Exam 15 2017, questions PDF

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Summer 2017 examination

MA107 Quantitative Methods (Mathematics)

Suitable for all candidates

Instructions to candidates This paper contains 5 questions. You may attempt as many questions as you wish, but only the best 4 questions will count towards the final mark. All questions carry equal numbers of marks. Answers should be justified by showing work. Please write your answers in dark ink (black or blue) only. Time Allowed

Reading Time:

None

Writing Time:

2 hours

You are supplied with:

Mathematics answer booklets

You may also use:

No additional materials

Calculators:

Calculators are not allowed in this examination

c LSE ST 2017/MA107 

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Question 1 A market for some good operates with the supply function q S (p) =

p 3 − , 10 2

and the inverse demand function p D (q ) = q 2 − 40q + 415. In this question we assume that an economically meaningful inverse demand function, p D (q), must be a decreasing function of q . (a)

Sketch the supply and demand curves for this market.

(b)

Find the inverse supply function, p S (q), and the demand function, q D (p), for this market. For what values of p and q are the supply and demand functions economically meaningful?

(c)

Find the equilibrium set for this market.

(d)

An excise (or per-unit) tax of T is imposed on the sale of goods in this market. If the price in the presence of the tax is pT , then the supply function in the presence of the tax is q S (pT − T ). By considering how the presence of the tax alters the supply curve, find

(i) the maximum tax, Tm , that can be imposed and (ii) the value of T that will reduce the quantity sold at equilibrium by half.

Question 2 (a)

Use row operations to determine all possible solutions to the system of equations x − 3y + z = 1

x − y − z = −1

x − 5y + az = b,

where a and b can be any numbers. Express any solutions you find in vector form. (b)

What does it mean to say that a function, f (x , y ), is homogeneous of degree r ? What does Euler’s theorem state for such a function? Show that the production function q(x , y ) =

√

xy ,

is homogeneous. What is its degree of homogeneity? Does this function give rise to increasing, constant or decreasing returns to scale? Verify that Euler’s theorem holds for this function.

c LSE ST 2017/MA107 

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Question 3 (a)

An asset can be purchased now for $100. It is believed that, if it were to be sold in t years’ time, it would sell for $V (t) where V (t ) = t 2 + 40t + 100. Assuming that continuously compounded interest at an instantaneous rate of 10% is applied, explain why the present value of selling this asset at time t is given by P (t ) = V (t )e−t/10 . Find the stationary points of P (t ) and determine their nature. Should the asset be purchased and, if so, when is the best time to sell it? [Hint: The fact that e < 3 may be useful.]

(b)

Two firms, Firm 1 and Firm 2, operate in one market. If they announce prices of p1 and p2 dollars per unit for their respective products, they will experience demands given by 1 1 q1 = 20 − p1 + p2 2 2

and

1 3 q2 = 20 + p1 − p2 , 2 2

respectively. It also costs each firm 20 dollars to produce one unit of their product. Suppose that Firm 2 knows that Firm 1 is going to announce a price, p1 . Explain why the profit function for Firm 2 is given by   1 3 Π2 (p2 ) = (p2 − 20) 20 + p1 − p2 , 2 2 as the known price p1 can be treated as a constant. Hence show that Firm 2 should choose a price p2 =

100 + p1 , 6

if they want to maximise their profit. Suppose further that Firm 1 assumes that Firm 2 will choose this value of p2 . What value should Firm 1 choose for p1 ?

c LSE ST 2017/MA107 

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Question 4 (a)

Find the stationary points of the function f (x , y ) = 3x 3 + 9x 2 − 72x + 2y 3 − 12y 2 − 126y + 19, and determine their nature.

(b)

A consumer has a utility function given by 1/3

u(x1 , x2 ) = x1 x22/3, and their budget constraint is p1 x1 + p2 x2 = M. Find the bundle of goods, (x1 , x2 ), that will maximise their utility subject to this constraint. If the price, p1 , was to decrease, how would this affect the bundle you found? (c)

Another consumer has a utility function given by u(x1 , x2 ) = x12 + x22, and their budget constraint is x1 + 4x2 = 16. By sketching the budget set and some contours u(x1 , x2 ) = c where c is a constant, find the bundle of goods, (x1 , x2 ), that will maximise their utility subject to this constraint.

Question 5 (a)

Verify that 3x 2 + x + 1 1 2x + 1 , = + 2 x +1 x(x 2 + 1) x and hence determine the integral Z 3x 2 + x + 1 dx . x(x 2 + 1)

(b)

Find the solution to the differential equation 3x 2 + x + 1 2x dy y (x) = , − 2 x +1 x dx when y (1) = π/2.

(c)

Find the general solution to the differential equation d2 f df − 6f (t) = 6t 2 + 16t + 19. + 2 dt dt What is the particular solution when f (0) = −1 and f ′ (0) = −2? END OF PAPER

c LSE ST 2017/MA107 

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