1. Consumer Optimization

Utility Maximization

Consumer optimization involves choosing consumption and leisure to maximize utility, subject to time and budget constraints. The representative consumer's utility function is:

• Utility Function: , where , .

• Parameters: and reflect preferences, tastes, and personality.

• Marginal Rate of Substitution (MRS): The rate at which the consumer is willing to trade leisure for consumption, given by:

• Time Constraint: (where is labor supplied, is leisure, is total available time).

• Budget Constraint: (where is wage, is profit income, is taxes).

Solving for Optimal Choices

• Combine the time and budget constraints to eliminate :

• Set for utility maximization:

• Solve for in terms of :

• Optimal consumption function:

• Consumption Demand Function:

• Leisure Demand Function:

• Labor Supply:

Example: If , , , , , , then .

2. Firm Optimization

Production Function and Profit Maximization

The representative firm chooses labor to maximize profits, given a Cobb-Douglas production function:

• Production Function: ,

• Profit Function: (assuming capital is fixed in the short run)

Marginal Product and Labor Demand

• Marginal Product of Labor (MPL):

• Profit maximization: set

• Labor Demand Function:

• Consumption Good Supply Function: Substitute into the production function:

Example: If , , , , then and .

3. Solving for the Competitive Equilibrium Outcome using the PPF - A Numerical Example

Production Possibility Frontier (PPF)

The PPF shows the maximum possible output combinations of two goods (here, consumption and leisure) given resources and technology. The general formula is:

• For Cobb-Douglas:

Numerical Example

• Given parameters: , , ,

• Exogenous variables: , ,

• Plug into the PPF:

• Optimal bundle is where (the slope of the PPF equals the marginal rate of substitution).

• Spreadsheet calculations show the optimal point is , , .

• Equilibrium wage:

• Equilibrium profits:

Summary Table: Key Equilibrium Values

Additional info: The notes provide a step-by-step derivation of consumer and firm optimization, and how these interact to determine the competitive equilibrium in a simple macroeconomic model. The numerical example illustrates the use of the PPF and equilibrium conditions in practice.