Solutions

  1. Certainty Equivalent can be defined as the guaranteed return that an investor would consider as equivalent to taking up a higher but riskier (uncertain) return. In other words, the utility that an investor gets at the Certainty Equivalent would be equal to the Expected utility from the Riskier asset. In this case, the individual enjoys a higher utility of the expected lottery prize compared to the Expected Utility of the Lottery, which is uncertain.

    Where,  is the expected prize of the lottery and  is the Lottery.

For the investor to accept the lottery instead of the expected prize money, the payoff from the Lottery needs to be higher by a Risk Premium, to compensate for the uncertainty. Thus, the Certainty Equivalent of the Prize Money, CE is defined as,
Thus, the Certainty Equivalent of the expected prize money is higher than the Lottery or conversely, the Certainty Equivalent of the Lottery is lesser than the expected prize money.

  1. Dominant Strategy Equilibrium: Dominant strategy equilibrium can be defined as the set of strategy that a set of players adopt irrespective of the strategies adopted by the other players. The equilibrium is the point where all the dominant strategies of all the players coincide.

    Nash Equilibrium: Nash Equilibrium can be defined as the set of strategic choices adopted by a set of players, in which all the players know the equilibrium strategies of the other players and do not stand to benefit (or increase the pay offs) by changing their strategies.

    A dominant strategy is a special case of Nash Equilibrium in that there is only one set of strategies where the players do not benefit by changing their strategies (dominant strategy). However, the dominant strategy equilibrium may not necessary be the Pareto Efficient outcome (i.e. the outcome with the highest pay-off / utility) and there could be better outcomes that do not form an equilibrium in a non cooperative game.

  2. The following are the key assumptions about consumer preferences. (1)  Preferences are complete:  this means that the consumer is able to compare and rank all possible combinations of resources; (2) Preferences are transitive:  this means that preferences are consistent, in that if bundle A is preferred to bundle B and bundle B is preferred to bundle C, then we should be able to conclude that bundle A is preferred to bundle C; (3) More is preferred to less:  this means that all goods are desirable, and that the consumer will always prefer to have more of a good; (4)  Diminishing marginal rate of substitution:  this means that indifference curves are convex, and that the slope of the indifference curve increases (becomes less negative) as we move down along the curve. Assumptions about consumer preferences that were violated.
    1. Preferences are transitive
    2. Preferences are complete
    3. More is preferred to less
  3. Marginal Rates of Substitution is defined as the rate at which a consumer prefers to give up one goods for a unit of another good while maintaining the same level of utility. This can be explained by a micro-economy that produces two commodities A and B and has two consumers who have the same budget line. Consumer 1 prefers Commodity A to Commodity B and Consumer 2 prefers Commodity B to Commodity A. The Production Frontier represents the available combination of Resources A and B in the economy.

    In the case when the MRS is different for consumers 1 and 2, the equilibrium consumption is given by points C1 and C2. In the scenario, Consumer 1’s preference would be skewed towards A and consumer 2’s preference would be skewed towards B to maximize their utilities. Thus, consumers 1 and 2 would not be competing for the same resource available in the economy and hence the maximizing utility would be attained at the sum of the utilized resources below the Production Frontier (total available resource in the economy).

  4. Production Externalities are defined as the additional social cost (caused due to production) that have to be borne by entities other than the producer of the good or service. A merger between two firms (one producing the social cost increasing good/service and other affected by the cost) can result in reduction (or elimination) of production externalities.

This phenomenon can be explained by considering the example of a steel mill producing steel with its production facility upstream and a fishery firm located downstream that depends on the fish in the river.

 

The steel mill produces steel (S) and discharges pollutants (D) in the river that affects the fish catch of the fisheries firm (F). The steel mill doesn’t take into consideration the social cost of production (i.e. the increased cost of fishing) to decide on the optimal amount of steel produced.
The Profit function for the Steel mill is given by,

 

The Profit function for the fisheries is given by

 

Where, and are the profit maximization prices

and are the respective cost functions of Steel Mill and Fisheries
The profit maximization of the steel mill would be achieved,

i.e. the Steel mill would operate at the level of Pollutants such that the Marginal Cost of Producing Steel w.r.t the amount of Pollutant D equals zero.

