Question

One way of measuring market concentration is through the use of the Herfindahl index, which is defined as \[ H=\sum_{i=1}^{n} s_{i}^{2} \] where $s_t=q_i/Q$ is firm $i^{\prime }$ s market share. The higher is $H$, the more concentrated the industry is said to be. Intuitively, more concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive pressure. We will assess the validity of this intuition using several models. a. If you have not already done so, answer Problem $15.2\mathrm{d}$ by computing the Nash equilibrium of this $n$ -firm Cournot game. Also compute market output, market price, consumer surplus, industry profit, and total welfare. Compute the Herfindahl index for this equilibrium. b. Suppose two of the $n$ firms merge, leaving the market with $n-1$ firms. Recalculate the Nash equilibrium and the rest of the items requested in part (a). How does the merger affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index? c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem $15.3,$ where Cournot duopolists face different marginal costs. Use your answer to Problem 15.3 to compute equilibrium firm outputs, market output, price, consumer surplus, industry profit, and total welfare, substituting the particular cost parameters $c_1=c_2=1/4.$ Also compute the Herfindahl index. d. Repeat your calculations in part (c) while assuming that firm 1's marginal cost $c_1$ falls to 0 but $c_2$ stays at 1/4. How does the cost change affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index? e. Given your results from parts (a)-(d), can we draw any general conclusions about the relationship between market concentration on the one hand and price, profit, or total welfare on the other?

Short answer

Based on the step-by-step solution provided above, formulate a short answer discussing the impact of market concentration on market outcomes in a Cournot model. Make sure to reference the general results obtained from parts (a) to (e). In a Cournot model, the Herfindahl index is used to measure market concentration. The analysis of market structures and scenarios in this exercise shows that market concentration has a significant impact on various market outcomes such as price, output, consumer surplus, total welfare, and firm profits. From parts (a) to (e), we observe that an increase in market concentration tends to result in higher prices, lower output levels, decreased consumer surplus, and possibly reduced total welfare. In assessing the impact of market concentration, the analysis of a merger (part b) and the impact of a cost change (part d) provided valuable insights. These scenarios demonstrated that increased market concentration (through a merger or cost change) leads to higher prices and potentially reduced consumer surplus and total welfare. However, firm profits can either increase or decrease depending on the scenario, cost structures, and initial market concentration levels. While a merger can lead to increased firm profits, a change in costs can result in varied profit outcomes for the firms involved. In conclusion, higher market concentration generally leads to unfavorable market outcomes for consumers in the form of higher prices and reduced consumer surplus. Additionally, total welfare may be negatively impacted by market concentration. Firm profits, however, can be positively or negatively affected, depending on the specific market scenario and factors affecting costs.

Step-by-step solution

Determine the demand and cost functions

Given the demand function: $P=a-Q$, and the cost function: $C_i=c{q}_i$.

Calculate the Nash equilibrium of this Cournot game

To find the Nash equilibrium, we first need to find the profit function for each firm: ${\pi }_i=P{q}_i-C_i$. Next, we differentiate the profit function with respect to $q_i$ and set it equal to zero. This will give us the reaction function for each firm, $q_i^{\ast }(q_{-i})$. Since there are n firms, we must sum the quantities of all firms: $Q=\sum _{i=1}^n{q}_i$. Finally, for our Nash equilibrium, substitute the reaction functions back into the demand function using the expression for $Q$. Solve for equilibrium quantities, $q_i^{\ast }$, and the equilibrium price, $P^{\ast }$.

Calculate consumer surplus, industry profit, and total welfare

Compute consumer surplus (CS) using the equation: $CS=\frac{1}{2}(a-P^{\ast })Q^{\ast }$. Compute industry profit ($\mathrm{\Pi }$) by summing the profits of all firms: $\mathrm{\Pi }=\sum _{i=1}^n{\pi }_i$. Compute total welfare (TW) by adding consumer surplus and industry profit: $TW=CS+\mathrm{\Pi }$.

Calculate the Herfindahl index

Compute the Herfindahl index (H) using the formula: $H=\sum _{i=1}^n{s}_i^2$, where $s_i=\frac{q_i}{Q}$ is the market share of firm i. **Part (b): Analyzing the impact of a merger on the market**

Determine the demand and cost functions of the merged firm and the remaining firms

Account for the merger by recalculating the cost function for the merged firm and the remaining $n-1$ firms.

Calculate the new Nash equilibrium with n-1 firms

Using the updated demand and cost functions, repeat the steps to find the new Cournot Nash equilibrium.

Recalculate market output, market price, consumer surplus, industry profit, and total welfare

Using the new Nash equilibrium outcomes, recompute the market output, market price, consumer surplus, industry profit, and total welfare.

Calculate the new Herfindahl index

Compute the new Herfindahl index using the updated market shares post-merger.

Compare the results before and after the merger

Compare the market outcomes such as price, output, profit, consumer surplus, total welfare, and the Herfindahl index to analyze the impact of the merger.

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**Part (c): Calculating equilibrium using Cournot duopolists with different marginal costs**

Set up the cost functions for firms 1 and 2 with given parameters

Using the provided cost parameters, set up the cost functions for both firms.

