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Technical Components: Math for microeconomics £6.05
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Technical Components: Math for microeconomics
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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.

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  • June 27, 2024
  • 49
  • 2022/2023
  • Lecture notes
  • Simon cowan
  • Mathematics needed for microeconomics

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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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