Rearranging the Differentiated Equilibrium Equation to Isolate the Partial Derivative

Consider a standard competitive market model where the quantity demanded is given by Qd = a - bP and the quantity supplied is Qs = c + dP. The parameters a, b, c, d are all positive constants. In equilibrium, the price P* is determined such that the market clears: a - bP* = c + dP*. What is the correct result of differentiating this entire equilibrium equation with respect to the demand parameter a, remembering that the equilibrium price P* is itself a function of a?

Identifying an Error in Comparative Statics

In a market model where the equilibrium price, P*, is determined by equating demand and supply functions (e.g., D(P*, a) = S(P*)), the correct way to find how P* changes with a demand parameter 'a' is to differentiate the equilibrium equation with respect to 'a' while treating P* as a constant. The reasoning is that P* is the variable being solved for, not an independent parameter.

Consider a general market model where equilibrium is defined by the equation D(P*, a) = S(P*). Here, D is quantity demanded, S is quantity supplied, P* is the equilibrium price, and a is a parameter affecting demand. To find how the equilibrium price changes when the parameter a changes (i.e., to find ∂P*/∂a), we must differentiate the entire equilibrium equation with respect to a, remembering that P* is a function of a. This results in the equation: (∂D/∂P) * (∂P*/∂a) + ∂D/∂a = (∂S/∂P) * (∂P*/∂a). Match each conceptual component of this process to its correct mathematical term from the equation.

Setting Up a Comparative Statics Analysis

The Functional Dependence of Equilibrium Price

To determine how a change in a demand-side parameter (let's call it 'a') affects the equilibrium price (P*) in a market, a specific analytical procedure must be followed. Arrange the following steps into the correct logical sequence for conducting this analysis.

In a general market model, the equilibrium condition can be expressed implicitly as F(P*, a) = 0, where P* is the equilibrium price and a is a parameter. To find the effect of a change in a on P*, the entire equation is differentiated with respect to a, remembering that P* is a function of a. This application of the chain rule results in the expression: (∂F/∂P*) * (∂P*/∂a) + ______ = 0. The term that correctly fills the blank, representing the direct impact of the parameter a on the equilibrium condition, is ____.

The Rationale for Differentiating Equilibrium Price

Interpreting Components of a Comparative Statics Derivative