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IGNOU MEC.-001 MICRO ECONOMIC THEORY FREE SOLVED ASSIGNMENT

ASSIGNMENT REFERENCE MATERIAL (2017-18) FREE

M.E.C.-001

MICRO ECONOMIC THEORY

 

Section A

Answer all the questions from this section.

Q1. Discuss the basic difference in approach adopted by Pigou and Pareto to deal with problems of welfare economics.

Ans. Arthur Cecil Pigou popularly known as A.C. Pigou had been one of the most influencing economists that this world has faced. He was an English economist and is known for his contribution in welfare economics. Pigouvian welfare analysis is after the name of him. He formulated some conditions for maximisation of social welfare under some assumption. Those assumptions are as follows:

• Rational consumer: Pigou assumes that every consumer is rational, i.e. for a given amount of income s/he always tries to maximise his or her utility.
• Equal capacity of consumers: This assumption says that all the consumers having equal real income get equal utility from consumption. Therefore, it can be said that all the consumers having same purchasing power are equally efficient in deriving satisfaction from consumption.
• Law of diminishing marginal utility is applicable to money also: The law of diminishing marginal utility is applicable to money also. Therefore, as the income rises, then marginal utility of income declines. An inference can be derived from it, i.e. marginal utility of money for rich people is less than that for poor people.
• National income is an indicator of welfare: Gross domestic product is a measure of national income. GDP means money value of all final goods and services produced and rendered by a country in a period of time, generally one year.

Pigou’s two welfare conditions

Condition 1: Higher national income means higher welfare

Pigou says that welfare will increase if national income (or GDP) increases and welfare achieves it maximum value when national output is maximised. Therefore, there is a positive relationship between GDP and welfare. In this way, it can be said that welfare (W) is directly proportional to national income/output (Q). Symbolically,

Condition 2: Distribution of national income/output is very important to maximise welfare

Since Pigou assumes that the marginal utility of money for rich people is less than the marginal utility of money for poor people, hence, transfer of income from rich people to poor people will improve/increase welfare.

Pigou’s diseconomies and welfare

Pigou observed some diseconomies which cause the welfare to decline. These diseconomies are air and water pollution, unemployment due to technical change, occupational disease (like Computer vision syndrome, Radiation sickness, etc.), child labour, industrial accidents (like Bhopal Gas Tragedy in 1984, India)

Vilfredo Pareto an Italian economist laid down three conditions of efficiency in an economy. These conditions are known as marginal conditions of Pareto optimality or Pareto-efficiency. These conditions are as follows:

Efficiency in Exchange/Consumption/ Equilibrium in consumption

Pareto says that efficiency in production is achieved when

Slopes of indifference curve of X = Slopes of indifference curve of Y = Slopes of budget line

Or

….(2)

where,

means marginal rate of substitution of labour for capital of a consumer X

means marginal rate of substitution of labour for capital of another consumer Y

andmean, prices of the goods A and B respectively.

Using the Edgeworth’s box the above condition can also be proved. The difference is that isoquant curves are replaced by indifference curves and labour, capital are replaced by the units of the goods A and B respectively.

Q2. Consider an industry with three firms each having marginal costs equal to zero. The inverse demand curve facing this industry is:

P(q1, q2, q3) = 60 – (q1, q2, q3).

(a) If each firm behaves as a cournot competitor, what is firm 1’s best response function?

(b) Calculate cournot equilibrium of this problem.

(c) Firms 2 and 3 decide to merge and form a single firm (MC is still zero). Calculate the new industry equilibrium and comment on combined profits from firms 2 and 3 considering pre and post merger profits.

Ans.

Section B

Answer all the questions from this section.

Q3. Suppose Ashok’s utility function is. His initial income when healthy is 36,000. However, there is a 50% chance that she will face financial loss on being taken ill and the income is likely to reduce by 20,000.

(a) Find the expected value of his income

(b) What expected utility he will have given the possible state of her health?

(c) What is the risk premium he will be willing to pay to cover the risk of sickness?

Ans. xxxxxxx

4. Consider the following game given in extensive form:

Section B

(i) Use backwards induction to compute equilibrium of the game.

Ans.xxxxxx

(ii) Write this game in normal form.

Ans.xxxxxx

(iii) How would you differentiate a static game from that of a dynamic game?

Ans. Although a static game is generally considered to be a special case of a dynamic game, the difference between the two is not always obvious. For example, if a machine implements a sophisticated programme which has been programmed once and for all but, once it is run, react dynamically to its environment and the actions played by the other players, is this a static or a dynamic game? In fact, in dynamic games it is generally assumed that players can extract some information from past moves and strategies, and take this into account to adjust their current and future moves. In static games, players have given knowledge (information assumptions, behaviour assumption) and have to make their decision once and for all. On the other hand, a game is dynamic if at least one player is allowed to use a strategy, which depends on past actions, moves, or strategies.

(iv) Suppose the following game is played for an infinite number of periods. If the players are discounting the future at the rates of δ_A and δ_B respectively, find the conditions under which they sustain the outcome (2, 2) in every period.

Ans. As this is an infinitely repeated  game, we will have to assume that future payoff are discounted. So as to obtain the present value (PV) of future profits.

