Do Advanced Microeconomics article
Budget: $10 – $30 USD
Question 1 (25%)
i) Which properties a consumer’s preferences have to satisfy. Please list them and
provide one example for each one of them.
Suppose that a consumer’s preferences are described by the following utility function:
?(?) = max {?ଵ, ?ଶ}
ii) Draw the consumer’s Indifference Curves for ?(?) = 1. Shade the area for which
?(?) ≤ 1. Clearly justify your answer.
iii) Derive the consumer’s Walrasian Demands for goods 1 and 2. Are they monotonic
to the arguments that they were supposed to? Show your work.
iv) Derive the consumer’s indirect Utility Function. Is it monotonic to the arguments
that it was supposed to? Show your work.
Question 2 (25%)
Suppose that a consumer’s preferences are described by the following utility function:
?(?) = ?ଵଶ?ଶ
i) Derive the Walrasian Demands for goods 1 and 2 and list their properties. Show
that indeed those properties hold true. Are the preferences convex? Show your
work.
ii) Are the First Order Conditions of the Utility Maximization Problem sufficient? (i.e.
do they ensure that the utility function is indeed maximized?)
iii) Derive their Indirect Utility Function and list its properties regarding monotonicity
and convexity/concacity. Show that the property regarding monotonicity holds true.
iv) Derive their Hicksian Demands for goods 1 and 2.
v) Derive their Expenditure Function. Is it monotonic to the argumments it should be
(i.e. the variables it should be monotonic with respect to)? Show your work.
Question 3 (25%)
Consider a firm with tecnnology described by the following production function:
?(?, ?) = (?? + ??)
ଵ
ఘ
Where k denotes the amount of capital and l denotes the amount of labor used in their
production, and ρ>0 is a positive parameter. Suppose that The cost of labor is w and the
cost of capital is r.
i) Calculate the elasticity of substitution (σ). What kind of returns to scale does this
technology represent? Show your work.
Consider now a firm with tecnnology described by the following production function:
?(?ଵ, ?ଶ, ?ଷ, ?ସ) = min {?ଵ + 2?ଶ, ?ଷ + 2?ସ}
ii) Derive the firm’s cost function.
Question 4 (25%)
Rob’s (agent 1) and Eve’s (agent 2) preferences can be described by the following utility
functions:
?ଵ(?) = max {?ଵ, ?ଶ} and ?ଶ(?) = ?ଵ + ?ଶ . Each one of the have equal initial endowments
? = (1,1).
i) Illustrate the situation explained above in an Edgeworth’s box. Clearly describe
why the Edgeworth’s box would look like this.
ii) In the competitive equilibrium, is there any relationship between the two prices? If
so, what is it? Clearly show every step of your derivations.
iii) What is the equilibrium allocation? Clearly show every step of your derivations.
iv) Is the competitive equilibrium Pareto efficient? Explain.
i) Which properties a consumer’s preferences have to satisfy. Please list them and
provide one example for each one of them.
Suppose that a consumer’s preferences are described by the following utility function:
?(?) = max {?ଵ, ?ଶ}
ii) Draw the consumer’s Indifference Curves for ?(?) = 1. Shade the area for which
?(?) ≤ 1. Clearly justify your answer.
iii) Derive the consumer’s Walrasian Demands for goods 1 and 2. Are they monotonic
to the arguments that they were supposed to? Show your work.
iv) Derive the consumer’s indirect Utility Function. Is it monotonic to the arguments
that it was supposed to? Show your work.
Question 2 (25%)
Suppose that a consumer’s preferences are described by the following utility function:
?(?) = ?ଵଶ?ଶ
i) Derive the Walrasian Demands for goods 1 and 2 and list their properties. Show
that indeed those properties hold true. Are the preferences convex? Show your
work.
ii) Are the First Order Conditions of the Utility Maximization Problem sufficient? (i.e.
do they ensure that the utility function is indeed maximized?)
iii) Derive their Indirect Utility Function and list its properties regarding monotonicity
and convexity/concacity. Show that the property regarding monotonicity holds true.
iv) Derive their Hicksian Demands for goods 1 and 2.
v) Derive their Expenditure Function. Is it monotonic to the argumments it should be
(i.e. the variables it should be monotonic with respect to)? Show your work.
Question 3 (25%)
Consider a firm with tecnnology described by the following production function:
?(?, ?) = (?? + ??)
ଵ
ఘ
Where k denotes the amount of capital and l denotes the amount of labor used in their
production, and ρ>0 is a positive parameter. Suppose that The cost of labor is w and the
cost of capital is r.
i) Calculate the elasticity of substitution (σ). What kind of returns to scale does this
technology represent? Show your work.
Consider now a firm with tecnnology described by the following production function:
?(?ଵ, ?ଶ, ?ଷ, ?ସ) = min {?ଵ + 2?ଶ, ?ଷ + 2?ସ}
ii) Derive the firm’s cost function.
Question 4 (25%)
Rob’s (agent 1) and Eve’s (agent 2) preferences can be described by the following utility
functions:
?ଵ(?) = max {?ଵ, ?ଶ} and ?ଶ(?) = ?ଵ + ?ଶ . Each one of the have equal initial endowments
? = (1,1).
i) Illustrate the situation explained above in an Edgeworth’s box. Clearly describe
why the Edgeworth’s box would look like this.
ii) In the competitive equilibrium, is there any relationship between the two prices? If
so, what is it? Clearly show every step of your derivations.
iii) What is the equilibrium allocation? Clearly show every step of your derivations.
iv) Is the competitive equilibrium Pareto efficient? Explain.