1.5 Mathematical proofs in economics

Mathematical proofs form the foundation of rigorous economic analysis. They enable economists to construct valid arguments, derive important theoretical results, and develop robust models. Understanding different proof types is crucial for establishing key theorems and properties in economics.

This topic explores various proof techniques used in economics, including direct proofs, proof by contradiction, and proof by induction. It also covers logical reasoning, key elements of economic proofs, and applications in microeconomics and macroeconomics. Best practices for proof writing and tools for constructing proofs are also discussed.

Types of mathematical proofs

• Mathematical proofs form the foundation for rigorous economic analysis and model development
• Understanding different proof types enables economists to construct valid arguments and derive important theoretical results
• Proofs in economics often combine multiple techniques to establish key theorems and properties

Direct proofs

Top images from around the web for Direct proofs

• 

  6.3 Understanding Consumer Theory – Principles of Microeconomics View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

• 

  Consumer Choice – Introduction to Microeconomics View original

  Is this image relevant?

• 

  6.3 Understanding Consumer Theory – Principles of Microeconomics View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

1 of 3

Top images from around the web for Direct proofs

• 

  6.3 Understanding Consumer Theory – Principles of Microeconomics View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

• 

  Consumer Choice – Introduction to Microeconomics View original

  Is this image relevant?

• 

  6.3 Understanding Consumer Theory – Principles of Microeconomics View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

1 of 3

• Demonstrate the truth of a statement by logical deduction from given premises
• Proceed step-by-step from known facts to reach the desired conclusion
• Often used to prove basic properties in consumer and producer theory
• Involve algebraic manipulation, application of definitions, and logical inference
• Strength lies in clarity and straightforward reasoning

Proof by contradiction

• Assume the opposite of what needs to be proven is true
• Derive a logical contradiction from this assumption
• Conclude that the original statement must be true
• Particularly useful for proving the non-existence of certain economic equilibria
• Can reveal deep insights about economic models by exploring counterfactuals

Proof by induction

• Consists of a base case and an inductive step
• Base case proves the statement for an initial value (often n=1)
• Inductive step shows that if the statement holds for k, it also holds for k+1
• Concludes the statement is true for all natural numbers
• Frequently used in dynamic economic models and recursive formulations
• Allows economists to generalize results across time periods or agent populations

Proof by contrapositive

• Proves the logically equivalent contrapositive statement instead of the original
• Original statement "If P, then Q" becomes "If not Q, then not P"
• Often simplifies the proof process for certain types of economic propositions
• Useful in game theory to prove strategy dominance or equilibrium properties
• Helps economists analyze the implications of violating certain conditions

Logical reasoning in proofs

• Logical reasoning forms the backbone of mathematical proofs in economics
• Enables economists to construct valid arguments and derive sound conclusions
• Critical for developing theoretical models and analyzing empirical evidence

Premises and conclusions

• Premises represent the starting assumptions or given information in a proof
• Conclusions are the logical consequences derived from the premises
• Identifying clear premises ensures the proof's foundation is well-established
• Drawing valid conclusions requires careful reasoning and avoiding logical leaps
• Economic models often start with premises about agent behavior or market structure

Validity vs soundness

• Validity refers to the logical correctness of an argument's structure
• A valid argument guarantees true conclusions if all premises are true
• Soundness requires both validity and true premises
• Economists must ensure their proofs are valid and based on realistic assumptions
• Critiquing economic arguments often involves examining both validity and soundness

Common logical fallacies

• Circular reasoning uses the conclusion as a premise
• Ad hominem attacks the person rather than the argument
• False dichotomy presents only two options when more exist
• Post hoc ergo propter hoc assumes correlation implies causation
• Hasty generalization draws broad conclusions from limited evidence
• Recognizing these fallacies helps economists construct stronger proofs and critique existing ones

