Tutor profile: Emily Y.
Subject: Applied Mathematics
Does the sequence created byan=n^2/n have a limit? Prove it does (in whichcase, find it) or show that it doesn’t.
Yes. It is 0. We use L’Hospital’s rule. Both the top and the bottom go toinfinity, but the top has derivative1and the bottom has derivative[log(2)2n] (recall that 2^n=e^[nlog(2)]), so the bottom increases much faster
Suppose an industry has an inverse demand curve given by P(Q) = 300 – Q, where P is the price in dollars and Q is the quantity. The marginal private cost of production is MPC(Q) = Q and the marginal negative external cost of production is given by MEC(Q) = 0.5Q. Find the socially optimal equilibrium level of output and price
The social optimal quantity is achieved when marginal social cost equals marginal social benefit. MSB = MSC ; the marginal social cost is the sum of marginal private cost MPC and marginal external cost MEC, so MSC = MEC + MPC 300– Q = 1.5Q Q* = 120 P* =300–120 =180
What two-year rate 5-years forward is implied by a 7-year spot rate of 2.75% and a 5-year spot rate of 1.25% if one assumes annual compounding?
The arbitrage relationship between these rates is (1 + r5)^5 * (1 + 2f5)^2 = (1+r7)^7 where r5 is the five-year annual spot rate, r7 is the seven-year annual spot rate and 2f5 is the 2-year spot rate 5-years forward. With some straightforward algebra we get 2f5 = sqrt[ (1 + r7)^7 / (1 + r5)^5] -1 So 2f5= 0.0659 or 6.59%.
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