1.
Mr. Smith has the following demand equation for a certain product, Q=30-2P
a)
At a price of $7, what is the point elasticity?
b)
Between price of $5 and $6, what is the arc elasticity?
c)
If the market is made up of 100 individuals with demand curves identical to Mr. Smith's, what
are the point and arc elasticities for the conditions specified in parts a and b?
1.
Mr. Smith has the following demand equation for a certain product: Q 30 2 P . a) At a price price of $7, what is the the point point elast elasticit icity? y?
dQ P dP Q
Slope of the demand curve:
dQ
2
dP At P $7 , Q 30 2(7) 16 7 2 0.875 16 At this this price, price, demand demand is price price inelasti inelasticc becau because se 0
1 . For a 1 percent increase in price,
quantity demanded decreases by 0.875 percent, all else equal. b) Between price of $5 and $6, what is the arc elasticity? At P $5 , Q 30 2(5) 20 At P $6 , Q 30 2(6) 18
Q Q1 P 1 P 2 18 20 5 6 arc 2 Q Q 6 5 20 18 0.579 P P 2 1 1 2 c) If the market is made up of 100 individuals with demand curves identical to Mr. Smith's, what are the point and arc elasticities for the conditions specified in parts a and b? Price elasticity of demand will be the same. The market market deman demand d equat equation ion is: Q 3000 200 P dQ 200 Slope of the demand curve: dP At P $7 , Q 3000 200(7) 1600 7 200 0.875 1600
2. Suppose the following demand function has been estimated: Qx = 60- 5Px+ 2.4 Py- 4Pz+ 10Y where Qx = is the quantity of rain coats demanded during a one-year period measured in thousands of units, Px = the price of a rain coat, Py= price of a jacket, Pz represents the price of an umbrella, and Y is the average annual income of prospective buyers, measured in thousands of dollars. Currently, Px = $60, Py= $50, Pz= $20, and Y = $40. a) Find the current demand curve equation, Use the information above to calculate the values for intercept and slope. Find the Qdx. b) Find the Inverse Demand and solve for TR and MR and plot a curve. Find the price and quantity at which Total Revenue Maximization occurs. c) Calculate Point Elasticity for the Demand Curve above and Calculate Arc Elasticity of Demand above for two price levels P1 = 60 and P2 = 58. And explain the relationship of arc elasticity of demand with TR and MR, Plot a graph to explain the r/s. d) Calculate Income elasticity of demand e)
Calculate Cross price elasticity of demand.
The Demand Function and the Demand Curve Suppose the following demand function has been estimated: Q X = 60- 5P X + 2.4 P Y - 4PZ + 10I where Q X = is the quantity of rain coats demanded during a one-year period measured in thousands of units, PX = the price of a rain coat, P Y= price of a jacket, P Z represents the price of an umbrella, and I is the average annual income of prospective buyers, measured in thousands of dollars. Currently, PX = $60, P Y= $50, P Z = $20, and I = $40. The current demand curve equation, given the existing values of non-price variables, is calculated using the following general formula: Q x = A + 1Px where A represents the sum of all the terms on the right-hand side of the demand equation except the own-price term - 5 P X. Use the information above to calculate the values for A = 500 and 1 = -5 so the demand curve equation Q x = 500 - 5 Px Now we can calculate the current value for the actual quantity demanded:
Q X = 500 - 5 (60) = 200
2. Total and Marginal Revenue and Objectives of the Firm Find the Inverse Demand (solving for the inverse demand gives us an expression of Px in terms of Qx) of the Demand curve above, TR and MR curve: Qx = 500 - 5 Px Px = 100 - .2 Qx TR = 100 Qx - .2 Q x 2 MR = 100 - .4 Qx a. Total Revenue Maximization MR = 0 100 - .4 Qx = 0 Qx = 250 and Px = 100 - .2 (250) = $50 b. Total Profit Maximization Assume MC = $10 and it is constant, where MC =
TC/Q
To Maximize Total Profit the firm should produce where 100 - .4 Qx = 10 Qx = 225 and Px = 100 - .2 (225) = $55
MR = MC
1 60
55
< 1
50
MC MR
D
200 225 250
500 Q
T R
0
250
500 Q
3. Price Elasticity of Demand Point Elasticity:
=dQ/dP x P/Q
Arc Elasticity:
= (Q1 - Q2)/(P1 - P2) x (P1 + P2)/(Q1 + Q2)
Relationship between Elasticity, TR and MR TR increases
TR decreases
> 1
Price decrease
= 1
(TR constant for small price changes either way)
< 1
Price increase
Price increase
Price decrease
Calculate Point Elasticity for the Demand Curve above:
= -5 (60/200) = 1.5
At this price the demand curve is relatively elastic == for a one percent increase in price, quantity demanded decreases by 1.4 percent, all else constant. Calculate Arc Elasticity of Demand above for two price levels P1 = 60 and P2 = 58. At P2 = 58, Qd is 210 units
=
(10/2) * (60 + 58)/(200 + 210) = 1.44
4. Income elasticity Point l = (Q d/I) (I/Q d) = 10 * 40/200 = 2 Raincoats are a normal good, for a 1 percent increase in income, quantity demanded of raincoats increases by 2 percent, all else equal.
5. Cross price elasticity Point xy = (Q x /Py) (Py/Q x) = 2.4 * 50/200 = 0.6 Raincoats and jackets are substitutes, for a one percent increase in the price of jackets, quantity demanded of raincoats increases by 0.6 percent, all else equal. Point xz= (Q x /Pz) (Pz/Q x) = - 4 * 20/200 = -.8 Raincoats and umbrellas are complements, for a one percent increase in the price of an umbrella, quantity demanded of raincoats decreases by 0.8 percent, all else equal.
