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Economics 101

Summer 2012

Answers to Homework #3

Due 6/12/12

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1. Suppose the market for corn in a country is described by the following demand and supply equations:

Demand: P = 100 – (1/2)Q

Supply: P = 10 + (13/10)Q

Use this information to answer the following set of questions.

a. What is the equilibrium price and quantity in this market? For your answers you may round to the nearest whole number.

b. What is the value of total revenue for farmers in this market?

Suppose the government institutes a price support in this market of $80 per unit of corn.

c. Given the price support, how many units of corn will consumers buy?

d. Given the price support, how many units of corn will the government buy?

e. Given the price support, what is the cost to the government of this program if storage costs are $10 per unit of corn stored?

f. Draw a diagram illustrating this price support program. Make sure you label your diagram clearly and completely.

Suppose the government cancels the price support program and, in its place, institutes a price guarantee program where the guaranteed price for corn is $80 per unit of corn.

g. Given the price guarantee, how many units of corn will consumers buy?

h. Given the price guarantee, what is the price per unit of corn that consumers will pay? What is the total expenditure on corn made by consumers?

i. Given the price guarantee, how many units of corn will the government purchase?

j. Given the price guarantee, what is the cost to the government of this program if storage costs are $10 per unit of corn stored?

k. Draw a diagram illustrating this price guarantee program. Make sure you label your diagram clearly and completely.

Answer:

a. To find the equilibrium use the demand and supply curves: 100 – (1/2)Q = 10 + (13/10)Q or Q = 50 units of corn. When Q is equal to 50 units of corn, the price of each unit of corn is $75.

b. Total revenue for farmers is equal to price times quantity. Or, total revenue is equal to ($75 per unit of corn)(50 units of corn) = $3750.

c. When the price of corn is $80 per unit of corn, consumers will purchase 40 units of corn. Use the demand curve to find this quantity: P = 100 – (1/2)(Q) or 80 = 100 – (1/2)Q or Q = 40 units of corn.

d. When the price of corn is $80 per unit of corn, suppliers will supply 54 units of corn. Use the supply curve to find this quantity: P = 10 – (13/10)Q or 80 = 10 – (13/10)Q or Q = 54 units of corn (this is rounded). Since consumers consume 40 units, this means that the government will purchase the surplus of 14 units.

e. The cost to the government is equal to the (price per unit of corn)(units of corn purchased by the government) + (units of corn purchased by the government)(storage costs per unit of corn). Or, the cost to the government is equal to ($80 per unit of corn)(14 units) + (14 units)($10 per unit of corn) = $1260.

f.


g. With a price guarantee of $80 per unit of corn consumers will purchase 54 units of corn. To see this recall that if the guaranteed price of corn is $80 per unit then suppliers will be willing to produce 54 units. With the price guarantee program the farmers will then sell all of this corn for whatever price they must in order to sell 54 units. Thus, consumers will buy the entire 54 units of corn.

h. With the price guarantee suppliers produce 54 units of corn. Consumers are only willing to pay $73 per unit of corn for this amount. To see this use the demand curve: P = 100 – (1/2)Q where Q = 54. Thus, P = 100 – (1/2)(54) = $73 per unit of corn. The total expenditure on corn made by consumers is ($73 per unit of corn)(54 units of corn) = $3942.

i. With a price guarantee program the government does not buy any of the good. There are no storage costs since the government has not purchased the good.

j. The cost to the government is the difference in the guaranteed price of $80 per unit of corn minus the price the corn actually sells for ($73 per unit of corn) times the number of units of corn sold. Thus, the cost to the government is ($80 per unit of corn - $73 per unit of corn)(54 units of corn) = $378. There are no storage costs.

k.



2. Suppose the market for candy bars can be described as follows:

  • When the price of candy bars is $1.00 per candy bar, 500 candy bars are demanded. When the price of candy bars increases by 10%, the quantity of candy bars demanded falls by 20%. The demand curve for candy bars is linear.
  • The supply curve for candy bars is linear and contains the points (Q, P) = (300, $.60) and (200, $.50).

a. What is the equation for the demand curve given the above information?

b. What is the equation for the supply curve given the above information?

c. What is the equilibrium price and quantity in the market for candy bars?

d. Suppose the government wants to institute an effective price ceiling in the market for candy bars. What must be true for this price ceiling to be effective?

e. Suppose the government wants to institute an effective price floor in the market for candy bars. What must be true for this price floor to be effective?


