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Q1. In a Cournot duopoly, two firms produce output Q₁ and Q₂, respectively. The market demand function is: P = 200 – Q where Q = Q₁ + Q₂ and the total cost for each firm is TC = 20Q. (a) Derive the reaction function of firm 1 and firm 2. (b) Find the Cournot-Nash equilibrium output for each firm. (c) Calculate the equilibrium market price and the total profit of the industry.

Ans. (a) Derivation of Reaction Functions Each firm seeks to maximize its profit, assuming the other firm’s output is fixed. For Firm 1: Firm 1’s profit (Π₁) is the difference between its total revenue (TR₁) and total cost (TC₁). TR₁ = P × Q₁ = (200 – Q₁ – Q₂)Q₁ = 200Q₁ – Q₁² – Q₁Q₂ TC₁ = 20Q₁ Π₁ = TR₁ – TC₁ = (200Q₁ – Q₁² – Q₁Q₂) – 20Q₁ = 180Q₁ – Q₁² – Q₁Q₂ To maximize profit, we differentiate Π₁ with respect to Q₁ and set it to zero: dΠ₁/dQ₁ = 180 – 2Q₁ – Q₂ = 0 2Q₁ = 180 – Q₂ Q₁ = 90 – 0.5Q₂ . This is Firm 1’s reaction function. For Firm 2: Since the cost functions for both firms are identical, Firm 2’s reaction function will be symmetric: Q₂ = 90 – 0.5Q₁ . This is Firm 2’s reaction function. (b) Cournot-Nash Equilibrium The Cournot-Nash equilibrium occurs where both firms are on their reaction functions simultaneously. We can solve this by substituting one reaction function into the other. Substitute Firm 2’s reaction function (Q₂) into Firm 1’s reaction function: Q₁ = 90 – 0.5(90 – 0.5Q₁) Q₁ = 90 – 45 + 0.25Q₁ Q₁ – 0.25Q₁ = 45 0.75Q₁ = 45 Q₁ = 45 / 0.75 Q₁ = 60 Now, substitute Q₁ = 60 into Firm 2’s reaction function: Q₂ = 90 – 0.5(60) Q₂ = 90 – 30 Q₂ = 60 Thus, in the Cournot-Nash equilibrium, each firm produces 60 units. (c) Equilibrium Price and Industry Profit Equilibrium Market Price: Total market output (Q) = Q₁ + Q₂ = 60 + 60 = 120 Substituting Q = 120 into the market demand function: P = 200 – Q = 200 – 120 P = 80 So, the equilibrium market price is 80. Total Industry Profit: First, calculate the profit for each firm: Profit of Firm 1 (Π₁) = P × Q₁ – TC₁ = 80 × 60 – 20 × 60 = (80 – 20) × 60 = 60 × 60 = 3600 Profit of Firm 2 (Π₂) = P × Q₂ – TC₂ = 80 × 60 – 20 × 60 = 3600 Total industry profit is the sum of the profits of both firms: Total Profit = Π₁ + Π₂ = 3600 + 3600 Total Profit = 7200

Q2. (a) Distinguish between short-run and long-run production functions. Illustrate how the law of variable proportions operates in the short-run? (b) Given the following Cobb-Douglas production function: Q = 2K⁰.⁴ L⁰.⁶ where Q is the output, K is capital and L is labour input. (i) Determine the returns to scale for this production function. (ii) Calculate the output elasticities of labour and capital. (iii) Derive the marginal rate of technical substitution.

Ans. (a) Short-run vs. Long-run Production and Law of Variable Proportions A production function describes the technical relationship between inputs and outputs. It can be differentiated into short-run and long-run based on the time horizon.

• Short-run Production Function: The short run is a period in which at least one input (typically capital) is fixed, while other inputs (like labour) are variable. The firm can only change its level of output by varying the quantity of the variable input. It is governed by the Law of Variable Proportions or the Law of Diminishing Returns .
• Long-run Production Function: The long run is a period in which all inputs of production are variable. The firm can adjust all factors, including its plant size, machinery, and scale of production. The long-run production function is concerned with Returns to Scale .

