Q: Given two points on a linear demand curve: $$(P_1$, Q_1$) = (10, 100)$$ and $$(P_2$, Q_2$) = (8, 140)$$, what is the point price elasticity of demand at $$P_1$$?

$$\frac{\Delta Q}{\Delta P} \times \frac{P_1$}{Q_1$} = \frac{40}{-2} \times \frac{10}{100} = -20 \times 0.1 = -2$$

Q: If the median income rises from $$25,000 to$$ $$45,000$$, what is the percentage change in income?

$$\frac{45,000 - 25,000}{25,000} \times 100 = 80\%$$

Question 1

A firm is deciding how to allocate its first $$100,000 of advertising spending among radio, TV, and internet. The marginal sales (in barrels) from each are: radio: 4750, TV: 750, internet: 5000. According to marginal analysis, where should the firm allocate this first 100$$,000?

Accepted Answer

To internet, since it has the highest marginal sales.

Question 2

Given two points on a linear demand curve: $$(P_1$, Q_1$) = (10, 100)$$ and $$(P_2$, Q_2$) = (8, 140)$$, what is the point price elasticity of demand at $$P_1$$?

Accepted Answer

$$\frac{\Delta Q}{\Delta P} 	imes \frac{P_1$}{Q_1$} = \frac{40}{-2} 	imes \frac{10}{100} = -20 	imes 0.1 = -2$$

Question 3

Suppose a per-unit tax raises $105 in revenue on 35 gallons of gasoline. What is the per-unit tax?

Accepted Answer

$$\frac{105}{35} = 3$$

Question 4

If a demand function is log-linear in price, such as $$\ln Q = a - b \ln P$$, what is the own price elasticity of demand?

Accepted Answer

$$-b$$

Question 5

A consumer has preferences such that he always uses two goods in the same proportion. What shape do his indifference curves take?

Accepted Answer

Right angles (L-shaped)

Question 6

If a consumer's utility function is $$U(X,Y) = X + Y$$, what can be said about the convexity of his preferences?

Accepted Answer

Preferences are weakly convex

Question 7

For the utility function $$U(X,Y) = X$$^{0.5}$$ Y$$^{0.5}$$, what is the marginal rate of substitution (MRS) between X and Y?

Accepted Answer

$$\frac{0.5 Y$$^{0.5}$$ X$$^{-0.5}$$}{0.5 X$$^{0.5}$$ Y$$^{-0.5}$$} = \frac{Y}{X}$$

Question 8

If a consumer's preferences are strictly monotonic and strictly convex, what does this imply about his indifference curves?

Accepted Answer

Indifference curves are downward sloping and convex to the origin

Question 9

Given the budget constraint $$P_X X + P_Y Y = I$$ and utility function $$U(X,Y) = X$$^{0.5}$$ Y$$^{0.5}$$, what is the optimal demand for X?

Accepted Answer

$$X^* = \frac{I}{2 P_X}$$

Question 10

For Cobb-Douglas preferences $$U(X,Y) = X$$^{a}$$ Y$$^{1-a}$$, what fraction of income does the consumer spend on good X?

Accepted Answer

$$a$$

Question 11

If the median income rises from $$25,000 to$$ $$45,000$$, what is the percentage change in income?

Accepted Answer

$$\frac{45,000 - 25,000}{25,000} 	imes 100 = 80\%$$

Question 12

If Bill Gates spends the same percentage of his income on food as you do, what does this imply about the income elasticity of demand for food?

Accepted Answer

Income elasticity is equal to 1

Question 13

For the utility function $$U(X,Y) = X \cdot Y$$, what is the Lagrangian for maximizing utility subject to $$P_X X + P_Y Y = I$$?

Accepted Answer

$$\mathcal{L} = X \cdot Y + \lambda (I - P_X X - P_Y Y)$$

Question 14

What is the economic interpretation of the Lagrange multiplier $$\lambda in $$consumer optimization problems?

Accepted Answer

It represents the marginal utility of income.

Question 15

If a consumer's optimal solution is a corner solution (consuming only one good), what does this imply about the marginal rate of substitution (MRS) and the price ratio?

Accepted Answer

MRS is greater than or less than the price ratio

Question 16

Suppose the demand function is $$Q = k P$$^{-\epsilon}$$, where $$\epsilon$$ is the constant price elasticity. If $$\epsilon = 2$$, what happens to quantity demanded if price doubles?

Accepted Answer

Quantity demanded falls to one-fourth its original value.

Question 17

If preferences are strictly monotonic, what can be said about the marginal utilities of both goods?

Accepted Answer

Marginal utilities are positive

Question 18

If a consumer's preferences are strictly convex, what does this imply about his attitude toward averages versus extremes?

Accepted Answer

Averages are strictly preferred to extremes

Question 19

If a consumer's utility function is $$U(X,Y) = X + Y$$, and his budget constraint is $$P_X X + P_Y Y = I$$, what is the optimal consumption bundle?

Accepted Answer

Consume only the good with the lowest price

Question 20

If a consumer's utility function is $$U(X,Y) = \min(X,Y)$$, what type of preferences does he have?

Accepted Answer

Perfect complements

Microeconomics Problem Set Guidance: Marginal Analysis, Elasticity, and Consumer Theory

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