Answer:

a. We are given one point on the demand curve: (Q, P) = (500, $1.00). We are given enough information that we can easily find a second point. If the price of candy bars increases by 10% and the initial price was $1.00, then the new price would be $1.10. When the price of candy bars increases by 10%, the quantity decreases by 20%: 20% of 500 is 100. So, we know a second point on the linear demand curve: (Q, P) = (400, $1.10). Use these two points to find the slope of the demand curve: (change in y)/(change in x) = (change in price)/(change in quantity) = -.1/100 = -.001. Plug this slope into the demand curve to get P = b - .001Q. Then, use one of the points on the demand curve to solve for b, the y intercept. Thus, P = 1.50 - .001Q.

b. Use the two given points to find the slope of the supply curve: slope = .1/100 = .001. Plug this slope into the supply curve to get P = .001Q + b. Then use one of the given points to find the y-intercept. The supply curve is P = .001Q + .30.

c. Use the demand and supply curves you found in steps (a) and (b) to solve for the equilibrium 1.50 - .001Q = .001Q + .3 or 1.2 = .002Q or Q = (1.2)/(.002) = 600. When Q is equal to 600, then P = $.90.

d. For a price ceiling to be effective, the price ceiling must be set at a price that is lower than the equilibrium price. In this case the equilibrium price is $.90 per candy bar so that implies that an effective price ceiling is only possible when the price ceiling is less than $.90 per candy bar.

e. For a price floor to be effective, the price floor must be set at a price that is greater than the equilibrium price. In this case the equilibrium price is $.90 per candy bar so that implies that an effective price floor is only possible when the price floor is greater than $.90 per candy bar.

3. Suppose that the government of Zanzi decides that there is a need to reduce cigarette smoking in their country. The cigarette market in Zanzi can currently be described by the following demand and supply equations:

Demand for cigarettes: Q = 1125 – 12.5P

Supply of cigarettes: Q = 1100P – 1100

The government proposes implementing a quantity control of 500 units: this quantity control would limit the number of cigarettes that could be sold in Zanzi to exactly 500 units. The government has asked you to evaluate this program by answering the following series of questions.

a. Before implementing the quantity control, what is the equilibrium price and equilibrium quantity of cigarettes in Zanzi?
Answer:

Use the supply and demand curves to find the equilibrium price and quantity:

1125 – 12.5P = 1100P – 1100

2225 = 1112.5P or P = $2

Q = 1125 – 12.5(2) = 1100 cigarettes
b. Before implementing the quantity control, what is the value of consumer surplus in the market for cigarettes in Zanzi?
Answer:

To calculate the consumer surplus we first need to determine the y-intercept for the demand curve: so, use the demand curve and set Q = 0 to find this y-intercept. Thus, 0 = 1125 – 12.5P or P = 90. Consumer surplus is thus equal to (1/2)($90/cigarette - $2/cigarette)(1100 cigarettes) = $48,400.


c. Before implementing the quantity control, what is the value of producer surplus in the market for cigarettes in Zanzi?

Answer:

To calculate the producer surplus we first need to determine the y-intercept of the supply curve: so, use the supply curve and set Q = 0 to find this y-intercept. Thus, 0 = 1100P – 1100 or P = 1. Producer surplus is thus equal to (1/2)(($2 /cigarette - $1/cigarette)(1100 cigarettes) = $550.

d. Suppose the government implements the quantity control. What price must consumers pay in order to only demand 500 cigarettes in Zanzi?
Answer:

To find the price consumers must pay in order to demand only 500 cigarettes use the demand equation and substitute Q = 500 into that equation. Thus, 500 = 1125 – 12.5P or P = $50.


e. Suppose the government implements the quantity control. What price must producers receive in order to only supply 500 cigarettes in Zanzi? (Round your answer to the nearest cent.)
Answer:

To find the price producers must receive in order to supply only 500 cigarettes use the supply equation and substitute Q = 500 into that equation. Thus, 500 = 1100P – 1100 or P = approximately $1.45.


f. Suppose the government implements the quantity control. What price will the government sell the right to sell a unit of cigarettes for in Zanzi if the government sets the quantity control at 500 cigarettes?
Answer:

If producers must receive a price of $1.45 per cigarette and demanders must pay a price of $50 per cigarette in order for only 500 cigarettes to be consumed, this implies that the price for the right to sell a unit of cigarettes in Zanzi must be equal to $50 - $1.45 or $48.55.


g. Suppose the government implements the quantity control. What is the value of consumer surplus with this program? What is the value of producer surplus with this program? What is the government’s revenue from this program?

