A research team has proposed a simple linear model to predict the demand (in thousands of units) for a new product based on its price (in dollars). The initial model is given by: \(D = a – bP\), where \(D\) is demand and \(P\) is price. However, preliminary data indicates that the relationship between price and demand exhibits curvature that is not captured by a linear model.
(a) Propose a refinement to the model by incorporating an additional term that accounts for the observed curvature.
(b) Justify how the inclusion of this term improves the accuracy and economic relevance of the demand prediction, considering aspects like market saturation or accelerated changes in demand.
Question
Refine the initial linear demand model \(D = a – bP\) by including an additional term that accounts for the curvature observed in the data. Provide the revised model and briefly explain the modification mathematically.
Hint
Hint
Hint
Consider adding a quadratic term, such as \(cP^2\), to capture non-linear behavior.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
A refined model is \(D = a – bP + cP^2\), where the term \(cP^2\) accounts for the curvature in the demand relationship.
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Answer Explanation
Answer Explanation
Question
Justify the improvements made in your refined model with reference to potential economic interpretations and data behavior.
Hint
Hint
Hint
Discuss how the quadratic term better captures phenomena such as market saturation or accelerated changes in demand.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The inclusion of the quadratic term \(cP^2\) allows the model to account for non-linear effects in demand, such as market saturation or variations in consumer behavior that lead to either diminishing or accelerating changes as price varies. This results in a more accurate and reliable prediction of demand.
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Mark Scheme
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Answer Explanation
Answer Explanation
