Covers math needed for microeconomics FHS paper: differentiation, Concavity and Convexity, Optimisation, Comparative statics and Envelope theorem, Integration.
Prepared by a first class Economics and Management student for the FHS Microeconomics paper.
Math for Microeconomics
Topics
1. Differentiation
First and second Derivatives, Taylor Approximations, first and second partial derivatives, Total
Differentiation, Implicit Differentiation
2. Concavity and convexity
Convex sets, Definitions of Concave/Convex functions, Cardinal and Ordinal Properties, Quasi-
concavity/Quasi-convexity
3. Optimisation
3.1 Unconstrained Optimisation
o First and Second Order Conditions for Uni-variable and Multi-variable Optimisation
3.2 Optimisation with Equality Constraints
o Substitution, Tangency Conditions, Lagrange Multiplier Method
3.3 Optimisation with Positivity Constraints
o Complementary Slackness Conditions, Khun-Tucker Conditions
4. Comparative statics and Envelope theorem
Applications of Implicit Differentiation, Envelop Theorem, Solved Examples
5. Integration
Anti-derivatives and areas
How math is applied in microeconomics
Differentiation
Required to calculate marginal utility
Thus, it is needed to obtain marginal rate of substitution (ratio of marginal utilities)
Similarly for marginal product and marginal rate of transformation
Needed to obtain (price) elasticities
Concavity and convexity
Once we have obtained the marginal utilities or any first order condition, we need to assess
whether the function that is being optimised is concave or convex
This allows us to know whether we have obtained a maximum or a minimum
This makes a big difference, especially for welfare related issues!
Applications in optimisation, preferences, second welfare theorem, risk attitudes
Optimisation
Unconstrained
o A firm who wants to set their price in a way that maximises their profit
o Some constrained optimisation problems can be simplified into unconstrained problems
Equality Constraints (as opposed to inequality constraints)
o Utility maximisation problem (budget constraint!)
o Cost minimisation
Positivity Constraints
o Central bank sets policy subject to an inflation target
o Public goods
,Comparative Statics & Envelope Theorem
Comparative Statics
o Evaluation of the effect of an external shock such as government policy
Envelope Theorem
o Policy analysis, for example the e↵ects of wealth redistribution on welfare
o Micro theory: e.g. Roy’s identity
o Some Macro consumption problems, for example consumption decisions in a
multiperiod optimisation problem with discounting
Integration
Needed to calculate consumer and producer surplus
Used to obtain Gini index (measure of inequality)
Differentiation
Derivatives
Notation means that function f takes in real number arguments and produces a real number
range (values)
Derivative notation:
Sometimes derivatives are called the gradient of the function
Interpretation: tells us direction of a function (increasing/ decreasing/ stationary point) and
magnitude of slope
Tangent lines
Tangent line through the point is
Equivalent to
Sometimes, derivatives are not well defined (function is not differentiable):
, We deal mainly with differentiable functions in economics
Second derivatives
Notation
Interpretation: tells us shape of the function (if it is concave/ convex)
Taylor approximation
Sometimes we will be interested on the behaviour of a function around a particular point
In those cases, we will be able to approximate the function using polynomials (which are very
easy to work with!)
, Taylor Approximation of Degree 1
o Obtained by substituting h = x – x0 into the above equation
o
o Residual gets smaller as (x → x0) aka (h → 0). Note that the residual is extremely small,
with the limit function going to zero even as the denominator also goes to zero.
o
o Therefore, Taylor approximation of degree 1 is just the tangent equation at x 0
Taylor Approximation of Degree 2
o To get an even better approximation, we use a second order polynomial
o
o
o We can get even better approximations with higher order polynomials in this pattern
Taylor approximations only work around the point of interest and get better at higher degrees
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