Exer ise 1
Consider the following game in strategi form, to be alled dis rete Hotelling game,
with two players that we denote by 1 and 2. Ea h player i has the same strategy set
Xi = {0, 1, 2, 3, 4}. In order to des ribe the payo fun tions onsider the 5 points of
Xi on the real line, to be referred to as verti es. The two players simultaneously and
independently hoose a vertex. If player 1 hooses vertex x1 and player 2 hooses vertex
x2, then the payo fi(x1, x2) of player i = 1, 2, is the number of verti es that is the losed
to him; if a vertex has equal distan e to both players, then this vertex ontributes only
1
2
a. Determine for ea h player his best-reply- orresponden e.
b. Determine the stri tly dominant strategies.
. Determine for ea h player strategies x, y su h that y dominates x weakly.
1
e. Determine the weakly Pareto ine ient multi-strategies.
Solution



x+ 1 if x < 2, 2 if x = 2, x− 1 if x > 2.
b. No player has a stri tly dominant strategies.
. For ea h player: 0 is weakly dominated by 2 and 4 is weakly dominated by 2. d. (2, 2). e. Do not exist.
1
Denition: let i be a player in a game in strategi form and let x, y be two strategies of this player.
One says `y dominates x weakly' if independently of the strategies of the other players, y gives player i at
least the payo that x gives and at least one time a higher payo.
Exer ise 2
1. A game in strategi form with two players is alled `stri tly ompetitive' if for all
multi-strategies (a, b) and (c, d) it holds that f 1(a, b) ≥ f 1(c, d) ⇔ f 2(a, b) ≤ f 2(c, d).
(a) Show that ea h zero-sum with two players game is stri tly ompetitive.
(b) Show that in a stri tly ompetitive game ea h multi-strategy is strongly pareto
e ient.
2. Are the following statements about games in strategi form true or false? Prove your
answer or give a ounter-example.
(a) If x is a full ooperative multi-strategy, i.e. a maximiser of the total payo
fun tion, then none of the strategies xi is strongly dominated.
(b) In ase ea h player has at least two strategies, there exists a strongly dominated
strategy.
( ) There exists a bi-matrix where no multi-strategy is weakly pareto e ient.
3. Determine the subgame perfe t nash equilibria of the following game in extensive
form:
1
L R
Solution
1a. For a zero-sum game we have f1 + f2 = 0. So f1(a, b) ≥ f1(c, d) ⇔ −f2(a, b) ≥ −f2(c, d) ⇔ f2(a, b) ≤ f2(c, d). Thus the game is stri tly ompetitive.
1b. By ontradi tion suppose a is strongly pareto-ine ient. Then there exists a pareto
improvement b of a. So then there exists a player, say 1, who at b has a greater payo, and
2
the other player does not have a smaller payo. Thus f1(b) > f1(a) and f2(b) ≥ f2(a). As the game is stri tly ompetitive, we have f1(b) ≤ f1(a), a ontradi tion.
2a. This is not true:
(
)
.
(
)
.
2 . This is not true als ea h fully ooperative strategy prole is weakly Pareto e ient.
(Note that su h strategy prole exists as the game is nite.)
3. Ba kward indu tion leads to 3 subgame perfe t equilibria: (L, r, b, A, E), (L, l, b, B, E) and (L, r, b, B, , E).
3
A contest game with complete (part a-c) and incomplete information (d-g). There are two agents 1,2i = competing for a prize P which is worth ( )iv P . Both agents
invest simultaneously a cost {1,2}ic ∈ in order to receive the prize.
Agent i’s probability of winning the prize is 1 2
ic
c c+ .
a) Determine the payoff function. b) Draw the game tree that represents the game as an extensive form game. c) Show that high investments ( 2ic = ) are a dominant strategy if the valuation
of the prize ( ) 6iv P > .
Now consider incomplete information. Suppose the agents either place a high value on the prize ( ) 9H
iv P = or a low value ( ) 3L iv P = . The common prior probabilities are given
in the following table:
1 Lv 0.16 0.24
1 Hv 0.24 0.36
d) Draw the game tree that reflects that players’ types are randomly determined. e) Calculate the expected payoff of a player with a high valuation for each of his
investment strategies. f) Describe the Bayesian-Nash equilibria of the game.
