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1.6: Solving the Consumer’s Problem

ECON 306 · Microeconomic Analysis · Spring 2020

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS20
microS20.classes.ryansafner.com

The Consumer's Problem: Review

  • The consumer's constrained optimization problem is:

  1. Choose: < a consumption bundle >

  2. In order to maximize: < utility >

  3. Subject to: < income and market prices >

The Consumer's Problem: Tools

  • We now have the tools to understand consumer choices:

  • Budget constraint: consumer's constraints of income and market prices

    • How the .red[market] trades off between two goods
  • Utility function: consumer's preferences to maximize
    • How the .green[consumer] trades off between two goods

The Consumer's Problem: Verbally

  • The consumer's constrained optimization problem:

choose a bundle of goods to maximize utility, subject to income and market prices

The Consumer's Problem: Mathematically

$$\max_{x,y} u(x,y)$$ $$s.t. p_xx+p_yy=m$$

  • This requires calculus to solve1. We will look at graphs instead!

1 See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.

The Consumer's Optimum: Graphically

  • Graphical solution: Highest indifference curve tangent to budget constraint
    • Bundle A!

The Consumer's Optimum: Graphically

  • Graphical solution: Highest indifference curve tangent to budget constraint

    • Bundle A!

  • B or C spend all income, but a better combination exists

    • Averages \(\succ\) extremes!

The Consumer's Optimum: Graphically

  • Graphical solution: Highest indifference curve tangent to budget constraint

    • Bundle A!

  • B or C spend all income, but a better combination exists

    • Averages \(\succ\) extremes!

  • D is higher utility, but not affordable at current income & prices

The Consumer's Optimum: Why Not B?

$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\\end{align*}$$

The Consumer's Optimum: Why Not B?

$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\ | MRS_{x,y} | &> | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &> | \frac{p_x}{p_y} | \\ | -2 | &> | -0.5 | \\\end{align*}$$

  • Consumer would exchange at 2Y:1X

  • Market exchange rate is 0.5Y:1X

The Consumer's Optimum: Why Not B?

$$\begin{align*} \text{indiff. curve slope} &> \text{budget constr. slope} \\ | MRS_{x,y} | &> | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &> | \frac{p_x}{p_y} | \\ | -2 | &> | -0.5 | \\\end{align*}$$

  • Consumer would exchange at 2Y:1X

  • Market exchange rate is 0.5Y:1X

  • Can spend less on y more on x and get more utility!

The Consumer's Optimum: Why Not C?

$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\\end{align*}$$

The Consumer's Optimum: Why Not C?

$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\ | MRS_{x,y} | &< | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &< | \frac{p_x}{p_y} | \\ | -0.125 | &< | -0.5 | \\\end{align*}$$

  • Consumer would exchange at 0.125Y:1X

  • Market exchange rate is 0.5Y:1X

The Consumer's Optimum: Why Not C?

$$\begin{align*} \text{indiff. curve slope} &< \text{budget constr. slope} \\ | MRS_{x,y} | &< | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &< | \frac{p_x}{p_y} | \\ | -0.125 | &< | -0.5 | \\\end{align*}$$

  • Consumer would exchange at 0.125Y:1X

  • Market exchange rate is 0.5Y:1X

  • Can spend less on x, more on y and get more utility!

The Consumer's Optimum: Why A?

$$\begin{align*} \text{indiff. curve slope} &= \text{budget constr. slope} \\\end{align*}$$

The Consumer's Optimum: Why A?

$$\begin{align*} \text{indiff. curve slope} &= \text{budget constr. slope} \\ | MRS_{x,y} | &= | \frac{p_x}{p_y} | \\ | \frac{MU_x}{MU_y} | &= | \frac{p_x}{p_y} | \\ | -0.5 | &= | -0.5 | \\\end{align*}$$

  • Consumer would exchange at same rate as market

  • No other combination of \((x,y)\) exists at current prices & income that could increase utility!

The Consumer's Optimum: Two Equivalent Rules

Rule 1

$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

  • Easier for solving math problems (slope of each curve)

The Consumer's Optimum: Two Equivalent Rules

Rule 1

$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

  • Easier for calculation (slopes)

Rule 2

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$

  • Easier for intuition (next slide)

The Consumer's Optimum: The Equimarginal Rule I

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \cdots = \frac{MU_n}{p_n}$$

  • Equimarginal Rule: consumption is optimized where the marginal utility per dollar spent is equalized across all \(n\) possible goods/decisions

  • You will always choose an option that gives higher marginal utility (e.g. if \(MU_x > MU_y)\)

    • But each option has a different cost, so we weight each option by its cost, hence \(\frac{MU_x}{p_x}\)

The Consumer's Optimum: The Equimarginal Rule II

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \cdots = \frac{MU_n}{p_n}$$

  • Why is this the optimum?

The Consumer's Optimum: The Equimarginal Rule II

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \cdots = \frac{MU_n}{p_n}$$

  • Why is this the optimum? Example: suppose you could get a higher marginal utility per $1 for \(x\) than for \(y\) (i.e. "more bang for your buck"!)

  • Not maximizing your utility!

    • Spend more on \(x\) and less on \(y\)!
    • This will raise \(MU_x\) and lower \(MU_y\)!

  • Continue until cost-adjusted marginal utilities are equalized

The Consumer's Optimum: The Equimarginal Rule III

  • Any optimum in economics: no better alternatives exist under current constraints

  • No possible change in your consumption that would increase your utility

Markets Equalize Everyone's MRS I

  • Markets make it so everyone faces the same relative prices

    • Budget constraint. slope, \(-\frac{p_x}{p_y}\)
    • Note individuals' incomes, \(m\), are certainly different!

  • A person's optimal choice \(\implies\) they make same tradeoff as the market

    • Their MRS \(=\) relative price ratio

  • markets equalize everyone's MRS

Markets Equalize Everyone's MRS II

Two people will very different income and preferences face the same market prices, and choose optimal consumption (points A and A') at an exchange rate of \(0.5Y:1X\)

Optimization and Equilibrium

  • If people can learn and change their behavior, they will always switch to a higher-valued option

  • If a person has no better choices (under current constraints), they are at an optimum

  • If everyone is at an optimum, the system is in equilibrium

Practice I

Example: You can get utility from consuming bags of Almonds \((a)\) and bunches of Bananas \((b)\), according to the utility function:

$$\begin{align*} u(a,b)&=ab\\ MU_a&=b \\ MU_b&=a \\ \end{align*}$$

You have an income of $50, the price of Almonds is $10, and the price of Bananas is $2. Put Almonds on the horizontal axis and Bananas on the vertical axis.

  1. What is your utility-maximizing bundle of Almonds and Bananas?
  2. How much utility does this provide? [Does the answer to this matter?]

Practice II, Cobb-Douglas!

Example: You can get utility from consuming Burgers \((b)\) and Fries \((f)\), according to the utility function:

$$\begin{align*} u(b,f)&=\sqrt{bf} \\ MU_b&=0.5b^{-0.5}f^{0.5} \\ MU_f&=0.5b^{0.5}f^{-0.5} \\ \end{align*}$$

You have an income of $20, the price of Burgers is $5, and the price of Fries is $2. Put Burgers on the horizontal axis and Fries on the vertical axis.

  1. What is your utility-maximizing bundle of Burgers and Fries?
  2. How much utility does this provide?

The Consumer's Problem: Review

  • The consumer's constrained optimization problem is:

  1. Choose: < a consumption bundle >

  2. In order to maximize: < utility >

  3. Subject to: < income and market prices >

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