Examen de Microeconomía II UPF 2015 · Mid-Term Exam May 2015

Exercise 1 (5 points). Jane's MRS for swimming and running

Jane derives utility from two activities: swimming and running. She feels completely indifferent between swimming 2 km and running 20 km (bundle A), and swimming 5 km and running 10 km (bundle B). At bundle A, Jane is ready to give up at most 5 km of running to swim an extra 1 km. At bundle B, she is willing to give up at most 1 km of running to swim an additional 1 km. What conclusion can be drawn regarding Jane's Marginal Rate of Substitution ($|\mathrm{MRS}|$)?

Exercise 2 (5 points). Monotonic transformation of Oliver's utility

Oliver's preferences over two goods are described by the utility function:

$$u = x_1^{1/3} x_2^{2/3} - 10$$

Which of the following functions represents a valid monotonic transformation representing Oliver's preferences? (Keep in mind that $u$ can take negative, zero, or positive values).

Exercise 3 (5 points). Billy's budget line with quantity tax

Billy has an income $m$ to divide between beer and natural juice. Under standard conditions, he consumes 10 beers. The authority decides to discourage excessive alcohol consumption by levying a per-unit tax $t$ on every beer purchased beyond the first 5 units. What happens to the shape of Billy's budget constraint?

Exercise 4 (5 points). Ana's expenditure on good 2 with perfect substitutes

Ana has preferences described by the utility function:

$$u(x_1, x_2) = \frac{x_1}{4} + \frac{3x_2}{4}$$

If Ana's total income is 20, how much money will she spend on good 2?

Exercise 5 (5 points). Ordinal interpretation of utility levels

Consider two bundles of goods, (bananas, peanuts). John Monkey's utility function gives values $U(2, 3) = 5$ and $U(3, 2) = 10$. What can be correctly stated about his preferences?

Exercise 6 (5 points). Strictly concave indifference curves

An individual's utility function is given by $U(x_1, x_2) = x_1^2 + x_2^2$. What does this function imply regarding the individual's consumption choices?

Exercise 7 (5 points). Conditions for strictly positive demand

According to consumer theory, the optimal demand for every good is guaranteed to be strictly positive (interior solution) if:

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Exercise 8 (5 points). Income and substitution effects for utility depending on one good

Considering consumption bundles of (bananas, peanuts), suppose John Monkey's utility function is $U(x_1, x_2) = 3x_1$, where $x_1$ represents bananas. If the price of bananas increases, what describes the resulting change in demand?

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Exercise 9 (5 points). Engel curve for perfect complements

For bundles composed of (bananas, peanuts), suppose John Monkey's utility function is $U(x_1, x_2) = e\min(x_1, x_2)$. What is the shape of the Engel curve for bananas?

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Exercise 10 (5 points). Jordi's heroin consumption and Giffen goods

Jordi has a drug addiction such that whenever the market price of heroin rises, he ends up buying more of it. Which of the following assertions is correct?

Exercise 11 (5 points). Marina's demand function with quasilinear-like preferences

Marina's preferences are given by the utility function $U(x_1, x_2) = x_1 x_2 + x_1$. What is her demand function for good 1, denoted $X_1(p_1, p_2, m)$?

Exercise 12 (5 points). Slutsky substitution effect for perfect substitutes

Yulia's preferences are represented by the utility function:

$$U(x_1, x_2) = x_1 + x_2$$

When prices are $p_1 = p_2 = 3$, Yulia chooses the bundle $(10, 10)$. If the price of good 1 increases to $p'_1 = 4$, what is the substitution effect on $x_1$ according to Slutsky's definition?

Exercise 13 (5 points). Ivan's inverse demand function for perfect complements

Ivan's preferences are described by the utility function:

$$U(x_1, x_2) = \min(x_1, 2x_2)$$

Which expression represents the inverse demand function for good 2, $p_2(x_2)$?

Exercise 14 (5 points). Revealed preference axioms (WARP and SARP)

In 2013, Mark had sufficient budget to afford bundles A and B, and he chose B. In 2014, he had enough budget to afford bundles A and C, and he chose C. In 2015, all three bundles (A, B, and C) are affordable to him. Which revealed preference axioms are consistent with Mark choosing bundle C in 2015?

Exercise 15 (5 points). Price effect decomposition for perfect substitutes

Suppose goods $x_1$ and $x_2$ are perfect substitutes. Which of the following statements is unconditionally true regarding the decomposition of a price change?

Exercise 16 (5 points). Utility with a satiation point

Elena's preferences are represented by the utility function:$$u(x_1, x_2) = -(x_1 - 5)^2 - (x_2 - 2)^2$$What can be deduced regarding her utility-maximizing choice?

Exercise 17 (5 points). Properties of quasilinear preferences

For an individual possessing quasilinear preferences, which of the following statements is true?

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Exercise 18 (5 points). Welfare change from a price decrease

If the price of good 2 decreases, can we guarantee that the individual will be strictly better off or at least as well off?