 

The profit maximization of the fisheries would be achieved,

 

If the two firms are merged, the profit function of the combined entity would be,

 

The profit maximization of the combined entity would be achieved,
i.e. the combined entity would operate at the level of Pollutants such that the Marginal Cost of Producing Steel w.r.t the amount of Pollutant D equals the Marginal Cost of Fishing w.r.t the amount of Pollutant D.

 

From the graph, the stand-alone still mill produces steel till the Marginal Cost is zero, equivalent to DSteel Mill of pollutants. However, in the merged entity, the steel mill produces steel till the Marginal Cost of Steel and Fishing are equal in magnitude (and opposite in sign), represented by DMerged Entity. Thus, the merger between the two firms reduces the amount of pollutant D and thus reduces the Production Externalities associated with steel production.
 

  1. The payoffs for both the aluminum producers are given in the payoff matrix below.
LAG/BAG Don’t Expand Expand
Don’t Expand 3,4 2,3
Expand 4,2 1,1
  1. A Dominant Strategy can be defined as a set of choices that a firm can adopt to maximize its payoffs irrespective of the strategic choices that the opponent adopts. LAG doesn’t have a dominant strategy, because it is beneficial (maximum pay-off) for LAG to adopt an ‘Expand’ strategy if BAG ‘Doesn’t Expand’ while it is beneficial to adopt a ‘Don’t Expand’ strategy if BAG ‘Expands’. However, BAG has a dominant strategy, because irrespective of whether LAG adopts an ‘Expand’ or ‘Don’t Expand’ strategy, BAG would be benefited by adopting a ‘Don’t Expand Strategy’.
  2. Nash Equilibrium can be defined as the set of strategic choices adopted by a set of players, in which all the players know the equilibrium strategies of the other players and do not stand to benefit (or increase the pay offs) by changing their strategies.
    In this case, the Nash Equilibrium would be attained when LAG adopts an ‘Expand’ and BAG adopts a ‘Don’t Expand’ strategy.
  3. BAG moves first. The sequential game’s extensive form is given below:
  4. As can be seen from the extensive form of the game in section (c), the equilibrium obtained is
    BAG – Expand and LAG – Don’t Expand. Also from the pay-off in the strategy equilibrium, BAG has a payoff of 3 compared to a payoff of 2 for LAG. Clearly, BAG has a first mover advantage.
  5. BAG moves first. The sequential game’s extensive form is given below:

    As can be seen from the extensive form of the game above, the equilibrium obtained is
    BAG – Expand and LAG – Don’t Expand. Hence the equilibrium strategy doesn’t change compared to the game in section (c). Also from the pay-off in the strategy equilibrium, BAG has a payoff of 3 compared to a payoff of 2 for LAG. Clearly, LAG does not have a first mover advantage.

 

  1. Answers to the questions by sections below:
    1. Value to buyers for high-quality laptops = 400
      Value to buyers for low-quality laptops = 100
      If the buyer assumes that a given laptop will be of high quality, the value to the buyer for any laptop =
      (Please note that this assumption holds if the buyers are risk neutral)
    2. The supply function for high quality laptops,
      At a price of 250, the available supply of high-quality laptops = 250 – 100 = 150

      The supply function for low quality laptops,
      At a price of 250, the available supply of high-quality laptops = 2*250 – 50 = 450

      Buyers are not correct in their assumption, since only 33% (=150/450) of the laptops available would be of high quality.

    3. Once the buyers know the odds of buying a high-quality laptops are lower than expected, the Willingness to Pay for any laptop reduces till it reaches an equilibrium state, where the market for high-quality laptops does not exist.
      At equilibrium price, the customers would be willing to pay only 100 for a laptop since the supply of low-quality laptops is 100%.

