Consumer and Firm Optimization, and Competitive Equilibrium with the PPF

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: $U(C, \ell) = \frac{C^{1-a}}{1-a} + \theta \frac{\ell^{1-a}}{1-a}$, where $\theta > 0$, $0 < a < 1$.

• Parameters: $a$ and $\theta$ 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:

$MRS_{\ell, C} = \theta \left( \frac{C}{\ell} \right)^a$

• Time Constraint: $N^* + \ell = h$ (where $N^*$ is labor supplied, $\ell$ is leisure, $h$ is total available time).

• Budget Constraint: $C = wN^* + \pi - T$ (where $w$ is wage, $\pi$ is profit income, $T$ is taxes).

Solving for Optimal Choices

• Combine the time and budget constraints to eliminate $N^*$:

$C = w(h - \ell) + \pi - T$

• Set $MRS_{\ell, C} = w$ for utility maximization:

$\theta \left( \frac{C}{\ell} \right)^a = w$

• Solve for $C$ in terms of $\ell$:

$C = w(h - \ell) + \pi - T$

$\theta \left( \frac{C}{\ell} \right)^a = w$

• Optimal consumption function:

$C \left[ 1 + \theta^{1/a} w^{-1/a} \right] = w h + \pi - T$

• Consumption Demand Function:

$C = \frac{w h + \pi - T}{1 + \theta^{1/a} w^{-1/a}}$

• Leisure Demand Function:

$\ell = \frac{\theta^{1/a}}{1 + \theta^{1/a} w^{-1/a}} [w h + \pi - T] w^{-1}$

• Labor Supply: $N^* = h - \ell$

Example: If $a = 0.5$, $\theta = 3$, $h = 1456$, $w = 20.475$, $\pi = 0$, $T = 0$, then $N^* = 920.75$.

2. Firm Optimization

Production Function and Profit Maximization

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

• Production Function: $Y = A K^{1-a} N^{a}$, $0 < a < 1$

• Profit Function: $\pi = Y - wN$ (assuming capital $K$ is fixed in the short run)

Marginal Product and Labor Demand

• Marginal Product of Labor (MPL):

$MP_N = (1-a)A K^{1-a} N^{a-1}$

• Profit maximization: set $w = MP_N$

• Labor Demand Function:

$N^* = \left[ \frac{(1-a) Z K^{1-a}}{w} \right]^{1/a}$

• Consumption Good Supply Function: Substitute $N^*$ into the production function:

$Y = K \left[ (1-a) \frac{Z}{w} \right]^{1/a}$

Example: If $a = 0.5$, $Z = 1$, $K = 71,000,000$, $w = 20.475$, then $N^* = 920.75$ and $Y = 24,932.5026$.

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:

• $C = Z F(K, h - \ell) - G$

• For Cobb-Douglas: $C = Z K^{1-a} (h - \ell)^a - G$

Numerical Example

• Given parameters: $\alpha = 0.3$, $a = 0.5$, $\theta = 3$, $h = 1456$

• Exogenous variables: $Z = 1$, $K = 71,000,000$, $G = 2,000,000$

• Plug into the PPF:

$C = 226.661(h - \ell)^{0.5} - 2,000,000$

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

• Spreadsheet calculations show the optimal point is $N^* = 920.75$, $Y^* = 24,932.5026$, $C^* + G = 24,932.5026$.

• Equilibrium wage: $w^* = (1-a) Z K^{1-a} / (N^*)^a = 20.475$

• Equilibrium profits: $\pi^* = Y^* - w^* N^* = 8,079.75$

Summary Table: Key Equilibrium Values

Variable	Symbol	Value
Labor Supply	$N^*$	920.75
Total Output	$Y^*$	24,932.5026
Consumption + Government	$C^* + G$	24,932.5026
Wage Rate	$w^*$	20.475
Profits	$\pi^*$	8,079.75

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.