Let  equals to each firms rate of discounted, where r is the rate of interest. In this condition Now, we have calculate Present Value (PV);

PV (High)                                                  …(1)

PV (High)                                             …(2)

By (1) & (2)  (1– ) PV (High) = 2

or PV (High)

and the alternative present value of deviating from this cooperative outcomes;

PV (low)                                                          …(3)

PV (low)                                        …(4)

By (3) and (4) (1– ) PV (low) = 4 +  – 4 = 4 (1– ) +

PV (low)

Therefore, the cooperative outcome will be maintained indefinitely for sustain the outcome (2, 2) or High, High, If, PV (High)  PV (low)

so, if the both players must have the same discount factor, then the answer is  2/3 will keep them at the equilibrium output (2, 2).

Q5. Derive the indirect utility function form the given direct utility function. Use Roy’s identity to construct demand functions for the two goods x₁ and x₂. Are these same as demand functions derived from the direct utility function?

Ans. Use Roy’s identity to construct demand functions

We have the following direct utility function:

…(1)

Subject to the following budget constraint

…(2)

Lagrange function

…(3)

First order conditions of maximisation are as follows:

…(4)

…(5)

…(6)

From (4) we have

…(7)

From (5) we have

…(8)

From (7) and (8) we have

Or

…(9)

From (6), (8) and (9) we have

y = m – 1                                                                         …(10)

The equations (9) and (10) are called Marshallian demand function.

Second order condition Bordered Hessian (H) determinant is defined as follows:

 

where

Note:  g = px₁ + _x2

Therefore, the value of H is calculated by solving the above determinant and we get:

Since, H > 0, hence, the values of x and y will give maximum value of the direct utility function.

 

the values of x and y derived in the equations (3) and (4) are optimal values, hence, putting the same into (1) we get optimum utility function denoted by

…(11)

The equation (11) is not called same as demand functions derived from the direct utility function.

Q6. Consider a world with two agents, A and B. there are two goods 1 and 2. The utility functions of A and B are given as  and. Their initial . Their initial endowments are W_A = (1,2) and W_B (2,1)

(a) Draw the Edgewroth Box for the agents considering their initial endowments and commodity consumptions.           

Ans.

(b) Find the contract curve through your Edgework Box.

Ans. In the contract curve the two indifference curves must touch only once. They cross-over only point of the corner of B’s indifference curve can be part of the contract curve, in contract curve X_B2 = X_B2.

(c) Find the demand functions of A and B for prices P₁ and P₂ and incomes m_A of A and m_B of B.

Ans. Maximising utility gives the value of initial endowments, we get

and

(d) Find the competitive equilibrium price and equilibrium allocation of this economy.

Ans. In equilibrium price

x_A + x_B = 4

(4 is the total endowment of good I)

After solving, we get

P_i = 1

Q7. Write short notes on the following:

(a) Hotelling’s lemma

Ans. Suppose we have a profit function defined as follows:

where

= Profit; P = Price of the product; w = wage rate; r = capital rent;
L = Labour; K = Capital and production function f(L, K). Also note that
w, r and P are parameters of the profit function.

The first order conditions of maximisation are as follows:

…(1)

…(2)

Suppose the optimum values of L and K are denoted by and respectively. Then the optimum value profit will be

…(3)

Therefore, the equation (1) is an indirect objective function.

Taking partial differentiation of (1) with respect to the parameter w

…(4)

(b) First welfare theorem

Ans. According to this theorem, the allocation of goods and inputs obtained in a general competitive equilibrium is economically efficient, i.e. such allocation is Pareto efficient. In other words, the first fundamental theorem of welfare tells us that “any competitive general equilibrium or Walrasian equilibrium is Pareto efficient.” This theorem is also known as “Invisible hand” theorem because the theorem states that a general competitive equilibrium exhausts all the possible mutual gains from trade or redistribution of the resources.

(c) Public goods

Ans. A public good means a good which is produced by a public agency and they are non-rival and non-excludable. A good is non-rival when the good can be consumed by two persons and one’s consumption does not interfere in the consumption of other. A non-excludable good means the good when once produced, people cannot be excluded from enjoying its benefits. Parks, pollution control, police, military, etc. are all public goods.

Market failure in case of Public Good

There is a market failure in case of public good. The market does not produce optimum output of a public good. This is because condition of efficiency in product mix is violated. The third condition of Pareto efficiency states that in case of two goods A and B and two consumers
X and Y, the marginal rate of transformation(slope of Production possibility curve) must be equal to marginal rate of substitutions (slope of indifference curves). Symbolically,

But in case of public good, the equilibrium condition is

Hence, there is Pareto inefficiency in case of a public good.

(d) VNM utility function

Ans. If X is a lottery yielding prize X_i with probability P_i: i = 1, 2, 3…, k and another lottery Y yielding prize Y_i with probability R_i: j = 1, 2, 3…, t, then iff

where, known as vNM utility index and it denotes the utility received from the prize X_i and similarlydenotes the utility received from the prize Y_i.

The expressions  in the above inequality are called expected

Thus, as per the theorem if the expected utility derived from a lottery X is greater than the expected utility derived a lottery Y, then lottery X is preferred to Y.

The converse of the theorem is that any agent obeying the above axioms and maximising the expectation of a function U, then U is called vNM utility function.