Answer:

Consumer surplus with this quantity control is equal to (1/2)($90/cigarette - $50/cigarette)(500 cigarettes) = $10,000.

Producer surplus with this quantity control is equal to (1/2)($1.45/cigarette - $1/cigarette)(500 cigarettes) = $112.50.

Government revenue from this quantity control program = ($50/cigarette - $1.45/cigarette)(500 cigarettes) = $24,275.

h. Suppose the government implements the quantity control. What is the deadweight loss due to this program?
Answer:

Deadweight loss from this quantity control program = (1/2)($50/cigarette - $1.45/cigarette)(1100 cigarettes – 500 cigarettes) = $14,565.


4. Coba is a small, closed economy. Coba’s domestic demand curve and domestic supply curve for coconuts is given by the following equations where Q is units of coconuts and P is the price per unit of coconuts:

Domestic Demand: P = 100 – (1/20)Q

Domestic Supply: P = (1/60)Q

a. Given the above information, what is the equilibrium price and quantity in the market for coconuts?


Answer:

100 – (1/20)Q = (1/60)Q

Q = 1500 units of coconuts

P = (1/60)(1500) = $25 per unit of coconuts


b. Calculate the value of consumer surplus, producer surplus, and total surplus in the market for coconuts in Coba if this market is closed to trade.
Answer:

CS no trade = (1/2)($100 per unit of coconuts - $25 per unit of coconuts)(1500 units of coconuts) = $56,250

PS no trade = (1/2)($25 per unit of coconuts - $0 per unit of coconuts)(1500 units of coconuts) = $18,750

TS no trade = $56,250 + $18,750 = $75,000


Suppose the world price per unit of coconuts is $10.
c. Coba opens its coconut market to trade. How many units of coconuts will be imported once this market is opened to trade?

Answer:

Use the domestic demand curve to determine the number of units of coconuts demanded in Coba at $10 per unit. Thus, 10 = 100 – (1/20)Q or Qdemanded = 1800 units of coconuts. Use the domestic supply curve to determine the number of units of coconuts supplied by domestic producers in Coba. Thus, 10 = (1/60)Q or Qsupplied = 600 units of coconuts. The difference between the quantity demanded domestically and the quantity supplied domestically is the number of units imported: number of units imported = 1800 – 600 = 1200 units of coconuts.

d. Given that Coba has opened its coconut market to trade, calculate the value of consumer surplus with trade, producer surplus with trade, and total surplus with trade for Coba in this market.
Answer:

CS with trade = (1/2)($100 per unit of coconuts - $10 per unit of coconuts)(1800 units of coconuts) = $81,000

PS with trade = (1/2)($10 per unit of coconuts - $0 per unit of coconuts)(600 units of coconuts) = $3,000

TS with trade = $84,000


e. Is trade in the coconut market beneficial to Coba? Explain your answer.
Answer:

Yes, when Coba opens its coconut market to trade the total surplus in this market increases from $75,000 to $84,000: trade is beneficial since it results in greater surplus.


f. In Coba, who favors opening the coconut market to trade and who opposes opening the coconut market to trade? Explain your answer.
Answer:

Domestic consumers favor opening this market to trade because with trade they get more coconuts at a lower price and this results in their consumer surplus rising from $56,250 to $81,000. Domestic consumers are better off when this market opens to trade.


Domestic producers oppose opening this market to trade because with trade they sell fewer domestically produced coconuts at a lower price and this results in their producer surplus decreasing from $18,750 to $3,000. Domestic producers are worse off when this market opens to trade.
Coba responds to domestic producer concerns by implementing a tariff that results in 600 units of coconuts being imported.
g. Given the imposition of this tariff, what is the new tariff price?

