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SECOND PUBLIC EXAMINATIONHonour School of Philosophy Politics and Economics

Honour School of Economics and ManagementHonour School of History and Economics

M THEM TIC L METHODS

TRINITY TERM 2011Thursday 9th June 2011 09:30 - 12:30

Please start the answer to each question on a separate page.

There are 8 questions in this papernswer five questions

Candidates may use their own calculators.Do not turn over until told that you may do so.

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1. An individual with utility function u(y) has a coefficient of absolute prudence givenby -ulll(y)/u (y), and a coefficient of relative prudence given by -yulll(y)/u (y).(a) Show that someone with the utility function u(y) = a by cexp -Ay) has a

constant coefficient of absolute prudence A.Given A > 0 what restrictions, if any, must be placed on the parameters a, b, cto ensure that this is the utility function of a risk-averse utility maximiser?

(b) f someone has a constant coefficient of relative prudence R, what functional formmust their utility function have in general? State dearly for what particularvalues of R your derivation is not valid.)Now given R > 1, what restrictions, if any, must be placed on parameters such asa, b c to ensure that this is the utility function of a risk-averse utility maximiser?

2. (I) Let sO be a continuously differentiable function defined on the real line with thefollowing properties:

s O) = 0 Is(x)l:S; 1 Vx, Is (x)l:S; 1 Vxbut l m x ~ o o s(x) and l m x ~ o o s (x) do not exist . [sin(x) is such a function.]Define the function f 0 on the real line by

f(x) = {Show that fO is continuous at OFind the value of f (O) from first principles.

if x aif x = a

Find an expression for f (x) when x O What can be said about l m x ~ o f (x)?Is fO differentiable everywhere? Is the derivative of fO continuous?

(II) Given any triangle, explain briefly why the longest side can be no longer thanhalf the perimeter.Two numbers, p and q, are independently drawn at random from the continuousuniform distribution on [0 1].Define x as min{p, q}, y as 1 - max{p, q}, and let z = 1 - x - y.

o min{p, q}I

max{p, q}I

1

x _. _____ z y -What is the probability that you can draw a triangle with sides of length x, y, z?

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3. Two agents work in a team and wish to maximise its payoff function

where b i 4, but they choose their actions independently.(a) Suppose that the agents interact once and choose their actions simultaneously.

i) Find the Nash equilibrium of the game.(ii) Determine whether the Nash equilibrium choices of Xl and X2 maximise V by

examining its Hessian.(b) Suppose instead that the agents interact continuously and adjust their actions

according to the differential equations. 8VXI ( xl

i) Find the general solution to this system in terms of the values of Xl and X2at time O

(ii) Describe the long-run behaviour of Xl and X2

4. (a) Suppose that the random variable X has the density function

)"+1 X_f X - x{ a f > 1- 0 otherwise where a > O

i) Verify that f is a density function.(ii) Find the moments E xr), specifying for which values of r they exist.

(iii) Find the expected value of X given that X is greater than k (with k > 1) if0 >1.

(b) Suppose that X and Y have the joint density function

{a a+1)

f(x, y = ~ x y l)"+2 i f x ~ 1 y ~ 1otherwisei) Find the marginal distributions of X and Y.

(ii) Are X and Y independent?(iii) Find the conditional density of X given Y = y and show that

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E XI Y = y = 1 for Go> 1.Verify that E X) = E E XI Y)).

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5. (I) You have been asked to design a solid cylinder of a given volume v by specifyingthe height h and the diameter of the circular cross-section d.(a) I f your aim is to minimise the surface area of the cylinder, what should you

choose for d in terms of h?(b) What if your aim is to maximise the surface area?[The volume of a cylinder is the height x the area of the circular cross-section;the surface area is the height x the circumference of the circular cross-section +the area of the two circular faces.]

(II) Find the general solution of the second-order linear difference equation3YH2 = 7Yt 1 - 2Yt, t = 0, 1,

What is the steady state? Is it stable?Now assume that o = 1 and Yl = k. What is

and how does it depend on k?I YHIm

t HX J t

For what values of k if any, does Yt converge to the steady state?

6. Two firms are trying to invent a new product. Firm A has probability of success peach month and firm B probability q, in both cases independent of the outcomes inother months and of the outcome of the other firm s efforts. Let X be the numberof months it takes firm A to succeed and Y the number of months it takes firm B tosucceed.(a) Explain why X has a geometric distribution, with Pr X = r) = p l py- l for

r = 1,2, ....(b) Find Pr X k for any k lc) Find the probability generating function of X.

(d) Find E X).e) Find the probability that it takes A at least as long as B to succeed.(f) Let Z = min{X, Y} be the time it takes for at least one firm to succeed. Find

Pr(Z:::: k) for any k, and hence show that Z also has a geometric distribution.g) Suppose that the first time either firm succeeds is month n. What is the proba

bility that it is A that succeeds in month n?

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7. (a) Suppose f : Rn ---t R is a concave function. Show that for any a E R the setS = {x : f x) a} is convex.

(b) A consumer has utili ty function U Xb X2 = -xII - xiI, where the quantities Xland X2 must be non-negative. Denote the prices of goods 1 and 2 by PI > 0 andP2 > O She wishes to choose Xl and X2 to minimise the cost of obtaining utilityat least u. Let e p,u) be the minimised cost, where p = PI,P2).i) Explain whether the Kuhn-Tucker first-order conditions for the consumer s

problem are a) necessary, and 3) sufficient for optimality.(ii) Find the optimal choices of Xl and X2 and e p, u).

(iii) Show that e(p, u is homogeneous of a certain degree in p, which you shouldspecify.

(iv) State Euler s theorem for homogeneous functions and verify that it holds for eas a function of p. Give this result an economic interpretation.

8. (a) For what values of a do the vectors

G), D, G)span R3?

(b) In a certain economy there are two possible states. Asset A sells for 7 per unitand each unit pays out 2 in state 1 and 3 in state 2. Asset B sells for 4 perunit and each unit pays out 1 in state 1 and 2 in state 2. If a unit of asset Cpays out 1 in each state, at what price must it sell if there is to be no arbitrage?

c) In a certain country there are two goods, 1 and 2. To produce one unit of good1 requires 0.1 units of good 1, 0.2 units of good 2 and 0.6 units of labour. Toproduce one unit of good 2 requires 0.8 units of good 1, 0.1 units of good 2 and0.3 units of labour. f it is desired to produce 130 units of each good, how muchlabour is required in total?

(d) In the country in the previous part, capitalists require a return of r 0 on thecost of non-labour inputs so prices satisfy the equationsPI = (1 r) (O.IPI 0.2p2) 0.6wP2 = 1 r) (0.8pI 0.lp2) 0.3w

What values can r take?

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