wants to choose an optimal consumption bundle of books (x) and pens (y).
    1. What two mathematical conditions should be satisfied at this optimal bundle?
    2. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(P_x = 2\text{,}\) \(P_y = 2\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    3. Suppose the price of \(y\) increases to 5. What happens to the budget set? Write a new equation and draw the new budget set.
    4. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(P_x = 2\text{,}\) \(P_y = 5\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    5. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) \(MU_y = \frac{1}{y}\text{,}\) and where \(P_x\) and \(P_y\) are parameters that have not been assigned numerical values. Notice that here, your solution will not be numerical.
    6. Graph your answer for \(y^*\text{.}\) What is this curve? Illustrate the two points you found in parts b and d.!
  6. A consumer wants to choose an optimal consumption bundle of books (x) and pens (y).
    1. What two mathematical conditions should be satisfied at this optimal bundle?
    2. Find the optimal bundle \((x^*, y^*)\) when \(I = 100\text{,}\) \(P_x = 2\text{,}\) \(P_y = 2\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    3. Suppose \(I\) increases to 120. What happens to the budget set? Write a new equation and draw the new budget set.
    4. Find the optimal bundle \((x^*, y^*)\) when \(I = 120\text{,}\) \(P_x = 2\text{,}\) \(P_y = 2\text{,}\) \(MU_x = \frac{1}{x}\text{,}\) and \(MU_y = \frac{1}{y}\text{.}\)
    5. Find the optimal bundle \((x^*, y^*)\) when \(MU_x = \frac{1}{x}\text{,}\) \(MU_y = \frac{1}{y}\text{,}\) and where \(I\text{,}\) \(P_x\) and \(P_y\) are parameters that have not been assigned numerical values. Notice that here, your solution will not be numerical.
    6. What does \(y^*\) represent? Illustrate the two \(y^*\) points you found in parts b and d. in a graph of this \(y^*\) curve (or curves).
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