Needed In 3 Hours: Question 3 Contesta Manufacturer Promises

Needed In 3 Hoursquestion 3 Contesta Manufacturer Promises To Give A

Question 3. Contest: A manufacturer promises to award a prize to one of two dealers who win a sales contest. The value of the prize is V = 10 for each dealer. The dealers compete by exerting effort levels xi. The probability of winning is modeled by the Tullock function: p₁(x₁, x₂) = x₁ / (x₁ + x₂) and p₂(x₁, x₂) = x₂ / (x₁ + x₂). Both dealers have the same constant marginal effort cost, c = 1.

1. Write down firm 1's expected payoff from participating in the sales contest.

2. Find the best response of firm 1.

3. Are effort levels strategic substitutes or complements? Explain.

4. By analogy, write the best response of firm 2.

5. The firms are symmetric in all respects. Using the symmetry, find the equilibrium sales efforts, x₁ and x₂.

6. What is the extent of rent dissipation (that is, what fraction of the prize value V is spent in rent-seeking efforts)?

7. Would dealers choose to participate in the sales contest? What are their expected equilibrium payoffs from participation in the contest?

8. Sketch the best-response functions or use Excel to draw them. Mark the Nash equilibrium.

Paper For Above instruction

The problem outlined involves analyzing the strategic effort choices of two competing dealers under a prize contest promised by a manufacturer. This scenario exemplifies a classic rent-seeking contest modeled through a probability function of effort efforts, specifically the Tullock contest function. The analysis involves deriving expected payoffs, best response functions, equilibrium efforts, and the degree of rent dissipation, which offers insight into strategic interactions in competitive efforts and resource allocation.

Introduction

Contests are common in economics and political science as they model situations where agents expend effort to secure a prize or advantage. The model under discussion involves two dealers competing for a prize V=10, with each exerting effort xi. The probability of winning is proportional to effort, according to the Tullock function. This setup allows us to analyze the strategic decisions of the dealers and their outcomes, including the effort levels, equilibrium states, and efficiency of the contest.

Expected Payoff for Firm 1

The expected payoff for dealer 1 depends on the probability of winning and the associated gains and costs. Dealer 1’s payoff is given by:

Π₁(x₁, x₂) = p₁(x₁, x₂)  V - c  x₁

Where p₁(x₁, x₂) = x₁ / (x₁ + x₂), V = 10, and c = 1. Substituting these values:

Π₁(x₁, x₂) = (x₁ / (x₁ + x₂)) * 10 - x₁

This payoff reflects the expected benefit from winning the prize minus the effort cost incurred.

Best Response of Firm 1

To find firm 1's best response, we differentiate Π₁ with respect to x₁ and set it to zero:

∂Π₁/∂x₁ = d/dx₁ [(x₁ / (x₁ + x₂)) * 10 - x₁]

Calculating the derivative:

∂Π₁/∂x₁ = 10 * [ (x₂) / (x₁ + x₂)^2 ] - 1

Setting this to zero for maximization:

10 * (x₂) / (x₁ + x₂)^2 = 1

Rearranged, the best response function of firm 1 is:

( x₁ + x₂ )^2 = 10 * x₂

Or explicitly solving for x₁:

x₁ = √(10 * x₂) - x₂

This function defines the best effort choice of firm 1 given a fixed effort level by firm 2.

Strategic Substitutes or Complements?

Effort levels are strategic substitutes if an increase in one firm's effort decreases the best effort level of the other (or vice versa). Examining the best response functions reveals that as x₂ increases, the optimal x₁ decreases, indicating efforts are strategic substitutes. This is typical in rent-seeking contests where excessive effort by one agent discourages additional effort by others due to diminishing marginal returns and increased costs.

Best Response of Firm 2

By symmetry, the best response function of firm 2 mirrors that of firm 1:

x₂ = √(10 * x₁) - x₁

Since the firms are symmetric, the equilibrium efforts are symmetric solutions where x₁ = x₂ = x*.

Symmetric Equilibrium Efforts

At equilibrium, set x₁ = x₂ = x*, so the best response functions become:

x = √(10  x) - x

This simplifies to the quadratic:

x = √(10  x) - x

or:

x + x = √(10  x)

Squaring both sides yields:

(2x)^2 = 10  x*

4x^2 = 10 x

Dividing both sides by x (assuming x > 0):

4 x* = 10

Resulting in:

x* = 2.5

Thus, the symmetric equilibrium effort level is x₁ = x₂ = 2.5.

Rent Dissipation Analysis

Rent dissipation refers to the total expenditure on effort relative to the prize value:

Total effort in equilibrium = x₁ + x₂ = 5

Since the total effort equals 5 and the prize value is 10, the fraction spent on rent-seeking is:

( x₁ + x₂ ) / V = 5 / 10 = 0.5

Approximately 50% of the prize is dissipated in effort expenditures, indicating significant rent-seeking inefficiency.

Participation and Expected Payoffs

Dealers benefit from participating if their expected payoff exceeds zero. The equilibrium expected payoff of each dealer is:

Π₁(x) = (x / (2x))  10 - x = (1/2)  10 - 2.5 = 5 - 2.5 = 2.5

Therefore, each dealer expects to earn an average payoff of 2.5 units upon participation, which is positive, incentivizing participation. The total rent-seeking effort confirms that resources are predominantly used for securing the prize rather than producing additional value, leading to inefficiency.

Conclusion

This analysis underscores the strategic nature of effort in rent-seeking contests. The efforts are strategic substitutes, with equilibrium efforts balancing the marginal benefits of increased effort against its costs. The significant rent dissipation suggests that contests may be inefficient, wasting resources without adding proportional value. The symmetric equilibrium efforts and positive expected payoffs confirm that rational firms will participate under these conditions.

References

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