Q: For a Cobb-Douglas utility function $$U(x, y) = x$$^{c alpha}$$y^{5c beta}$$ with $$5c alpha + 5c beta = 1$$, what is the general formula for the Marshallian demand for $$x$?

$$x^* = 5c frac{5c alpha I}{p_x}$$

Q: For a Cobb-Douglas utility function $$U(x, y) = x$$^{5c alpha}$$y^{5c beta}$$, what is the Engel curve for $$x$?

$$x^* = 5c frac{5c alpha I}{p_x}$$

Question 1

Suppose a consumer has a Cobb-Douglas utility function $$U(x, y) = x$$^{0.5}$$y^{0.5}$$, income $$I = 100$$, price of $$x is$$ $$p_x = 4$$, and price of $$y is$$ $$p_y = 2. $$What is the optimal bundle $$(x^*, y^*)$ the consumer will choose?

Accepted Answer

$$(x^*, y^*) = (12.5, 25)$$

Question 2

Given the budget constraint $$4b + 2c = 100$$, what is the slope of the budget line in terms of $$c ($$coffee) and $$b ($$basketball games)?

Accepted Answer

$$-2$$

Question 3

If a consumer’s utility function is $$U(x, y) = x + y$$ and $$p_x = 1$$, $$p_y = 2$$, and $$I = 6$$, what is the optimal consumption bundle?

Accepted Answer

$$(x^*, y^*) = (6, 0)$$

Question 4

For perfect complements with utility $$U(x, y) = \min(2x, y)$$, $$p_x = 1$$, $$p_y = 1$$, and $$I = 21$$, what is the optimal bundle?

Accepted Answer

$$(x^*, y^*) = (7, 14)$$

Question 5

If the marginal utility per dollar spent on good $$x is $$greater than that for good $$y$$, what should the consumer do to maximize utility?

Accepted Answer

Buy more of $$x$$ and less of $$y$$

Question 6

For a Cobb-Douglas utility function $$U(x, y) = x$$^{c alpha}$$y^{ 5c beta}$$ with $$ 5c alpha +  5c beta = 1$$, what is the general formula for the Marshallian demand for $$x$?

Accepted Answer

$$x^* =  5c frac{ 5c alpha I}{p_x}$$

Question 7

If a consumer’s utility function is $$U(x, y) = \min(2x, y)$$, what is the relationship equation between $$x$$ and $$y at $$the optimal bundle?

Accepted Answer

$$y^* = 2x^*$$

Question 8

If the price of Snickers bars is less than $$1$$ and M&Ms cost $$1$$, which good will a consumer with $$U(m, s) = m + s$$ buy?

Accepted Answer

Only Snickers bars

Question 9

Which of the following best describes the indifference curves for perfect complements?

Accepted Answer

Right angles

Question 10

If a consumer’s income doubles, how does the optimal consumption of each good change for a Cobb-Douglas utility function $$U(x, y) = x$$^{0.5}$$y^{0.5}$?

Accepted Answer

Both $$x^*$$ and $$y^*$$ double

Question 11

For perfect substitutes, if $$\frac{MU_x}{p_x} \u003e \frac{MU_y}{p_y}$$, what is the optimal consumption bundle?

Accepted Answer

All income is spent on $$x$$

Question 12

Given $$U(x, y) = x$$^{0.5}$$y^{0.5}$$, $$I = 200$$, $$p_x = 10$$, $$p_y = 2$$, what is the optimal amount of $$x$?

Accepted Answer

$$x^* = 10$$

Question 13

Which of the following is NOT a property of the optimal bundle for standard convex preferences?

Accepted Answer

It is always a corner solution

Question 14

If $$U(x, y) = x$$^{0.5}$$y^{0.5}$$, $$I = 100$$, $$p_x = 4$$, $$p_y = 2$$, and $$p_x$$ increases to $$10$$, what happens to the optimal $$x^*$?

Accepted Answer

$$x^*$$ decreases

Question 15

For a utility function $$U(x, y) = x + y$$, if $$p_x = p_y$$, what can be said about the consumer’s choice?

Accepted Answer

The consumer is indifferent between all bundles on the budget line

Question 16

Which equation represents the condition for utility maximization for standard convex preferences?

Accepted Answer

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$

Question 17

If a consumer’s utility function is $$U(x, y) = \min(2x, y)$$, $$p_x = 1$$, $$p_y = 1$$, and $$I = 21$$, what is the maximum utility the consumer can achieve?

Accepted Answer

$$14$$

Question 18

If a consumer is at a bundle inside the budget constraint, what should they do to maximize utility?

Accepted Answer

Move to a bundle on the budget constraint

Question 19

For a Cobb-Douglas utility function $$U(x, y) = x$$^{ 5c alpha}$$y^{ 5c beta}$$, what is the Engel curve for $$x$?

Accepted Answer

$$x^* =  5c frac{ 5c alpha I}{p_x}$$

Consumer Choice: Maximizing Utility Subject to Income Constraints

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