# Cobb-Douglas Utility Maximization

## Utility Function (in the form x^{α}y^{1-α} or αlog(x)+(1-α)log(y))

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

If necessary, apply a monotonic transformation so the utility function is in the form αlog(x)+(1-α)log(y).

Lagrangian:

First order conditions:

Solving the system:

# Income and Substitution Effects (Cobb-Douglas), Slutsky Compensation

## Utility Function (in the form x^{α}y^{1-α} or αlog(x)+(1-α)log(y))

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky transfer is ___ dollars.

If necessary, apply a monotonic transformation so the utility function is in the form αlog(x)+(1-α)log(y).

Original Lagrangian:

First order conditions:

Solving the system:

To calculate the substitution effect using Slutsky compensation, we must find the bundle that the individual would purchase if given enough income to afford the original bundle at the final prices.

Next, with the new income and the final prices, we set up another Lagrangian:

First order conditions:

Solving the system:

Next, to find the final bundle, we use the final income and the final prices to set up another Lagrangian:

First order conditions:

Solving the system:

# Endowment Economies (Cobb-Douglas), Slutsky Compensation

## Utility Function (in the form x^{α}y^{1-α} or αlog(x)+(1-α)log(y))

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

If necessary, apply a monotonic transformation so the utility function is in the form αlog(x)+(1-α)log(y).

Original Lagrangian (the income is the amount the individual receives from selling the endowments at their original prices):

First order conditions:

Solving the system:

To calculate the substitution effect using Slutsky compensation, we must find the bundle that the individual would purchase if given enough income to afford the original bundle at the final prices.

First order conditions:

Solving the system:

To find the income effect, we set up another Lagrangian using the final prices and the income the individual receives from selling the endowments at their original prices:

First order conditions:

Solving the system:

Next, to find the final bundle, we use the final prices and the income obtained from selling the endowment at the final prices to set up another Lagrangian:

First order conditions:

Solving the system:

# Quasilinear Utility Maximization

## Utility Function (in the form alog(x)+by)

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

Lagrangian:

First order conditions:

Solving the system:

# Income and Substitution Effects (Quasilinear), Slutsky Compensation

## Utility Function (in the form alog(x)+by)

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky transfer is ___ dollars.

Original Lagrangian:

First order conditions:

Solving the system:

To calculate the substitution effect using Slutsky compensation, we must find the bundle that the individual would purchase if given enough income to afford the original bundle at the final prices.

Next, with the new income and the final prices, we set up another Lagrangian:

First order conditions:

Solving the system:

Next, to find the final bundle, we use the final income and the final prices to set up another Lagrangian:

First order conditions:

Solving the system:

# Endowment Economies (Quasilinear), Slutsky Compensation

## Utility Function (in the form alog(x)+by)

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

Original Lagrangian (the income is the amount the individual receives from selling the endowments at their original prices):

First order conditions:

Solving the system:

First order conditions:

Solving the system:

To find the income effect, we set up another Lagrangian using the final prices and the income the individual receives from selling the endowments at their original prices:

First order conditions:

Solving the system:

Next, to find the final bundle, we use the final prices and the income obtained from selling the endowment at the final prices to set up another Lagrangian:

First order conditions:

Solving the system:

# General Equilibrium (Cobb-Douglas)

## Person 1's Utility Function (in the form x^{α}y^{1-α} or αlog(x)+(1-α)log(y))

## Person 2's Utility Function (in the form x^{β}y^{1-β} or βlog(x)+(1-β)log(y))

## Endowments

Person 1 consumes ___ units of good x and ___ units of good y.

Person 2 consumes ___ units of good x and ___ units of good y.

The relative price (py/px) is ___.

If necessary, apply a monotonic transformation so the utility functions are in the form αlog(x)+(1-α)log(y) and βlog(x)+(1-β)log(y).

Individual 1's Lagrangian:

First order conditions:

Solving the system:

Individual 2's Lagrangian:

First order conditions:

Solving the system:

Next, we use equations 4, 5, 9, and 10 to solve for the relative price and each individual's optimal bundle. Since the demand for good x must add up to the total endowment of good x, we have the following:

# Labor Supply (Cobb-Douglas)

## Utility Function (in the form C^{α}R^{1-α} or αlog(C)+(1-α)log(R))

## Budget Constraint

The consumer will supply ___ hours of labor and purchase ___ units of consumption.

The consumer's reservation wage is ___ dollars per hour.

If necessary, apply a monotonic transformation so the utility function is in the form αlog(C)+(1-α)log(R).

Lagrangian:

First order conditions:

Solving the system:

To find the reservation wage, we set up a Lagrangian with an arbitrary wage and find the wage at which the individual consumes an amount of leisure that is exactly equal to the time endowment:

First order conditions:

# Labor Supply Income/Substitution/Endowment Effects (Cobb-Douglas), Slutsky Compensation

## Utility Function (in the form C^{α}R^{1-α} or αlog(C)+(1-α)log(R))

## Budget Constraint

The consumer originally supplies ___ hours of labor and purchases ___ units of consumption.

