By Pokuri Roshan
1. Explain the concept of Nash equilibrium. How it is related to (a) Dominant strategy equilibrium and (b) Subgame perfection?
Ans. In a ‘n’ player normal form game G = {S1, S2,…..……,Sn; u1, u2,..……….,un}, the strategies (s*1,s*2,…….,s*n) constitute a Nash equilibrium if, for each player I, si is player i’s best response to the strategies specified for the (n1) other players, (s*1,s*2,………, s*i1, s*i+1,…….,s*n) :
or U1 (s*1,s*2,………, s*i1, s*i, s*i+1,…….,s*n) ≥ U1 (s*1,s*2,………, s*i1, s*i, s*i+1,…….,s*n) where i = 1,2,…,n
or, for every feasible strategy si in Si; that is s*i solves
max U1 (s*1,s*2,………, s*i1, s*i, s*i+1,…….,s*n)
s*2ЄS*i
Nash equilibrium is strategically stable and selfenforcing because no single player wants to deviate from her predicted strategy.
Dominant Strategy – A player in a simultaneous move game may have nay finite number of pure strategies at her disposal. One of these strategies called as dominant strategies if it outperforms all of her other strategies, no matter what any other player does.
In the normal form game G = {S1, S2,…..……,Sn; u1, u2,..……….,un}, let s1i and s11i be feasible strategies for the player I (i.e s1i and s11i are members of Si). Strategy s11i strictly dominates strategy s1i, if for each possible combination of other players’ strategies, i’s payoff from playing s1i is strictly less than i's payoff from playing s11i. Symbolically,
U1 (s1,s2,………, si1, s1i, si+1,…….,sn) < U1 (s1,s2,………, si1, s11i, si+1,…….,sn)
for each (s1,s2,………, si1, si+1,…….,sn) that can be constructed from other players’ strategy spaces (S1,S2,………, Si1, Si+1,…….,Sn). Strategy s11i is called strictly dominant strategy for player i. The dominance is said to be weak when there is a weak inequality (≤ rather than <) in the above equality.
Subgame perfection – A subgame is a part of the game, which starts from a singleton information set and stretches up to the end of the game. A configuration of strategies, which induces Nash equilibrium in every subgame of a game, is called subgame perfect Nash equilibrium. This is explained in the example given below:
Tough(4,1)
Tough
Accommodating
Coke (3,1)
Pepsi
Tough (0,3)
Accommodating
Accommodating
(1,2)
Coke will always play Accommodating, and as Pepsi can foresee this. Pepsi will also play accommodating as well. Therefore the Nash equilibrium of the subgame is (Accommodating, Accommodating) and PSNE (Enter Accommodating, Accommodating) is the only SPNE (subgame perfect Nash equilibrium) of the game. Thus, the concept of SPNE is able to eliminate noncredible threat.
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2. What is indirect utility function? How will you derive an indirect utility function from a direct utility function? Explain Roy’s identity.
Ans. Let qi denotes commodity I and pi is the price of that commodity. Let y denotes money income of the consumer. Suppose vi = pi/y. The budget constraint written as:
n
1 =∑ viqi (1)
i=1
Since optimal solutions in the demand functions are homogeneous of degree zero in income and prices, nothing essential is lost by this transformation to ‘normalised’ prices. The utility function U = f(q1,qn) together with equation (1) gives the following first order conditions of utility maximization:
fi – λvi = 0 for all I – 1,….n
n
and 1 =∑ viqi (2)
i=1
Ordinary demand functions obtained by solving equation (2): qi = Di (v1,….,vn)
The indirect utility function g(v1,…..vn) is defined by
U = f[D1(v1,….,vn), Dn(v1,….,vn) = g(v1,….,vn) (3)
It gives the maximum utility as a function of normalized prices. The indirect utility function reflects a degree of optimization and market prices.
Applying the composite function rule of calculus to equation (3), we get
n n
gj = ∑ fi (∂qi/∂vi) = λ∑ vi (∂qi/∂vj) j = 1,…..n (4)
i=1 i=1
where the second equalities are based on equation (2). Partial differentiation of equation (1) with respect of vj yields
n
∑ vi (∂qi/∂vj) =  qj j = 1,…..n
i=1
Thus, equation (4) implies
qj =  (gi/ λ) j = 1,…..n (5)
which is called Roy’s identity. Optimal commodity demands are related to the derivatives of the indirect utility function and optimal value of the Lagrange multiplier (i.e, the marginal utility of income). Substituting equation (5) into the last equation of equation (2) gives
n
λ = ∑ vigi
i=1
n
and qj = gi / ∑ vigi j = 1,…..n
i=1
to provide an alternative form of Roy’s identity.