Law of Variable Proportions:

This law applies in the short run. It states that as we add more and more units of a variable factor (e.g., labour) to a fixed factor (e.g., capital), the marginal product (MP) of the variable factor will eventually decline. The law has three stages:

• Stage 1: Increasing Returns. Initially, as more labour is added, the marginal product increases due to specialization and division of labour. The total product (TP) increases at an increasing rate.
• Stage 2: Diminishing Returns. After a point, adding more labour leads to a decline in marginal product, though it is still positive. This is because the fixed factor becomes crowded. The total product (TP) increases at a decreasing rate and reaches its maximum. A rational producer always operates in this stage.
• Stage 3: Negative Returns. If too much labour is added, the factors start getting in each other’s way, and the marginal product becomes negative. The total product (TP) starts to fall.

(A diagram showing the Total Product, Average Product, and Marginal Product curves should accompany this answer, illustrating the three stages.)

(b)

Cobb-Douglas Production Function Analysis

The given production function is: Q = 2K⁰.⁴ L⁰.⁶

(i) Returns to Scale:

To determine the returns to scale, we sum the exponents: 0.4 + 0.6 = 1.0.

Since the sum of the exponents is equal to 1, this production function exhibits

constant returns to scale

. This means that if all inputs (K and L) are increased by a certain proportion (e.g., doubled), the output (Q) will also increase by the same proportion (doubled).

(ii) Output Elasticities of Labour and Capital:

For a Cobb-Douglas production function, the exponents directly represent the output elasticities of the respective inputs.

• Output Elasticity of Capital (Eₖ): This is the exponent of capital, which is 0.4 . It means that if capital increases by 1%, holding all else constant, output will increase by 0.4%.
• Output Elasticity of Labour (Eₗ): This is the exponent of labour, which is 0.6 . It means that if labour increases by 1%, holding all else constant, output will increase by 0.6%.

(iii) Marginal Rate of Technical Substitution (MRTS):

The MRTS measures the rate at which capital can be substituted for labour while keeping the level of output constant. It is the ratio of the marginal product of labour (MPₗ) to the marginal product of capital (MPₖ).

MRTSₗₖ = MPₗ / MPₖ

First, we calculate the marginal products:

MPₗ = dQ/dL = 2K⁰.⁴

(0.6)L⁰.⁶⁻¹ = 1.2K⁰.⁴L⁻⁰.⁴

MPₖ = dQ/dK = 2L⁰.⁶

(0.4)K⁰.⁴⁻¹ = 0.8L⁰.⁶K⁻⁰.⁶

Now, calculate the MRTS:

MRTSₗₖ = (1.2K⁰.⁴L⁻⁰.⁴) / (0.8L⁰.⁶K⁻⁰.⁶)

MRTSₗₖ = (1.2 / 0.8)

(K⁰.⁴K⁰.⁶) / (L⁰.⁶L⁰.⁴)

MRTSₗₖ = 1.5

(K¹ / L¹)

MRTSₗₖ = 1.5 (K/L)

Q3. (a) Explain how the divergence between marginal private cost and marginal social cost lead to market inefficiency. Illustrate. (b) Discuss the role of Coase theorem in correcting externalities. What are its limitations?

Ans. (a) Divergence between Marginal Private Cost (MPC) and Marginal Social Cost (MSC) Definitions:

• Marginal Private Cost (MPC): This is the cost to a producer of producing one additional unit of a good. It includes only the costs that the firm bears, such as labour, raw materials, and capital.
• Marginal Social Cost (MSC): This is the total cost to society of producing one additional unit of a good. It includes both the marginal private cost (MPC) and the marginal external cost (MEC). That is, MSC = MPC + MEC .

Market Inefficiency:

Market inefficiency arises when there is a divergence between MPC and MSC, typically due to the presence of

externalities

. A negative externality (e.g., pollution) occurs when production or consumption imposes a cost on a third party not directly involved in the transaction.

In this case, the firm only considers its private costs (MPC) and ignores the social costs like pollution. Therefore,

MSC > MPC

.

