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Emily Y.

Teaching assistant at UC Berkeley

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Applied Mathematics

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Question:

Does the sequence created byan=n^2/n have a limit? Prove it does (in whichcase, find it) or show that it doesn’t.

Emily Y.

Answer:

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

Economics

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Question:

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

Emily Y.

Answer:

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

Finance

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Question:

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?

Emily Y.

Answer:

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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