Answer:

We know that 600 units of coconuts are imported when Coba imposes this tariff. That is, the quantity supplied domestically plus 600 imported units of coconuts equals the quantity demanded domestically. Symbolically we can write this as

Qs + 600 = Qd

Solving both the supply and demand equations for Q we can substitute for Qs and Qd as follows.

P = 100 – (1/20)Qd

Qd = 2000 – 20P

P = (1/60)Qs

Qs = 60P


60P + 600 = 2000 – 20P

P = 1400/80 = $17.50 per unit of coconuts


h. Given the imposition of this tariff, what is the new consumer surplus with the tariff, producer surplus with the tariff, and license holder revenue from the tariff?
Answer:

CS with tariff = (1/2)($100 per unit of coconuts - $17.50 per unit of coconuts)(1650 units of coconuts) = $68,062.50

PS with tariff = (1/2)($17.50 per unit of coconuts - $0 per unit of coconuts)(1050 units of coconuts) = $9187.50

License Holder Revenue = ($17.50 per unit of coconuts - $10 per unit of coconuts)(600 units of coconuts) = $4,500


5. Consider the market for widgets in Westonia, a small closed economy whose domestic demand and supply curves for widgets are described by the following equations:

Domestic Demand: P = 10 – Q

Domestic Supply: P = 2 + (1/3)Q

where P is the price per unit of widgets and Q is units of widgets. You are also told that the world price of widgets is $3 per unit of widgets.

a. Suppose that Westonia opens the widget market to trade. Will Westonia import or export widgets? Fully explain your answer.
Answer:

To determine whether Westonia will be an importer or exporter we first need to determine the quantity demanded at the world price of $3 per unit of widgets. Using the demand equation we have 3 = 10 – Q or Qdomestic demand = 7 units of widgets. Using the supply equation we have 3 = 2 + (1/3)Q or Qdomestic supply = 3 units of widgets. At the world price of $3 the quantity demanded domestically exceeds the quantity supplied domestically: Westonia will import the difference or 4 units of widgets.

b. Suppose the government of Westonia imposes an import quota of 5 units of widgets. Describe the effect of this quota on the market for widgets in Westonia.

Answer:

An import quota of 5 units of widgets will have no effect on the market for widgets in Westonia since the quota amount set by the government is greater than the number of units of widgets that Westonia would choose to import when this market opens to trade. For an import quota to be effective in this example, the import quota would need to be set at an amount less than the amount of imports that Westonia would import at the world price of the good.


c. Suppose the import quota is set by the government of Westonia at 2 units of widgets. What will be the price of widgets in Westonia due to the imposition of this quota?
Answer:

Qs + 2 = Qd

3P – 6 + 2 = 10 – P

P = $3.50 per unit of widgets

d. Given the import quota described in part (c), what is the deadweight loss associated with this quota?
Answer:

DWL = (1/2)($3.50 per unit of widgets - $3 per unit of widgets)(4.5 units of widgets – 3 units of widgets) + (1/2)($3.50 per unit of widgets - $3 per unit of widgets)(7 units of widgets – 6.5 units of widgets)

DWL = $0.375 + $0.125 = $0.50
6. Answer the next set of questions about various kinds of elasticity.

a. You are given a demand curve, P = 10 – Q, and asked to calculate the arc elasticity of demand between the quantities of 1 unit and 2 units. Show the general equation you plan to use before putting in numbers. Also indicate if price elasticity of demand is elastic, inelastic, or unit elastic given the value you calculated.

Answer:

Price elasticity of demand =│(% change in Q demanded)/(% change in P)│= │[( Q2 – Q1)/(Q2 + Q1)]/ [(P2 – P1)/(P2 + P1)] |

When Q = 1, P = 9 and when Q = 2, P = 8. Use these two points and the above equation to calculate the price elasticity of demand using the arc formula.

