Question

Suppose that the market demand for a product is given by \(Q_{D}=A-B P .\) Suppose also that the typical firm's cost function is given by \(C(q)=k+a q+b q^{2}\) a. Compute the long-run equilibrium output and price for the typical firm in this market. b. Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem. c. Describe how changes in the demand parameters \(A\) and \(B\) affect the equilibrium number of firms in this market. Explain your results intuitively. d. Describe how the parameters of the typical firm's cost function affect the long-run equilibrium number of firms in this example. Explain your results intuitively.

Short answer

In summary, the long-run equilibrium output, price, and the number of firms in the market are affected by different parameters from the market demand function and the cost function for a typical firm. By analyzing and solving the demand and cost functions, we can derive the optimal output, price, and a function for the equilibrium number of firms in the market. The impacts of changes in demand and cost function parameters on the equilibrium number of firms are generally intuitive, with higher demand or lower costs leading to more firms operating in the market. However, the relative magnitudes of changes in these parameters will depend on the specific interactions between the demand curve, cost function, and market conditions.

Step-by-step solution

a. Compute the long-run equilibrium output and price for the typical firm in this market.

To compute the long-run equilibrium output and price for the typical firm in this market, we first need to find the marginal cost (MC) for a firm. This is found by taking the derivative of the cost function (C(q)) with respect to q. Suppose that the market demand is given by \(Q_{D} = A - B P\), and the cost function for the typical firm is given by \(C(q) = k + a q + b q^{2}\). Differentiating the cost function with respect to q, we get: \(\text{MC} = \frac{dC(q)}{dq} = a + 2bq\) Now, in the long-run equilibrium, the firm maximizes its profit, which implies that marginal cost (MC) is equal to the market price (P). Therefore, we have: \(P = a + 2bq\) Since we have the market demand function, we can substitute the price in terms of q to obtain the optimal output for the firm: \(Q_{D} = A - B(a + 2bq)\) Solving for q, we get the optimal output for a firm: \(q^{*} = \frac{A - Ba}{2Bb + B}\) We can now plug \(q^{*}\) back into the price function to determine the long-run equilibrium price: \(P^{*} = a + 2bq^{*} = a + 2b\left(\frac{A - Ba}{2Bb + B}\right)\)

b. Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem.

Now that we have the optimal output for a typical firm (\(q^{*}\)) and the market demand function (\(Q_{D} = A - BP\)), we can calculate the equilibrium number of firms (n) in this market as follows: First, note that in equilibrium, the total quantity supplied in the market will be equal to the total quantity demanded. The total quantity supplied is the product of the quantity produced by each firm and the number of firms (n). Thus, we have: \(nq^{*} = Q_{D}\) Now, we can plug the expressions we derived in part (a) for \(q^{*}\) and \(Q_{D}\): \(n\left(\frac{A - Ba}{2Bb + B}\right) = A - B\left(a + 2b\left(\frac{A - Ba}{2Bb + B}\right)\right)\) Solving for n, we get the equilibrium number of firms in this market as a function of all the parameters in this problem: \(n = \frac{(2Bb + B)(A - Ba)}{A(2Bb - B) + B^{2}a}\)

c. Describe how changes in the demand parameters \(A\) and \(B\) affect the equilibrium number of firms in this market. Explain your results intuitively.

We can analyze the impact of changes in the demand parameters \(A\) and \(B\) on the equilibrium number of firms by looking at the expression we derived for n: \(n = \frac{(2Bb + B)(A - Ba)}{A(2Bb - B) + B^{2}a}\) 1. Impact of change in \(A\): From the expression for n, we can see that the equilibrium number of firms in the market is positively related to the demand parameter A. This implies that when A increases, the number of firms in the market will increase. This can be explained intuitively as an increase in A represents a shift of the demand curve to the right which means that consumers demand more products at each price level. As a result, market equilibrium changes and the number of firms needed to meet the new market demand also increases. 2. Impact of change in \(B\): The impact of B on the number of firms is not as straightforward. The term \((2Bb + B)\) on the numerator has a positive relationship with B, while the denominator term, \(A(2Bb - B) + B^{2}a\), has a negative relationship with B. Therefore, the overall impact of B on the equilibrium number of firms will depend on the values of A, a, and b. However, the main intuition can be derived from B's impact on the slope of the demand curve. An increase in B makes the demand curve steeper (more inelastic), leading to a larger decrease in the quantity demanded when price increases. Hence, the overall change in the number of firms due to a change in B will depend on the market's response to changes in price.

d. Describe how the parameters of the typical firm's cost function affect the long-run equilibrium number of firms in this example. Explain your results intuitively.

