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# Calculus-II

Course Code: CAL-211 Title: Calculus-II Credit Hours: 03 Prerequisite: Calculus-I

## Objectives:

To prepare the students with the essential tools of calculus to apply the concepts and the techniques in Economics.

## Partial & Total Differentiation

• Partial Differentiation.
• Marginal Physical Product of Labor and Capital.
• Marginal Utility Functions.
• Money Market Analysis, Demand for Money Function Analysis using Partial Derivatives.
• Income Elasticity of Demand and Cross Price Elasticity of Demand using Partial Derivatives.

## Second and Higher Order Derivatives.

• Profit Maximization Condition using Second Derivative.
• Young’s Theorem.
• Partial Derivatives Application on Three Input Production Function Differentials versus Derivatives.
• Point Elasticity using Differentials, Income and Price Elasticity of Demand using Differentials.

## Concept of Total Differentials.

• Savings Function and Total Differentials.
• Specific Utility Function and Total Differentials.

## Concept of Total Derivatives.

• Specific Production Function with time-dependent Labor and Capital.
• Exponential Functions and Growth.
• Rules of Differentiation of Exponential and Logarithmic Functions.
• Optimal Timing: A Problem of Wine Storage, A Problem of Timber Cutting, A Problem of Land Purchase for Speculation.
• Finding the Rate of Growth using Exponential and Logarithmic Functions.
• Point Elasticity.
• Rate of Growth of Per Capita Employment. Rate of Growth of Profit.

## Optimization

Concept of Optimization:

## Calculus Approach to Optimization

• 1st Order and 2nd Order Conditions.
• Average Cost Minimization Analysis.

## Matrix Approach to Optimization (2nd Order)

• Hessian Test
• Profit Maximization Analysis
• Profit Maximization of Technically Related Goods
• Monopolistic Firm Producing Related Goods
• Firm Producing Substitute Goods
• Short Run Production Function Analysis

## Optimization of More than one Choice Variable

• Economic Application
• Multi-product Firm
• Multi-plant Firm
• Price Discrimination by Monopoly
• Price Discrimination by Monopsony
• Input Decision of a Firm
• Profit Maximization of Two-Product Firm.

## Comparative-Static Aspects of Optimization

• Rationale for Constrained Optimization.
• Finding Stationary Values using Substitution-Elimination Method.
• Finding Stationary Values Method of Lagrange Multiplier.
• Interpretation of the Lagrange Multiplier.
• Second Order Condition: The Bordered Hessian. Convexity and Concavity using Second Order Derivative.
• Economic Applications:
• Utility Maximization
• Law of Equi-marginal Utility using Lagrangian Multiplier
• Production Function Maximization
• Logarithmically Transformed Production Function.

## Integral Calculus

• Dynamics and Integration.
• The Nature of Indefinite Integrals.
• Basic Rules of Integration.
• The Substitution Rule
• Rule of Integration by Parts.
• Economic Applications of Integrals:
• Finding Total Functions from Marginal Functions
• Investment & Capital Formation
• Utility Function
• Definite Integrals
• Major Properties of Definite Integral
• A Definite Integral as an Area Under a Curve
• Improper Integrals

## Consumer Analysis using Integrals

• Applications of Definite Integrals:
• Consumer’s Reservation Price using Differential Equations
• Consumer Willingness to Pay using Integrals
• Consumer Surplus using Integrals
• Differential of Consumer Surplus using Integrals
• Producer & Social Surplus using Integrals
• Producer Surplus with Tax using Integrals.

## Firm Related Microeconomic Applications of Integrals

• Wine Storage Problem with Storage Costs.
• Breakeven Analysis of a Firm. Scale of Production Analysis.
• Mobile Production Analysis.
• Firm’s Average Profit Analysis.
• Firm’s Average Cost Analysis.
• Average Production Analysis.
• Average Inventory Analysis.
• Profit over Useful Life of Machine.
• Oil Extraction Analysis, Dynamics of Labour Cost.

## Miscellaneous Microeconomic Applications of Integrals

• Efficiency Analysis using Integrals.
• Cost Effectiveness Decision using Differential Equations.
• Depreciation Analysis using Integrals.
• Retirement Annuity Analysis using Integrals.
• Wage Differential using Integrals.
• Present Value of Cash Flow.
• Present Value of a Perpetual Flow.

## Macroeconomic Applications of Integrals

• Gini Index using Integrals.
• Depletion of Energy Resources Analysis using Integrals.
• Present Value of Gold Mine using Integrals.
• Balance of Trade Analysis using Integrals.
• Demand Analysis of Oil using Integrals.
• Demand Analysis of Oil using Integrals.
• Domar Growth Model: Framework, Solution and Numerical.

## Recommended Books:

• Chiang A.C. Fundamental Methods of Mathematical Economics McGraw Hill –Latest
• Weber E. Jean, Mathematical Analysis, Business and Economic Application (latest edition).
• Bradley, T. Essential mathematics for economics and business. Latest edition. John Wiley & Sons.
• Dowling, E. T. Theory and Problems of Introduction to Mathematical Economics. Latest Edition, McGraw-Hill, New York.
• Hoffman, L. D., & Bradley, G. L. Calculus for business, economics, and the social and life sciences. McGraw-Hill. Latest edition.
• Kolman, B., & Beck, R. E. (1995). Elementary linear programming with applications. Academic Press.
• Sydsæter, K. and Hammond, P. Essential Mathematics for Economic Analysis. Latest edition., Pearson Education Limited, England.