The information it gives in contrast to utility function are described by a set of duality theorems. The direct utility function determined by the indirect is same as the direct utility function that determined is indirect. Duality in consumption forges a much closer link between demand and utility functions.
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Section B
Medium Answer questions
3 Derive the elasticity of substitution from the CobDouglas production function q=f(K,L) = AKaLb.
Ans The CobbDouglas production function is given by:
q=f(K,L) = AKaLb ,
where A, a and b are all positive constants.
Isoquants resulting from the functional form have convex shape. The CobbDouglas function can exhibit any degree of returns scale depending on the values of a and b. Suppose all inputs were increased by a factor m. Then,
f(mK,mL) = A(mK)a(mL)b = Ama+b KaLb (
= ma+b f(K,L)
Hence, in the CobbDouglas function
• a+b = 1, implies constant returns to scale
• a+b > 1, implies increasing returns to scale
• a+b < 1, implies decreasing returns to scale
To determine the elasticity of substitution in CobbDouglas production function, let us use Allen’s definition
σ = ∂q/∂L.( ∂q/k) / q (∂2q/∂L∂K)
When q = AKaL1a
σ = [(1a).(q/L).a(q/K)] / [q2.{(1a)(a)/LK)}] = 1
The CobbDouglas function is linear in logarithms, i.e,
log q = log A + a log K + b log L.
As a result, the constant a in above equation is the elasticity of output with respect to capital output, and b is the elasticity of output with respect to labour input. To get the result, take
eq,K = ∂q/∂K . (K/q)
= ∂ log q / ∂ log K) such that
eq,K = a from above eqn. Similarly, we get eq,L = b
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4. What are the recommendations of Coase to solve the problem of externalities?
Ans. Ronald Coase asserted that under perfect competition private and social cost will be equal and addressed a well known problem of market externalities, known as the ‘Spillover Effect’. This occurs when someone other than the buyer must share the benefits or costs of a product. The classic example of such a perception is pollution. Factories can either treat pollution – which costs money or dump it into the air or water for free. If they choose to dump, they may save their customers some money, but citizens who live near the factory will also pay a price in higher death and disease rates, less fertile land, environmental catastrophes, etc. Sometimes the spillover effect is both positive and negative. An airport benefits its customers, but it also subjects the local neighborhood to various externalities eg. noise pollution.
Coase argued that individuals could organise bargains so as to bring about an efficient outcome and eliminate externalities without without government intervention. The government should restrict its role to facilitating bargaining among the affected groups or individuals and to enforcing any contracts that result. This result, often known as the “Coase Theorem” requires that:
• property rights are well defined;
• the number of people involved is small; and
• bargaining costs are very small
Only if all three conditions, apply then individual bargaining will solve the problem of externalities. All that is needed is a common law or statutory rule, which assigns rights over the externality to one party or another. The market/pricing mechanism will then function in the same way as it does for the ordinary goods and services over which rights clearly defined.
Thus Coase’s theorem has helped in the emergence of numerous policy formulations in the welfare imperatives of societies.
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5. Explain Arrow’s Impossibility Theorem.
Ans. The democratic procedure for reaching a social choice or group decision is the expression of their preferences by individuals through free voting. Social choice will be determined by the majority rule. But Arrow has demonstrated through his impossibility theorem that consistent social choices cannot be made without violating the consistency or transitivity condition. The social choice on the basis of majority rule may be inconsistent even if individual preferences are consistent. Arrow first considers a simple case of two alternative social states and proves that in this case group decision or social choice through a majority rule yields a social choice, which can satisfy all the five conditions. But when there are more than two alternatives, majority rule fails to yield a social choice without violating at least one of the five conditions. Thus, Arrow’s theorem says that if the decisionmaking body has at least two members and at least three options to decide among, then it is impossible to design a social choice function that satisfies all these conditions at once.
Alternative Social States
X Y Z
A 3 2 1
B 1 3 2
C 2 1 3
Ranking of Alternatives by Individuals and Social Choices
In this table three individuals A, B and C who constitutes the society have voted for three alternative social states X, Y and Z by writing 3, against the most preferred alternative, 2 for the next preferred alternative and 1 for the least preferred alternative. Majority rule leads to inconsistent social choices because on the one hand, X has been preferred to Z by the majority and on the other hand, Z has also been preferred to X by majority, which is contradictory or inconsistent.
