Economics 101

Economics 101

Summer 2013

Answers to Homework #3

Due Tuesday, June 11, 2013

Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework (legibly). Make sure you write your name as it appears on your ID so that you can receive the correct grade. Late homework will not be accepted so make plans ahead of time. Please show your work. Good luck!

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1. This problem consists of two separate problems using the price elasticity of demand concept.

1. Suppose that you know that the market demand curve for a product is given by the equation P = 100 – 2Q. Furthermore you know that initially 40 units are demanded in this market when it is in equilibrium. Then, some event causes the equilibrium to change so that only 35 units are demanded in this market. From this information you are asked to calculate the price elasticity of demand using the arc elasticity concept. Finally you are asked to identify whether demand is elastic, unit elastic, or inelastic when quantity changes from 40 units to 35 units.
2. Suppose you know that the price elasticity of demand for good X has a value of 2. Suppose that the price in the market is initially $10 and the quantity demanded is 100 units. If price in this market decreases by 10%, what will be the percentage change in the quantity demanded given the above information?

Answer:

1. To answer this question you will want to find the price associated with the quantity demanded of 40 units: P = 100 – 2Q = 100 – 2(40) = $20 per unit. You will also want to find the price associated with the quantity demanded of 35 units: P = 100 – 2Q = 100 – 2(35) = $30.  Now, we have a (Q1, P1) and a (Q2, P2) that we can use in our arc elasticity formula for price elasticity of demand.

Price elasticity of demand = │[(Q2 – Q1)/Q2 + Q1)]/[(P2 – P1)/(P2 + P1)] │

Price elasticity of demand = │[(35 – 40)/(35 + 40)]/[(30 – 20)/(30 + 20)]

Price elasticity of demand = 1/3 and since this value is less than one we can conclude that the demand curve is inelastic between

1. From the information we know that the price elasticity of demand =2; we also know that the price elasticity of demand = the absolute value of [(% change in the quantity demanded of good X)/(% change in the price of good X)]. Thus, 2 = the absolute value of [(the % change in the quantity demanded of good X)/-10%]. Or, the % change in the quantity demanded of good X is 20%. The quantity demanded of good X will increase by 20% (the quantity demanded will now be 120 units) since the price of the good is inversely related to the quantity demanded of the good.

1. Suppose the market demand and supply of widgets is given by the following equations:

Market Demand for Widgets: P = 100- Q

Market Supply of Widgets: P = 3Q + 20

where P is the price per unit and Q is the quantity demanded.

1. What is the equilibrium price and equilibrium quantity of widgets?
2. Describe what happened to the supply curve due to this change in production costs. What is the equation for the new supply curve?

Suppose that production costs increase in the market for widgets such that at every quantity the cost has now increased by $20.

1. Given the change in production costs described above, calculate the new equilibrium price and equilibrium quantity in the market for widgets.
2. Intuitively what do you think happened to total expenditure in this market given the increase in production costs? Explain your answer.
3. Calculate total expenditure in the market for widgets initially and total expenditure in the market for widgets after the increase in production costs.
4. Does your answer in (e) support or confirm your answer in (d)?
5. Calculate the value of the price elasticity of demand between these two points of equilibrium using the arc elasticity of demand formula.

Answer:

1. 100 – Q = 3Q + 20

4Q = 80

Qe = 20

Pe = 100 – 20 = $80 per unit

1. The new supply curve will shift to the left but be parallel to the initial supply curve: the two curves will have the same slope. Thus, the new supply curve will be P = b’ + 3Q. We also know that the supply curve has shifted vertically up by 20 units since costs have risen at each quantity by $20. Thus, the new y-intercept of the supply curve will be equal to the initial y-intercept plus 20: the new supply curve will be P = 40 + 3Q.
2. 3Q = 40 = 100 – Q

4Q = 60

Qe’ = 15

Pe’ = 100 – 15 = $85 per unit

1. Total expenditure should decrease since the price is increasing in the elastic region of the demand curve. We know we are in the elastic region of the demand curve for any price greater than $50 since (50, $50) is the midpoint of the demand curve.
2. Total expenditure initially = ($80 per unit)(20 units) = $1600

Total expenditure after change in production costs = ($85 per unit)(15 units) = $1275

1. Yes, provided in (d) you predicted that total expenditure would fall.
2. Price Elasticity of Demand = │{[(Q2 – Q1)/(Q2 + Q1)]/[(P2 – P1)/(P2 + P1)]}│

Price Elasticity of Demand = {[(15 – 20)/35]/[(85 – 80)/165]} = [(1/7)/(5/165)] ≈ 4.7

1. Suppose the market for doughnuts has five consumers and each consumer’s demand for doughnuts can be described by the equation Pd = 5 – Qd where Pd is the price per doughnut and Qd is the quantity of doughnuts demanded.

1. What is the market demand curve for doughnuts?

Suppose the market demand and market supply curves for coffee are given by the following equations where Pc is the price per cup of coffee and Qc is the quantity of cups of coffee:

Market Demand for Coffee: Pc = 5 – (1/20)Qc

Market Supply of Coffe: Pc = 1 + (1/60)Qc

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