Sign Analysis of the Price Change Resulting from a Demand Shock (∂P*/∂a)
The direction of the change in equilibrium price following a demand shock can be determined by analyzing the sign of the partial derivative ∂P*/∂a. The denominator of the fraction representing this derivative, (∂S/∂P) - (∂D/∂P), is positive, as the supply curve's slope (∂S/∂P) is positive and the demand curve's slope (∂D/∂P) is negative. Given that the numerator is also positive by the definition of the demand function, the entire fraction is positive. Consequently, ∂P*/∂a > 0, which mathematically confirms that a positive demand shock (an increase in the parameter 'a') results in an increased equilibrium price.
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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?
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Sign Analysis of the Equilibrium Quantity Change from a Demand Shock (∂Q*/∂a)
In a competitive market model, the effect of a positive demand shock (represented by an increase in a parameter 'a') on the equilibrium price (P*) is given by the expression:
∂P*/∂a = (∂D/∂a) / [(∂S/∂P) - (∂D/∂P)]
where D is quantity demanded and S is quantity supplied. Analyze this expression under the following unusual market conditions:
- The demand curve is downward-sloping (∂D/∂P < 0).
- The supply curve is also downward-sloping (∂S/∂P < 0).
- The demand shock parameter 'a' has a positive effect on quantity demanded (∂D/∂a > 0).
Based on these conditions, what can be definitively concluded about the sign of ∂P*/∂a?
Price Effect of a Demand Shock
Evaluating an Analyst's Claim on Price Effects
Consider the following statement about a competitive market model where P* is the equilibrium price and 'a' is a parameter that positively shifts the demand curve: 'A positive shock to demand (an increase in 'a') causes the equilibrium price to rise. This occurs because the expression for the price change, ∂P*/∂a, has a positive numerator, and its denominator, (∂S/∂P) - (∂D/∂P), is also positive since both the slope of the supply curve (∂S/∂P) and the slope of the demand curve (∂D/∂P) are positive.' Is this entire statement, including its reasoning, true or false?
In a standard competitive market model, the change in equilibrium price (P*) resulting from a shift in the demand curve (due to a parameter 'a') is given by the expression:
∂P*/∂a = (∂D/∂a) / [(∂S/∂P) - (∂D/∂P)]
Match each mathematical component from the expression with its correct economic interpretation and sign.
Deriving the Price Impact of a Demand Shock
In the standard analysis of how a demand shock (represented by a parameter 'a') affects equilibrium price (P*), the denominator of the expression for ∂P*/∂a is (∂S/∂P) - (∂D/∂P). Given that the supply curve is upward-sloping and the demand curve is downward-sloping, the sign of this denominator is definitively ______.
To mathematically determine the direction of the change in equilibrium price (P*) following a positive shock to demand (represented by a parameter 'a'), one must analyze the sign of the expression for ∂P*/∂a. Arrange the following logical steps into the correct sequence used to prove that the equilibrium price increases in a standard competitive market.
In a standard competitive market model, a positive shock to demand causes the equilibrium price to increase. The magnitude of this price increase is determined by the expression ∂P*/∂a = (∂D/∂a) / [(∂S/∂P) - (∂D/∂P)], where 'a' is the demand shock parameter, P is price, D is quantity demanded, and S is quantity supplied. Holding the size of the initial demand shock (the numerator) constant, the resulting price increase will be largest under which of the following conditions?
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