Exercise 1:
Consider a firm in a perfectly competitive market with cost function C(q)
= 50 + 2q + 2q².
a) At a market price of P= $12/unit, the firm produces q = 4 units. Is it
maximizing profits? If not, how should it modify its output?
Cost function : C(q) = 50 + 2q + 2q2
Market price : 12$ per unit
Current production : 4 units
For the firm to maximize its profits. They need to produce a quantity
where Mr is equal to MC
MR : P = 12$ per unit
MC : C(q) = 50 + 2q + 2q2 -> MC = 2 + 4q
Put in q = 4 in the cost function and simplify the expression
MC (4) = 2 + 4 x (4)
MC (4) = 2 + 16 = 18
MC (4) = 18
-> Answer = 18$ per unit
We can see that the MC ($18$/ Unit) is greater than the MR ($12$/Unit).
This means the firm is not maximizing its profits when producing 4 units.
To maximize its profits, the firm should reduce production and lower the
marginal cost. When the marginal cost equals MR, the firm will maximize
its profits.
b) What is the minimum price at which the firm earns positive profits?
Cost function: C(q) = 50 + 2q +2q2
Average cost function: AC(q) = C(q) / q
Calculating average cost function:
C(q) / q = 50 + 2q +2q2 / q
AC(q) = (50 / q) + 2 + 2q
Minimize AC
To minimize AC, we need to set the derivative equal to zero, which
provides the output quantity q at which the AC is minimized.
d(AC) / dq = -(50 / q2) + 2
-(50 / q2) + 2 = 0
50 / q2 = 2
q2 = 50 / 2 = 25
q=5
Calculate minimum AC
AC(q) = (50 / q) + 2 + 2q
AC(5) = (50 / 5) + 2 + 2(5)
AC(5) = 10 + 2 + 10 = 22
-> Answer : The minimum price where the firm earns a profit is 22$. This
is the minimum AC so they must at least charge 22$ to cover costs.
c) What quantity will the firm produce in the long run?
In the long run, the firm produces at a price where P = MC and where AC
is minimized. To make sure it operates with the most effective output
level, minimizing costs and maximizing profits.
In exercise 1 b, both MC(5) and AC(5) equals 22, this means q = 5, the
firm minimizes its average costs. The price they will sell at is 22$.
Exercise 2:
Consider a competitive industry, with the following demand and supply
equations :
Qd = 4,000 – 80P
Qs = 1,000P
Consider a typical firm in this perfectly competitive market with cost
function is
210 +q²/100
a) Find the market price, market output, the output supplied by a typical
firm, and the profits of each firm.
Market price: Qd = Qs
-> P = 4000 – 80P = 1000P
P = 4000 = 1080 P
P=
P = 3.70
The market price is at 3.70$
Market output: Qs = 1000 P
->We then substitute P by the market price which is 3.70$.
We get: 1000 x 3.70 = 3700 units
The market output is 3 700 units.
Output supplied by a typical firm:
We use the marginal cost which means that: marginal cost = C’ (q)
We also know that the cost function here is: 210 +q²/ 100.
We can compute: 2q/100 = q/50
q* P = MC
q* 3.70 = q/50
q* 3.70 x 50 = 185
We can conclude that the output supplied by one typical firm is 185 units.
The profits of each firm :
For the profit, we have: π = R – C (q)
= P x q* - C (q)
Π = 3.70 x 185 – (210 + 185 2/100)
Π = 684.5 - 552.25
= 132.25
We can say that the profit of each firm will be = 132.25$
b) What is the break-even price of a typical firm in this market? What can
you say about profits at that price?
Break-even price <=> Mc = ATC
<=>
I can cancel out one q from nominator and denominator which gives me:
<=>
<=>
When i do the calculation, I obtain:
= 2q 2 = 2100 + q2
So, we have q2 = 2100
q=
q = 144.91
The break-even price of a typical firm in this market is 144.91$. We can
say that the firm will earn 0 profits. In fact, the break-even price is the
price at which the firm’s total revenue is equal to total cost. There is no
profits at the break-even price.
c) What is the shutdown price of a typical firm in the short run? What can
you say about profits at that price?
For Shutdown price, the shutdown price is the price at which a firm covers
its variable costs (VC) but not fixed costs. The firm's variable cost is:
Pshutdown = q/100= MC/2
At q=0, Pshutdown=0 meaning firms may operate at very low prices provided fixed
costs are ignored. In this problem, the shutdown price is the minimum price where
AVC is covered.
For the exact Pshutdown , firms avoid P<2.90 since they cannot cover total costs.
Exercise 3:
Consider a competitive industry, with the following demand and supply
equations:
Qd = 450 – 3P
Qs = 2P - 100
a) Compute the market equilibrium price and quantity.
At equilibrium, Qd = Qs
450 – 3P = 2P – 100
5P = 550
Substituting P=110 into the equation
Qd = 450 –3 (110) = 120
Equilibrium Price = 110 $ and equilibrium quantity = 120 units
b) Compute the consumer surplus and the producer surplus.
Consumer Surplus (CS) : is the maximum price consumers are willing to
pay
P = 150
So CS = 0,5 x 120 x (150 – 110) = 2400
Producer Surplus (PS) = is the minimum price producers are willing to
accept
P = 50
So PS = 0,5 x 120 x (110 – 50) = 3600
-> Consumer surplus = 2400 and Producer Surplus = 3600
c) Suppose the price is fixed at $90/unit. What does this mean for total
welfare? Illustrate graphically the consequences.
We start by determining Qd and Qs at P = 90
Quantity demanded = 450 – 3P = 450-3 x 90 = 180
Quanity supplied = 2P – 100 = 2x 90 – 100 = 80
Result : Qd > Qs means there is excess demand (shortage)
-> Qd -Qs =180−80= 100 units. The consumer increases (A + D + C + E
instead of A),
the producer surplus decreases (B - C- E - G instead of B + D) and
the total welfare decreases (A + B + D - G instead of A + B + D).
d) Suppose the price is fixed at $120/unit. What does this mean for total
welfare? Illustrate graphically the consequences.
When the price is fixed at $120 it's below the equilibrium price.
This creates an excess supply as the quantity supplied exceeds
the quantity demanded.
Quantity demanded : Qd = 450 - 3P = 450 - 3 x 120 = 90
Quantity supplied : Qs = 29 - 100 = 2 x 120 –100 = 140
When the price is fixed at $120 it's below the equilibrium price.
This creates an excess supply as the quantity supplied exceeds
the quantity demanded.
The excess is 140 - 90 = 40 units. The total welfare decreases (A + B
instead of A + B +
C).
