Assume that the manufacturing of plastic (PVC) windows is a perfectly competitive industry. The market demand for plastic windows is described by a linear demand function: Q = (6000 -50P)/9. There are 50 manufacturers of plastic windows. Each manufacturer has the same production costs, described by the function TC=50+q2+10q. Answer the next questions:
Describe the equilibrium in the short run.
Describe the new short run equilibrium if every producer needs to pay 40 ("protection money") to mafia every period
Describe the new short run equilibrium if every producer needs to pay 4 from every window to local bureaucrats (“safety certification" fee).
To describe the equilibrium in the short run, we need to determine the market price and quantity of plastic windows. In a perfectly competitive industry, the equilibrium is achieved where the market demand equals the total supply.
Given the demand function Q = (6000 - 50P)/9 and the presence of 50 manufacturers, each with the same production costs described by the function TC = 50 + q^2 + 10q, we can find the equilibrium.
1. Equilibrium in the Short Run:
In the short run, each manufacturer will consider their own profit maximization. They will produce the quantity where marginal cost (MC) equals the market price (P). To find the equilibrium price, we equate market demand with the total supply:
Market Demand (Q) = Total Supply (aggregate quantity from all manufacturers)
Substituting the demand function Q = (6000 - 50P)/9 and the number of manufacturers (50), we get:
(6000 - 50P)/9 = 50 * q
Simplifying and rearranging the equation, we find:
6000 - 50P = 450 q
Now, let's find the marginal cost (MC) for a typical manufacturer. We can calculate the derivative of the total cost (TC) function:
MC = d(TC)/dq = 2q + 10
Setting MC equal to P, we have:
2q + 10 = P
Now we can solve this system of equations to find the equilibrium price (P) and quantity (Q):
6000 - 50P = 450 q
2q + 10 = P
Solving the system, we find:
P ≈ $137.04
Q ≈ 188.89 units
Therefore, in the short run equilibrium, the market price of plastic windows would be approximately $137.04 per unit, and the quantity produced and demanded would be around 188.89 units.
2. New Short Run Equilibrium with Mafia "Protection Money":
If every producer needs to pay $40 as "protection money" to the mafia every period, this cost would be considered an additional expense in their total cost function. Therefore, the new total cost (TC) function would be:
TC = 50 + q^2 + 10q + 40
Following a similar process as before, we calculate the new marginal cost (MC) function:
MC = d(TC)/dq = 2q + 10
Setting MC equal to P and solving the equations, we find:
P ≈ $141.30
Q ≈ 184.35 units
In this case, the market price would increase to approximately $141.30 per unit, and the quantity produced and demanded would decrease to around 184.35 units due to the additional cost imposed by the mafia.
3. New Short Run Equilibrium with "Safety Certification" Fee:
If every producer needs to pay $4 from each window to local bureaucrats as a "safety certification" fee, this cost would be added to their total cost (TC) function. Thus, the new total cost function becomes:
TC = 50 + q^2 + 10q + 4q
Again, we calculate the new marginal cost (MC) function:
MC = d(TC)/dq = 2q + 10 + 4
Setting MC equal to P and solving, we find:
P ≈ $140
Q ≈ 185 units
In this scenario, the market price would be approximately $140 per unit, and the quantity produced and demanded would be around 185 units due to the added expense of the "safety certification" fee.
Please keep in mind that these calculations are approximate and rounded to simplify the presentation.
To describe the equilibrium in the short run, we need to determine the market price and quantity where the supply and demand curves intersect.
1. Equilibrium in the Short Run:
In a perfectly competitive market, the equilibrium occurs when the market price is such that the quantity demanded is equal to the quantity supplied. Therefore, we can set the demand function equal to the supply function.
Demand function: Q = (6000 - 50P) /9
Supply function: Q = q
Setting them equal, we have:
(6000 - 50P) /9 = q
To find the equilibrium price, we need to determine the quantity supplied. The supply function tells us that the quantity supplied is equal to q. However, the production costs function is given by TC=50+q^2+10q.
Production costs: TC = 50 + q^2 + 10q
Now, we can differentiate the production costs function with respect to q and set it equal to zero to find the quantity supplied.
dTC/dq = 2q + 10 = 0
2q = -10
q = -5
However, since a negative quantity does not make sense, we disregard it.
So, in the short run equilibrium, the quantity supplied by each manufacturer is q = 0. With this information, we can substitute q = 0 into the demand function to find the equilibrium price:
Q = (6000 - 50P) / 9
0 = (6000 - 50P) / 9
6000 - 50P = 0
50P = 6000
P = 120
Therefore, in the short run equilibrium, the market price is P = 120 and the quantity supplied and demanded by each manufacturer is q = 0.
2. New Short Run Equilibrium with a "Protection Money" Payment:
If each producer has to pay 40 as "protection money" to the mafia every period, their production costs will increase by 40. Therefore, the new production costs function will be:
TC = 50 + q^2 + 10q + 40
TC = 90 + q^2 + 10q
To find the new short run equilibrium, we follow the same steps as before. Differentiating the new production costs function:
dTC/dq = 2q + 10 = 0
2q = -10
q = -5 (disregarding negative quantity)
With the new production costs, we substitute q = 0 into the demand function to find the equilibrium price:
Q = (6000 - 50P) / 9
0 = (6000 - 50P) / 9
6000 - 50P = 0
50P = 6000
P = 120
Therefore, in the new short run equilibrium with a "protection money" payment, the market price remains the same at P = 120, and the quantity supplied and demanded by each manufacturer is q = 0.
3. New Short Run Equilibrium with a "Safety Certification" Fee:
If each producer has to pay 4 from every window to local bureaucrats as a "safety certification" fee, their production costs will increase by 4q.
The new production costs function becomes:
TC = 50 + q^2 + 10q + 4q
TC = 50 + q^2 + 14q
Differentiating the new production costs function:
dTC/dq = 2q + 14 = 0
2q = -14
q = -7 (disregarding negative quantity)
Substituting q = -7 into the demand function to find the equilibrium price:
Q = (6000 - 50P) / 9
-7 = (6000 - 50P) / 9
-63 = 6000 - 50P
50P = 6063
P = 121.26 (rounded to two decimal places)
Therefore, in the new short run equilibrium with a "safety certification" fee, the market price is approximately P = 121.26, and the quantity supplied and demanded by each manufacturer is q = -7 (disregarding negative quantity).