# Микроэкономика: задачник (~2 курс)

Notations:

A# — a basic (in most cases easy) problem

B# — a moderately complex problem

C# — a kind of more cumbersome problem

Topic list:

1. CONSUMER CHOICE
2. FIRM
3. MARKET STRUCTURES

1. CONSUMER CHOICE

Problem A1
Write down utility functions for the following situations:
a) Consumer is indifferent between having one bottle of mineral water and one bottle of kvass.
b) Consumer is indifferent between having one 2L (2 liter) bottle of kvass and four 0.5L bottles of kvass.
c) An oligarch always consumes a Rolls-Royce with a driver. Additional cars or drivers without the counterpart do not give him any happiness.
d) His girlfriend prefers a yacht with 45 crew members. Additional yachts or crew without the counterpart do not give her any happiness.
e) Mike dislikes to consume apples with coffee together.
f) Alice feels best when she has 50 square meters of living space and 2 kids. Any other numbers here decrease her wellbeing.

Problem A2
There are two goods: x1 and x2. Their prices are ₽p1 and ₽p2. Income is fixed at level ₽m.
a) Draw a budget line. Calculate its intercepts with x1-axis and x2-axis (always try to do it in future).
b) Derive equation of budget line. What is its slope (how can you tell it)?
c) Show a feasable set on the graph.

Problem A3
Let's continue Problem A2. The government introduces a per-unit subsidy on x1 for everybody, ₽s.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive equation of new budget line. Reconcile it with a graph.

Problem A4
Let's continue Problem A2. The government introduces a per-unit tax on x1 for everybody, ₽t.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive equation of new budget line. Reconcile it with a graph.

Problem A5
Let's continue Problem A2. The government introduces a per-unit subsidy on x1 for those who consume more than W units of x1, ₽s.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive relevant equations for the new budget line. Reconcile it with a graph.

Problem A6
Let's continue Problem A2. The government introduces a lump-sum subsidy for everybody, ₽S.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive equation of new budget line. Reconcile it with a graph.

Problem A7
Let's continue Problem A2. The government introduces a lump-sum tax on everybody, ₽T.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive equation of new budget line. Reconcile it with a graph.

Problem A8
Let's continue Problem A2. The government gives everybody T units of x1 for free.
a) Draw a new and old budget lines. Calculate their intercepts with x1-axis and x2-axis.
b) Derive equation of new budget line. Reconcile it with a graph.

Problem A9
Derive MRS (marginal rate of substitution) for the following cases. Explain its meaning.
a) U(x1, x2)=x10.6·x21.2
b) U(x1, x2)=x10.5·x20.5
c) U(x1, x2)=x10.5+3ln(5x2)
d) U(x1, x2)=2x1+3x2
e) U(x1, x2)=12x1+0.2x2
f) U(x1, x2)=min{x1,x2}
g) U(x1, x2)=min{5x1,2x2}

Problem B1
There are two goods: x1 and x2. Price of x1 decreases. Show on the graph (budget lines+indifferent curves) the location of optimal bundles before and after price decrease for the following cases. Illustrate substitution and income effects for x1.
a) x1 is normal
b) x1 is inferior
c) x1 is ordinary
d) x1 is Giffen
e) x1 and x2 are substitutes
f) x1 and x2 are complements
g) x2 is normal
h) x2 is inferior

Problem B2
Derive a shortcut for a smooth well-behaved utility function: in the interior solution MRS equals price ratio.
Make a graphic illustration for it, where indifference curve tangents budget line in the optimal bundle.

Problem B3
a) What is income-consumption curve? Derive it graphically. Pay attention to whether the good is normal or inferior.

b) What is Engel curve? Derive it graphically. Pay attention to whether the good is normal or inferior.

Problem C1
A consumer buys two goods: x1 and x2. Price of x1 is ₽5, price of x2 is ₽1. Income is ₽100. Utility function is:
U(x1, x2)=x10.5·x22.
Easy questions:
a) Find the optimal choice of the consumer under given conditions.
b) Derive demand on x1 and x2.

Medium questions:
c) Whether x1 is normal or inferior? Is it Giffen or ordinary?
d) Repeat analysis for x2.
e) Whether x1 and x2 are substitutes or complements?
f) Derive all relevant demand elasticities for x1.
g) Compute substitution and income effects with respect to x1 if its price increases up to ₽10. Illustrate on graph.

