1 - Mathematical Basics

Question 1

How does the Lagrangian of

f (x, y) s.t. g (x, y) = c

look like?

Answer 1

ℒ (x, y, λ) = f (x, y) - λ [g (x, y) - c]

Question 2

Why do we use the Lagrangian method in economics?

Answer 2

• The Lagrangian method is a method of constrained optimization.
• In economic theory, especially in micro-founded models, we often assume that agents optimize an objective function (e.g. utility, profit, cost function,…). Usually agents face constraints that have to be taken into account in the maximization problem.
• The Lagrangian method can deal with such optimization problems.

Question 3

What is the intuition behind setting up the Lagrangian?

Answer 3

• Since you are maximizing one function subject to another function as constraint, you are looking for the point where both are tangent to each other (=maximizes the function).
• If they are tangent, their gradients are parallel (but not the same length -> λ is that scaling factor).
• Therefore: ∇ f (x, y) = λ ∇ g( x, y)

Question 4

What is the interpretation of the optimized lagrange multiplier λ*?

Answer 4

It is the rate of change for the maximum value.

I.e. if the constraint is relaxed by 1 unit, the maximum value increases by λ* units.

Question 5

What is the Taylor approximation?

Answer 5

A Taylor series approximation uses a Taylor series to represent a number as a polynomial that has a very similar value to the number in a neighborhood around a specified x value.

Question 6

What is the formula used to approximate a function f(x) near point x₀?

Answer 6

The Taylor approximation formula to approximate f(x) near point x₀ is:

Question 7

What is the product rule? (when differentiating)

Answer 7

( fg )′ = f′g + fg′

Question 8

What is the quotient rule? (when differentiating)

Answer 8

( f / g )′ = [f′g − fg′] / g²

Question 9

What is the chain rule? (when differentiating)

Answer 9

f[g(x)] = f’ [g (x)] ⋅ g’ (x)

Question 10

Use the product rule for logarithms to transform the following function:

ln(xy) = ?

Answer 10

ln(xy) = ln(x) + ln(y)

Question 11

Use the quotient rule for logarithms to transform the following function:

ln(x/y) = ?

Answer 11

ln(x/y) = ln(x) − ln(y)

Question 12

Use the log of power rule for logarithms to transform the following function:

ln(xʸ) = ?

Answer 12

ln(xʸ) = y ln(x)

Question 13

ln(e) = ?

Answer 13

ln(e) = 1

Question 14

ln(1) = ?

Answer 14

ln(1) = 0

Question 15

ln(1/x) = ?

Answer 15

ln(1/x) = − ln(x)

Question 16

What is log-linearization?

Answer 16

Log-linearization is a first-order Taylor expansion, expressed in percentage terms rather than in levels differences.

In Economics, since units are not always well defined or consistent, we prefer to think in terms of percentage deviations from reference values (which will usually be the steady-state of a model)

Question 17

What is the the cookbook procedure for log-linearizing? (3 steps)

Answer 17

1. Take logs
2. Do a first order Taylor series expansion about a point (usually a steady state)
3. Simplify so that everything is expressed in percentage deviations from steady state

Question 18

Why do we use log-linearization in the field of macroeconomics?

Answer 18

1. Log-linearization = way to linearize a model. Solving a nonlinear model can be very difficult and is sometimes impossible.
2. Usually, we analyze the effects of small shocks to the model, therefore we can linearize around the steady state (with small approximation error) i.e., the error term for impulse response functions, and for second moments of macroeconomic variables is (almost) insignificant.
3. log-linearization is popular, because the resulting variables will be “unit-free”, i.e., the resulting log-deviations are percentage deviations from steady state.

Question 19

What is the differentiation of ln(x)?

Answer 19

Question 20

How else can X̂ₜ be written mathematically?

Answer 20

X̂ₜ is the log deviation of variable Xₜ from its steady state X:

= ln(Xₜ) - ln(X) = ln(Xₜ / X)

≈ (Xₜ - X) / X

Question 21

What does X̂ₜ stand for (in words)?

Answer 21

It is the log deviation (= natural log) of variable Xₜ from its steady state X.

Question 22

Why can the expectation be rewritten like this?

Answer 22

Because an expectation is a probability weighted sum, which in this case we have represented with the sum of all states of the world multiplied by their respective probability of occuring.

Question 23

What is the law of iterated expectations?

Answer 23

In period (t+1) there is a bigger information set than in t, and the LIE implies that the smaller information set always dominates.

Question 24

What is the formula for infinite geometric series?

Answer 24