ECON2121-WE01

INTERMEDIATE METHODS FOR ECONOMICS AND FINANCE

2020

SECTION A

1. a) We have the following matrices and vectors:

i.   Evaluate the rank of A , and calculate AA′ , where A′ is the transpose of A. (20 marks)

ii.   Find the inverse of matrix B ,  and the inverse 15/1B′ , where B′ is the transpose of B. (20 marks)

iii.  Use Cramer's rule to solve the following linear system: (15 marks)

Bx = d

b) Suppose that there are two states of the world and two securities A and B.  The unit price of A is £9, and that of B is £8.  The return vectors of A and B are and respectively.

i.   Calculate the state prices. (20 marks)

ii.   Suppose a new security C has a unit price of £17 and a return vector of . Show by one example that arbitrage is possible in this market. (15 marks)

iii.  What would be the correct unit price for C such that there is no arbitrage opportunity?

Assume that the prices of A and B are unchanged. (10 marks)

2. a)  Find the eigenvalues and eigenvectors of matrix D, where (20 marks)

b)  In a two-industry economy, Industry 1 uses £0.6 of its own product and £0.25 of commodity 2 to produce a £1 worth of commodity 1; Industry 2 uses £0.5 of its own product and £0.4 of commodity 1 to produce a £1 worth of commodity 2.

i. Write the input matrix for this economy. (10 marks)

ii. Suppose the final demands for commodity 1 and 2 are £100 and £150 respectively. Find the solution of output levels. (20 marks)

c) Solve for income Y , the interest rate r , and the exchange rate e  in the following IS-LM model of an open economy.

(50 marks)

3. A consumer has the following quasi-linear utility function

u(x, y) = 2√x + y

and a budget constraint of px x + pyy  = M , with prices px   = 1, py   = 2, and income M  =  10.

a)  Write down the utility maximisation problem and the corresponding Lagrangian function. (5 marks)

b)  Solve the first order conditions for optimisation. (15 marks)

c)  Verify the second order sufficient conditions. (15 marks)

d)  What is the maximum utility level? If income decreases by 0.2 unit, what is the approximate reduction in maximum utility level? (15 marks)

e)  Suppose x  is food consumption and the consumer is on a diet with the restriction that x  ≤ 2.25. Write down the new utility   maximization problem, and the corresponding Lagrangian function. (5 marks)

f)   Solve the new utility maximisation problem and find the new maximum utility level. (45 marks)

SECTION B

1. Find the general solution of the following differential equations

a)    ṫ(x) − x = x 2 ln t           witℎ  t  > 0 (35 marks)

b)   ̇(x) = x + 3tet (30 marks)

c)    (1 − t2 )̇(x) + 4tx  = t (35 marks)

In each case find also the particular solution(s) passing through the point (t,x)=(1,1/4)

2. Consider the logistic equation

with rand K strictly positive constants.

a)  Compare this equation with the Haavelmo’s growth model and find the parameters  condition under which the two coincide. (20 marks)

b)  Find the equilibria and then use a phase diagram to characterize geometrically (qualitatively) their stability. (20 marks)

c)  Verify that the equilibria are hyperbolic and confirm analytically their stability behaviour. (30 marks)

d)  Comment about the change in the dynamics if time is no more continuous but now a discrete variable. (30 marks)

3. Consider the following system of differential equations:

k = rk − C

ℎ = a(C − ℎ)

where k, c and h are three variables indicating physical capital, consumption and habit stock respectively; rand a are two strictly positive constants indicating the interest rate and the persistence/intensity of habits respectively. Moreover k(0)=k0 and h(0)=h0 are given.

a)  Provide an economic explanation of the two differential equations. (20 marks)

b)  Suppose that the consumption-habit   ratio, c/h, is equal to an exogenously given constant m. What can you say about the (asymptotic) growth rate of physical capital and habits? How does it change for different values of m? (40 marks)

c)  Draw the integral curves of physical capital and habit and show again how different they are by choosing different value of m. (40 marks)