Exam 2015, questions EC1005 PDF

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Exam Question Paper College/ Institute

CBASS

Department

Economics and Finance

Module Code

EC1005

Module Title

Mathematics for Economics and Finance

Month

May

Paper Type

2016

Full

Duration Question Instructions

Year

2 hours plus 5 minutes reading time Answer all questions from section A and any two questions from section B

Are Calculators Permitted?

Calculators may be used but all working must be shown. Casio fx 82, Casio fx 83, Casio fx 85 Any letters after these numbers are permitted.

Permitted Reference Materials Required Stationary

NA ---

SECTION A – Answer all questions. 1) The supply and demand functions of a good are given by P

= -6Q + 320 P = 42Q +210

Respectively, where P is the price in pounds sterling and Q is the number of goods. i)

Find the market equilibrium.

[3]

ii)

Sketch both functions and market equilibrium on the same graph [2]

iii)

Calculate the consumer and producer surpluses at equilibrium and sketch both on separate graphs. [4]

iv)

Suppose the government imposes a fixed tax of £40 per good. Redraw you sketch of part i) above and modify it to show the effect of tax. Find the coordinates of new equilibrium position. How much tax per good is paid by the consumer and how much by the company? [7]

v)

What is the price elasticity of demand at the pre-tax and post-tax equilibria found in parts ii) and iv) above? [4]

2) Consider the demand curve Q=430-8P+3Y-5Pa relating to the quantity of a good to its price and to average customer income Y and the price of an alternative good Pa. Is the good normal? Is the alternative good complementary? Give reasons for your answers. [2] 3) Find the total derivative, df/dt , when -3

-2

f(x,y)= x +4y 2

4

where x=3t and y=-5t , by i)

Substitution of x and y

ii)

Using the chain rule formula for a total derivative

[3]

[4] iii)

Show that your results for a) and b) are equivalent.

[3]

4) Determine the rate of interest required for a principal of £3100 to produce a future value of £10000 after 8 years compounded annually? [4] 3

5

5) Consider the production function Q = 5 K L . Calculate the marginal product of labour and the marginal product of capital. Hence, or otherwise, find the marginal rate of technical substitution (MRTS) and explain what this means by drawing an isoquant on an (L,K) graph. [4] 6) Sam decides to invest £15,000 at the start of every year in a sinking fund offering 8% compounded annually. After how many years does the value of the investment first

exceed £200,000?

[10]

SECTION B - Answer any TWO questions.

7) Find the first and second derivatives of the following functions. Simplify your answers as much as possible:

i) ii)

y=(3x+√x)

2

[5]

− 7x

y=

[5]

ln x

iii)

2 (− x)

y = (x e

5

)

[5]

Evaluate the following integrals: iv)

∫ (xe ) dx 3

− x

[5]

−2

43x2 + 5 v)

∫1 x

3

+ 5x

dx

[5]

8) A production function generally depends on the number of units of capital, K, and number of units of labour, L. Consider the production function: 2

2

Q(K,L) = L + 4KL + 2K + 33L + 77K The company is subject to a budget of £117 (all of which must be spent) with capital costs of £21 per unit and labour costs of £11 per unit. Write down the constraint equation. Find the coordinates (Ko,Lo) and value Q of maximum production by:

[2]

a) Rearranging your constraint equation and substituting into the above production function to eliminate one of the variables, and then optimising this function. [10] b) Using the method of Lagrange multipliers.

[10]

Suppose the company now has a budget of £110 (all of which must be spent). Use your

results in part b to estimate new value Q of maximum production.

[3]

9).i) Solve the following system of equations, either by Gaussian elimination and back substitution, or by finding the inverse matrix. (NB: you will get no marks for ad hoc methods of solving simultaneous equations.)

9 4

3 4x

7

3 4y

=8

z

3

1 1 1

[15]

1 ii) Show that

1

is an eigen vector of

2 −3

24

− 12

A= − 2

15 − 8

−2

18 − 11

and find the corresponding eigenvalue.

[3]

Show that the eigenvalues of A are given by the solution of the characteristic equation

-λ3+λ2+9λ-9=0 and hence find the other two eigenvalues.

[7]...