Consumer and Firm Optimization: Key Concepts

Consumer Optimization

The budget constraint (BC) is defined as: P₁X₁ + P₂X₂ = M, where the slope is -P₁/P₂ = dX₂/dX₁. The consumer optimizes consumption decisions by reaching the highest satisfaction level given their resources. Any point on the BC other than the equilibrium is non-optimal, as it won't be on the highest indifference curve.

The Marginal Rate of Substitution (MRS) is -MU₁/MU₂, representing the slope of the indifference curve.

Firm Optimization

Profit Maximization

Profit = Py - w

Profit maximization occurs when marginal revenue (MR) equals marginal cost (MC).

• When MC = MR, the firm achieves maximum efficiency.
• When MC < MR, the firm is inefficient and should increase production.
• When MC > MR, the firm is inefficient and should reduce production as costs exceed revenue.

Iso-profit

Iso-profit is represented as y = (w₁x₁)/p + Profit + (x₂w₂)/p, with a slope of dy/dx₁ = w₁/p. The firm optimizes inputs and production to achieve the highest possible iso-profit function.

Impact of Price Changes

Suppose the output price (p) decreases, how would it affect x₁* and y*? Now suppose the factor price w₁ decreases, how would it affect x₁* and y*?

Note that x₁* depends on both output price (p) and its own factor price (w₁):

• A lower output price (p) leads to lower x₁*.
• A lower factor price (w₁) leads to higher x₁*.

Similarly, y* depends on both output price (p) and factor price (w₁):

• A lower output price (p) leads to lower y*.
• A lower factor price (w₁) leads to higher y*.

Cobb-Douglas Production Function

A Cobb-Douglas production function is: Y = L^β * K^α

• Constant Returns to Scale (CRS): Output increases proportionally to input changes (β + α = 1).
• Decreasing Returns to Scale (DRS): Output increases by less than the proportional change in inputs (β + α < 1).
• Increasing Returns to Scale (IRS): Output increases by more than the proportional change in inputs (β + α > 1).

Technical Rate of Substitution (TRS)

The Technical Rate of Substitution (TRS) is the slope of an isoquant function. It represents how much of input 1 must be given up to get more of input 2 while producing the same output (y). The TRS is negatively sloped, monotonic, and convex, with a flatter slope as you move downward in the isoquant.