MFE 11 - Tips

Question 1

What are some strategic errors students make every year?

Answer 1

Not enough hours at the coal face.
2. Grossly uneven distribution of effort over time. Spread your effort
equally from October right through to early January.
3. Over-emphasis on some Weeks, under-emphasis on others. All
Weeks (Weeks 6-10) carry equal weight in the exam. The low-
hanging fruit lie just on the border of your current comfort zone.
4. Neglect of definitions, concepts, and terminology, in favour of
algorithms. There are many marks for both.
5. Neglect of explanation, in favour of getting the right answer

Question 2

Summarise what was learned in ‘Week 1: From Pure Maths to Economic Maths’

Answer 2

• Treat numbers with understanding, as a human speaking to fellow humans, taking account of what job the number is doing, and how to
  communicate it.
• Be aware of units of measurement; they will help you to achieve the objective above, and to avoid writing nonsense.
• Mathematical functions are input-output machines and they have a central role in the maths of economics.
• Remember the special functions exp and ln
• Functions of two variables cannot be graphed on two-dimensional
  sheets of paper, but we can represent them using contour maps; this works with the height function h(x,y) , a production function F(K,L), and a utility function U(A,B).

Question 3

How can exp (A+B) also be expressed?

Answer 3

exp (A) x exp (B) NOT ln A + ln B

Question 4

How can exp (A) x exp (B) also be expressed?

Answer 4

exp (A+B)

Question 5

How can ln(AB) also be expressed?

Answer 5

ln A + ln B

Question 6

How can ln A + ln B also be expressed?

Answer 6

ln(AB)

Question 7

How can ln(A^n) also be expressed?

Answer 7

n x ln A

Question 8

How can n x ln(A) also be expressed?

Answer 8

ln (A^n)

Question 9

State the quotient rule

Answer 9

(u/v)’ = (vu’ - uv’)/v^2

Question 10

State the chain rule

Answer 10

[f(g(x))]’ = g’(x)f’(g(x))

Question 11

What’s the ln x formula?

Answer 11

ln x = The integral of dt/t with bounds of x as upper and 1 as lower

Question 12

What’s ln x differentiated?

Answer 12

1/x

Question 13

State the small increment approximation formula

Answer 13

C(Q+h) ~= C(Q)+htimes C’(Q)

Question 14

What’s the formula for elasticity?

Also include other forms of the formula under certain conditions

Answer 14

• ε(QP) = P/Q * dQ/dP
• If P>0 and Q>0 this may also be written d ln Q / d ln P

Question 15

Which functions have a constant rate of increase (or decrease)?

State the name & the form and show how it’s a constant rate

Answer 15

The so-called ‘linear’ functions [strictly speaking these are not ‘linear’ by the official definition (except when a=0), even though their graphs are straight lines. They are ‘affine’], of the form Y(t)-=a+bt, where and are parameters. Then Y(t)’=b

Question 16

Which functions have a constant rate of growth (or decay)?

State the name & the form and show how it’s a constant rate

Answer 16

The exponential functions, of the form Y(t) -= A exp (bt), where A and B are parameters, and A=Y(0). Then Ychevron=b

Question 17

Which functions have a constant elasticity?

Answer 17

The power functions, of the form Y(X)=aX^theta, where ‘a’ and theta are parameters. Then epsilonY,X = theta

Question 18

Evaluate d/dx ln(107x)

Answer 18

1/x

Question 19

If ur ever asked to show why d/dx ln ax = 1/x, what’s the best way to do so?

Show how this method proves the point

Answer 19

The easiest way to work it out is to use the rule ln AB = ln A + ln B:
ln Ax = ln A + ln x, so
d/dx ln Ax = d/dx (ln A + ln x) = 0 {remember that ln A would be a fixed number} + 1/x = 1/x.