In a competitive market, firms produce where the price (which equals marginal social benefit, MSB) is equal to their marginal private cost (P = MPC). However, the socially optimal level of production is where the price is equal to the marginal social cost (P = MSC).

Since MSC > MPC, the level of output in the market equilibrium (Q_market) is greater than the socially optimal level (Q_optimal). This results in

overproduction

and an inefficient allocation of resources, leading to a

deadweight loss

and market failure.

Illustration:

A steel factory pollutes a river in the production process. The factory’s MPC includes the cost of labour and steel but not the cost of cleaning the river or the damage to fisheries. These are external costs. Therefore, the factory’s MSC is higher than its MPC, and it produces more steel than is socially optimal.

(A diagram showing the MPC, MSC, and demand curves, illustrating the difference between the market equilibrium and the socially optimal equilibrium, and the resulting deadweight loss, would enhance this explanation.)

(b)

The Coase Theorem and its Limitations

Role of the Coase Theorem:

The Coase theorem, developed by Ronald Coase, provides a potential private, market-based solution to the problem of externalities. The theorem states that:

“If property rights are well-defined and transaction costs are zero or low, private parties can bargain to an efficient outcome to resolve an externality, regardless of the initial allocation of property rights.”

This means that an efficient outcome can be achieved even without government intervention (like taxes or subsidies). The parties will “internalize” the externality by compensating each other.

Example:

If the factory has the right to pollute, the fishers can pay the factory to reduce pollution. They will pay as long as the benefit to them from reduced pollution is greater than the cost of the payment. If the fishers have the right to clean water, the factory can either install pollution-control technology or pay the fishers to accept some pollution. In both cases, the negotiation will lead to an efficient level of pollution.

Limitations of the Coase Theorem:

While powerful in theory, the Coase theorem has several practical limitations:

• Transaction Costs: In the real world, transaction costs are rarely zero. These include the costs of negotiating, writing, and enforcing contracts, which can be prohibitively high.
• Assignment of Property Rights: In many cases, property rights are not clearly defined (e.g., clean air). Assigning property rights can be difficult and contentious.
• Holdout Problem: When many parties are involved, one party may refuse to cooperate, holding out for a better deal, which can cause the negotiation to fail.
• Free-Rider Problem: If an externality affects many people (like air pollution), it is difficult to coordinate everyone to contribute to the payment for a solution. Some may hope to benefit without paying.
• Information Asymmetry: Parties may not have perfect information about the costs of damage or the costs of abatement, making efficient bargaining difficult.

Due to these limitations, government intervention often becomes necessary to correct externalities.

Q4. (a) An individual earns ₹ 100 in period 1 and ₹ 50 in period 2. The rate of interest is 20%. Write down the intertemporal budget line and calculate the intercepts on both axes. Draw the budget line showing the endowment point. (b) Suppose the rate of interest falls to 10%. Draw the new budget line. If the individual was initially a lender, discuss whether his welfare improves or declines after the fall in interest rate.

Ans. (a) Intertemporal Budget Line and Endowment Point Given: Income in period 1 (M₁): ₹100 Income in period 2 (M₂): ₹50 Interest rate (r): 20% or 0.2 The intertemporal budget constraint shows how much an individual can consume over their lifetime. It can be written in present value or future value terms. The budget constraint in future value is: (1+r)C₁ + C₂ = (1+r)M₁ + M₂ Substituting the values: (1+0.2)C₁ + C₂ = (1+0.2)100 + 50 1.2C₁ + C₂ = 120 + 50 1.2C₁ + C₂ = 170 This is the equation of the intertemporal budget line. Intercepts on Axes:

• Horizontal Intercept (Max C₁): This occurs when C₂ = 0. The person consumes their entire lifetime income in period 1. 1.2C₁ = 170 => C₁ = 170 / 1.2 = 141.67 . So, the horizontal intercept is (141.67, 0). This is the present value of the endowment.
• Vertical Intercept (Max C₂): This occurs when C₁ = 0. The person consumes their entire lifetime income in period 2. C₂ = 170 . So, the vertical intercept is (0, 170). This is the future value of the endowment.

