Rearranging the Differentiated Equilibrium Equation to Isolate the Partial Derivative
After differentiating the market equilibrium equation with respect to a parameter, such as the demand intercept 'a', the subsequent algebraic step involves rearranging the resulting equation. This manipulation serves to isolate the partial derivative of the equilibrium price with respect to that parameter (e.g., ∂P*/∂a), expressing it in terms of the other partial derivatives within the model.
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CORE Econ
Introduction to Microeconomics Course
Ch.8 Supply and demand: Markets with many buyers and sellers - The Economy 2.0 Microeconomics @ CORE Econ
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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 - bPand the quantity supplied isQs = c + dP. The parametersa, b, c, dare all positive constants. In equilibrium, the priceP*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 parametera, remembering that the equilibrium priceP*is itself a function ofa?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,Dis quantity demanded,Sis quantity supplied,P*is the equilibrium price, andais a parameter affecting demand. To find how the equilibrium price changes when the parameterachanges (i.e., to find∂P*/∂a), we must differentiate the entire equilibrium equation with respect toa, remembering thatP*is a function ofa. 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, whereP*is the equilibrium price andais a parameter. To find the effect of a change inaonP*, the entire equation is differentiated with respect toa, remembering thatP*is a function ofa. 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 parameteraon the equilibrium condition, is ____.The Rationale for Differentiating Equilibrium Price
Interpreting Components of a Comparative Statics Derivative
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Sign Analysis of the Price Change Resulting from a Demand Shock (∂P*/∂a)
In a market model, after differentiating the equilibrium condition with respect to a demand parameter 'a', you arrive at the following equation:
(∂D/∂P) * (∂P*/∂a) + (∂D/∂a) = (∂S/∂P) * (∂P*/∂a)
Where P* is the equilibrium price, D is the demand function, and S is the supply function. Which of the following correctly isolates the partial derivative (∂P*/∂a) to show how the equilibrium price changes with respect to the parameter 'a'?
In a market analysis, after differentiating the equilibrium condition with respect to a demand parameter 'a', the following equation is obtained:
(∂D/∂P) * (∂P*/∂a) + (∂D/∂a) = (∂S/∂P) * (∂P*/∂a)
Arrange the subsequent algebraic steps into the correct logical sequence to isolate the term ∂P*/∂a, which represents the change in equilibrium price with respect to the parameter.
Isolating a Variable of Interest
Analyzing a Flawed Algebraic Derivation
Consider a standard market model where the equilibrium condition has been differentiated with respect to a demand parameter 'a', resulting in the equation:
(∂D/∂P) * (∂P*/∂a) + (∂D/∂a) = (∂S/∂P) * (∂P*/∂a)where P* is the equilibrium price, D is the demand function, and S is the supply function.
True or False: The correct algebraic rearrangement to isolate the term representing the change in equilibrium price (∂P*/∂a) is:
∂P*/∂a = (∂D/∂a) / [ (∂D/∂P) - (∂S/∂P) ]In a standard market model, the process of finding how a change in a demand parameter 'a' affects the equilibrium price (P*) begins with differentiating the equilibrium condition. This results in the following equation:
(∂D/∂P) * (∂P*/∂a) + (∂D/∂a) = (∂S/∂P) * (∂P*/∂a)The next goal is to algebraically rearrange this equation to solve for
∂P*/∂a. Match each equation representing a step in this rearrangement with the algebraic principle that justifies it.In a market analysis, the equation
(∂D/∂P) * (∂P*/∂a) + (∂D/∂a) = (∂S/∂P) * (∂P*/∂a)is derived to understand how a parameteraaffects the equilibrium priceP*. The next algebraic step is to group terms and factor out∂P*/∂a. Complete the following rearranged version of the equation by filling in the blank:(∂D/∂a) = [______] * (∂P*/∂a)Rationale for Isolating the Equilibrium Price Derivative
In a market model, an economist is analyzing how a change in a supply-side parameter, 't' (such as a per-unit tax), affects the equilibrium price, P*. After differentiating the market equilibrium condition with respect to 't', the following equation is derived:
(∂D/∂P) * (∂P*/∂t) = (∂S/∂P) * (∂P*/∂t) + (∂S/∂t)
Where D is the demand function and S is the supply function. Which of the following correctly isolates the partial derivative (∂P*/∂t) by rearranging the equation?
Evaluating Algebraic Manipulations in Comparative Statics