Wednesday, September 27, 2006
Econs tml, maths on friday.. You may or may not be glad to hear that I have offered alternative explanation for the chapter on market failure: monopoly.
Some of you might undoubtedly wonder why we to achieve allocative efficiency, we require MB=MC. Here's a mathematical reason why.
Let's start by defining satisfaction, S.
S=B-C
Satisfaction of each individual in any partaking any action is calculated by taking total benefit involved and substracting from it, the total cost incurred. Remember that B and C are functions that are defined based on the variable Q, quantity. Next, we want to find the position where we get to maximise the value of S. Remember differentiation? We can find the maximum point of any curve by differentiating this function and equating the differentiated function to zero. So here goes.
d(S)/d(Q) = d(B)/d(Q) - d(C)/d(Q)
d(S)/d(Q) = 0
0= d(B)/d(Q) - d(C)/d(Q)
Therefore,
d(B)/d(Q) = d(C)/d(Q)
In econs, d(B)/d(Q) and d(C)/d(Q) are known as marginal benefit and marginal costs. See now?
Also remember that for a monopolist, his marginal revenue curve is less than the average cost curve? Yeah I now show a maths proof why.
Recall that total revenue is given by (Price)(Quantity). Also, verify that Price is a function of Quantity, yes? Here the Price referred to the by the equation is also the average cost curve. Observe that it is a straight line sloping downwards. So we can do a simple formulation:
P = -KQ ; Where P is price, K is an arbitrary constant, and Q is quantity.
Lets go back to the total revenue equation, substituting the Price equation in.
TR = -KQ^2
To find marginal revenue, we differentiate this function.
d(TR)/d(Q) = -2KQ
MR = -2KQ
Observe that the gradient of marginal revenue curve is -2K, and that the gradient of the average revenue curve is only -K. Of course, the marginal revenue curve is steeper..
Please do not be alarmed by the maths here. I used only Sec 4 A-maths and nothing more. If you are panicking over this alien maths, then do be very worried. Because its Maths exam on Friday.
Hope you have gained insight.
Jing Hao