A market for micro credit
Consider a loan market to finance investment projects. All projects cost €1. All projects are either good or bad. Only investors know whether their project is good or bad. The bank knows that of all projects a share p are good projects and a share 1 p− are bad projects. Each project yields either a positive benefit V or zero benefits to the investor. The probability of a positive benefit is higher for good projects. We denote this by g bP P> . Banks are competitive and risk neutral. A loan contract specifies a
repayment R that is repaid to the bank only if the project gives a positive benefit. The opportunity cost of funds (recall project costs are 1) to the bank is 0r > . Suppose that (*) (1 ) 0 (1 )g bP V r PV r− + > > − + .
a) State what condition (*) says in your own words. b) Find the equilibrium level of R for a competitive credit market. How does this
depend on , , , , and g bP P p V r .
c) Discuss which set of projects are financed. Argue that the loan market breaks down completely conditional on some combination of , , , , and g bP P p V r .
d) What instrument could the banks use to establish at least a partial market.
Consumer Theory
An infinitely lived agent owns 1 unit of a commodity that he consumes over his lifetime. The
commodity is perfectly storable and he will receive no more than he has now. Consumption of
the commodity in period t is denoted xt, and his lifetime utility function is given by
t
t
, where 10 .
a.) Write down a general consumer’s utility maximization problem. (1 point)
b.) Show that the consumer will consume all of his commodity endowment at the optimum.
(1 point)
c.) Calculate consumer’s optimal consumption level in each period. Provide intuition for this
consumption pattern.
(Hint 1: Feel free to use the Lagrangian function;
Hint 2: Recall from your high school math course that 0
1
1
t
t
(3 points)
d.) What is the marginal utility of having a little bit greater endowment of the commodity at the
optimum?
(1 point)
e.) Now assume that the consumer will live only T years. Calculate consumer’s optimal
consumption level in each period in this case. How does the consumption pattern compare to
your solution in c.)? Explain (4 points)
Producer Theory
Provide a brief answer to each of the following questions.
a.) Derive the profit function for the single-output technology whose production function is
given by 1 2f z z z . The prices of inputs z1 and z2 are w1 and w2, respectively.
(3 points)
b.)
b.) Corn (C) is produced from labor (L) using a decreasing returns to scale technology of the
formC AL , where A is a scale parameter and 0,1 . How is the parameter related
to the price elasticity of the corn supply curve? (2 points)
c.) Ethanol (E) is produced from corn (C) and labor (L) using a Leontief technology
min ,E aC bL ,
where a and b are technological parameters. Draw the inverse ethanol supply curve and
determine its price elasticity. (2 points)
d.) When the ratio of goods consumed, xi/xj, is independent of income (m) for all i and j (i.e.,
0i jx x m ), then the ratio of any two income elasticities is always equal to 1 (i.e.,
1i j ) . True/false? Show your work. (3 points)
Consumer Theory An infinitely lived agent owns 1 unit of a commodity that he consumes over his lifetime. The commodity is perfectly storable and he will receive no more than he has now. Consumption of the commodity in period t is denoted xt, and his lifetime utility function is given by
( ) t t
= β , where 10 << β .
a.) Write down a general consumer’s utility maximization problem. (1 point) Answer:
0
0
=
∞
=
=
>
∑
∑
b.) Show that the consumer will consume all of his commodity endowment at the optimum. (1 point) Answer:
Because marginal utility in any period t is positive, 0 t
tx β
> , the consumer will consume all of his
endowment. c.) Calculate consumer’s optimal consumption level in each period. Provide intuition for this consumption pattern. (Hint 1: Feel free to use the Lagrangian function;
Hint 2: Recall from your high school math course that 0
1 1
λ − = ⇒ =
t t
∞ ∞ ∞
= = =
= = = = ⇒ = − −
∑ ∑ ∑
t t t tx x x xβ β β λ −= ⇒ = − ⇒ >
d.) What is the marginal utility of having a little bit greater endowment of the commodity at the optimum? (1 point) Answer:
* 1 1
λ β
= −
e.) Now assume that the consumer will live only T years. Calculate consumer’s optimal consumption level in each period in this case. How does the consumption pattern compare to your solution in c.)? Explain (4 points)
Answer: The only difference compared to question c.) is
1 1 1
t t t t
− − −
= = =
− − = = = = ⇒ = − −
∑ ∑ ∑
λ β −
= ⇒ = ⇒ > −
.
