Question

Suppose that the market demand for a particular product is given by $$Q_{D}=-2 P+13$$ and the industry supply curve by $$Q_{s}=2 P^{2}-12 P+21$$. What are the equilibrium prices for this market? Which of these prices is stable by the Walrasian criterion?

Short answer

The equilibrium price P = 4 is stable according to the Walrasian stability criterion because the slope of the excess demand function is negative at this price. The equilibrium price P = 2 is not stable because the slope of the excess demand function is positive at this price.

Step-by-step solution

Set the demand function equal to the supply function

In order to find the equilibrium price, we need to set the market demand function equal to the market supply function. Mathematically, this is represented as: $$Q_{D} = Q_{S} \Rightarrow -2P + 13 = 2P^{2} -12P + 21$$

Rearrange and solve for P

Next, we want to rearrange the equation and solve for P. To do this, let's move all the terms to the left side of the equation and then solve for P: $$2P^{2} - 12P + 21 + 2P - 13 = 0 \Rightarrow 2P^{2} - 10P + 8 = 0$$ To solve for P, we can factor the quadratic equation: $$(2P - 4)(P - 2) = 0$$ This gives us two possible equilibrium prices: $$P = 2, \ P = 4$$

Determine the excess demand function

To apply the Walrasian stability criterion, we need to find the excess demand function. Excess demand is the difference between the demand and the supply: $$ED = Q_{D} - Q_{S} \Rightarrow -2P + 13 - (2P^{2} -12P + 21)$$ Simplifying the expression, we get: $$ED = -2P^{2} + 10P - 8$$

Check the slope of the excess demand function at the equilibrium prices

To determine which equilibrium price is stable using the Walrasian criterion, we need to check the slope of the excess demand function at each of the equilibrium prices we found earlier. We can do this by finding the first derivative of the excess demand function with respect to P: $$\frac{d(ED)}{dP} = -4P + 10$$ Now, we'll evaluate the derivative at our two equilibrium prices: $$\frac{d(ED)}{dP}\Big|_{P=2} = -4(2) + 10 = 2$$ $$\frac{d(ED)}{dP}\Big|_{P=4} = -4(4) + 10 = -6$$

Determine the stable equilibrium price using the Walrasian criterion

According to the Walrasian stability criterion, if the slope of the excess demand function is negative at the equilibrium price, then the price is stable. From our calculations, we found: $$\frac{d(ED)}{dP}\Big|_{P=2} = 2$$ $$\frac{d(ED)}{dP}\Big|_{P=4} = -6$$ Since the excess demand function has a negative slope at P = 4, we can conclude that this equilibrium price is stable according to the Walrasian criterion. The equilibrium price P = 2 is not stable because the slope of the excess demand function is positive.

Key concepts

Market Demand

Market demand represents the total quantity of a product that all consumers in a market are willing and able to purchase at various price points. It is crucial to understand that market demand is influenced by factors such as consumer preferences, income levels, and prices of related goods. In a demand function, which is often represented mathematically, price is typically a key variable. For instance, the demand function provided is \(Q_D = -2P + 13\). Here:

• \(Q_D\) is the quantity demanded.
• \(-2P\) indicates that for every unit increase in price, the quantity demanded decreases by 2 units, showcasing the law of demand.
• The constant term, 13, can be interpreted as the intercept where demand would theoretically exist if the price were zero.

Understanding the demand function helps predict how changes in price affect the market, guiding businesses and policymakers in decision-making.

Market Supply

Market supply represents the total quantity a supplier is willing to produce and sell when various price levels are considered. The supply function is crucial in determining how much of a good is produced and made available in the market at different prices. In the given exercise, the supply function is expressed as \(Q_s = 2P^2 - 12P + 21\). Here's a breakdown of the terms:

• \(Q_s\) stands for the quantity supplied.
• The term \(2P^2\) suggests that the supply tends to increase as the price increases. This is due to the quadratic relationship where higher prices lead to higher productivity or supply capacity in this context.
• \(-12P\) indicates a linear component affecting supply negatively when prices increase, possibly accounting for costs.
• The constant term, 21, serves as the supply level at baseline price conditions.

The supply function provides insights into how producers respond to price changes and can be used alongside demand to find equilibrium in the market.

Excess Demand

Excess demand occurs when the quantity demanded in the market exceeds the quantity supplied. This often results in upward pressure on prices as buyers compete for limited goods. Conversely, if excess demand is negative, it implies excess supply, leading to potential price decreases as sellers compete for sales.The excess demand function is the difference between market demand and market supply. For our exercise, this is represented as \(ED = Q_D - Q_S = -2P + 13 - (2P^2 - 12P + 21)\), simplifying to \(-2P^2 + 10P - 8\). Important takeaways include:

• The function can help predict price movements by showing whether the market needs adjustment to reach equilibrium.
• Positive excess demand indicates a potential for prices to rise since demand outstrips supply.
• Negative values suggest too much supply, resulting in possible price reductions.

Analyzing excess demand is essential to understand whether the market is in balance and how prices are likely to change.

Walrasian Stability Criterion

The Walrasian Stability Criterion helps determine which equilibrium prices are stable by examining the slope of the excess demand function at those points. Essentially, it posits that for an equilibrium price to be stable, the derivative of the excess demand function \( \frac{d(ED)}{dP} \) must be negative at that price. For the given exercise, we determined:

• At \(P = 2\), the slope \(\frac{d(ED)}{dP} = 2\) is positive, indicating instability at this price. The market tends to move away from this equilibrium.
• At \(P = 4\), the slope \(\frac{d(ED)}{dP} = -6\) is negative, suggesting this is a stable equilibrium. Here, price will adjust back toward equilibrium if shifted.

Stability analysis is critical in economic models as it helps predict how the market responds to external shocks and whether it adjusts naturally over time. Identifying stable and unstable equilibria assists decision-makers in forecasting market dynamics.