Proof techniques in economics

• Proof techniques in economics combine mathematical rigor with economic intuition
• Allow economists to establish key properties of models and theoretical results
• Different techniques serve various purposes in economic analysis and model development

Existence proofs

• Demonstrate that a particular economic object or solution exists
• Often use fixed-point theorems (Brouwer, Kakutani) to prove equilibrium existence
• May involve constructive methods or abstract existence arguments
• Critical for establishing the validity of economic models and solution concepts
• Include proofs of market clearing prices or Nash equilibria in game theory

Uniqueness proofs

• Show that a solution or equilibrium is the only one satisfying given conditions
• Often use proof by contradiction or direct methods
• Important for ensuring determinacy in economic models
• Establish the stability and predictability of economic outcomes
• Involve proving properties like single-crossing conditions or contraction mappings

Constructive vs non-constructive proofs

• Constructive proofs provide a method to find or construct the object being proved
• Non-constructive proofs show existence without providing a specific example
• Constructive proofs often more useful for applied economic analysis
• Non-constructive proofs can establish theoretical results when explicit solutions are difficult
• Choice between methods depends on the economic question and available mathematical tools

Key elements of economic proofs

• Economic proofs combine mathematical rigor with economic intuition and interpretation
• Understanding these elements helps in constructing and analyzing economic arguments
• Proper use of these components ensures clarity and validity in economic reasoning

Assumptions and axioms

• Represent fundamental starting points for economic models and proofs
• Include rationality assumptions, market structure, information sets
• Must be clearly stated and justified in the context of the economic problem
• Often based on empirical observations or theoretical simplifications
• Critical for determining the scope and applicability of economic results

Definitions and notation

• Provide precise meanings for economic concepts and variables
• Establish a common language for communicating ideas and results
• Include formal mathematical definitions (utility functions, production sets)
• Use standard economic notation ($\pi$ for profit, $U$ for utility)
• Clarity in definitions prevents misunderstandings and ambiguities in proofs

Theorems and lemmas

• Theorems represent major results or conclusions in economic theory
• Lemmas are smaller supporting propositions used to prove larger theorems
• Structure proofs by breaking complex arguments into manageable steps
• Often build on previous results to develop more advanced economic insights
• Examples include the Fundamental Theorems of Welfare Economics or Envelope Theorem

Applications in microeconomics

• Microeconomic proofs focus on individual decision-making and market interactions
• Establish key properties of consumer behavior, firm decisions, and market outcomes
• Combine mathematical techniques with economic intuition to derive important results

Consumer theory proofs

• Prove properties of utility functions (continuity, monotonicity, convexity)
• Derive demand functions from utility maximization
• Establish Slutsky equation decomposing price effects
• Prove existence and uniqueness of consumer equilibrium
• Demonstrate properties of indirect utility and expenditure functions

Producer theory proofs

• Prove properties of production functions (returns to scale, marginal products)
• Derive cost functions and supply curves
• Establish duality between production and cost functions
• Prove profit maximization conditions for competitive firms
• Demonstrate properties of factor demand functions

Equilibrium proofs

• Prove existence and uniqueness of competitive equilibrium
• Establish First and Second Welfare Theorems
• Demonstrate properties of oligopoly equilibria (Cournot, Bertrand)
• Prove existence of general equilibrium using fixed-point theorems
• Analyze stability and comparative statics of market equilibria

Applications in macroeconomics

• Macroeconomic proofs deal with aggregate economic phenomena and long-run growth
• Establish properties of dynamic models and policy implications
• Often involve intertemporal optimization and dynamic programming techniques

Growth model proofs

• Prove existence and uniqueness of steady-state equilibrium in Solow model
• Demonstrate convergence properties in neoclassical growth models
• Establish optimal savings rates in Ramsey-Cass-Koopmans model
• Prove existence of balanced growth path in endogenous growth models
• Analyze transitional dynamics and speed of convergence

