MFE 11 - Tips Flashcards
What are some strategic errors students make every year?
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
Summarise what was learned in ‘Week 1: From Pure Maths to Economic Maths’
- 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).
How can exp (A+B) also be expressed?
exp (A) x exp (B) NOT ln A + ln B
How can exp (A) x exp (B) also be expressed?
exp (A+B)
How can ln(AB) also be expressed?
ln A + ln B
How can ln A + ln B also be expressed?
ln(AB)
How can ln(A^n) also be expressed?
n x ln A
How can n x ln(A) also be expressed?
ln (A^n)
State the quotient rule
(u/v)’ = (vu’ - uv’)/v^2
State the chain rule
[f(g(x))]’ = g’(x)f’(g(x))
What’s the ln x formula?
ln x = The integral of dt/t with bounds of x as upper and 1 as lower
What’s ln x differentiated?
1/x
State the small increment approximation formula
C(Q+h) ~= C(Q)+htimes C’(Q)
What’s the formula for elasticity?
Also include other forms of the formula under certain conditions
- ε(QP) = P/Q * dQ/dP
- If P>0 and Q>0 this may also be written d ln Q / d ln P
Which functions have a constant rate of increase (or decrease)?
State the name & the form and show how it’s a constant rate
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
Which functions have a constant rate of growth (or decay)?
State the name & the form and show how it’s a constant rate
The exponential functions, of the form Y(t) -= A exp (bt), where A and B are parameters, and A=Y(0). Then Ychevron=b
Which functions have a constant elasticity?
The power functions, of the form Y(X)=aX^theta, where ‘a’ and theta are parameters. Then epsilonY,X = theta
Evaluate d/dx ln(107x)
1/x
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
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.
Identify the accident black spot to do with differentiating vs approximating & marginal values
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!
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?
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
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?
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
What’s a critical point?
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.
What does it mean if one is ‘radically or knightean uncertain’ about an event?
unwilling to attach probability
numbers to the event
What’s the ‘law of multiplication’ with probability?
P(AB) = P(A) x P(B|A) and P(B) x P(A|B)
If all lecturers are grumpy or lazy, and 70% are grumpy, and
60% are lazy, what proportion are grumpy and lazy?
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
Define conditional probability
P(A|B) -= P(AB) / P(B) as long as P(B) doesnt equal 0
State the Law of Multiplication in Conditional Probability
P(AB) = P(A) x P(B|A)
State the Total Probability Theorum
Include a realistic scenario where this might be used
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’
State Bayes’ Theorum of ‘inverse probability’
Include a realistic scenario where this might be used
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.
