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Chapter 2: Problem 57

Both the supply and demand equations for a certain product have negative slope. Can there be an equilibrium price? Explain.

Short Answer

Expert verified
Yes, there can be an equilibrium price when both supply and demand equations have negative slopes. The equilibrium price depends on the intercepts of the supply and demand equations (\(b_s\) and \(b_d\)), and as long as the demand intercept (\(b_d\)) is greater than the supply intercept (\(b_s\)), an equilibrium price will exist where the quantity supplied equals the quantity demanded.

Step by step solution

01

Define Supply and Demand Equations

Supply and demand equations are given as functions of the price (P) for a certain product. A supply equation represents the quantity that the producers of the product are willing to supply at a certain price, while the demand equation depicts the quantity purchasers are willing to buy at a certain price. Mathematically, these equations are represented as: Supply: \(Q_s = m_sP + b_s\) Demand: \(Q_d = m_dP + b_d\) where \(m_s\) and \(m_d\) are the slopes of the supply and demand equations, and \(b_s\) and \(b_d\) are the intercepts.
02

Equilibrium Concept

An equilibrium price occurs when the quantity supplied is equal to the quantity demanded. At this price, producers are willing to produce the same quantity that purchasers are willing to buy, minimizing shortages or surpluses in the market. To find the equilibrium price, we can set the supply equation equal to the demand equation: \(Q_s = Q_d\)
03

Analyzing Slopes for Equilibrium Price

Given that both supply and demand equations have negative slopes, it means that as price increases, the quantity supplied and demanded decreases. The negative slope of the supply equation, \(m_s < 0\), implies that producers will supply fewer goods when the price is higher, while the negative demand slope, \(m_d < 0\), means that consumers will demand less when the price is higher.
04

Equilibrium Price Condition

For an equilibrium price to exist, the supply and demand curves must intersect at a certain point. Since both curves have negative slopes, they will have the possibility to intersect, depending on the intercepts \(b_s\) and \(b_d\). If the demand intercept (\(b_d\)) is greater than the supply intercept (\(b_s\)), which typically represents a higher initial demand than supply, the two curves will intersect at an equilibrium point, giving us an equilibrium price.
05

Conclusion

Yes, there can be an equilibrium price when both supply and demand equations have negative slopes. The equilibrium price depends on the intercepts of the supply and demand equations, and as long as the demand intercept is greater than the supply intercept, an equilibrium price will exist where the quantity supplied equals the quantity demanded.

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