Question

Two farmers, Nduccio and John, graze their animals on a common land. They have the choice to use the common land frequently (heavy use H) or just occasionally (light use L). Their interaction can be represented by the following simultaneous-move game (where Nduccio chooses the row and John chooses the column):

	L	H
L	40,40	20,55
H	55,20	30,30

a) Find the Nash equilibrium of the game

Assume that the same game is repeated infinitely. The utility function of a player is now given by ∑t δt Ut where δ∈(0,1) is a discount factor, Ut is the payoff of the stage game and t∈(0,∞) is time.

Consider the following strategy:

• Start playing L

• Play L as long as any player chooses L

• Whenever any player chooses H switch to H and play it forever.

b) Can (L, L) be a subgame perfect Nash equilibrium if both players use this strategy and both have a discount factor equal to 0.7? Explain your answer

c) Can (L, L) be a subgame perfect Nash equilibrium if both players use this strategy and both have a discount factor equal to 0.1? Explain your answer

d) What is an externality? Explain why externalities create market inefficiencies.

Answer 1

P1/ P2	L	H
L	(40,40)	(20,55•)
H	(55*,20)	(30*,30•)

A) pure strategy NE : ( H, H)

B) δ= .7

Then present discounted value of cooperation payoffs, when both Cooperate to play L only,

V^{​​​​​​C} = 40+40*.7 + 40*.7² + 40*.7³+............ & So on

= 40/(1-.7) = 40/.3 = 133.33

Now if any player defects, then instantaneous payoff = 55, & next period onwards both play only NE of of single stage game & gets 30

So V^{​​​​​​d} = 55+30*.7 + 30*.7²+30*.7³+.............. & So on

= 55+30*.7/(1-.7)

= 55+70

= 125

Thus as V^{​​​​​​c} > V^{​​​​​​d}

so (L,L) can be sustained as SPNE

c) now if discount factor =.1

Then V^{​​​​​​c} = 40+40*.1 + 40*.1²+ ........

= 40/(1-.1) = 40/.9 = 44.44

V^{​​​​​​D} = 55+30*.1 + 30*.1² +.........

= 55+30*.1/(1-.1)

= 55+3.33

= 58.33

Now as V^{​​​​​​d} > V^{​​​​​​c}, so now

(L,L) can't be sustained as SPNE in repeated game.

D) externality is defined as benefit / loss to the third party, who is not directly involved in the production / consumption activity , that generates this externality as the by product .

externality create inefficiency bcoz the market production is always different from that of the socially optimum level