Endowment Point:

This is the combination of income the individual receives, i.e., (M₁, M₂). Here, the endowment point is

E = (100, 50)

. This point always lies on the budget line.

Drawing the Budget Line:

On a graph, measure C₁ on the horizontal axis and C₂ on the vertical axis. Mark the intercepts (141.67, 0) and (0, 170) and connect them with a straight line. Also mark the endowment point E(100, 50) on this line. The slope of the line is -(1+r) = -1.2.

(b)

Effect of a Change in Interest Rate

New Budget Line:

New interest rate (r’) = 10% or 0.1.

The equation of the new budget line is:

(1+0.1)C₁ + C₂ = (1+0.1)100 + 50

1.1C₁ + C₂ = 110 + 50

1.1C₁ + C₂ = 160

The intercepts of the new budget line are:

• Horizontal Intercept (C₂=0): C₁ = 160 / 1.1 = 145.45
• Vertical Intercept (C₁=0): C₂ = 160

The new budget line also passes through the endowment point E(100, 50) and is flatter than the original line, as its slope is now -(1+r’) = -1.1.

Effect on a Lender’s Welfare:

A

lender

is someone who consumes less than their income in period 1 (C₁ < M₁) and saves. On the graph, their optimal consumption bundle would be located to the left of the endowment point E.

When the interest rate falls from 20% to 10%, the budget line pivots around the endowment point E and becomes flatter. For a lender, this change reduces their opportunity set. The consumption combinations available to them at the original (higher) interest rate are no longer available at the new (lower) rate.

Because the return on saving is lower, the lender cannot reach as high an indifference curve as before. Therefore,

a lender’s welfare declines

when the interest rate falls. They are unambiguously worse off because their original optimal bundle is now unaffordable.

Q5. What is asymmetrical information? How does asymmetrical information lead to market failure?

Ans. Asymmetric Information is a situation in which one party in an economic transaction possesses more or superior information than the other party. This imbalance of information prevents the market from functioning efficiently and often leads to market failure. Asymmetric information gives rise to two main problems: 1. Adverse Selection:

• This problem arises before the transaction and relates to “hidden information.” In this situation, the less-informed party is more likely to end up trading with individuals with whom trading is least desirable for them.
• Example: The used car market. The seller of a car knows more about its true quality (whether it’s a good “peach” or a bad “lemon”) than the buyer. The buyer, being uncertain about the quality, is only willing to pay an average price. This price might be acceptable to owners of lemons, but not to owners of peaches. As a result, good cars are driven out of the market, leaving only bad cars. This shrinks the market or causes it to fail entirely.
• A similar problem exists in insurance markets, where high-risk individuals (who know their risk status) are more likely to buy insurance, driving up premiums and pushing low-risk individuals out of the market.

2.

Moral Hazard:

• This problem arises after the transaction and relates to “hidden action.” It occurs when one party, after entering a transaction, changes their behavior in a way that is detrimental to the other party, because they do not bear the full consequences of their actions.
• Example: Insurance. Once a person has purchased car insurance, they may drive less carefully because they know that the financial cost of an accident will be borne by the insurance company.
• Another example is the employer-employee relationship. If an employee is hired at a fixed salary and is difficult to monitor, they may shirk (not work hard) instead of being diligent.

Market Failure:

Because of these two problems, markets with asymmetric information result in an inefficient allocation of resources. Mutually beneficial trades may not occur due to adverse selection, and resources may be misused due to moral hazard. This all creates a

deadweight loss

, reducing total economic surplus and thus constituting a market failure.

Q6. Find the optional consumption bundle in case of quantity tax and an equivalent income tax. In which of the two tax regimes is the consumer better off?

