## Consumer's Optimal Choice

• Budget determines what I can buy.

• Utility function (preferences) determine how I value those affordable alternatives.

• Which bundle do I buy?

## Consumer's Optimal Choice

• The bundle with the highest utility among the affordable.

• We call this bundle the Rational Constrained Choice.

## Three main cases of Optimal Choice.

1. Tangency Solution: When preferences are well behaved (smooth, convex, ...), then at the optimal bundle: $MRS = \frac{−p_1}{p_2}$ (for example Cobb-Douglass preferences)

2. Corner solutions or "boundary optimum": if $MRS > \frac{−p_1}{p_2}$ or $MRS < \frac{−p_1}{p_2}$ always (for example: perfect substitutes)

3. Kink optimality: if preferences are "kinky" (for example: perfect complements)

## Optimal Choice - Tangency Solution (intuitively)

• Suppose your preferences look like Cobb-Douglass (smooth, convex).

• You have a BC and you are considering buying $(x_1', x_2')$ such that $x_1'>0, x_2'>0$ on the BC.

• Suppose that only thing you know is that your MRS, at the bundle $(x_1', x_2')$, is higher in magnitude than $p_1 / p_2$. That is, the associated indifference curve is steeper than the budget constraint.

• Should you buy the bundle $(x_1', x_2')$?

## Optimal Choice - Tangency Solution (intuitively)

• No.

• If MRS is steeper than BC, it means that at that point you value $x_1$ more than the market. So...

• Buy more of that good 1.

• But how much more?

• Up to a point in which you and the market value $x_1$ the same (relative to $x_2$)

• First optimality condition: $MRS = \frac{−p_1}{p_2}$

## Optimal Choice - Tangency Solution (math method 1)

Steps to find the optimal bundle (aka the demanded bundle) for tangency cases:

1. Identify clearly the utility function.

2. Calculate the $MRS$, it will be a function of $x_1, x_2$ and (possibly) on some parameters of the utility function.

3. Set the tangency condition: $MRS = - \frac{p_1}{p_2}$ call this Equation 1.

4. Identify the budget constrain and call it Equation 2.

5. Equation 1 and Equation 2 form a 2-equation-2-unknowns system, so you can solve for the two unknowns: $x_1$ and $x_2$.

## Optimal Choice - Tangency Solution (math method 1) - Example

Let's apply these steps to the case of Cobb-Douglas preferences: $U(x_1, x_2) = x_1^{0.5} x_2^{0.5}$

• Tangency : $MRS = - \frac{x_2}{x_1}$. Equate MRS to: $- \frac{p_1}{p_2}$ (Eq1)

• Budget constraint : $p_1 x_1 + p_2 x_2 = m$ (Eq2)

• [ solve for x1 and x2 in the system of two equations -- details in doc camera ]

• Optimal bundle: $x_1^{*} = \frac{1}{2} \frac{m}{p_1}$ and $x_2^{*} = \frac{1}{2} \frac{m}{p_2}$

## Optimal Choice - Tangency Solution - Cobb-Douglas Function

• Note if you have Cobb-Douglass utility, $U = x_1^{a} x_2^{b}$, you can always use method 1.

• Exercise: apply method 1 to this utility function: $U = x_1^{a} x_2^{b}$

• An alternative to method 1, is a more general method called the Lagrange Method that we will cover later.

## Case 2: Optimal bundle in corner solutions

The most typical case of this type of solution is with perfect substitutes preferences.

Steps to finding the optimal bundle when x_1 and x_2 are perfect substitutes:

1. Calculate the $MRS$, it will be a function of $x_1, x_2$ and (possibly) on some parameters of the utility function.

2. Compare its magnitude to the price ratio: $\frac{p_1}{p_2}$.

3. If $|MRS| > \frac{p_1}{p_2}$, then all income is spent on good 1: $x_1 = m / p_1$ and $x_2 = 0$

4. If $|MRS| < \frac{p_1}{p_2}$, then all income is spent on good 2: $x_2 = m / p_2$ and $x_1 = 0$

5. If $|MRS| = \frac{p_1}{p_2}$ any bundle that exhaust income will be optimal.

## Finding the optimal bundle (perfect substitutes) - Example!

1. Say, $u = 2 x_1 + x_2$

2. $MRS = - 2 / 1 = - 2$

3. Compare |MRS| to price ratio: 2 vs. $\frac{p_1}{p_2}$.

4. If $\frac{p_1}{p_2} < 2$, then: $x_1 = m / p_1$ and $x_2 = 0$

5. If $\frac{p_1}{p_2} > 2$, then: $x_1 = 0$ and $x_2 = m / p_2$

6. If $\frac{p_1}{p_2} = 2$, any $(x_1, x_2)$ such that $p_1 x_1 + p_2 x_2 = m$ is optimal.

## Perfect substitutes

• See graphs in document camera

• See graphs on EconGraphs

## Case 3: Optimal bundle in "kink" solutions

Most cases of "kink" solutions appear because of "perfect complement" preferences.

