Model Answer
0 min readIntroduction
In economics, understanding the relationship between cost, supply, and price is fundamental to analyzing market behavior. The supply curve represents the quantity of a good or service that producers are willing to offer at various prices. The cost function, which details the costs incurred by a firm at different output levels, is a crucial determinant of the supply curve. Given the cost function C(y) = y² + 1, where y represents output, this answer will derive the inverse supply curve and illustrate its characteristics, demonstrating the link between production costs and market supply.
Derivation of the Supply Curve
The supply curve is derived from the firm’s cost structure. Specifically, a firm in a competitive market will supply the quantity where marginal cost (MC) equals price (P). Therefore, to derive the supply curve, we first need to calculate the marginal cost.
1. Calculating Marginal Cost (MC)
Marginal cost is the change in total cost resulting from producing one additional unit of output. Mathematically, it is the derivative of the cost function with respect to output (y).
C(y) = y² + 1
MC = dC(y)/dy = 2y
2. Deriving the Supply Function
In a perfectly competitive market, firms maximize profit by producing where P = MC. Therefore, the supply function is given by:
P = 2y
3. Deriving the Inverse Supply Curve
The inverse supply curve expresses price as a function of quantity. To obtain this, we rearrange the supply function to solve for P:
y = P/2
Therefore, the inverse supply curve is: P = 2y
Shape of the Supply Curve
The inverse supply curve, P = 2y, is a linear function with a positive slope. This indicates that as the quantity supplied (y) increases, the price (P) also increases. The slope of the supply curve is 2, meaning that for each additional unit of output, the price must rise by 2 to cover the increasing marginal cost of production. This is a typical upward-sloping supply curve observed in competitive markets.
Graphical Representation
The supply curve can be visualized as a straight line originating from the origin with a slope of 2. At y=0, P=0. As y increases, P increases linearly. This implies that the firm is willing to supply more only at higher prices, reflecting the increasing cost of production.
| Quantity (y) | Price (P) |
|---|---|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
Conclusion
In conclusion, given the cost function C(y) = y² + 1, the derived inverse supply curve is P = 2y. This represents a positively sloped, linear supply curve, indicating a direct relationship between price and quantity supplied. The increasing marginal cost of production, as reflected in the cost function, drives the upward slope of the supply curve, demonstrating a fundamental principle of microeconomic theory. This analysis highlights how a firm’s cost structure directly influences its supply decisions in a competitive market.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.