Question

Suppose that the market demand for a particular product is given by $$Q_{D}=-1 P+13$$ and the industry supply curve by $$Q_{s}=2 \mathrm{P}^{2}-12 \mathrm{P}+21$$ What are the equilibrium prices for this market? Which of these prices is stable by the Walrasian criterion?

Short answer

Answer: The stable equilibrium price is $$P_1 = \frac{11 - \sqrt{57}}{4}$$.

Step-by-step solution

Set market demand equal to market supply

To find the equilibrium prices, set the demand function equal to the supply function and solve for $$P$$: $$-P + 13 = 2P^2 - 12P + 21$$

Re-arrange the equation and solve for $$P$$

Rearrange the equation to solve for $$P$$: $$2P^2 - 11P + 8 = 0$$ This is a quadratic equation and we can find the roots by using the Quadratic Formula: $$P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ For our equation, $$a = 2, b = -11, c = 8$$: $$P = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(8)}}{2(2)}$$

Calculate the equilibrium prices

Calculate the equilibrium prices using the quadratic formula: $$P = \frac{11 \pm \sqrt{121 - 64}}{4}$$ $$P = \frac{11 \pm \sqrt{57}}{4}$$ This gives us two possible equilibrium prices: $$P_1 = \frac{11 - \sqrt{57}}{4}$$ $$P_2 = \frac{11 + \sqrt{57}}{4}$$

Find the slope of the excess demand curve

To find the slope of the excess demand curve, we must differentiate the excess demand function with respect to $$P$$: Excess demand function is given by $$Q_D - Q_S$$, so: $$\frac{d(Q_D - Q_S)}{dP} = \frac{d((-P + 13) - (2P^2 - 12P + 21))}{dP}$$ $$= \frac{d(-2P^2 + 11P - 8)}{dP}$$ After differentiating, we find the slope of the excess demand curve to be: $$-4P + 11$$

Evaluate which equilibrium price is stable by the Walrasian criterion

Determine the slope of the excess demand curve at the equilibrium prices: For $$P_1$$: $$-4\left(\frac{11 - \sqrt{57}}{4}\right) + 11 = -11 + \sqrt{57}$$ For $$P_2$$: $$-4\left(\frac{11 + \sqrt{57}}{4}\right) + 11 = -11 - \sqrt{57}$$ By the Walrasian criterion, an equilibrium price is stable if the slope of the excess demand curve is positive at that price. In our case, the slope of the excess demand curve is positive for $$P_1$$ (\(( -11 + \sqrt{57} > 0 )\)), and negative for $$P_2$$ (\(( -11 - \sqrt{57} < 0 )\)). Hence, the stable equilibrium price is $$P_1 = \frac{11 - \sqrt{57}}{4}$$.

Key concepts

Market Demand

In economics, market demand is the total quantity of a good or service that all consumers in the market are willing and able to purchase at different price levels. It's a fundamental concept that reflects the purchasing habits and preferences of consumers and is visually represented by the demand curve on a graph.

The market demand can be affected by various factors such as income levels, tastes, and prices of other goods. A typical demand curve slopes downward to the right, indicating that as the price of the good decreases, the quantity demanded increases. This inverse relationship is known as the law of demand.

In our exercise, the given market demand function is \( Q_D = -P + 13 \), which is a simple linear equation where \( P \) represents the price, and \( Q_D \) is the quantity demanded. As the price increases by one unit, the quantity demanded decreases by the same amount, illustrating the negative relationship between price and quantity demanded.

Industry Supply Curve

Conversely, the industry supply curve represents the total quantity of a good or service that producers in the market are willing and able to sell at various price levels. It is a graphical illustration of the relationship between price and quantity supplied and generally slopes upward to the right.

This upward slope indicates that producers are willing to supply more of the good as the price rises, due to the potential for increased revenue and profit. Factors affecting the industry supply curve include production costs, technology advancements, and the number of sellers in the market.

For our exercise, the industry supply curve is defined by the equation \( Q_S = 2P^2 - 12P + 21 \), which is a quadratic function specifying how the quantity supplied \( Q_S \) changes with the price \( P \). Unlike linear supply curves, this quadratic curve allows for more complex relationships between price and quantity supplied.

Walrasian Criterion

The Walrasian criterion, also known as Walras' law, is a principle in economics that pertains to the stability of equilibrium. According to this criterion, an equilibrium price is considered stable if any excess demand (or supply) will induce price movements that will return the market to equilibrium.

To apply the Walrasian criterion, one typically looks at the slope of the excess demand curve. If the slope is positive (demand increases when price decreases), any deviation from the equilibrium would cause adjustments bringing the market back to equilibrium. This means that consumers are reacting to price changes in a way that drives the market towards the equilibrium point.

In our exercise, the stable equilibrium price is determined by examining the slope of the excess demand curve at each potential price. We concluded that the first equilibrium price \( P_1 = \frac{11 - \sqrt{57}}{4} \) is stable because at this price, the slope of the excess demand curve is positive, satisfying the Walrasian criterion. In contrast, the second price does not meet this condition, indicating it is an unstable equilibrium.