Ans. This question compares the effects on consumer welfare of two types of taxes: a quantity tax and an equivalent lump-sum income tax. Analysis: Let’s consider a consumer choosing between two goods, X and Y. 1. Initial Situation (No Tax): The consumer chooses their optimal consumption bundle at point A, on the highest possible indifference curve U₁, given their budget line. 2. Quantity Tax: Suppose the government imposes a tax on good X. This increases the price of good X. As a result, the budget line pivots inwards and becomes steeper. The consumer now chooses a new optimal bundle B on a new, lower indifference curve U₂. The government collects a certain amount of revenue (say, R) from this tax. 3. Equivalent Income Tax: Now, consider a lump-sum income tax that raises the same amount of revenue R for the government. A lump-sum tax reduces the consumer’s income directly but does not change the relative prices of the goods. This causes a parallel downward shift of the original budget line. Because this tax raises the same revenue R, this new budget line will pass through point B. Comparison and Conclusion:

• Under the quantity tax, the consumer is at point B, on indifference curve U₂.
• Under the lump-sum income tax, the consumer faces a budget line that passes through point B but has the same slope as the original budget line (i.e., is less steep than the quantity tax line).
• Because the indifference curve U₂ is tangent to the steep (quantity tax) line at point B, the flatter (lump-sum tax) line must cut through U₂.
• This means the consumer can move along the lump-sum tax budget line away from point B and reach a new optimal point C on a higher indifference curve, say U₃.

Conclusion:

The consumer is

better off under the lump-sum income tax

. This is because:

• The quantity tax has both an income effect and a substitution effect . The substitution effect distorts relative prices by encouraging the consumer to shift away from the taxed good, causing an additional inefficiency or deadweight loss .
• The lump-sum income tax has only an income effect . It does not distort relative prices, allowing the consumer to allocate their reduced purchasing power more efficiently.

In short, for the same revenue, a tax that distorts relative prices (quantity tax) is worse for the consumer than a tax that does not (lump-sum tax).

Q7. A firm faces the following demand curve and total cost function: P = 200 – 2Q, TC = 500 + Q². Calculate the profit maximizing output and price if the firm operates in a perfectly competitive market.

Ans. The key point in this question is that the firm operates in a perfectly competitive market . A perfectly competitive firm is a price taker , meaning it faces a horizontal (perfectly elastic) demand curve at the price determined by the market. The given demand curve, P = 200 – 2Q, is the market demand curve, not the firm’s. The firm’s decisions are not directly affected by this curve. The profit-maximization rule for a perfectly competitive firm is P = MC (Price = Marginal Cost). Furthermore, in the long run in a perfectly competitive market, free entry and exit drive firms to earn zero economic profit. This occurs when the price is equal to the minimum point of the Average Cost (AC) curve. At minimum AC, we have MC = AC. Therefore, we can find the firm’s long-run equilibrium output and price by finding the point where P = MC = min(AC). 1. Calculate Marginal Cost (MC) and Average Cost (AC): Total Cost (TC) = 500 + Q² Marginal Cost (MC) = d(TC)/dQ = 2Q Average Cost (AC) = TC/Q = (500 + Q²)/Q = 500/Q + Q 2. Find the output (Q) at minimum Average Cost: The minimum of the AC curve is where MC = AC. 2Q = 500/Q + Q 2Q – Q = 500/Q Q = 500/Q Q² = 500 Q = √500 = 10√5 ≈ 22.36 Thus, the firm’s profit-maximizing (and zero-profit) output level is approximately 22.36 units. 3. Find the equilibrium price (P): In perfect competition, P = MC. We substitute the equilibrium value of Q into the MC function: P = 2Q = 2(10√5) = 20√5 ≈ 44.72 So, the price in the long-run competitive equilibrium would be approximately ₹44.72. At this price and quantity, the firm’s profit will be zero, as expected in long-run competitive equilibrium: Profit = TR – TC = P Q – (500 + Q²) Profit = (20√5 10√5) – (500 + (10√5)²) = (200 * 5) – (500 + 500) = 1000 – 1000 = 0. Conclusion:

• Profit maximizing output (Q): 10√5 (≈ 22.36) units
• Price level (P): 20√5 (≈ 44.72)

Q8. What do you understand by the term ‘Consumer Surplus’? Explain with diagram, how it is calculated.