Business cycle proofs

• Prove properties of DSGE models (existence, uniqueness of equilibrium)
• Demonstrate impulse response functions to various shocks
• Establish conditions for determinacy in New Keynesian models
• Prove existence of sunspot equilibria in models with self-fulfilling expectations
• Analyze persistence and amplification mechanisms in RBC models

Monetary policy proofs

• Prove neutrality of money in classical models
• Demonstrate effectiveness of monetary policy in New Keynesian frameworks
• Establish optimal monetary policy rules under commitment and discretion
• Prove existence and uniqueness of equilibrium under various policy regimes
• Analyze time consistency issues and inflation bias in monetary policy

Proof writing best practices

• Effective proof writing is crucial for communicating economic ideas and results
• Following best practices ensures clarity, rigor, and persuasiveness in economic arguments
• Helps readers understand and verify the logical steps in economic reasoning

Structure and organization

• Begin with a clear statement of the theorem or proposition to be proved
• Outline the proof strategy before diving into details
• Use a logical flow of ideas, building from simpler to more complex steps
• Divide long proofs into lemmas or intermediate results
• Conclude with a clear statement that the desired result has been established

Clarity and conciseness

• Use precise mathematical language and economic terminology
• Explain key steps and intuition behind complex manipulations
• Avoid unnecessary detours or redundant information
• Balance between providing enough detail and maintaining readability
• Use appropriate notation and define all variables and functions clearly

Common pitfalls to avoid

• Circular reasoning or assuming what needs to be proved
• Skipping important steps or making unjustified leaps in logic
• Misusing mathematical symbols or economic concepts
• Overgeneralizing results beyond their valid scope
• Failing to address all cases or conditions in the proof

Tools for economic proofs

• Economic proofs often require a diverse set of mathematical tools
• Mastery of these tools enables economists to tackle complex theoretical problems
• Understanding when and how to apply these tools is crucial for effective proof construction

Set theory in proofs

• Use set operations to define economic concepts (budget sets, production possibilities)
• Prove properties of preference relations using set-theoretic notions
• Apply fixed-point theorems to prove existence of equilibria
• Utilize set-valued functions in general equilibrium theory
• Employ measure theory in advanced microeconomic and financial models

Calculus in proofs

• Use differentiation to derive first-order conditions for optimization
• Apply integration to calculate consumer and producer surplus
• Employ multivariable calculus for constrained optimization problems
• Utilize differential equations in dynamic economic models
• Use calculus of variations in optimal control problems

Linear algebra in proofs

• Apply matrix operations in input-output models and production theory
• Use eigenvectors and eigenvalues in dynamic stability analysis
• Employ linear programming techniques in resource allocation problems
• Utilize vector spaces to represent complex economic systems
• Apply linear transformations in econometric proofs and estimation techniques

Critiquing and evaluating proofs

• Critical evaluation of economic proofs is essential for advancing theoretical knowledge
• Helps identify strengths, weaknesses, and potential improvements in economic arguments
• Develops skills in logical reasoning and economic intuition

Identifying gaps in logic

• Look for missing steps or unjustified assumptions in the proof
• Check if all cases and conditions are adequately addressed
• Verify that conclusions follow logically from premises and intermediate steps
• Examine the validity of mathematical operations and transformations
• Ensure that economic interpretations are consistent with mathematical results

Assessing assumptions

• Evaluate the realism and relevance of economic assumptions
• Consider the sensitivity of results to changes in key assumptions
• Examine the generalizability of assumptions to different economic contexts
• Assess whether simplifying assumptions overly restrict the model's applicability
• Consider alternative assumptions that might lead to different conclusions

Recognizing circular reasoning

• Identify instances where the conclusion is implicitly assumed in the premises
• Check for hidden assumptions that may lead to circular arguments
• Examine definitions to ensure they don't presuppose what needs to be proved
• Verify that each step in the proof builds on previously established results
• Consider whether the proof genuinely adds new information or merely restates assumptions