We can analyze the impact of the cost function parameters, a and b, on the equilibrium number of firms by looking at the expression we derived for n: \(n = \frac{(2Bb + B)(A - Ba)}{A(2Bb - B) + B^{2}a}\) 1. Impact of change in \(a\): From the expression for n, we can observe that the equilibrium number of firms in the market is negatively related to the cost parameter a. This means that when a increases, the number of firms in the market decreases. This result can be explained intuitively as an increase in a represents an increase in the marginal cost of production for the firm, making it more expensive to produce additional units. As a consequence, fewer firms will operate profitably in the market, and the equilibrium number of firms will decrease. 2. Impact of change in \(b\): The impact of b on the number of firms is more complex. The term \((2Bb + B)\) on the numerator shows a positive relationship with b, while the denominator term, \(A(2Bb - B) + B^{2}a\), has a positive relationship with b as well. This means that the overall effect of b on the number of firms will depend on the relative magnitudes of these effects. The key intuition behind this result lies in the fact that an increase in b represents an increase in the curvature of the cost function. As a result, the shape of the cost function changes, which in turn alters the relationship between output, price, and the number of firms in the long-run equilibrium.

Key concepts

Long-Run Equilibrium

Long-run equilibrium in a market occurs when firms are producing at a level where no new firms wish to enter, and existing firms do not want to leave the market. This state is characterized by firms making normal profit, which is the minimum amount needed to keep them operational in the industry. In long-run equilibrium, marginal cost (MC) equals market price (P), and average total cost (ATC) is also equal to price.

This means that each firm is operating efficiently, covering all their costs, including opportunity costs. There are no long-term economic profits due to the perfect competition structure. Over time, adjustments are made as firms enter or exit the market, driven by potential profits or losses, until equilibrium is reached.

Hence, in long-run equilibrium, market forces have adjusted output levels, costs, and prices to a stable point where demand equals supply, and firms are satisfied with both their production and financial returns.

Demand Function

The demand function represents the relationship between the quantity demanded of a good and its price, typically expressed as: \[ Q_{D} = A - BP \] where \(Q_{D}\) is the quantity demanded, \(P\) is the price of the product, \(A\) is the intercept (representing demand at a price of zero), and \(B\) is the slope of the demand curve (indicating how much demand decreases as price increases).

A higher \(A\) indicates greater initial demand at zero or low prices, while a steep \(B\) means even small price increases will significantly reduce quantity demanded.

Understanding the demand function is crucial in determining market equilibrium, as it sets the stage for matching supply quantities provided by firms at particular price levels, analyzed in conjunction with cost functions.

Cost Function

A cost function reflects the total cost a firm incurs as it produces varying quantities of output, often represented as: \[ C(q) = k + aq + bq^2 \] where \(k\) is fixed cost, \(a\) is the variable cost per unit of output, and \(bq^2\) represents increasing marginal costs with higher output levels, or diminishing returns.

Analyzing the cost function helps identify the firm's marginal cost (MC), crucial for setting prices in long-run equilibrium, derived as \(MC = a + 2bq\).

It's important to note that changes in the coefficients \(a\) and \(b\) affect how firms evaluate profitability. High fixed or variable costs can make some levels of production less attractive, influencing the optimal production quantity and consequently the market's long-term firm structure.

Number of Firms

The number of firms in a market is determined by how market demand and costs align to set a feasible production and cost-sharing system across firms. In long-run equilibrium, it's calculated by considering demand satisfaction and individual firm's capacity derived from their cost functions.
The equilibrium number of firms (\( n \)) depends on all involved parameters: demand (\( A, B \)) and cost firms bear (\( a, b \)), expressed as: \[ n = \frac{(2Bb + B)(A - Ba)}{A(2Bb - B) + B^2a} \] This formula highlights that any alterations in demand or cost influence the number of viable firms.
- Increasing \( A \) generally allows for more firms as it indicates higher market demand.
- Higher \( B \) might reduce them by steepening the demand curve's responsiveness to price increases.
- Changes in cost parameters like \( a \) and \( b \) reflect on costs per additional unit produced, leading to adjustments in firm numbers needed to equilibrate supply and demand.