Intermediate Microeconomics with Calculus

Introduction

Intermediate microeconomics extends basic economic principles by introducing mathematical tools, particularly calculus, to analyze consumer behavior, production decisions, and market outcomes. This mathematical framework allows economists to derive more precise predictions about economic phenomena and better understand the marginal relationships that underpin economic decisions.

Consumer Theory

Consumer theory examines how individuals allocate their limited income among various goods and services to maximize their satisfaction or utility. Calculus plays a crucial role in modeling this optimization problem.

Utility Maximization

A consumer's utility function U(x, x) represents their satisfaction from consuming goods x and x. The consumer faces a budget constraint I = px + px, where I is income and p, p are prices.

The utility maximization problem can be expressed as:

Using the Lagrangian approach:

The first-order conditions are:

Price Elasticity of Demand

Price elasticity measures the responsiveness of quantity demanded to price changes. Using calculus, we define price elasticity of demand as:

A perfectly elastic demand has (horizontal demand curve), while a perfectly inelastic demand has = 0 (vertical demand curve).

Production Theory

Production theory examines how firms combine inputs to produce outputs, with a focus on cost minimization and profit maximization.

Production Functions

A production function Q = f(K, L) relates output quantity (Q) to inputs of capital (K) and labor (L). The marginal products are:

The marginal rate of technical substitution (MRTS) measures the rate at which one input can be substituted for another while maintaining output:

Cost Minimization

Firms seek to minimize costs for a given output level. This problem can be expressed as:

Using the Lagrangian method: C = rK + wL + [Q - f(K, L)]

The first-order conditions yield:

Dividing these equations gives the cost-minimization condition:

Profit Maximization

A competitive firm maximizes profit = PQ - C(Q), where P is the market price of output. The first-order condition is:

This implies that profit maximization occurs at the output level where price equals marginal cost. For a monopoly, the problem becomes:

The first-order condition gives:

Market Equilibrium

Market equilibrium analysis examines how supply and demand interact to determine prices and quantities.

Supply and Demand Analysis

Mathematically, a market equilibrium occurs where quantity demanded equals quantity supplied:

Producer and Consumer Surplus

Consumer surplus is the difference between what consumers are willing to pay and what they actually pay:

Producer surplus is the difference between the market price and the minimum price at which producers would supply:

General Equilibrium Theory

General equilibrium theory examines the simultaneous equilibrium in all markets. The Edgeworth box is a useful tool for analyzing exchange efficiency in a two-person, two-good economy.

Pareto Efficiency

An allocation is Pareto efficient if no one can be made better off without making someone worse off. In an exchange economy, the contract curve shows all Pareto efficient allocations:

Welfare Economics

Welfare economics evaluates the economic well-being of individuals and society as a whole.

Social Welfare Functions

A social welfare function W(U, U, ..., U_n) combines individual utilities into a social welfare measure. Common forms include:

• Utilitarian: W = U_i
• Rawlsian: W = min(U, U, ..., U_n)
• Benthamite: W = w_iU_i (weighted utilitarian)

Conclusion

Intermediate microeconomics with calculus provides a robust framework for analyzing economic decisions and market outcomes. The mathematical approach allows for more precise predictions and policy analysis in both individual markets and the broader economy. The concepts of optimization, marginal analysis, and equilibrium form the foundation of modern economic theory and its applications to real-world problems.