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GobbetsPacket_2_Problem_Set

# GobbetsPacket_2_Problem_Set - Gobbet 2 Consumer...

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Gobbet # 2: Consumer Demand (These problems are adapted from Callan and Thomas: Environmental Economics and Management , 2004) Consider the following demand schedule for bottles of water: PRICE (P) Quantity Demanded by Consumers (bottles/month) \$0.50 1,100 1.00 1,050 1.50 1,000 2.00 950 2.50 900 3.00 850 3.50 800 4.00 750 4.50 700 5.00 650 Plot the demand curve in product space (that is with prices along the vertical axis and quantity along the horizontal axis). Derive the linear demand and inverse demand equations from the table. Solution: To derive the demand and inverse demand equations you can use the point-slope formula. The demand equation describes quantity as a function of price . The appropriate equation will have the form: 11.50 1,150 Demand Quantity Price

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Q d = m P + b where m is the slope, and b is the intercept. (The subscript d on Q indicates that we are looking at demand) The inverse demand equation describes price as a function of quantity . At various points in this course we will also refer to the inverse demand function as the marginal benefit function. This is because the “price” indicates the maximum amount of money that an individual would pay to obtain an extra unit of the commodity, or equivalently, the benefit to the consumer of the extra unit of the commodity. The appropriate equation will have the form: P = m Q d + b This is the form that is typically graphed, as it makes intuitive sense. At high prices, consumers demand less quantity, while at low prices, consumers demand more quantity. You can derive either equation first, but it may make more sense to derive the inverse demand equation first, since this is what we graph. Recall that the point-slope formula is ( Y 2 – Y 1 ) / (X 2 – X 1 ) = m , where m is the slope So given any two (X,Y) points, you can find the slope of a line. For inverse demand, P corresponds with Y , and Q d corresponds with X , so the formula can be written as: ( P 2 – P 1 ) / (Q d 2 – Q d 1 ) = m (this flips if we are finding the demand equation) The points we use need to be in (Q d ,P) form. Using the first two rows of the demand schedule we can take these points: (1,100, 0.50) and (1,050, 1.00) (remember that this technique will work with any two points) We can calculate the slope as follows: (1.00 – 0.50) / (1,050 – 1,100) = (0.50 / - 50) = - 0.01 = m Plugging m into the inverse demand equation, we have: P = -0.01 Q d + b Now we need to find b . This is done by selecting any point (Q,P) , substituting it into the equation, and solving. Using the point (1,100, 0.50), we have: 0.50 = (- 0.01 * 1,100) + b 0.50 = (-11) + b
11.50 = b We can now write the inverse demand equation; P = - 0.01 Q d + 11.50 You may want to check that this equation corresponds to other points on the demand schedule. For example, if Q d =800, does P=\$3.50 using this inverse demand equation? As a matter of practice, it is always good to cross-check your results in this manner. Now that we have the inverse demand equation, we can obtain the demand equation by simply rearranging terms to fit the form: Q d = m P + b Starting with; P = - 0.01 Q d + 11.50 - 11.50 - 11.50 (subtract 11.50 from both sides) P – 11.50 = - 0.01 Q d ---------- ---------- - 0.01 - 0.01 (divide both sides by - 0.01) ((1 / (-0.01)) * P) – (11.50 / - 0.01) = Q d - 100 P + 1,150 = Q d (simplify) Q d = - 100 P + 1,150 (rewrite) The last line is the demand equation.

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