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1.4 — Utility Maximization

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Constrained Optimization

Constrained Optimization I

  • We model most situations as a constrained optimization problem:

  • People optimize: make tradeoffs to achieve their objective as best as they can

  • Subject to constraints: limited resources (income, time, attention, etc)

Constrained Optimization II

  • One of the most generally useful mathematical models

  • Endless applications: how we model nearly every decision-maker

consumer, business firm, politician, judge, bureaucrat, voter, dictator, pirate, drug cartel, drug addict, parent, child, etc

  • Key economic skill: recognizing how to apply the model to a situation

Remember!

Constrained Optimization III

  • All constrained optimization models have three moving parts:

Constrained Optimization III

  • All constrained optimization models have three moving parts:
  1. Choose: < some alternative >

Constrained Optimization III

  • All constrained optimization models have three moving parts:
  1. Choose: < some alternative >

  2. In order to maximize: < some objective >

Constrained Optimization III

  • All constrained optimization models have three moving parts:
  1. Choose: < some alternative >

  2. In order to maximize: < some objective >

  3. Subject to: < some constraints >

Constrained Optimization: Example I

Example: A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.

  1. Choose:

  2. In order to maximize:

  3. Subject to:

Constrained Optimization: Example II

Example: How should FedEx plan its delivery route?

  1. Choose:

  2. In order to maximize:

  3. Subject to:

Constrained Optimization: Example III

Example: The U.S. government wants to remain economically competitive but reduce emissions by 25%.

  1. Choose:

  2. In order to maximize:

  3. Subject to:

Constrained Optimization: Example IV

Example: How do elected officials make decisions in politics?

  1. Choose:

  2. In order to maximize:

  3. Subject to:

The Utility Maximization Problem

  • The individual's utility maximization problem we've been modeling, finally, is:
  1. Choose: < a consumption bundle >

  2. In order to maximize: < utility >

  3. Subject to: < income and market prices >

The Utility Maximization Problem: Tools

  • We now have the tools to understand individual choices:

  • Budget constraint: individual’s constraints of income and market prices

    • How market trades off between goods
    • Marginal cost (of good x, in terms of y)
  • Utility function: individual’s objective to maximize, based on their preferences

    • How individual trades off between goods
    • Marginal benefit (of good x, in terms of y)

The Utility Maximization Problem: Verbally

  • The individual's constrained optimization problem:

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

The Utility Maximization Problem: Mathematically

maxx,y0u(x,y) s.t.pxx+pyy=m

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

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

The Individual's Optimum: Graphically

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

The Individual'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

The Individual'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

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

The Individual's Optimum: Why Not B?

indiff. curve slope>budget constr. slope

The Individual's Optimum: Why Not B?

indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5

  • Consumer views MB of x is 2 units of y

    • Consumer’s “exchange rate:” 2Y:1X
  • Market-determined MC of x is 0.5 units of y

    • Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not B?

indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5

  • Consumer views MB of x is 2 units of y

    • Consumer’s “exchange rate:” 2Y:1X
  • Market-determined MC of x is 0.5 units of y

    • Market exchange rate is 0.5Y:1X
  • Can spend less on y, more on x for more utility!

The Individual's Optimum: Why Not C?

indiff. curve slope<budget constr. slope

The Individual's Optimum: Why Not C?

indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5

  • Consumer views MB of x is 0.125 units of y

    • Consumer’s “exchange rate:” 0.125Y:1X
  • Market-determined MC of x is 0.5 units of y

    • Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not C?

indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5

  • Consumer views MB of x is 0.125 units of y

    • Consumer’s “exchange rate:” 0.125Y:1X
  • Market-determined MC of x is 0.5 units of y

    • Market exchange rate is 0.5Y:1X
  • Can spend less on y, more on x for more utility!

The Individual's Optimum: Why A?

indiff. curve slope=budget constr. slope

The Individual's Optimum: Why A?

indiff. curve slope=budget constr. slopeMUxMUy=pxpy0.5=0.5

  • Marginal benefit = Marginal cost

    • Consumer exchanges at same rate as market
  • No other combination of (x,y) exists that could increase utility!

At current income and market prices!

The Individual's Optimum: Two Equivalent Rules

Rule 1

MUxMUy=pxpy

  • Easier for calculation (slopes)

The Individual's Optimum: Two Equivalent Rules

Rule 1

MUxMUy=pxpy

  • Easier for calculation (slopes)

Rule 2

MUxpx=MUypy

  • Easier for intuition (next slide)

The Individual's Optimum: The Equimarginal Rule

MUxpx=MUypy==MUnpn

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

  • Always choose an option that gives higher marginal utility (e.g. if MUx<MUy), consume more y!

    • But each option has a different price, so weight each option by its price, hence MUxpx

An Optimum, By Definition

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

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

Practice I

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

u(a,b)=abMUa=bMUb=a

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:

u(b,f)=bfMUb=0.5b0.5f0.5MUf=0.5b0.5f0.5

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?

Constrained Optimization

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