A Certain Type of Rare Gem Serves as a Status $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$

question 22

Short Answer

A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
where $y =$ Demand (in thousands) and $x =$ Retail price per carat (dollars).
This model was fit to data collected for a sample of 12 rare gems.
If the experts are correct in their assumptions about the relationship between price and demand, which of the following should be true?
A) $\beta _ { 2 } > 0$
B) $\beta _ { 2 } < 0$
C) $\beta _ { 1 } > 0$
D) $\beta _ { 1 } < 0$

• A) \(\beta _ { 2 } > 0\)
• B) \(\beta _ { 2 } < 0\)
• C) \(\beta _ { 1 } > 0\)
• D) \(\beta _ { 1 } < 0\)

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Definitions:

Marginal Revenue

Income gained by selling an additional unit of a product or service.

Marginal Cost

The increased expenditure incurred from producing one more unit of a product or service.

Demand Curve

A visual diagram that illustrates how the quantity of a product demanded by buyers correlates with its price.

Marginal Revenue Curve

A graphical representation showing the additional income generated from the sale of one more unit of a good or service.