Problem C2
A consumer buys two goods: x1 and x2. Price of x1 is ₽5, price of x2 is ₽1. Income is ₽100. Consumer always prefers 1 unit of x1 for 4 units of x2.
Easy questions:
a) Find the optimal choice of the consumer under given conditions.
b) Derive demand on x1 and x2.

Medium questions:
c) Whether x1 is normal or inferior? Is it Giffen or ordinary?
d) Repeat analysis for x2.
e) Whether x1 and x2 are substitutes or complements?
f) Compute substitution and income effects with respect to x1 if its price decreases down to ₽2. Illustrate on graph.

Problem C3
A consumer buys two goods: x1 and x2. Price of x1 is ₽5, price of x2 is ₽1. Income is ₽100. Consumer always prefers to have 1 unit of x1 with 2 units of x2.
Easy questions:
a) Find the optimal choice of the consumer under given conditions.
b) Derive demand on x1 and x2.

Medium questions:
c) Whether x1 is normal or inferior? Is it Giffen or ordinary?
d) Repeat analysis for x2.
e) Whether x1 and x2 are substitutes or complements?
f) Compute substitution and income effects with respect to x1 if its price decreases down to ₽2. Illustrate on graph.

2. FIRM

Problem 2A_1
a) Recall definitions of TC, FC, VC, ATC, AVC, AFC, MC.
b) Draw standard cost curves of everything listed in a). Уделите особое внимание их относительному расположению.
c) Recall the following formulae: TC=VC+FC, MC=TC'=VC', TC=wL+rK, ATC=AVC+AFC, ATC=TC/Q, AVC=VC/Q, AFC=FC/Q. Think about them. Is it clear why they are calculated so?

Problem 2A_2
a) Что является аргументом у функций TC, FC, VC, ATC, AVC, AFC, MC? Почему так?
b) Почему графики функций TC, FC, VC, ATC, AVC, AFC, MC имеют такой вид (в стандартном случае)?
c) Derive graphically the fact that AC and AVC intersect MC at their minima.
d) Derive algebraically the fact that AC and AVC intersect MC at their minima.

Problem 2A_3
a) What is MRTS (marginal rate of technical substitution)? Write its formula, explain it.
b) What is production function? Каковы ее аргументы? Что она показывает?
c) What is isoquant? Recall that MRTS is a slope of isoquant.
d) A map of isoquants is an example of what is called a 'level set' in maths. Why is it so?
e) What is isocost? Write its formula, explain it. What is its slope? Derive it.

Problem 2B_1
Derive a shortcut for a smooth well-behaved production function: in the interior solution MRTS equals factor price ratio.
Make a graphic illustration for it, where an isoquant tangents isocost in the optimal choice.

Problem 2B_2
Recall mathematical definition of returns to scale. What returns to scale the following functions exhibit?
a) Q(L, K)=L0.6·K1.2
b) Q(L, K)=L0.5·K0.5
c) Q(L, K)=L0.5+7ln(K)
d) Q(L, K)=L+3K
e) Q(L, K)=min{2L,5K}

Problem 2B_4
What is MRTS for the following production functions?
a) Q(L, K)=L0.6·K1.2
b) Q(L, K)=L0.5·K0.5
c) Q(L, K)=L0.5+7ln(K)
d) Q(L, K)=L+3K
e) Q(L, K)=min{2L,5K}

Problem 2B_5
Discuss how we can incorporate short-run and long-run into theory of a firm. Speak about formulae, graphs, cost structure.

Problem 2C_1
A firm is a price-taker on the factor market. Production function is Q(L, K)=L0.5·K2.
a) Find demand on labor and capital. Denote wage and rent by w and r.
b) Derive TC, FC, VC, ATC, AVC, AFC, MC.

Problem 2C_2
A firm is a price-taker on the factor market. Production function is Q(L, K)=L+K.
a) Find demand on labor and capital. Denote wage and rent by w and r.
b) Derive TC, FC, VC, ATC, AVC, AFC, MC.

Problem 2C_3
A firm is a price-taker on the factor market. Production function is Q(L, K)=2L+ln(K).
a) Find demand on labor and capital. Denote wage and rent by w and r.
b) Derive TC, FC, VC, ATC, AVC, AFC, MC.