 

  1. Providing a guarantee would make the consumers perceived odds of buying a high-quality laptop, increasing the customer’s willingness to pay for any laptop. This can ensure that the reservation price to ensure a supply of high-quality laptops (i.e. can result in a market for high-quality laptops).
  1. Answers to the questions by sections below:
    1. Philippa’s expected income would be
      Expected Utility =
    2. With the insurance, Philippa’s expected income would be
      30%*(50000+34000)+70%*(100000-16000) = 84000
      Expected Utility =
    3. Philippa’s utility function shows that he/she is a risk-averse individual. The insurance premiums reduce the expected income and hence the utility also decreases from 291.55 Utils to 289.83 Utils. Hence Philippa is worse off with the insurance.
    4. With the revised insurance, Philippa’s expected income would be
      30%*(50000+20000)+70%*(100000-10000) = 84000
      Expected Utility =
      Hence the expected Income and the Utiity  remains the same as in case b and lesser compared to case a.
    5. Contracting Insurance between a risk-neutral insurer and a risk-averse agent would constitute a Pareto optimal situation. This can be explained by the fact that the insurer being risk-averse would have a constant marginal utility across all states of risk. On the other hand, the risk-averse agent would have a different marginal utility at all states. In a pareto efficient condition, the marginal utility should be the same for both the insurer and the agent across all states, indicating that the insurer would absorb fully absorb the risk (uncertainty) in the agent’s income (or pay-off)
  2. Answers to the questions by sections below:
    1. Without the special promotion, I can purchase 4 pizzas (£ 10 each) or 40 units of other goods (£ 1 each). With the promotion, I can purchase 4 Pizzas and 15 units of other goods or 40 units of other goods. The budget curve is represented by the locus of the different purchasing options bound by the maximum income.
      Mathematically, the budget line is defined as,

With intercept of  and slope of

  1. Buying behavior is defined by the point where the preferred available indifference curves is tangential to the budget line. In the figure below, the original scenario and the scenario with the promotion, the buying behaviors are defined by the tangential point to the Indifference Curves (A and A’ respectively)
  1. If the shape of the indifference curves are different (different MRS), the buying behavior would be different as illustrated in the diagram below. A’’ is the new optimal behavior for the consumer for the new indifference curve.
  2. With the promotion, the optimal buying behavior is defined as the point in the new budget line tangential to the available preferred indifference curve. As discussed in section (a), the number of pizzas that can be purchased as per the budget line is more than the original scenario without the promotion. As per the assumptions of consumer preferences, the more the quantity, the higher the utility. Thus, in the case of the promotion, the utility at the optimal behavior is higher than the original scenario. Thus the consumer is better off with the promotion.
  1. Answers to the questions by sections below:
    1. The Edgeworth Box is a representation of the distribution of resources between two or more actors in a micro economy.
    2. The Endowment point is defined as the point of the initial distribution of resources namely coffee and biscuits. At the point of endowment, the Indifference curves intersect each at more than one point, and the point of endowment lies on one of the intersecting points.
      For Chris, Coffee and Biscuits are perfect substitutes indicating a Marginal Rate of Substitution of 1. For Pat, Coffee and Biscuits are complementary, indicating a MRS of 0 for the horizontal segment and Infinity for the vertical segment of the Indifference curve. It is to be noted that these Indifference curves are for a defined utility.

      For Pat to achieve maximum utility for a given number of biscuits, he needs to consume the equal number of cups of coffee. And for Chris, since the two commodities are substitutable, Chris can trade a cup of coffee for a biscuit and otherwise. Thus, there will be a re-distribution of the resources till the Marginal Rate of Substitution of both Chris and Pat become equal. Or in other words, the indifference curves become tangential to each other, at which point there cannot be further redistribution to derive incremental utility for both Chris and Pat.

    3. The contract curve is the locus of points at various levels of distribution of the commodities coffee and biscuits, at which the Marginal Rates of Substitution of Pat and Chris are equal.
      Since Pat’s preferences are that coffee and biscuits are complementary, the Contract Curve would follow the utility maximizing points where Pat’s Biscuits equals the number of cups of coffee he gets. The Contract curve is shown below.
    4. The Utility Frontier is the set of all possibilities of the distribution of commodities that provides the maximum utility for Chris given the utility levels of Chris. The Utility Frontier is a derivative of the Contract Curve, mapped between the utility levels of Pat and Chris as a function of the distribution of Coffee and Biscuits between Pat and Chris. The contract curve represents the point of maximum utility for Pat and Chris for a particular distribution and thus the points on the Utility Frontier are Pareto Efficient.