Question

If the long-run cost function is \(c(y)=y^{2}+1,\) what is the long-run supply curve of the firm?

Short answer

The long-run supply curve of the firm is \( y = \frac{P}{2} \).

Step-by-step solution

Understand the Cost Function

We are given the long-run cost function of the firm, which is \( c(y) = y^2 + 1 \), where \( y \) represents the quantity of output produced. Our task is to find the long-run supply curve, for which we need to understand the relationship between marginal cost and price.

Calculate the Marginal Cost

The marginal cost (MC) is the derivative of the cost function with respect to \( y \). So, we differentiate the cost function \( c(y) = y^2 + 1 \) with respect to \( y \) to get: \[ MC = \frac{dc(y)}{dy} = 2y. \]

Set Marginal Cost Equal to Price

In the long-run, the firm will supply the quantity where the price \( P \) equals the marginal cost (MC). This condition provides the supply curve. Thus, we set \( 2y = P \) to determine \( y \) in terms of \( P \).

Solve for the Supply Curve

Solving the equation \( 2y = P \) for \( y \), we find the supply curve: \[ y = \frac{P}{2}. \] This equation represents the firm's supply curve in the long run.

Key concepts

Long-Run Cost Function

In economics, a long-run cost function is a crucial tool for understanding how the costs of a firm change relative to its output over a longer period. A cost function like \( c(y) = y^2 + 1 \) describes how total costs \( c(y) \) vary as output \( y \) changes. The term \( y^2 \) captures the variable costs, where costs increase with higher output. The constant \( +1 \) represents fixed costs, incurred regardless of the level of production.
In the long run, all input factors are variable, offering flexibility in production. This is different from the short-run, where some inputs are fixed. Understanding the long-run cost function helps firms make strategic decisions about production levels that will minimize their costs while meeting market demands. It aids in planning how to allocate resources efficiently as the firm expands or contracts its production capacity.

Marginal Cost

Marginal cost (MC) is pivotal in decision-making for firms, representing the additional cost incurred from producing one more unit of output. To compute it from a cost function \( c(y) = y^2 + 1 \), we take its derivative with respect to \( y \). This gives us the marginal cost function \( MC = 2y \).
Marginal cost provides vital insights:

• It helps determine the level of output where the firm maximizes its profit.
• A comparison between marginal cost and price guides how much output a firm should produce.

In essence, firms will seek to produce until the cost of making an additional unit equals the revenue it brings in. This helps in achieving cost-effective operations, avoiding excess production that incurs losses.

Supply Curve Derivation

The supply curve represents how much of a good a firm is willing to supply at various price levels. To derive it from a long-run cost perspective, we set the marginal cost equal to the market price since the firm in a competitive market will produce until \( MC = P \).
With the cost function \( c(y) = y^2 + 1 \) and \( MC = 2y \), equate the marginal cost to price: \( 2y = P \). Solving for \( y \), the supply curve is derived as \( y = \frac{P}{2} \).
This equation tells us:

• The quantity supplied \( y \) is directly proportional to the price \( P \).
• A rise in price leads to more quantity supplied, following the law of supply.

This understanding helps firms adjust their production in response to price changes in long-run scenarios, ensuring they efficiently meet market demands.