Arrow has derived three consequences to explain his impossibility theorem. According to Consequence I, whenever the two individuals prefer X to Y, then irrespective of the rank of the third alternative Z, society will prefer X to Y. According to Consequence II, if in a given social choice, the will of individual prevails against the opposition of individual B, then the will of A will certainly prevail in case individual B is different or agrees with A. According to Consequence III, if individuals A and B have exactly conflicting interests in the choice between two alternatives X and Y, then the society will be indifferent between X and Y. It is interesting to note that the simple proof of the impossibility theorem follows from Consequence III. For instance, if individual A prefers X to Y and individual B prefers Y to Z and if society opts for X, then A will be a dictator in as much as her choice will always be social choice. Thus, Arrow’s theorem says that ‘if the decisionmaking body has at least two members and at least three options to decide among, then it is impossible to design a social choice function that satisfies all these conditions at once. Arrow, therefore concludes that it is impossible to derive a social ordering of different conceivable alternative social states on the basis of the individual ordering of those social states without violating at least one of the value judgments expressed in the five conditions of social choices. This is in essence his impossibility theorem.
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6. Discuss the concept of asymmetric information. Explain the relation among moral hazard, adverse selection and signaling, giving suitable examples.
Ans. Asymmetric information signifies a situation in which one party involved in transaction with another, has more or superior knowledge and information than the other. This is often the case between buyer and seller, where seller has more knowledge than buyer. However, the opposite condition can also happen at times. The situation can potentially be harmful as the party with more information can take advantage of other’s lack of knowledge and thereby exploit the other party.
Asymmetric information is the source of two major problems such as the following:
• Moral Hazard  This reflects on the immoral behavior of a party with asymmetric information subsequent to a transaction. For example, some people commit arson purposely to reap benefits from the fire insurance. After being insured, a driver may drive recklessly as damage in the event of accident will be borne by the insurance company.
• Adverse Selection – In this case, the party displays immoral behavior by taking advantage of the knowledge or asymmetric information prior to transaction. For example, some people secure life insurance aware of the languishing health. A bank may fail to observe the riskreturn characteristics of a project. Consequently, it will extend credit facilities to bad projects while rationing credit to good projects.
The presence of asymmetric information creates an adverse selection problem: if consumers cannot tell the quality of a product and are willing to pay only an average price for it, then this price is more attractive for sellers who have bad products than to seller who have good products (hence the term adverse selection). Consequently, more bad products (i.e lemons) will be offered than good products. Now, if consumers are rational, they should anticipate this adverse selection and expect that at any given price, a randomly chosen product is more likely to be a lemon than a good product. Of course, these expectations imply a lower willingness to pay for products and so the proportion of goods products that is actually offered falls further. Eventually, this process may lead to a complete breakdown of the market.
• Signalling – Initiative taken by individuals of informed category in markets characterized by adverse selection to help identify their true risk category. For example, In the lemons model if sellers of a good product could find some activity that was less costly for them than for seller of a bad product, then it might pay them to undertake this activity as a signal of good quality product. The buyers, too, would learn that the signal was associated with good quality product.
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7. Do you agree with the proposition that a riskaverse person will optimally buy full insurance if the insurance is actuarially fair? Give reasons in support of your answer.
Ans. Yes a riskaverse person will optimally buy full insurance if the insurance is actually fair as the expected utility of such consumer rises with the purchase of insurance although expected wealth is unchanged. Since insurance increases the consumer’s welfare, she is willing to pay some positive price in excess of the actuarially fair premium to defray risk. Thus, the agent is trying to equate the marginal utility of wealth across states. Because, the utility of average wealth is greater than the average utility of wealth for a risk averse agent. The agent wants to distribute wealth evenly across states of the world, rather than concentrate wealth in one state. She will attempt to maintain wealth at the same level in all states of the world, assuming that she can costlessly transfer wealth between states of the world (which is what actuarially fair insurance allows the agent to do).
Let a riskaverse person’s decided to buy insurance by taking the initial endowment wo and L is the amount of the loss from an accident.
Pr(1 – p):U(.) = U(wo),
Pr(p) : U9.) = (wo – L)
If insured, the endowment is: Pr(1 – p) : U(.) = U(wo – pA)
Pr(p):U(.) = U(wo – pA + A – L)
Expected utility is uninsured is : E(UI = 0) = (1 – p)U(wo) + pU(wo – L)
Expected utility if insured is : E(UI = 1) = (1 – p)U(wo – pA) + pU(wo – L + A – pA))
The optimal policy that the agent should purchase, differentiate above eqn with respect to A,
= p(1 – p)U’(wo – pA) + p(1 – p)U’(wo – L + A – pA) = 0
U’(wo – pA) = U’(wo – L +A – pA)
A = L,
which implies that wealth is wo – L in both states of the world (insurance claim or no claim)
