The following relations describe the supply and demand for posters.

Qd = 65,000 – 10,000 P
Qs = -35,000 + 15,000P

Where Q is the quantity and P is the price of a poster, in dollars.
a. Complete the following table.

Price Qs Qd Surplus or Shortage
$6.00
5.00
4.00
3.00
2.00
1.00

b. What is the equilibrium price?

Chapter 4 Page 108 Problem 2

The demand function for a cola-type soft drink in general is Q = 20-2P, where Q stands for quantity and P stands for price.

1. Calculate point elasticities at prices of 5 and 9. Is the demand curve elastic or inelastic at these points?
2. Calculate arc elasticity at the interval between P = 5 and P = 6.
3. At which price would a change in price and quantity result in approximately no change in total revenue? Why?

a. Is just a metter of plugging in numbers.
When P = $6, Qd (demand) = 5000 and Qs = 55,000, for a surplus of 50,000.
When P=$4, 4, Qd = 25,000 and Qs = 25,000
etc
b. Clearly the equilibrium price is $4, since that is where Qs=Qd

For problem 2, review the definitions of point and arc elasticites and compute the approprate values.

Revenue is
R = P*Q = 20P - 2P^2.
dR/dP = 0 when 20 - 4P = 0
P = $5 is the price at which small price changes do not affect revenue

The Equilibrium price is $4

a. Complete the following table:

Price Qs Qd Surplus or Shortage
$6.00 20,000 5,000 Surplus of 15,000
$5.00 10,000 15,000 Shortage of 5,000
$4.00 5,000 25,000 Shortage of 20,000
$3.00 0 35,000 Shortage of 35,000
$2.00 -5,000 45,000 Shortage of 50,000
$1.00 -10,000 55,000 Shortage of 65,000

b. The equilibrium price is $4. This is the price at which the quantity demanded (Qd) is equal to the quantity supplied (Qs).

For problem 2:

1. To calculate the point elasticities at prices of 5 and 9, we use the formula:

E = (% change in quantity demanded) / (% change in price)

At P = 5:
Q = 20 - 2(5) = 10
E = [(ΔQ / Q) / (ΔP / P)]
= [(10 - 0) / 10] / [(5 - 0) / 5]
= (10/10) / (5/5)
= 1 / 1
= 1

At P = 9:
Q = 20 - 2(9) = 2
E = [(ΔQ / Q) / (ΔP / P)]
= [(2 - 0) / 2] / [(9 - 0) / 9]
= (2/2) / (9/9)
= 1 / 1
= 1

The demand curve is unit elastic at these points, as the elasticity is equal to 1.

2. To calculate the arc elasticity at the interval between P = 5 and P = 6, we use the formula:

E = [(ΔQ / Q) / (ΔP / P)]
= [(Q2 - Q1) / ((Q1 + Q2)/2)] / [(P2 - P1) / ((P1 + P2)/2)]

At P = 5:
Q = 20 - 2(5) = 10

At P = 6:
Q = 20 - 2(6) = 8

E = [(8 - 10) / ((10 + 8)/2)] / [(6 - 5) / ((5 + 6)/2)]
= (-2 / 9) / (1 / 5.5)
= (-2 / 9) * (5.5 / 1)
= -1 * 5.5 / 9
= -5.5 / 9
≈ -0.61

The arc elasticity at this interval is approximately -0.61.

3. To find the price at which a change in price and quantity results in approximately no change in total revenue, we need to find the price that maximizes total revenue.

Total revenue (R) is given by: R = P * Q = 20P - 2P^2

To maximize R, we differentiate R with respect to P and set it equal to zero:

dR/dP = 20 - 4P = 0
4P = 20
P = 5

Therefore, the change in price that results in approximately no change in total revenue is at a price of $5.

a. To complete the table, we need to plug in different values of price (P) into the supply (Qs) and demand (Qd) functions to calculate the quantities and determine if there is a surplus or shortage.

Let's start with the first row of the table, where the price is $6.00:
Qs = -35,000 + 15,000P
Qs = -35,000 + 15,000 * 6
Qs = -35,000 + 90,000
Qs = 55,000

Qd = 65,000 - 10,000P
Qd = 65,000 - 10,000 * 6
Qd = 65,000 - 60,000
Qd = 5,000

To determine if there is a surplus or shortage, we compare Qs and Qd:
Surplus/Shortage = Qs - Qd
Surplus/Shortage = 55,000 - 5,000
Surplus/Shortage = 50,000

Now, you can follow the same steps for the remaining rows of the table, plugging in the corresponding values of price and calculating Qs, Qd, and the surplus/shortage.

b. The equilibrium price is the price at which the quantity demanded (Qd) equals the quantity supplied (Qs). In other words, it is the price at which there is neither a surplus nor a shortage.

From the table, we can see that when P = $4.00, Qd = Qs = 25,000, which means there is no surplus or shortage. Therefore, the equilibrium price is $4.00.

For problem 2:
1. To calculate point elasticities, we need to use the formula: elasticity = (dQ/dP) * (P/Q)

At P = 5:
dQ/dP = -2 (since the derivative of Q = 20 - 2P is -2)
Q = 20 - 2 * 5 = 20 - 10 = 10

Elasticity at P = 5: (-2) * (5 / 10) = -1

At P = 9:
dQ/dP = -2 (same as above)
Q = 20 - 2 * 9 = 20 - 18 = 2

Elasticity at P = 9: (-2) * (9 / 2) = -9

Based on the calculated elasticities, we can say that the demand curve is elastic at P = 5 (elasticity = -1), and it is even more elastic at P = 9 (elasticity = -9).

2. To calculate the arc elasticity between P = 5 and P = 6, we use the formula:
arc elasticity = ((Q2 - Q1) / ((Q2 + Q1) / 2)) / ((P2 - P1) / ((P2 + P1) / 2))

For P = 5:
Q = 20 - 2 * 5 = 10

For P = 6:
Q = 20 - 2 * 6 = 8

Arc elasticity = ((8 - 10) / ((8 + 10) / 2)) / ((6 - 5) / ((6 + 5) / 2))
Arc elasticity = (-2 / (18 / 2)) / (1 / (11 / 2))
Arc elasticity = (-2 / 9) / (2 / 11)
Arc elasticity = (-2 / 9) * (11 / 2)
Arc elasticity = -22 / 9

Therefore, the arc elasticity between P = 5 and P = 6 is approximately -22/9.

3. To find the price at which a change in price and quantity results in approximately no change in total revenue, we need to find the price (P) that makes the derivative of the revenue (R) function equal to 0.

Revenue function: R = P * Q = 20P - 2P^2

To find the derivative:
dR/dP = 20 - 4P

Set dR/dP = 0 and solve for P:
20 - 4P = 0
4P = 20
P = 5

Therefore, at a price of $5, a change in price and quantity would result in approximately no change in total revenue.