Ans. Consumer Surplus is the difference between the total amount that consumers are willing and able to pay for a good or service (indicated by the demand curve) and the total amount that they actually do pay (the market price). It is a measure of the net benefit or welfare that consumers receive from participating in the market. In simple terms, it is the “extra” value consumers get from a good over and above the price they paid for it. If you are willing to pay ₹500 for a book but you get it for ₹300, your consumer surplus is ₹200. Explanation with a Diagram: Consumer surplus can be easily calculated and illustrated with the help of a diagram. 1. Draw Demand and Supply Curves: On a standard graph, plot Price on the Y-axis and Quantity on the X-axis. Draw a downward-sloping Demand Curve (D) , which represents the quantity consumers are willing to buy at various prices. Draw an upward-sloping Supply Curve (S) . 2. Identify the Equilibrium: The point where the demand and supply curves intersect is the Equilibrium Point (E) . The price corresponding to this point is the Equilibrium Price (P ) and the quantity is the Equilibrium Quantity (Q ) . In the market, all consumers pay the price P and a total quantity Q is bought. 3. The Area of Consumer Surplus:

• The demand curve represents the willingness to pay or marginal utility for each additional unit of a good.
• Consumer surplus is the sum of all the individual surpluses that each consumer receives.
• In the diagram, this is represented by the area below the demand curve , above the equilibrium price line (P ) , and out to the quantity Q .

Calculation:

This area is typically a triangle. The area of the triangle (which equals the consumer surplus) can be calculated as:

Consumer Surplus = ½ × Base × Height

Where:

• Base is the equilibrium quantity (Q*).
• Height is the difference between the Y-intercept of the demand curve (the maximum price anyone is willing to pay) and the equilibrium price (P*).

The concept of consumer surplus is very important in evaluating policy decisions, such as the impact of taxes and subsidies, as it measures how economic changes affect the welfare of consumers.

Q9. What is social welfare function? Explain its importance in Welfare Economics.

Ans. A Social Welfare Function (SWF) is a theoretical concept that aggregates the utility (or welfare) of all individuals in a society into a single measure of overall social welfare. It is a function that ranks different social states (alternative descriptions of the economy) as less desirable, more desirable, or indifferent. Mathematically, it can be represented as W = f(U₁, U₂, U₃, …, Uₙ), where W is social welfare and U₁, U₂, … Uₙ are the utilities of the individuals in society. Importance in Welfare Economics: The social welfare function plays a central role in Welfare Economics, the branch of economics that evaluates the allocation of resources and its impact on the well-being of individuals. Its importance lies in the following points: 1. A Framework for Policy Evaluation: The SWF provides a criterion for economists and policymakers to judge the outcomes of different economic policies (e.g., tax systems, subsidies, or regulations). A policy is considered an improvement if it increases the value of the social welfare function. 2. Choosing Between Efficiency and Equity: Pareto efficiency only states that it is impossible to make someone better off without making someone else worse off. It does not help us choose among the many Pareto-efficient outcomes. A SWF, by incorporating considerations of equity, helps in choosing the “best” point on the utility possibility frontier. 3. Making Interpersonal Utility Comparisons Explicit: To aggregate individual utilities, one must make assumptions about how to compare one person’s happiness to another’s. The SWF makes these value judgements explicit. Different SWFs represent different ethical viewpoints:

• Utilitarian SWF: W = U₁ + U₂ + … + Uₙ. Aims to maximize the total sum of utility, regardless of its distribution.
• Rawlsian SWF: W = min(U₁, U₂, …, Uₙ). Aims to maximize the utility of the worst-off person in society.
• Egalitarian SWF: Emphasizes an equal distribution of utilities.

4.

Basis for Normative Economics:

Positive economics describes “what is,” while normative economics prescribes “what should be.” The SWF is a fundamental tool of normative economics, providing a formal way to make recommendations about which economic policies and outcomes are most socially desirable.

In essence, the social welfare function acts as a bridge between abstract ethical considerations and concrete policy analysis, allowing economists to rigorously evaluate the social desirability of economic outcomes.

