Question

If \(S_{1}(p)=p-10\) and \(S_{2}(p)=p-15,\) then at what price does the industry supply curve have a kink in it?

Short answer

No kink occurs as the functions do not equal each other.

Step-by-step solution

Understand the Supply Functions

The problem provides two supply functions \(S_1(p)\) and \(S_2(p)\). These functions define the quantity supplied at a given price \(p\). The key task is to determine when these two functions, which could represent different segments of an industry supply curve, intersect to form a kink.

Set the Supply Functions Equal

To determine the kink in the supply curve, we need to find the price \(p\) at which the quantity supplied by both functions is equal. This means solving the equation \(S_1(p) = S_2(p)\), which gives us \(p - 10 = p - 15\).

Solve for Price

To find the price where the kink occurs, we solve the equation \(p - 10 = p - 15\). Simplifying this equation gives us \(p - p = 10 - 15\), leading to \(0 = -5\). Since the math leads to an incorrect statement, it implies that this point of pricing might need reconsideration.

Interpret the Result

The result of \(0 = -5\) indicates that there is no value of \(p\) where the two supply functions intersect; hence there is no kink formed by the given functions. The problem setup may involve segments that do not physically switch at the same price.

Key concepts

Understanding the Supply Function

In intermediate microeconomics, the supply function is a critical concept that helps us understand how much of a product a seller is willing to supply at a given price. You can think of a supply function as a mathematical way to represent a producer's willingness to supply based on price changes. In the given exercise, we have two supply functions:

• \( S_1(p) = p - 10 \) – This supply function means that the quantity supplied increases as the price rises. When the price is slightly over 10, the supply becomes positive.
• \( S_2(p) = p - 15 \) – Similar to the first, this function starts supplying quantities when the price exceeds 15.

Supply functions help establish how different suppliers react to changes in prices, capturing variations in their minimal price acceptable to start supplying goods. Observing these functions aligns closely with understanding supply elasticity.

Exploring the Industry Supply Curve

The industry supply curve aggregates all individual supply functions within a market, representing the total quantity supplied by all producers as a function of the product's price. It's like the big picture where you sum up all quantities supplied by each firm at every price level.
In the problem, the two supply functions \( S_1(p) = p-10 \) and \( S_2(p) = p-15 \) each represent different suppliers within the industry. If these functions come together to construct the industry supply curve, it indicates that there should usually be a price where the behavior of the aggregate supply changes, often causing a 'kink' in the curve. A kink occurs at the point where one firm stops supplying, and another starts, which usually happens at different price landmarks for different firms.
In this exercise, however, since the solving process yielded a contradiction (\(0 = -5\)), it suggests that these two suppliers do not contribute to a kink within the industry supply curve as they do not create a crossover in supply quantities at any particular price.

Finding the Price Equilibrium

Price equilibrium is a fundamental concept in economics that occurs when market supply equals market demand at a given price. Essentially, it’s the sweet spot where the quantity of goods supplied matches the goods demanded. Although this exercise primarily deals with the supply side, understanding equilibrium helps contextualize supply interactions.
When analyzing where the industry supply curve might kink due to different functions, it inherently relates to different suppliers reaching their equilibrium price. However, if no identical price exists where such an intersection happens, like in this case, it means no real price can balance these two supply functions equally.
Thus, with the problem showing no real solution through the equation \(0 = -5\), this further implies that the industry’s equilibrium would occur under different market conditions, possibly involving more complex interactions than just these two functions. Recognizing where equilibrium is maintained helps us align industry supply capabilities with market demand more effectively.