Problem 2C_4
What is firm's expansion path? Derive it graphically. Pay attention to whether it is short-run or long-run.

3. MARKET STRUCTURES

Problem 3A_1
Что является аргументом MR? Что является аргументом MC? Почему так?

Problem 3A_2
Derive algebraically condition that MR=MC. For which market structures is it applicable?

Problem 3A_3
Why a firm wishes to increase production while MR is greater or equal than MC but not vice versa?

Problem 3A_4
How to choose an optimum if MR=MC in several points?

Problem 3A_5
Why the profit of a firm falls if the number of firms increases?

Problem 3A_6
Define the following: normal form game and game in extensive form, backward induction, Nash equilibrium, Pareto optimum, pure strategies and mixed strategies, simultaneous move and sequential move, dominating and dominated strategy, reaction function (best response).

Problem 3A_7
Is Cournot equilibrium a NE? How is it incorporated into reaction functions?

Problem 3A_8
What is a first-mover advantage? Can this term be used for monopolistic competition?

Problem 3A_9
Give one most important reason for each pair what distinguishes perfect competition, monopolistic competition, oligopoly and monopoly from each other.

Problem 3A_10
Can there be positive profits in the short-run in the case of (make graphs and explain intuitively):
a) perfect competition
b) monopolistic competition
c) oligopoly
d) monopoly
Repeat for the long-run.

Problem 3A_11
What is a natural monopoly? Remember about AC shape.

Problem 3B_1
What is a symmetric firms case?

Problem 3B_2
How do these industries adjust to per-unit tax on producer? Explain using graphs.
a) perfect competition
b) monopolistic competition
c) monopoly
Make separate back-to-back graphs for firm and insutry where applicable.
Discuss what happens to firm and industry output, market price, profit, number of firms.

Problem 3B_3
How do these industries adjust to per-unit subsidy on producer? Explain using graphs.
a) perfect competition
b) monopolistic competition
c) monopoly
Make separate back-to-back graphs for firm and insutry where applicable.
Discuss what happens to firm and industry output, market price, profit, number of firms.

Problem 3B_4
Market demand is Q(P)=100-P. Find equilibrium firm's output, market price and quantity, profit of a firm for the cases listed below.
a) perfect competition, MC of a typical firm is ₽10
b) monopoly, MC is ₽30
c) 2 firms which set output simultaneously, MC of each firm is ₽10.
d) 2 firms which set output simultaneously, MC of firm 1 is ₽10, MC of firm 2 is ₽15.
e) 3 firms which set output simultaneously, MC of each firm is ₽10.
f) 3 firms which set output simultaneously, MC of firm 1 and 2 is ₽10, MC of firm 3 is ₽15.
g) 12 firms which set output simultaneously, MC of each firm is ₽10.
h) 2 firms, firm 1 chooses output before firm 2. MC of each firm is ₽10.
i) 2 firms, firm 1 chooses output before firm 2. MC of firm 1 is ₽10, MC of firm 2 is ₽15.
j) 2 firms which set prices simultaneously, MC of each firm is ₽10.
k) 2 firms which set prices simultaneously, MC of firm 1 is ₽10, MC of firm 2 is ₽15.
Sketch reaction functions and equilibrium where applicable.

Problem 3B_5
There are two firms with differentiated product. Market demand is Q(P1,P2)=100-P1+P2. Find equilibrium prices if they set them simultaneously. Marginal cost of each firm is ₽10.

Problem 3C_1

Problem 3C_2
Does symmetric firms shortcut work for Stackelberg or just for Cournot?

Problem 3C_3
How to distinguish algebraically corner and interior solutions?

Problem 3C_4
Assume symmetric Cournot with n firms. Prove algebraically that profit of a firms tends to zero and industry output tends to perfect competitive level when n tends to infinity. That is, show that perfect competition is a limiting case of Cournot, i.e. Cournot transforms to perfect competition when the number of firms is very large.

Обновлено: 24.01.2023
Создано: 24.12.2022

К списку всех статей

Некоторые ВУЗы и программы, студентам которых была предоставлена квалифицированная помощь репетитора по математике, статистике, макро- и микроэкономике и прочим наукам с экономическим, финансовым и математическим уклоном.