Economics study guide question help!Watch
I have a microeconomics test this week and I am extremely stuck on a few of the questions about graphing monopolies that are on my study guide. Please help me out, it would be much appreciated!
In the small town of Smallville, Ski Tech is the only firm in the glove industry. If any
other firm tries to start up a glove company in Smallville the owners of that firm are subjected to
painful physical torture, so no other firms are considering entry to the market.
Ski Tech has marginal cost equal to MC: P = 2 + q.
Total cost is TC: P = 4 + 2q + 1/2q2
Demand for gloves in Smallville is given by P = 12 - 2q
(a) What is the marginal revenue function?
(b) Find the price and profit maximizing quantity produced by Ski Tech. Draw a graph, labeling
price, quantity, intercepts and other points of interest.
(c) Compute profit at the profit maximizing quantity.
(d) Compute consumer surplus and producer surplus for this market. Shade these
areas on the graph. If there is any DWL compute and show this as well.
Marginal Revenue = d(TR)/dQ
If we differentiate our total revenue equation (12q-2q^2) we get 12-4q, which is our marginal revenue function.
(b) Profit = TR - TC
Profit = 12q-2q^2 - (4+2q+1/2q^2)
To maximise profit we need to differentiate our profit function and then set equal to zero and find the optimal quantity (q*). We may also want to check that this is a maximum by taking the second differential, substituting in q* and finding that our value is negative.
When I calculate this I get q*=2.
We can check this, because we also know that the profit maximising quantity occurs when marginal revenue equals marginal cost (MR=MC) and we have functions for both of these:
12-4q = 2+q
Which verifies that q*=2.
If we substitute q*=2 into the price function then we find that P*=8
I'll leave you to draw the graph.
(c) We simply substitute q* back into our profit function:
Profit = 12(2) - 2(2)^2 - (4+2(2)+(2)^2)/2))
Which I find to be 6.
(d) If you draw your graph then you find that Consumer Surplus is the area above the price line but below the demand curve and the producer surplus is the area below the price line above the supply curve.
With some rough calculations I derive CS = 4 and PS = 8, but I'm not 100% about the final value so check.
Hope this helps,