The consumer now supplies ___ hours of labor and purchases ___ units of consumption.

The substitution effect is ___ hours of labor and ___ units of consumption.

The income effect is ___ hours of labor and ___ units of consumption.

The endowment effect is ___ hours of labor and ___ units of consumption.

The consumer's reservation wage is ___ dollars per hour.

If necessary, apply a monotonic transformation so the utility function is in the form αlog(C)+(1-α)log(R).

Original Lagrangian:

First order conditions:

Solving the system:

To find the substitution effect using Slutsky compensation, we must find the amount of leisure and consumption the individual would choose if given enough income to consume the original amounts at the final wage/price.

First order conditions:

Solving the system:

To find the income effect, we set up another Lagrangian using the final wage/price and the original full income:

First order conditions:

Solving the system:

Next, we set up a Lagrangian using the final wage/price and the final full income:

First order conditions:

Solving the system:

To find the reservation wage, we set up a Lagrangian with an arbitrary wage and find the wage at which the individual consumes an amount of leisure that is exactly equal to the time endowment. Note that the price will cancel out, so we will also use an arbitrary price, p:

First order conditions:

# Insurance Markets (Logarithmic)

## Utility Function (in the form π_{g}log(c_{g})+π_{b}log(c_{b}))

## Budget Constraint

The individual's expected utility in the absence of insurance is ___ utils.

The individual's certainty equivalent is ___ dollars.

The actuarially fair price of the insurance is ___ dollars.

To find the individual's expected utility, we simply substitute all of the corresponding values into the utility function:

To find the certainty equivalent, we solve for when the logarithmic utility function equals the expected utility level:

# Intertemporal Choice (Logarithmic)

## Utility Function (in the form log(c_{t})+βlog(c_{t+1}))

## Budget Constraint

The individual spends ___ dollars in period t and ___ dollars in period t+1.

The individual saves ___ dollars in period t.

Lagrangian:

First order conditions:

Solving the system:

# Welfare Effects of Price Changes (Cobb-Douglas)

## Utility Function (in the form x^{α}y^{1-α} or αlog(x)+(1-α)log(y))

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The change in consumer surplus is ___ dollars.

The compensating variation is ___ dollars.

The equivalent variation is ___ dollars.

# Perfect Substitutes Utility Maximization

## Utility Function (in the form ax+by)

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

# Income and Substitution Effects (Perfect Substitutes), Slutsky/Hicks Compensation

## Utility Function (in the form ax+by)

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky/Hicks transfer is ___ dollars.

# Endowment Economies (Perfect Substitutes), Slutsky/Hicks Compensation

## Utility Function (in the form ax+by)

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

# Perfect Complements Utility Maximization

## Utility Function (in the form min(ax, by))

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

# Income and Substitution Effects (Perfect Complements), Slutsky/Hicks Compensation

## Utility Function (in the form min(ax, by))

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky/Hicks transfer is ___ dollars.

# Endowment Economies (Perfect Complements), Slutsky/Hicks Compensation

## Utility Function (in the form min(ax, by))

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

# Constant Elasticity of Substitution Utility Maximization

## Utility Function (in the form (αx^{-ρ}+(1-α)y^{-ρ})^{-1/ρ})

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

The elasticity of substitution is ___.

# Income and Substitution Effects (Constant Elasticity of Substitution), Slutsky Compensation

## Utility Function (in the form (αx^{-ρ}+(1-α)y^{-ρ})^{-1/ρ})

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky transfer is ___ dollars.

The elasticity of substitution is ___.

# Endowment Economies (Constant Elasticity of Substitution), Slutsky Compensation

## Utility Function (in the form (αx^{-ρ}+(1-α)y^{-ρ})^{-1/ρ})

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

The elasticity of substitution is ___.

# Quadratic Utility Maximization

## Utility Function (in the form ax-bx^{2}+cy-dy^{2})

## Budget Constraint

The optimal bundle is ___ units of good x and ___ units of good y.

The individual's bliss point is ___ units of good x and ___ units of good y.

# Income and Substitution Effects (Quadratic), Slutsky Compensation

## Utility Function (in the form ax-bx^{2}+cy-dy^{2})

## Original Budget Constraint

## Final Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The Slutsky transfer is ___ dollars.

The individual's bliss point is ___ units of good x and ___ units of good y.

# Endowment Economies (Quadratic), Slutsky Compensation

## Utility Function (in the form ax-bx^{2}+cy-dy^{2})

## Budget Constraint

The original bundle is ___ units of good x and ___ units of good y.

The final bundle is ___ units of good x and ___ units of good y.

The substitution effect is ___ units of good x and ___ units of good y.

The income effect is ___ units of good x and ___ units of good y.

The endowment effect is ___ units of good x and ___ units of good y.

The individual's bliss point is ___ units of good x and ___ units of good y.

Contact information: sxshi@uchicago.edu