Intermediate Macroeconomics

1
Intermediate Macroeconomics

• Natalya Brown

2
Topics Covered

• National Income Accounting
• Long-run Economic Growth
• Money, Prices and Asset Markets
• Business Cycle Theory
• The IS-LM Model
• Exchange Rates and International Trade
• Monetary and Fiscal Policy

3
Mathematical Tools

• Review

4
Overview

• Derivatives
• Unconstrained Optimization
• Constrained Optimization
• Series
• Functional Forms
• Elasticities
• Exponents

5
Derivatives
The (first) derivative of a function y f(x) at
x0 is denoted
Higher-order derivatives are denoted
The nth derivative of the function f(x) is
derived by differentiating the function n times.
6
Rules of Differentiation
Power Rule
Addition Rule
Product Rule
7
Rules of Differentiation (Contd)
Quotient Rule
Chain Rule
Inverse Rule
8
Rules of Differentiation (Contd)
(Natural) Log Rule
Exponential Rule
Index Rule
9
Multivariate Differentiation
Suppose that y is a function of several
variables. y f(x1,x2,,xn)
The partial derivative of f with respect to xi is
denoted
All other variables are held constant.
Example The Marginal Product of Labour
Suppose
then
10
Cross-partial Derivatives
This is found by first taking the derivative of
the function with respect to x1, then taking the
derivative of the result with respect to x2. The
order of differentiation does not matter.
11
Unconstrained Optimization
Given a function y f(x), you may be asked to
find the value of x maximize or minimize the
value of the function.
First-Order (Necessary Condition)
If the function is differentiable, this
conditions must be satisfied at a maximum or
minimum.
12
Second-Order (Sufficient) conditions
So the function has a maximum at x1/2.
Beware that you may be at a local maximum/minimum
and not a global max or minimum. It is always a
good idea to know the shape of the function.
13
Constrained Optimization
Given a function z f(x,y), you may be asked to
find the values of x and y that maximize or
minimize the value of the function subject to a
constraint g(x,y) c.
The problem
Lagrangian Function
14
First-Order (Necessary Conditions)
Second-Order (Sufficient) conditions
The nature of the functions that we will use
throughout will ensure that SOCs are satisfied.
We will not check SOCs.
15
Series
A sequence of terms

• Arithmetic
• There is a constant difference, d, between
  successive terms
• Geometric
• There is a constant ratio, r, between successive
  terms

16
Sum of Series

• Finite
• Sum of an Arithmetic Series
• Sum of a Geometric Series
• Infinite Sum of a Geometric Series

17
Functional Forms

• Often used functions
• Cobb-Douglas
• Leontief
• Quasilinear

18
Elasticities

• The elasticity of y with respect to x is equal to

19
Exponents
If Then the elasticity of y with respect to x
is equal to ß.