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

Expert verified
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

01

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: $$Q_{D} = Q_{S} \Rightarrow -2P + 13 = 2P^{2} -12P + 21$$
02

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: $$P = 2, \ P = 4$$
03

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: $$ED = -2P^{2} + 10P - 8$$
04

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: $$\frac{d(ED)}{dP}\Big|_{P=2} = -4(2) + 10 = 2$$ $$\frac{d(ED)}{dP}\Big|_{P=4} = -4(4) + 10 = -6$$
05

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.

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Most popular questions from this chapter

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