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Advanced Microeconomics. Need all questions to be answered, with workings/intuition to derive the final answer.

(1) Suppose that a consumer has a complete and transitive preference relation over R`+ .

Which of the following statements are false? Circle all that apply.

(a) If the preference relation over R`+ is continuous, then it can be represented by

a continuous utility function u : R`+ ? R.

(b) If the preference relation over R`+ is locally nonsatiated, then it can be represented by a continuous utility function u : R`+ ? R.

(c) If the preference relation over R`+ is strictly monotonic, then it can be represented by a continuous utility function u : R`+ ? R.

(d) The preference relation over R`+ is strictly monotonic if it is locally nonsatiated.

(e) The preference relation over R`+ is locally nonsatiated if it strictly monotonic.

(2) Suppose that a firm wants to produce some level of output y > 0 according to the

production function f : R2+ ? R+ , which is given by

?

y = f (K, L) = KL, where K ? 0 is capital and L ? 0 is labor. Further suppose that r > 0 is the rental rate

for capital and w > 0 is the wage rate for labor, and that the firm wants to minimize

its costs. Which of the following statements are true? Circle all that apply.

(a) The conditional factor demand for capital is always positive.

(b) The conditional factor demand for labor is always positive.

(c) The conditional factor demand for capital is always greater than the conditional

factor demand for labor.

(d) The conditional factor demand for capital is always less than the conditional factor

demand for labor.

(e) The conditional factor demand for capital is always equal to the conditional factor

demand for labor.

(3) Suppose that a consumer must choose from R2+ , faces prices p = (p1 , p2 ) 0, and has

income w > 0. Her indirect utility function v : R2++ × R++ ? R is given according to

w

.

v((p1 , p2 ), w) = ?

?

( p1 + p2 )2

Which of the following statements are false? Circle all that apply.

(a) The demand for good 1 is more income elastic than the demand for good 2.

2 (b) The demand for good 2 is more income inelastic than the demand for good 1.

(c) The demand for good 2 is more income elastic than the demand for good 1.

(d) The demand for good 1 is more income inelastic than the demand for good 2.

(e) The income elasticity of demand for good 1 is always equal to the income elasticity

of demand for good 2.

(4) Suppose that a consumer has a preference ordering over R`+ that is represented by

the utility function u : R`+ ? R, which is given according to

u(x1 , x2 , . . . , x` ) = `

Y xi = x1 × x2 × · · · × x` . i=1 Which of the following statements are true? Circle all that apply.

(a) The marginal rate of substitution between any two goods is independent of the

consumption of those same two goods.

(b) The marginal rate of substitution between any two goods is not independent of

the consumption of those same two goods.

(c) The marginal rate of substitution between any two goods is independent of the

consumption of any other goods.

(d) The marginal rate of substitution between any two goods is not independent of

the consumption of any other goods.

(e) The marginal rate of substitution between any two goods is necessarily equal to

the marginal rate of substitution between any two other goods. (5) Consider a firm that requires more than one costly inputs to produce a single output.

Suppose that the firm?s production function is continuous, strictly increasing, and

strictly quasiconcave. Which of the following statements are false for a cost-minimizing

firm? Circle all that apply.

(a) The firm?s costs are always strictly increasing in output.

(b) The firm?s costs are always constant per unit of output.

(c) The firm?s marginal costs are always strictly increasing in output.

(d) The firm?s marginal costs are always constant at every level of output.

(e) The firm?s marginal costs are always zero. 3 (6) Suppose that a firm produces some level of output y ? 0 according to the production

function f : R2+ ? R+ , which is given by

y = f (K, L) = K + L,

where K ? 0 is capital and L ? 0 is labor. Further suppose that r = 10 is the rental

rate for capital and w = 5 is the wage rate for labor, and that the firm is perfectly

competitive. Which of the following statements are true? Circle all that apply.

(a) The market output price is less than one.

(b) The market output price is greater than two.

(c) The market output price is less than three.

(d) The market output price is greater than four.

(e) The market output price is less than five.

(7) Suppose that a consumer has a complete and transitive preference relation over R`+ .

Further suppose that the consumer faces prices p = (p1 , p2 , . . . , p` ) 0 and has income

w > 0. Which of the following statements are false? Circle all that apply.

(a) If is continuous, then the consumer exhausts her budget.

(b) If is locally nonsatiated, then the consumer exhausts her budget.

(c) If is strictly monotonic, then the consumer exhausts her budget.

(d) If is convex, then the consumer exhausts her budget.

(e) If is strictly convex, then the consumer exhausts her budget.

(8) Suppose that a preference relation on R`+ is complete and transitive. Let and ?

denote the strict and indifference relations on R`+ , respectively, which can be derived

from . Which of the following statements are true? Circle all that apply.

(a) The strict relation is complete.

(b) The indifference relation ? is complete.

(c) The strict relation is transitive.

(d) The indifference relation ? is transitive.

(e) The strict relation and the indifference relation ? are neither complete nor

transitive. 4 (9) Suppose that a consumer has a lexicographic preference ordering over R2+ , i.e., for

any x, y ? R2+ , x y if x2 > y2 , or if x2 = y2 and x1 ? y1 . Which of the following

statements are false? Circle all that apply.

(a) The preference is complete and transitive.

(b) The preference is complete, transitive, and continuous.

(c) The preference is complete, transitive, and locally nonsatiated.

(d) The preference is complete, transitive, and strictly monotonic.

(e) The preference is neither complete nor transitive. (10) Suppose that a monopolist is choosing some quantity q ? 0 to maximize profits, which

are given by p(q)q ? c(q), where p : R+ ? R++ is the inverse market demand function,

and c : R+ ? R+ is the firm?s cost function. Which of the following statements are

true? Circle all that apply.

(a) The firm always sets the quantity where marginal revenue equals marginal cost.

(b) The firm always sets the quantity where price equals marginal cost.

(c) The firm always sets the quantity where demand is elastic.

(d) The firm always sets the quantity where demand is inelastic.

(e) The firm always sets the quantity where demand is unit elastic. 5

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DATE ANSWEREDOct 14, 2020

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