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Chapter 2: Problem 1

Suppose the demand curve for a product is given by \(Q=300-2 P+4 I,\) where \(I\) is average income measured in thousands of dollars. The supply curve is \(Q=3 P-50\) a. If \(I=25,\) find the market-clearing price and quantity for the product. b. If \(I=50,\) find the market-clearing price and quantity for the product. c. Draw a graph to illustrate your answers.

Short Answer

Expert verified
a) For I = $25,000, the market-clearing price and quantity are $75 and 175 units, respectively.\n b) For I = $50,000, the market-clearing price and quantity are $130 and 340 units, respectively.

Step by step solution

01

Find the equilibrium for I=25

At market-clearing or equilibrium, the quantity supplied equals the quantity demanded. Hence, the two equations \(Q=300-2P+4I\) and \(Q= 3P-50\) should be equated. First, replace the I in the demand equation with 25 and solve for P. \[300-2P+4*25=3P-50, \] which simplifies to \[400 - 2P = 3P - 50.\] Solving this for P gives a market-clearing price of $75.
02

Obtain the Market-Clearing Quantity for I=25

Substitute this price back into either the supply or demand equation to find the quantity. \[ Q = 3*75 - 50 = 175. \] Therefore, when average income is $25,000, the market-clearing price is $75, and the market-clearing quantity is 175 units.
03

Find the equilibrium for I = 50

Now, substitute I = 50 into the demand equation and equate with the supply equation. \[300-2P+4*50=3P-50.\] Solving for P, it is found that the market-clearing price is $130.
04

Obtain the Market-Clearing Quantity for I=50

Then substitute this price back into either the supply or demand equation to find the quantity. \[ Q = 3*130 - 50 = 340.\] Therefore, when average income is $50,000, the market-clearing price is $130, and the market-clearing quantity is 340 units.
05

Explain the Graph

Draw the demand and supply curves on the graph with quantity on the x-axis and price on the y-axis. When I = 25, we have an equilibrium point at (175, 75), while I = 50 leads to a new equilibrium point at (340, 130). On increasing the income, the demand curve shifts to the right, leading to increased equilibrium price and quantity.

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Most popular questions from this chapter

Example 2.9 (page 76 ) analyzes the world oil market. Using the data given in that example: a. Show that the short-run demand and competitive supply curves are indeed given by \\[ \begin{array}{l} D=36.75-0.035 P \\ S_{C}=21.85+0.023 P \end{array} \\] b. Show that the long-run demand and competitive supply curves are indeed given by \\[ \begin{array}{l} D=45.5-0.210 P \\ S_{C}=16.1+0.138 P \end{array} \\] c. In Example 2.9 we examined the impact on price of a disruption of oil from Saudi Arabia. Suppose that instead of a decline in supply, OPEC production increases by 2 billion barrels per year (bb/yr) because the Saudis open large new oil fields. Calculate the effect of this increase in production on the price of oil in both the short run and the long run.

The rent control agency of New York City has found that aggregate demand is \(Q_{D}=160-8 P .\) Quantity is measured in tens of thousands of apartments. Price, the average monthly rental rate, is measured in hundreds of dollars. The agency also noted that the increase in \(Q\) at lower \(P\) results from more three-person families coming into the city from Long Island and demanding apartments. The city's board of realtors acknowledges that this is a good demand estimate and has shown that supply is \(Q_{s}=70+7 P\) a. If both the agency and the board are right about demand and supply, what is the free-market price? What is the change in city population if the agency sets a maximum average monthly rent of \(\$ 300\) and all those who cannot find an apartment leave the city? b. Suppose the agency bows to the wishes of the board and sets a rental of \(\$ 900\) per month on all apartments to allow landlords a "fair" rate of return. If 50 percent of any long-run increases in apartment offerings comes from new construction, how many apartments are constructed?

In Example 2.8 (page 74 ), we discussed the recent decline in world demand for copper, due in part to China's decreasing consumption. What would happen, however, if China's demand were increasing? a. Using the original elasticities of demand and supply (i.e., \(\left.E_{s}=1.5 \text { and } E_{D}=-0.5\right),\) calculate the effect of a 20 -percent increase in copper demand on the price of copper. b. Now calculate the effect of this increase in demand on the equilibrium quantity, \(Q^{*}\) c. As we discussed in Example 2.8 , the U.S. production of copper declined between 2000 and 2003 Calculate the effect on the equilibrium price and quantity of both a 20 -percent increase in copper demand (as you just did in part a) and of a 20 -percent decline in copper supply.

In Example 2.8 we examined the effect of a 20 -percent decline in copper demand on the price of copper, using the linear supply and demand curves developed in Section \(2.6 .\) Suppose the long-run price elasticity of copper demand were -0.75 instead of -0.5 a. Assuming, as before, that the equilibrium price and quantity are \(P^{*}=\$ 3\) per pound and \(Q^{*}=18 \mathrm{mil}\) lion metric tons per year, derive the linear demand curve consistent with the smaller elasticity. b. Using this demand curve, recalculate the effect of a 55-percent decline in copper demand on the price of copper.

In \(2010,\) Americans smoked 315 billion cigarettes, or 15.75 billion packs of cigarettes. The average retail price (including taxes) was about \(\$ 5.00\) per pack. Statistical studies have shown that the price elasticity of demand is \(-0.4,\) and the price elasticity of supply is 0.5 a. Using this information, derive linear demand and supply curves for the cigarette market. b. In \(1998,\) Americans smoked 23.5 billion packs of cigarettes, and the retail price was about \(\$ 2.00\) per pack. The decline in cigarette consumption from 1998 to 2010 was due in part to greater public awareness of the health hazards from smoking, but was also due in part to the increase in price. Suppose that the entire decline was due to the increase in price. What could you deduce from that about the price elasticity of demand?
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