Download ECON-115 Solutions to Two Problems

Deadweight Loss Problem
Assume a straight-line,
downward sloping inverse
demand curve: p = 100 – q.
Marginal Cost = 20.
I What is the allocatively
efficient price?
2. What is the profit
maximizing price?
Solution
1. The ALLOCATIVELY EFFICIENT price is always where
price = marginal cost. That is the point at which all profitable
transactions have taken place. (Pareto Optimal)
P = MC means that Price = 20.
2. There are two steps to finding the profit maximizing price.
STEP 1: Determine the Marginal Revenue (MR) Function
STEP 2: Equate MR to MC
Step 1: Determine the Marginal Revenue Function.
1a. First, determine the revenue function (See 1a above).
Revenue = price x quantity. Price = 100 – q. Therefore,
3. What is the efficiency loss
(deadweight loss) that results
from charging the higher
profit-maximizing price?
Revenue = p x q = (100 – q) x q = 100q – q2
1b. Take the derivative to determine MARGINAL revenue.
Factor Rule d/dx Cx = C. Therefore d/dq (100q) = 100
Power Rule d/dx xn = nxn-1 Therefore d/dq( q2) = 2q.
Marginal Revenue (MR) = 100 – 2q
Step 2: Use the Profit Maximizing Rule, MR = MC
MC = 20 = 100 – 2q. Therefore q = 40. Calculating the profit
maximizing price by using the demand equation:
p = 100 – q = 100 – 40 = 60, the profit maximizing price
3. To calculate the efficiency (or welfare) loss due to pricing at
the profit maximizing level, draw diagram representing the
consumer surplus when the P = MC and when the P is set profit
maximizing point of MR = MC. Use the values for P and Q at
both the allocatively efficient level and profit maximizing level.
Then calculate the area of lost due to profit maximizing pricing.
That’s the deadweight loss (DWL)
Allocatively Efficient Level: P = 20, Q = 80
Profit Maximizing Level = P = 60, = 40
Price
Deadweight Loss is defined by the Yellow Triangle
60
40
DWL
D
At MC = P; Q = 80 and P = 20
At MC = MR; Q = 40 and P = 60
20
40
0
40
DWL = the triangle Base = 40, Height = 40
Area = 40 x 40 x ½ = 800.
80
Quantity
Scale Economies Problem
A firm has a cost curve, C = 50 + 2q + 1/2q2.
Solution
1. Average Cost is simply Total Costs divided
by Quantity (q). Therefore:
1. Write the equation for Average Costs (AC):
C/q = 50/q + 2q/q + 1/2q2/q = 50/q + 2 + 1/2q
2. What are the AC values for q = 4, 8, 10 and
12 respectively?
3. Take the derivative of C and determine the
marginal cost curve. Use that function to
estimate the value of MC at points q = 4, 8, 10
and 12 respectively.
2. Plug in the q values into the AC formula to
calculate the various AC values.
q = 4; 50/4 + 2 + ½(4) = 16.5
q = 8; 50/8 + 2 + ½(8) = 12.25
q = 10; 50/10 + 2 + ½(10) = 12
q = 12; 50/12 + 2 + ½(12) = 12.17
3. C = 50 + 2q + 1/2q2
dC/dq = d/dq (50) + d/dq(2q) + d/dq(1/2q2 )
4. With the AC and MC values in 2 and 3,
compute the index of scale economies, S. At
what values of q are economies of scale
present? What about diseconomies of scale?
And at what value are constant returns to scale
exhibited?
Therefore the estimated values of MC is dC/dq
= 0 + 2 + 2*(1/2q1) = 2 + q
q = 4; 2 + 4 = 6
q = 8; 2 + 8 = 10
q = 10; 2 + 10 = 12
q = 12; 2 + 12 = 14
4. The Index of Scale Economies = AC/MC. If
the Index > 1, scale economies exist. If it is <1,
then diseconomies of scale exist. If it = 1, then
there are constant returns to scale present.
When the Index > 1, Average Cost is greater
than Marginal Cost, which means at that point,
for each additional unit produced average cost
is falling. This is technically what is meant by
scale economies: average cost is declining as
more quantity is being produced.
When the Index is < 1, the MC is greater than
AC. At this point, for each addition unit
produced, average cost is rising. This is the
technical definition of diseconomies of scale.
q = 4; AC/MC = 16.5/6 = 2.75 > 1. Therefore
scale economies are present.
q = 8; AC/MC = 12.25/10 = 1.225 > 1.
Therefore scale economies are present.
q = 10; AC/MC = 12/12 = 1. Therefore
constant returns to scale are present.
q = 12; AC/MC = 12.17/14 = .869 < 1.
Therefore diseconomies of scale are present.