Compute equilibrium firm outputs, market output, and price

Using the given cost functions, calculate the equilibrium output for both the firms, market output, and price by solving the reaction functions.

Calculate consumer surplus, industry profit, and total welfare

Compute consumer surplus, industry profit and total welfare based on the calculated equilibrium outputs and price.

Calculate the Herfindahl index for this scenario

Compute the Herfindahl index using the given market shares for this setup.

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**Part (d): Analyzing the impact of a cost change on the market outcome**

Update the cost functions for firms 1 and 2 with new parameters

Update the cost function for firm 1 with the new marginal cost.

Recalculate equilibrium firm outputs, market output, and price

Using the updated cost functions, calculate the new equilibrium output for both the firms, market output, and price.

Recalculate consumer surplus, industry profit, and total welfare

Compute the updated consumer surplus, industry profit, and total welfare based on the new equilibrium outputs and price.

Calculate the new Herfindahl index

Compute the new Herfindahl index using the updated market shares post-cost change.

Compare the results before and after the cost change

Compare the market outcomes such as price, output, profit, consumer surplus, total welfare, and the Herfindahl index to analyze the impact of the cost change.

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**Part (e): General conclusions about the relationship between market concentration and market outcomes**

Analyze the results from parts (a) to (d)

Review the results obtained in previous parts to analyze how market concentration affected price, output, profit, consumer surplus, and total welfare.

Determine general conclusions

Based on the analysis in the previous step, draw general conclusions about the relationship between market concentration and various market outcomes such as price, output, profit, consumer surplus, and total welfare.

Key concepts

Market Concentration

Market concentration measures how much control a few firms have over the total market's production and sales. When we talk about the Herfindahl index, we're dealing with a specific way to quantify market concentration. By squaring each firm's market share and summing these squares, we get a number that shows us just how concentrated a market is. A higher Herfindahl index indicates fewer firms holding more market power.

For instance, in a market where many firms exist with each holding a tiny fraction of market share, the Herfindahl index will be low, signaling a highly competitive landscape. In contrast, if a single firm dominates, the index soars, pointing to a concentrated and potentially less competitive market. This calculation helps us understand the level of competition and, by extension, the potential for consumer choice and pricing strategies within an industry.

Nash Equilibrium

The concept of Nash equilibrium is central to understanding strategic decision-making in economics. It occurs when all participants in a game, like firms in a Cournot game, choose their best response strategy while anticipating the choices of others. In this state, no one firm can benefit by changing its strategy unilaterally.

When we calculate the Nash equilibrium for firms competing in quantity (a Cournot game), we're looking for the quantity each firm will produce such that no firm can increase its profit by producing more or less, given the production levels of its competitors. This equilibrium is crucial for predicting outcomes in an oligopolistic market where firms' decisions are interdependent.

Cournot Game

The Cournot game is a model where firms compete on the amount of output they produce, influencing the market price of their goods. Named after the French economist Antoine Augustin Cournot, the model assumes that each firm chooses its output level based on the output levels of competitors, aiming to maximize its own profit. Finding the Nash equilibrium in a Cournot game helps us predict how much each firm will produce and the resulting market price.

In our textbook exercise, we calculated the Nash equilibrium for an n-firm Cournot game and examined how this equilibrium changes with a merger (market concentration), different costs, and other variables, highlighting how sensitive industry outcomes can be to the strategic choices of firms.

Consumer Surplus

Consumer surplus is the difference between what consumers are willing to pay for a good or service and what they actually pay. It represents the net benefit or satisfaction consumers receive from purchasing at a market price that is less than the maximum price they're willing to pay. In economic models, it's often illustrated as the area under the demand curve and above the price.

In the context of our Cournot game, consumer surplus can be affected by changes in market structure and firm behavior, like mergers or cost variations. As the exercise shows, when firms merge or costs change, there can be significant impacts on market price and output, which, in turn, affect consumer surplus. It's essential to understand this concept to grasp the broader implications of market competition on consumers.

Industry Profit

Industry profit, often referred to simply as 'profit,' is the total earnings of firms in a market after accounting for production costs. It's an essential measure of the financial health of an industry and an indicator of its competitive dynamics.

The industry profit calculation in the exercise, based on the Nash equilibrium of a Cournot game, sheds light on how market concentration, through actions like mergers, can influence a firm's profitability. When analyzing different market scenarios, one key observation is that changes in market structure, for example, a reduction in the number of firms, often lead to higher industry profits due to less competitive pressure and potentially higher market prices.

Total Welfare

Total welfare is the sum of consumer surplus and industry profit, providing a combined measure of the benefits received by consumers and firms from market transactions. It is a comprehensive gauge of the overall well-being that a market creates for its participants.

The exercise illustrates how total welfare is influenced by the strategic behaviors of firms within an industry. In economic terms, total welfare is maximized when resources are allocated most efficiently, which often occurs in competitive markets where no single firm has significant market power. Consequently, changes in market concentration through mergers and varying costs can have a profound impact on total welfare by shifting the balance between consumer surplus and industry profit.