Differentiating the Equilibrium Equation with Respect to the Demand Parameter 'a'

Comparative Statics in a Linear Market Model

A market is described by a general demand function Q_d = D(P, a) and a supply function Q_s = S(P, c), where 'P' is the price, 'a' is a demand-side parameter, and 'c' is a supply-side parameter. To find how the equilibrium price (P*) changes in response to a small change in the demand parameter 'a', you must use implicit differentiation. Arrange the following steps in the correct logical order to solve for the partial derivative ∂P*/∂a.

Comparative Statics using Implicit Differentiation

Consider a market where the demand function is Q_d = D(P, a) and the supply function is Q_s = S(P). The market is in equilibrium. Assume that the demand curve is downward-sloping (the partial derivative of D with respect to P is negative) and the supply curve is upward-sloping (the derivative of S with respect to P is positive). Also, assume that an increase in the parameter 'a' causes an outward shift in the demand curve (the partial derivative of D with respect to 'a' is positive). Based on an analysis of the equilibrium condition, what can be concluded about the effect of a small increase in 'a' on the equilibrium price, P*?

Consider a competitive market where the demand function is given by Qd = a - bP and the supply function is Qs = c + dP. All parameters (a, b, c, d) are positive constants. At equilibrium, the price (P*) is determined where quantity demanded equals quantity supplied. This equilibrium price is an implicit function of the model's parameters. What is the partial derivative of the equilibrium price with respect to the demand intercept 'a', denoted as ∂P*/∂a?

Implicit Differentiation in a Non-Linear Market

Comparing Methods for Comparative Statics

Consider a market where the demand function is given by Q_d = D(P, a) and the supply function is Q_s = S(P). The market is in equilibrium where D(P*, a) = S(P*). To find how the equilibrium price, P*, changes with respect to the demand parameter 'a', one can use implicit differentiation. The resulting expression for the partial derivative ∂P*/∂a is given by the formula: ∂P*/∂a = (∂D/∂a) / (dS/dP - ∂D/∂P).

A competitive market is described by the demand function Qd = a - bP and the supply function Qs = c + dP, where all parameters are positive. The equilibrium price, P*, is implicitly defined by the condition where quantity demanded equals quantity supplied. To determine how the equilibrium price responds to a shift in the supply curve, one must find the partial derivative ∂P*/∂c. Arrange the following steps in the correct logical order to perform this calculation.

Evaluating Methods for Comparative Statics

Justification of Implicit Differentiation in Comparative Statics

In a market model with demand Q_d = D(P, a) and supply Q_s = S(P, c), the equilibrium condition is D(P*, a) = S(P*, c). Using implicit differentiation to find how the equilibrium price (P*) changes with respect to the supply parameter 'c', we arrive at the expression:

∂P*/∂c = (∂S/∂c) / (∂D/∂P - ∂S/∂P)

Match each mathematical term from this expression with its correct economic interpretation.

Consider a market where the demand function Q_d = D(P, a) depends on price P and a parameter 'a', and the supply function Q_s = S(P) depends only on price. The market equilibrium is defined by the condition D(P*, a) = S(P*), where P* is the equilibrium price that implicitly depends on 'a'. To determine how a change in 'a' affects the equilibrium price, we differentiate the entire equilibrium equation with respect to 'a'. Applying the appropriate differentiation rule to the left side of the equation, D(P*, a), yields the expression: (∂D/∂P) * (∂P*/∂a) + ______. What is the missing term?

A competitive market is described by the demand function Qd = a - bP and the supply function Qs = c + dP, where all parameters (a, b, c, d) are positive. The equilibrium price (P*) and quantity (Q*) are implicitly defined by the condition where quantity demanded equals quantity supplied. Match each partial derivative, which represents a specific comparative static analysis, with its correct mathematical expression.

Critique of a Comparative Statics Analysis

In a competitive market model, demand is given by Qd = a - bP and supply is given by Qs = c + dP. All parameters (a, b, c, d) are positive constants. The equilibrium price (P*) and quantity (Q*) are implicitly defined by the condition where quantity demanded equals quantity supplied. Using the appropriate differentiation technique, the partial derivative of the equilibrium quantity with respect to the supply intercept 'c' (∂Q*/∂c) is ____.

In a market with a demand function Q_d = D(P, a) and a supply function Q_s = S(P), the equilibrium quantity Q* is determined by the equilibrium price P*, which itself is a function of the parameter 'a'. To find how the equilibrium quantity changes in response to a change in 'a' (i.e., to find ∂Q*/∂a), one can use the relationship Q* = S(P*(a)). Which of the following correctly represents the application of the chain rule to find ∂Q*/∂a?

A student is analyzing a competitive market with the demand function Qd = a - bP and the supply function Qs = c + dP, where all parameters are positive. To find how the equilibrium price (P*) changes in response to a shift in demand, they need to calculate the partial derivative ∂P*/∂a.

They set up the equilibrium condition: a - bP* = c + dP*

Then, they differentiate both sides with respect to 'a' and arrive at the following step: d/da(a) - d/da(bP*) = d/da(c) + d/da(dP*) 1 - 0 = 0 + 0

This leads to the incorrect conclusion that 1 = 0. What is the fundamental conceptual error in the student's application of differentiation?

Comparative Statics in a Non-Linear Market Model

In a competitive market model where equilibrium price (P*) is implicitly defined by the intersection of a downward-sloping demand curve and an upward-sloping supply curve, the partial derivative of the equilibrium price with respect to the demand curve's horizontal intercept (∂P*/∂a) is negative.