Question

Assume two large economies using the instrument of monetary policy to maximise their welfare function which includes an output and an inflation target. Consider the following two equilibria in the setting of national monetary policy: (a) Nash equilibrium and (b) cooperative equilibrium. Explain the meaning of each equilibrium and show graphically the above two equilibria. In your diagram include the iso-welfare curves for each country and the reaction functions. Which equilibrium leads to a higher welfare level and why? In your analysis you may assume that domestic monetary policy is transmitted abroad negatively

Answer 1

First let us understand what is Nash equillibrium and Co operative Equillibrium.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative gameinvolving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

In terms of game theory, if each player has chosen a strategy, and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and their corresponding payoffs constitutes a Nash equilibrium.

For example Alice and Bob are in Nash equilibrium if Alice is making the best decision she can, taking into account Bob's decision while his decision remains unchanged, and Bob is making the best decision he can, taking into account Alice's decision while her decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long as the other parties' decisions remain unchanged.

Nash equilibrium has been used to analyze hostile situations like war and arms races.

Let us understand with an example:

The coordination game is a classic (symmetric) two player, two strategy game, with an example payoff matrix shown to the right. The players should thus coordinate, both adopting strategy A, to receive the highest payoff; i.e., 4. If both players chose strategy B though, there is still a Nash equilibrium. Although each player is awarded less than optimal payoff, neither player has incentive to change strategy due to a reduction in the immediate payoff.

Player 2 Player 1	Player 2 adopts strategy A	Player 2 adopts strategy B
Player 1 adopts strategy A	4 4	3 1
Player 1 adopts strategy B	1 3	2 2

A sample coordination game showing relative payoff for player 1 (row) / player 2 (column) with each combination.

A famous example of this type of game was called the stag hunt; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat (4 utility units) than the latter (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, he will fail in hunting (0 utility units), whereas if they both hunt it they will split the payload (2, 2). The game hence exhibits two equilibria at (stag, stag) and (rabbit, rabbit) and hence the players' optimal strategy depend on their expectation on what the other player may do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they suspect that the other will hunt the rabbit, they should hunt the rabbit. This game was used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.

Nash Equilibrium (NE) assumes that players always make a best response to what other players are doing. However, this assumption does not always hold. Consider the Prisoner’s Dilemma, in which two prisoners can choose either to defect or cooperate with payoffs.

  Cooperate   Defect

Cooperate (3,3) (0,5)

Defect (5,0) (1,1)

Although the only best response here is to play Defect no matter what the other player does, people often do play (Cooperate, Cooperate). There are a number of other games in which Nash equilibrium does not predict actual behavior well.

Now, let us learn What is Social welfare function:

In welfare economics, a social welfare function is a function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.

The social welfare function is analogous to the consumer theory of indifference-curve–budget constraint tangency for an individual, except that the social welfare function is a mapping of individual preferences or judgments of everyone in the society as to collective choices, which apply to all, whatever individual preferences are for (variable) constraints on factors of production. One point of a social welfare function is to determine how close the analogy is to an ordinal utility function for an individual with at least minimal restrictions suggested by welfare economics, including constraints on the number of factors of production.

There are two major distinct but related types of social welfare functions:

• A Bergson–Samuelson social welfare function considers welfare for a given set of individual preferences or welfare rankings.
• An Arrow social welfare function considers welfare across different possible sets of individual preferences or welfare rankings and seemingly reasonable axioms that constrain the function.

Now let us understand What is Iso welfare curve with the help of a diagram.

Now, let us understand this with another more detailed example:

Considering the above two equillibriums the social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.