suppose a monopolist produces and sells a product ona 2 diferent markets. demand function on the two markets are repectively i=market 1 / ii=market 2
Pi= 200-2Qi Pii=180-4Qii
cost function is C=20(Qi+Qii)
A) what Quantities and price that maximize the firm's profit
B) how much profit is lost if price discrimination becomes illigal?
C)discuss the consequences on the optimal quantities, prices and profits of the introduction of a tax of 5 per unit sold in market 1
==i just seem to be getting very very weird answers === please i need help!!!
Always Always Always, set MC=MR
MC with respect to Qi and/or Qii is 20.
In market i:
TR is Pi*Qi = 200Qi - 2Qi^2. MR is the first derivitive. So MR = 200-4Qi
MR=MC = 200-4Qi = 20. Solve for Qi.
Repeat for market ii. With Qi and Qii known, solve for total profit
Now for the tricky part. Derive the demand functions in a combined market (where price discrimination is illegal). P = 200-2Q for Q<= 10 and P=180 - 6*(Q-10) for Q> 10. If we extend this latter line back to the origion, the demand equation becomes 240-6Q for Q>10. So, MR is 240-12Q. Again, solve for optimal Q, and then total profit.
For Part C, I need some clarification. Is the 5 per-unit tax apply in scenario A with price discrimination or scenario B without price discrimination. If B, can the tax be tacked on to the price, so that in market 1, consumers pay 5 more than in market 2. Or does the price after taxes still need to be the same in both markets.
I hope this helps.
To solve this problem, we need to find the profit-maximizing quantities and prices for the monopolist in each market. We can do this by maximizing the monopolist's profit function, given the demand and cost functions.
A) To find the quantities and prices that maximize the firm's profit, we need to determine the profit-maximizing level of output in each market. The profit function can be expressed as follows:
Profit = Total Revenue - Total Cost
For Market 1:
Total Revenue (TR) = Price (P) * Quantity (Q)
Total Cost (TC) = Cost (C)
Profit (π) = TR - TC
For Market 2:
Total Revenue (TR) = Price (Pii) * Quantity (Qii)
Total Cost (TC) = Cost (C)
Profit (pii) = TR - TC
Let's start with Market 1:
TR1 = Pi * Qi = (200 - 2Qi) * Qi = 200Qi - 2Qi^2
TC = C = 20(Qi + Qii) = 20Qi + 20Qii
π1 = TR1 - TC = 200Qi - 2Qi^2 - 20Qi - 20Qii
To find the profit-maximizing quantity in Market 1, we can take the derivative of the profit function (π1) with respect to Qi and set it equal to zero:
dπ1/dQi = 200 - 4Qi - 20 = 0
-4Qi = -180
Qi = 45
So, the quantity that maximizes profit in Market 1 is Qi = 45.
Now, let's move on to Market 2:
TR2 = Pii * Qii = (180 - 4Qii) * Qii = 180Qii - 4Qii^2
TC = C = 20(Qi + Qii) = 20Qi + 20Qii
pii = TR2 - TC = 180Qii - 4Qii^2 - 20Qi - 20Qii
To find the profit-maximizing quantity in Market 2, we can take the derivative of the profit function (pii) with respect to Qii and set it equal to zero:
dpii/dQii = 180 - 8Qii - 20 = 0
-8Qii = -160
Qii = 20
So, the quantity that maximizes profit in Market 2 is Qii = 20.
Now that we have the profit-maximizing quantities in each market, we can calculate the corresponding prices. Using the demand functions, we can substitute the optimal quantities into them to find the prices:
For Market 1:
Pi = 200 - 2Qi = 200 - 2(45) = 200 - 90 = 110
For Market 2:
Pii = 180 - 4Qii = 180 - 4(20) = 180 - 80 = 100
Hence, the profit-maximizing quantities and prices for the monopolist are Qi = 45, Qii = 20, Pi = 110, and Pii = 100.
B) If price discrimination becomes illegal, the monopolist would have to charge the same price in both markets. In this case, they would set a single price that maximizes their profit across both markets.
To find the profit-maximizing quantity and price under price discrimination becoming illegal, we need to set up a new profit function considering the combined demand from both markets:
TR = (Pi + Pii) * (Qi + Qii) = (200 - 2Qi + 180 - 4Qii) * (Qi + Qii)
TC = C = 20(Qi + Qii)
Profit = TR - TC
To maximize profit, we can take the derivative of the new profit function with respect to Qi and Qii, and set both derivatives equal to zero:
dProfit/dQi = -2Qi - 4Qii + 200 - 2Qi + 180 - 4Qii = 0
dProfit/dQii = -4Qii - 2Qi - 4Qii + 180 - 4Qi + 200 = 0
Solving these two equations simultaneously will give us the profit-maximizing quantities and the common price. However, since you mentioned that you are getting very weird answers, it's possible that there might be an error in the given equations or calculations. Make sure you double-check your calculations and equations and ensure they are correctly represented.
C) To analyze the introduction of a tax of $5 per unit sold in Market 1, we need to adjust the demand function and recalculate the profit-maximizing quantities, prices, and profits.
With the tax, the adjusted demand function in Market 1 becomes:
Pi = 200 - 2Qi - Tax = 200 - 2Qi - 5
Now, you can follow the same steps as in part A to find the profit-maximizing quantities and prices in both markets.
Remember to update the profit function, total revenue, and the derivative equations accordingly. Once you have the new quantities and prices, you can calculate the new profits.
If you are really getting very weird answers, it's possible that there might be an error in the given equations, or there might be an issue with the calculations. Double-check the equations, verify the calculations, and ensure that you are transcribing the equations correctly.
If you are still struggling, it might be helpful to consult with a professor, TA, or seek additional resources to clarify any concepts or errors that you might have encountered.