Elasticity of Demand = (1/3)/(1/17) = 17/3

Demand is elastic since the value of the price elasticity of demand is larger than one.

b. You are given a demand curve, P = 10 – Q, and asked to calculate the arc elasticity of demand between the quantities of 9 units and 8 units. Show the general equation you plan to use before putting in the numbers. Also, indicate if demand is elastic, inelastic, or unit elastic given the value you calculated.
Answer:

Price elasticity of demand =│(% change in Q demanded)/(% change in P)│= │[( Q2 – Q1)/(Q2 + Q1)]/ [(P2 – P1)/(P2 + P1)] |

When Q = 9, P = 1 and when Q = 8, P = 2. Use these two points and the above equation to calculate the price elasticity of demand using the arc formula.

Elasticity of Demand = (1/17)/(1/3) = 3/17

Demand is inelastic since the value of the price elasticity of demand is less than one.
c. You are given a demand curve, P = 10 – Q, and asked to calculate the point elasticity of demand when price is equal to $7. Show the general equation you plan to use before putting in numbers. Also, indicate if demand is elastic, inelastic, or unit elastic given the value you calculated.
Answer:

Point elasticity of demand = (-1/slope)(P/Q)

When price is equal to $7, quantity demanded is equal to 3. So, point elasticity of demand = (1/1)(7/3) = 7/3. Demand is elastic since the value of the price elasticity of demand is greater than one.

d. Currently bus tickets in your town are $1 per bus ride. You know that the demand for bus rides in your town is linear and that at a price of $1 per bus ride 50,000 rides a year are demanded. You also know that for every $0.50 increase in the price of a bus ticket demand decreases by 10%. Your community’s transportation board wishes to increase bus revenue and is considering changing the price of a bus ticket by $0.25. Should the price increase to $1.25 or decrease to $0.75 if increased revenues is the sole goal of the transportation board? Provide a full, detailed explanation of how you determined your answer.

Answer:

First you need to find the demand curve from the given information. We know that the point (Q, P) = (50,000, $1.00) is a point on the demand curve. We know from the information that the point (Q, P) = (45,000, $1.50) is also on the demand curve. Use these two points to find the slope of the demand curve. Then, use one of the points to find the y-intercept of the demand curve. The demand curve is P = 6 - .0001Q.


Then, use this demand curve to decide whether prices should be raised or lowered to increase revenue from bus transportation. The quick answer to this question is to recognize that the y-intercept of the demand curve is 6 and this tells you that at a price of $1 per bus ride you are well below the midpoint of the demand curve (where total revenue is maximized). Since P = 1 is less than the price at the midpoint this tells us that the community is currently pricing in the inelastic region of the demand curve and will be able to enhance their revenue by raising the price per bus ride.
The long answer to this question uses the demand equation to calculate (Q, P) coordinates for prices $1.25 and $0.75. So, when P = $1.25, Q = 47,500. When P = $0.75, Q = 52,500. Now calculate total revenue for each of these prices to determine which price results in the greater total revenue. Total revenue when price is $1.25 is $59,375. Total revenue when price is $0.75 is $39,375. The transportation board will increase its revenue if it increases the price of a bus ride to $1.25.

7. Consider the market for soft drinks. You know that the demand curve in this market contains the points (Q, P) = (100,000, $1.00) and (50,000, $2.00). You also know that the demand curve is linear. The linear supply curve contains the points (Q, P) = (0, $0) and (50,000, $1). The government decides they want to decrease consumption of soft drinks from its current equilibrium level to 50,000 drinks. You are asked to analyze this situation and provide answers to the following questions.

a. How big an excise tax will the government need to implement in order to reduce consumption to 50,000drinks?

Answer:

When the quantity demanded is 50,000 then the price consumers must pay to demand this amount is $2. When the quantity supplied is 50,000 then the price producers must receive in order to supply this amount is $1. Therefore the excise tax must increase the price of a soft drink by $1 per unit in order to align the quantity demanded with the quantity supplied at 50,000 units. So, the excise tax will be $1 per unit of soft drinks.

b. Given the above scenario, calculate the tax revenue the government will collect when they implement an excise tax that reduces consumption to 50,000 drinks.
Answer:

Tax revenue = (Tax per unit)(Quantity with tax) = ($1 per unit)(50,000 units) = $50,000


c. What is the deadweight loss associated with this excise tax?
Answer:

DWL = (1/2)($2 per unit - $1 per unit)(75,000 – 50,000) = $12,500







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