Q10. Consider the following normal game form in which Player 1 and Player 2 each have two strategies: Player 2 X Y Player 1 X | 3,7 | 8,4 y | 6,5 | 4,8 (Here, the first entry in each cell is Player 1’s payoff and the second is Player 2’s payoff). (a) Find the Nash equilibrium of the game. (b) Write the extensive form of the game assuming Player 1 moves first.

Ans. (a) Finding the Nash Equilibrium A Nash Equilibrium is a set of strategies where no player can increase their payoff by unilaterally changing their strategy, given the other players’ strategies. We will search for a pure strategy Nash Equilibrium by identifying the best responses of each player. 1. Player 1’s Best Responses:

• If Player 2 plays ‘x’, Player 1 chooses between ‘X’ (payoff 3) and ‘y’ (payoff 6). He will choose ‘y’ since 6 > 3.
• If Player 2 plays ‘Y’, Player 1 chooses between ‘X’ (payoff 8) and ‘y’ (payoff 4). He will choose ‘X’ since 8 > 4.

2.

Player 2’s Best Responses:

• If Player 1 plays ‘X’, Player 2 chooses between ‘x’ (payoff 7) and ‘Y’ (payoff 4). He will choose ‘x’ since 7 > 4.
• If Player 1 plays ‘y’, Player 2 chooses between ‘x’ (payoff 5) and ‘Y’ (payoff 8). He will choose ‘Y’ since 8 > 5.

Result:

A pure strategy Nash Equilibrium is a cell where both players are playing their best responses simultaneously.

• In cell (X, x), Player 1’s best response is ‘y’, not ‘X’. Not a NE.
• In cell (X, Y), Player 2’s best response is ‘x’, not ‘Y’. Not a NE.
• In cell (y, x), Player 2’s best response is ‘Y’, not ‘x’. Not a NE.
• In cell (y, Y), Player 1’s best response is ‘X’, not ‘y’. Not a NE.

Since there is no cell where both strategies are mutual best responses,

there is no Pure Strategy Nash Equilibrium in this game

. (A mixed strategy Nash equilibrium exists, but the question typically implies pure strategies unless specified otherwise).

(b)

Extensive Form of the Game

When Player 1 moves first, the game becomes a sequential game that can be represented as a game tree.

1.

Initial Node:

The game starts at a decision node where

Player 1

makes a move.

2.

Player 1’s Branches:

Two branches emerge from this node, representing Player 1’s possible actions:

‘X’

and

‘y’

.

3.

Player 2’s Nodes:

Each branch ends at a new decision node. These are for

Player 2

.

• If Player 1 chooses ‘X’, the game moves to the top node.
• If Player 1 chooses ‘y’, the game moves to the bottom node.

4.

Player 2’s Branches:

From each of Player 2’s decision nodes, two branches emerge, representing his actions:

‘x’

and

‘Y’

.

5.

Terminal Payoffs:

At the end of each final branch of the tree, we write the corresponding payoffs for the players.

Representation of the Game Tree:

• Start: Player 1’s Node.
  • Player 1 chooses ‘X’ –> Player 2’s Node.
    • Player 2 chooses ‘x’ –> Terminal Payoff: (3, 7)
    • Player 2 chooses ‘Y’ –> Terminal Payoff: (8, 4)
  • Player 1 chooses ‘y’ –> Player 2’s Node.
    • Player 2 chooses ‘x’ –> Terminal Payoff: (6, 5)
    • Player 2 chooses ‘Y’ –> Terminal Payoff: (4, 8)

This extensive form clearly illustrates the sequential structure of the game, including the information of who moves when and what the players know at each stage.

Q11. Write short notes on any two of the following: (a) Value Judgements (b) Homogeneous Production Function (c) Quasi-Linear Preferences

Ans. (a) Value Judgements Value judgements are statements or decisions based on personal opinions, beliefs, and ethical values. They describe how things “ought to be” rather than “what is.” In economics, these judgements distinguish normative economics from positive economics.