Question 20

Identify the accident black spot to do with differentiating vs approximating & marginal values

Answer 20

Given an economic function like C(Q), the cost of producing Q …
… the marginal cost is defined as the derivative C’(Q).
This may be approximated by C(Q+1) - C(Q)…
… which is sometimes used as the definition of marginal cost when the concept is discussed at an elementary level.
But do not take the approximation for the definition!

Question 21

I have an asset whose growth rate is constant, at Achevron=0.1, where
time is measured in years. If the asset is worth $1000 now, how
much is it worth in one year’s time?

What would student A do vs Student Q?

Answer 21

Student A: The rate of growth is (A(1)-A(0))/A(0) = 0.1, so A(1) = 1.1A(0) = $1100
Student Q. That’s a possible way to measure growth, but it isn’t
what’s meant by Achevron.
If Achevron = 0.1 for all t, then to solve the equation 1/A(t) * dA/dt=0.1, A(t) must have the exponential path A(t)=A(0)exp(0.1t).
That implies that A(1) = A(0) exp(0.1) ~= $1105.17

Question 21

If there is no investment, the value of the
capital stock has a constant growth rate of -0.15. That is, Kchevron = -0.15 where time is measured in years. If K now stands
at $1 tn, and there is no investment, what will K be in one year?

Answer 21

Students A - P, R – Z: ‘In one year, will be $1 x (1-0.15) = $0.85 tn
Student Q: ‘If K has a constant growth rate (as opposed to constant rate of increase), it must take the form K(t) = K(0) exp (Kchevront), where kchevron = -0.15.
So K(t+1) = K(0) exp (-0.15(t+1)) = K(0) exp (-0.15t-0.15)
Since K(t) = 1 tn, we get K(t+1) = exp (-0.15) ~= 0.861 tn

Question 21

What’s a critical point?

Answer 21

A critical point of a function f:R->R of a single variable is a value x_* at which the function is flat; that is, f’(x)=0.
A critical point of a function f:R^n -> R of several variables is a point x_ at which the function is flat; that is, [df/dxi]x_=x_* for all such that 1=< i >= n
If we introduce the abbreviation f𝒊=- df/dxi, we can abbreviate this as:
x_* is a critical point for f:R^n -> R if fi(x_*) for all 𝒊 such that 1=< i >= n
That is, if all n FOC hold.

Question 22

What does it mean if one is ‘radically or knightean uncertain’ about an event?

Answer 22

unwilling to attach probability
numbers to the event

Question 23

What’s the ‘law of multiplication’ with probability?

Answer 23

P(AB) = P(A) x P(B|A) and P(B) x P(A|B)

Question 24

If all lecturers are grumpy or lazy, and 70% are grumpy, and
60% are lazy, what proportion are grumpy and lazy?

Answer 24

Law of Addition: P(GnL) = P(G) + P(L) - P(GL)

Since ‘all lecturers are grumpy or lazy’ we have P(GUL) = 1, so 1 = 0.7 + 0.6 - P(GL) = 1.3 - P(GL) -> P(GL) = 0.3

Question 25

Define conditional probability

Answer 25

P(A|B) -= P(AB) / P(B) as long as P(B) doesnt equal 0

Question 26

State the Law of Multiplication in Conditional Probability

Answer 26

P(AB) = P(A) x P(B|A)

Question 27

State the Total Probability Theorum

Include a realistic scenario where this might be used

Answer 27

P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + P(A|B3)P(B3)

*Think of a specific example, for instance A -= ‘loan is repaid’,
B1, B2, B3 -= ‘economy does well / badly / terribly’

Question 28

State Bayes’ Theorum of ‘inverse probability’

Include a realistic scenario where this might be used

Answer 28

P(Bx|A) [= P(A|Bx)P(Bx) / P(A)] = P(A|Bx)P(Bx) / P(A|B1)P(B1) +… P(A|Bx)(P(Bx)

For instance, the probability that the economy did terribly, given that
the loan was repaid.
Remember that in Bayes’ theorem we have the TPT in the
denominator, and one term of the denominator in the numerator.