Steps to find the optimal bundle under "perfect complement" preferences:

1. Identify clearly the utility function: $U = \textrm{min} \{ \frac{x_1}{\alpha}, \frac{x_2}{\beta} \}$, for $\alpha, \beta > 0$

2. Calculate the optimal consumption path: $\frac{x_1}{\alpha} = \frac{x_2}{\beta}$. Call this Equation 1.

3. Identify the budget constraint and call it Equation 2.

4. Equation 1 and Equation 2 form a 2-equation-2-unknowns system, so you can solve for the two unknowns: $x_1$ and $x_2$.

## Finding the optimal bundle - Perfect complements - Numerical Example

1. $U = \textrm{min} \{ \frac{x_1}{2}, x_2 \}$.

2. Optimal consumption path: $\frac{x_1}{2} = x_2$. This is Equation 1.

3. Budget Constraint $m = p_1 x_1 + p_2 x_2$

4. Optimal bundle: $x_1^{*} = \frac{m}{p_1 + p_2/2}$ and $x_2^{*} = \frac{m}{2 p_1 + p_2}$

## Optimal choice with Lagrange's Method

• We are back to case 1 or "tangency solution".

• Conditions:

• Utility function is differentiable and preferences are convex,

• $MRS(0,y) = infinity$ and $MRS(x,0) = 0$

• E.g. Cobb-Douglas satisfies these conditions.

• You can always use the Lagrange's method.

## Optimal choice with Lagrange's Method - Steps

1. Set Problem: $\textrm{maximize} \quad U(x_1, x_2)$ subject to: $m = p_1 x_1 + p_2 x_2$

2. Write Lagrange's function

• $L = U(x_1, x_2) - \lambda ( p_1 x_1 + p_2 x_2 - m)$
3. Differentiate with respect to $x_1, x_2, \lambda$, equate to zero.

• $\frac{\partial L}{\partial x_1} = \frac{\partial U}{\partial x_1} - \lambda p_1 = 0$
• $\frac{\partial L}{\partial x_2} = \frac{\partial U}{\partial x_2} - \lambda p_2 = 0$
• $\frac{\partial L}{\partial \lambda} = - p_1 x_1 - p_2 x_2 + m = 0$
4. Solve system of three equations.

## Optimal choice with Lagrange's Method - Example

1. Consider: $\quad U(x_1, x_2) = x_1^{0.5} + x_2^{0.5}$

2. Problem: $\quad \textrm{maximize} \quad x_1^{0.5} + x_2^{0.5}$ subject to: $m = p_1 x_1 + p_2 x_2$

3. Lagrange's function: $\quad L = x_1^{0.5} + x_2^{0.5} - \lambda ( p_1 x_1 + p_2 x_2 - m)$

4. Differentiate with respect to $x_1, x_2, \lambda$, equate to zero.

• $\frac{\partial L}{\partial x_1} = 0.5 x_1^{-0.5} - \lambda p_1 = 0$
• $\frac{\partial L}{\partial x_2} = 0.5 x_2^{-0.5} - \lambda p_2 = 0$
• $\frac{\partial L}{\partial \lambda} = - p_1 x_1 - p_2 x_2 + m = 0$
5. Solution good 1: $\quad x_1^* = \frac{ m }{ p_1 + p_2 } \frac{ p_2 }{ p_1 }$

6. Solution good 2: $\quad x_2^* = \frac{ m }{ p_1 + p_2 } \frac{ p_1 }{ p_2 }$

## Practice all these cases with the lagrange's Method.

1. $U(x_1, x_2) = x_1^{1/2} x_2^{1/2}$

2. $U(x_1, x_2) = x_1^{1/4} x_2^{3/4}$

3. => $U(x_1, x_2) = x_1^{a} x_2^{b}$

4. $U(x_1, x_2) = x_1^{a} x_2^{1-a}$

5. $U(x_1, x_2) = (1/4) ln(x_1) + (3/4) ln(x_2)$

6. $U(x_1, x_2) = a x_1 + b ln(x_2)$

## Tangency does not work with non-convex preferences

• Be careful with tangency conditions.