Deriving the Supply Curve from the Cost Function: A Comprehensive Guide

In economics, the relationship between the cost function and the supply curve is fundamental for understanding how firms operate in a competitive market. This article will walk through the process of deriving the supply curve from a given cost function. Let's use the provided cost function: cy 10y^2 - 1000, where cy is the total cost, y is the quantity produced, and the cost function exhibits an increasing cost situation.

Step 1: Finding the Marginal Cost (MC)

The first step is to find the marginal cost (MC), which is the derivative of the total cost with respect to y. Marginal cost represents the additional cost incurred from producing one more unit of the product.

MC frac{dc}{dy} frac{d}{dy}(10y^2 - 1000) 20y

From this calculation, we see that the marginal cost is directly proportional to the quantity produced.

Step 2: Determining the Supply Curve

In a competitive market, a firm will supply output where the price P is equal to the marginal cost MC. Therefore, we set the price equal to the marginal cost and solve for the quantity y:

P MC 20y

Rearranging this equation, we get the supply curve:

y frac{P}{20}

This equation indicates that for any given price P, the firm is willing to supply frac{P}{20} units of output.

Conclusion

The supply curve for the firm in this example is y frac{P}{20}. This relationship helps us understand how the firm will respond to changes in market prices. For instance, if the price P is set at 1000 dollars, the quantity supplied will be 50 units of output.

Moreover, the revenue generated from selling these 50 units can be calculated as 50 times 1000 50000 dollars. At the same time, the total cost of producing these 50 units can be found by substituting y 50 back into the cost function:

"10(50)^2 - 1000 25000 - 1000 24000"

So, the firm's total cost is 24000 dollars, and the profit is 50000 - 24000 26000 dollars.

To summarize, the supply curve y frac{P}{20} provides a clear relationship between the price of a good and the quantity the firm is willing to supply, which is crucial for understanding market dynamics.

Additional Insights

Understanding the supply curve is essential for making informed business decisions. It helps firms plan production levels to maximize profits and adapt to changes in market conditions. As output increases, the cost function exhibits an increasing cost situation, indicating diminishing marginal returns. This information is valuable for strategic planning and financial forecasting.

By following this step-by-step guide, you can derive the supply curve from any given cost function, providing a solid foundation for market analysis and operational management.