Because the consumer lives shorter, he consumes more in each period relative to c.)
Producer Theory
Provide a brief answer to each of the following questions. Answer:
a.) Derive the profit function for the single-output technology whose production function is given by ( ) 1 2f z z= +z . The prices of inputs z1 and z2 are w1 and w2, respectively. (3 points) The classic approach via Lagrangian will not work (convince yourself!). If you look at the production function, you’ll see that the inputs are perfect substitutes. Therefore their use will depend on relative prices. If ( )1 1 1 1w w z f z< ⇒ ⇒
If ( )1 1 2 2w w z f z> ⇒ ⇒
If ( )1 1w w z f z= ⇒ ⇒ I only show the first case. The rest can be done similarly.
1 1 1
0 4 2 4 42
p z w z FOC
p p p p pw z w w w wz
π
π
⇒ −
− = ⇒ = ⇒ ⇒ − =
b.) b.) Corn (C) is produced from labor (L) using a decreasing returns to scale technology of the
form C ALε= , where A is a scale parameter and ( )0,1ε ∈ . How is the parameterε related to the price elasticity of the corn supply curve? (2 points) Answer: Solved in class
c.) Ethanol (E) is produced from corn (C) and labor (L) using a Leontief technology ( )min ,E aC bL= ,
,
C E
C E
E EE aC bL C L a b
w wE ETC w C w E w w E a b a b
w wTCMC E a b
w wSupply MC p p a b
= = ⇒ = =
= + = + = +
∂ = = +
∂
= ⇒ = +
Because the MC is constant, the inverse supply curve is horizontal:
d.) When the ratio of goods consumed, xi/xj, is independent of income (m) for all i and j (i.e., ( ) 0i jx x m∂ ∂ = ), then the ratio of any two income elasticities is always equal to 1 (i.e.,
1i jε ε = ) . True/false? Show your work. (3 points)
Answer:
0 1
j ji i ij ji ij ij i i j i jj i j
j j j
j ji i i i j i j i j i j i i j i j j
j
x xx x xm mx xx xx xx x x xx m x m m x mm m m m m x x x
x xx xx x x x x x x x m m m m
ε ε
∂∂∂∂∂ − − − ∂ ∂ ∂ ∂= = = = ∂
⇒ − = ⇒ = ⇒ = ⇒ =
.
1 2
i i
− +
.
b) The tree is
c) If the stakes P are low it would not pay to incur high cost. Notice that the game is symmetric. Given that the other player choses high investment it pays to invest the high amount if 1 1
2 32 1 6.P P P− ≥ − ⇔ ≥ Given that the other player choses low investment it pays to invest the high amount if 2 1
3 22 1 6.P P P− ≥ − ⇔ ≥ Hence, there is a dominant strategy for both player to invest 2ic = if the prize is larger than 6.
d) The tree under b) is preceded by a choice of nature where the branches are LL, LH, HL,HH. Player 1 cannot distinguish in the subsequent game between LL and LH and not between HL and HH. Player 2 cannot distinguish in the subsequent game between LL and HL and not between LH and HH.
e) For example we have for HHhh payoffs 1 1 2 2( 9 2, 9 2)− − .
f) For example we have for LLlh payoffs 1 2 3 3( 3 1, 3 2)− − .
g) The Bayesian Nash equilibrium is characterised by the beliefs given in the Table. This implies that a low value investor (type L) believes that the other investor is of the low (high) type with probability 0.4 (0.6). The high type has the same believes. The equilibrium strategy for player i is “h” if she is of type H. else “l” and this is regardless of the belief. A market for micro credit
Answer a) Good projects are worthwhile, while bad projects are not b) Banks should break even (competitive market). Hence,
(1 ) (1 )
1 (1 )
g b
g b
pP p P
+ − = +
+ ⇔ =
+ −
Since banks cannot distinguish good and bad projects, all projects get the loan should the investor want it.
c) Notice now that investors go ahead iff V R≥ . Otherwise the market breaks down. It is not worthwhile for anyone to invest if V R< . This will be the case if R is sufficiently large, i.e. if or are small or bP p r is large.
d) Banks could require some fraction of self-financing to separate bad and good projects.
h
1