Explanation Given: The price-demand equation is D ( x ) = 25 − 0.004 x 2 and the price-supply equation is S ( x ) = 5 + 0.004 x 2 . Definition used: Consumer’s Surplus : “If ( x ¯ , p ¯ ) is a point on the graph of the price-demand equation p = D ( x ) for a particular product, then the consumer’s surplus CS at a price level of p ¯ is, C S = ∫ 0 x ¯ [ D ( x ) − p ¯ ] d x . Which is the area between p = p ¯ and p = D ( x ) from x = 0 to x = x ¯ . The consumer’s surplus represents the total savings to consumers who are willing to pay at a more than p ¯ but are still able to buy the product for p ¯ ”. Producer’s Surplus : “If ( x ¯ , p ¯ ) is a point on the graph of the price-supply equation p = S ( x ) , then the producer’s surplus PS at a price level of p ¯ is, P S = ∫ 0 x ¯ [ p ¯ − S ( x ) ] d x . Which is the area between p = p ¯ and p = S ( x ) from x = 0 to x = x ¯ . The producer’s surplus represents the total gain to producers who are willing to supply at a lower price that p ¯ but are still able to supply units at p ¯ ”. Calculation: Consider the price-demand equation D ( x ) = 25 − 0.004 x 2 and the price-supply equation S ( x ) = 5 + 0.004 x 2 . For the equilibrium price p ¯ , the price-demand equation and the price-supply equation must be equal. That is, D ( x ) = S ( x ) 25 − 0.004 x 2 = 5 + 0.004 x 2 20 − 0.008 x 2 = 0 x 2 = 20 0.008 On further simplification, x = 2500 x = 50 2 = 50 Therefore, the demand at the equilibrium price is x ¯ = 50 . The equilibrium price is p ¯ = D ( x ¯ ) . Substitute x = 50 in D ( x ) = 25 − 0.004 x 2 to obtain the equilibrium price, p ¯ ( x ) = 25 − 0.004 ( 50 ) 2 = 25 − 0.004 ( 2500 ) = 25 − 10 = 15 Thus, the equilibrium price is p ¯ = 15 . Use the above mentioned definition to obtain the consumer’s surplus at the equilibrium price

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ISBN: 9781323174869

Author: Barnett

Publisher: PEARSON EDUCATION (COLLEGE)

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Chapter 13.2, Problem 78E

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To find: The consumer’s surplus and the producer’s surplus and sketch the graph.

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Chapter 13.2, Problem 77E

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