• Positive statement: It is factual and can be tested or verified with data. Example: “Raising the minimum wage will increase unemployment.”
• Normative statement: It is based on a value judgement. Example: “The government should raise the minimum wage to help the poor.” This statement involves an ethical choice between efficiency and equity.

Concepts like Social Welfare Functions (e.g., Utilitarian vs. Rawlsian), the desirability of income distributions, and the role of government intervention are key examples of value judgements in economics. These judgements play a crucial role in shaping the goals of economic policy.

(b) Homogeneous Production Function

A production function Q = f(K, L) is said to be homogeneous of degree ‘n’ if, when all inputs (K and L) are multiplied by a constant (λ > 0), the output is multiplied by λⁿ.

Mathematically: f(λK, λL) = λⁿ f(K, L) = λⁿQ

The degree of homogeneity ‘n’ indicates the

returns to scale

:

• n = 1: Constant returns to scale. Doubling inputs leads to a doubling of output. The Cobb-Douglas function Q = AKᵃL¹⁻ᵃ is an example.
• n > 1: Increasing returns to scale. Doubling inputs leads to more than a doubling of output.
• n < 1: Decreasing returns to scale. Doubling inputs leads to less than a doubling of output.

An important property of homogeneous production functions is Euler’s Theorem, which is significant in the theory of distribution.

(c) Quasi-Linear Preferences

Quasi-linear preferences are a specific type of consumer preference that can be represented by a utility function that is linear in one good and non-linear in the other.

A common form is: U(x, y) = v(x) + y

Here, utility is linear in good y (often considered the ‘numeraire’ or money) and strictly concave in good x.

Key Characteristics:

• Indifference Curves: The indifference curves for these preferences are parallel vertical shifts of one another.
• No Income Effect: A crucial implication is that (for an interior solution) the demand for good x does not depend on income. Any change in income only affects the consumption of the linear good, y. This greatly simplifies analysis.
• MRS Dependency: The marginal rate of substitution (MRS) depends only on the quantity of good x (MRS = v'(x)).

Due to these properties, quasi-linear preferences are frequently used in partial equilibrium analysis, and in the modeling of public goods and externalities.

Q12. Differentiate between any two of the following: (a) Weakly Preferred and Strongly Preferred Set (b) Monopoly and Monopolistic Competition (c) Signaling and Screening

Ans. (a) Weakly Preferred and Strongly Preferred Set These two concepts in consumer theory describe a consumer’s preferences for other bundles relative to a given consumption bundle. Let there be an initial consumption bundle X.

• Weakly Preferred Set: This includes all consumption bundles that are considered by the consumer to be at least as good as bundle X. This set contains all bundles that are better than X (on a higher indifference curve) and all bundles that are equally good as X (on the same indifference curve as X, including X itself).
• Strongly Preferred Set: This includes all consumption bundles that are strictly preferred to bundle X by the consumer. This set excludes the bundles lying on the same indifference curve as X. It only includes bundles that are on a distinctly higher indifference curve.

In short, the weakly preferred set contains “better or equal” bundles, while the strongly preferred set contains only “better” bundles.

(b) Monopoly and Monopolistic Competition

Both are market structures of imperfect competition, but they have significant differences.

Example: Monopoly – the local electricity company; Monopolistic Competition – restaurants, salons.

(c) Signaling and Screening

Signaling and screening are both strategies used to deal with the problem of asymmetric information (specifically, adverse selection). The difference lies in which party takes the action to reveal information.

• Signaling: This is an action taken by the informed party to reveal their private information to the uninformed party . To be effective, a signal must be costly, and its cost must be lower for the “high-quality” type than the “low-quality” type. Example: A talented worker gets an expensive master’s degree (the signal) to prove their ability to an employer (the uninformed party).
• Screening: This is an action taken by the uninformed party to induce the informed party to reveal their private information. The uninformed party designs a menu of choices, and the informed party’s choice reveals their “type”. Example: An insurance company (uninformed party) offers two types of policies: a high-deductible, low-premium policy and a low-deductible, high-premium policy. Low-risk customers will choose the former, and high-risk customers will choose the latter, thus “screening” themselves.