Deriving the Price Impact of a Demand Shock
Starting from the market equilibrium condition S(P) = D(P, a), where 'a' is a parameter that positively shifts demand (meaning an increase in 'a' increases the quantity demanded at any given price), mathematically derive the expression for the change in equilibrium price with respect to 'a' (∂P*/∂a). Then, explain step-by-step why this expression must be positive under standard market assumptions (i.e., an upward-sloping supply curve and a downward-sloping demand curve).
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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?
Analyzing Price Changes in the Market for Electric Scooters