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)
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
Choose: < some alternative >
In order to maximize: < some objective >
Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >
Example: A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.
Choose:
In order to maximize:
Subject to:
Example: How should FedEx plan its delivery route?
Choose:
In order to maximize:
Subject to:
Example: The U.S. government wants to remain economically competitive but reduce emissions by 25%.
Choose:
In order to maximize:
Subject to:
Example: How do elected officials make decisions in politics?
Choose:
In order to maximize:
Subject to:
Choose: < a consumption bundle >
In order to maximize: < utility >
Subject to: < income and market prices >
We now have the tools to understand individual choices:
Budget constraint: individual’s constraints of income and market prices
Utility function: individual’s objective to maximize, based on their preferences
choose a bundle of goods to maximize utility, subject to income and market prices
maxx,y≥0u(x,y) s.t.pxx+pyy=m
† See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.
Graphical solution: Highest indifference curve tangent to budget constraint
B or C spend all income, but a better combination exists
Graphical solution: Highest indifference curve tangent to budget constraint
B or C spend all income, but a better combination exists
D is higher utility, but not affordable at current income & prices
indiff. curve slope>budget constr. slope
indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5
Consumer views MB of x is 2 units of y
Market-determined MC of x is 0.5 units of y
indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5
Consumer views MB of x is 2 units of y
Market-determined MC of x is 0.5 units of y
Can spend less on y, more on x for more utility!
indiff. curve slope<budget constr. slope
indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5
Consumer views MB of x is 0.125 units of y
Market-determined MC of x is 0.5 units of y
indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5
Consumer views MB of x is 0.125 units of y
Market-determined MC of x is 0.5 units of y
Can spend less on y, more on x for more utility!
indiff. curve slope=budget constr. slope
indiff. curve slope=budget constr. slopeMUxMUy=pxpy0.5=0.5
Marginal benefit = Marginal cost
No other combination of (x,y) exists that could increase utility!†
† At current income and market prices!
MUxMUy=pxpy
MUxMUy=pxpy
MUxpx=MUypy
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!
Any optimum in economics: no better alternatives exist under current constraints
No possible change in your consumption that would increase your utility
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.
Example: You can get utility from consuming Burgers (b) and Fries (f), according to the utility function:
u(b,f)=√bfMUb=0.5b−0.5f0.5MUf=0.5b0.5f−0.5
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.
Keyboard shortcuts
↑, ←, Pg Up, k | Go to previous slide |
↓, →, Pg Dn, Space, j | Go to next slide |
Home | Go to first slide |
End | Go to last slide |
Number + Return | Go to specific slide |
b / m / f | Toggle blackout / mirrored / fullscreen mode |
c | Clone slideshow |
p | Toggle presenter mode |
t | Restart the presentation timer |
?, h | Toggle this help |
o | Tile View: Overview of Slides |
Esc | Back to slideshow |
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)
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
Choose: < some alternative >
In order to maximize: < some objective >
Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >
Example: A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.
Choose:
In order to maximize:
Subject to:
Example: How should FedEx plan its delivery route?
Choose:
In order to maximize:
Subject to:
Example: The U.S. government wants to remain economically competitive but reduce emissions by 25%.
Choose:
In order to maximize:
Subject to:
Example: How do elected officials make decisions in politics?
Choose:
In order to maximize:
Subject to:
Choose: < a consumption bundle >
In order to maximize: < utility >
Subject to: < income and market prices >
We now have the tools to understand individual choices:
Budget constraint: individual’s constraints of income and market prices
Utility function: individual’s objective to maximize, based on their preferences
choose a bundle of goods to maximize utility, subject to income and market prices
maxx,y≥0u(x,y) s.t.pxx+pyy=m
† See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.
Graphical solution: Highest indifference curve tangent to budget constraint
B or C spend all income, but a better combination exists
Graphical solution: Highest indifference curve tangent to budget constraint
B or C spend all income, but a better combination exists
D is higher utility, but not affordable at current income & prices
indiff. curve slope>budget constr. slope
indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5
Consumer views MB of x is 2 units of y
Market-determined MC of x is 0.5 units of y
indiff. curve slope>budget constr. slopeMUxMUy>pxpy2>0.5
Consumer views MB of x is 2 units of y
Market-determined MC of x is 0.5 units of y
Can spend less on y, more on x for more utility!
indiff. curve slope<budget constr. slope
indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5
Consumer views MB of x is 0.125 units of y
Market-determined MC of x is 0.5 units of y
indiff. curve slope<budget constr. slopeMUxMUy<pxpy0.125<0.5
Consumer views MB of x is 0.125 units of y
Market-determined MC of x is 0.5 units of y
Can spend less on y, more on x for more utility!
indiff. curve slope=budget constr. slope
indiff. curve slope=budget constr. slopeMUxMUy=pxpy0.5=0.5
Marginal benefit = Marginal cost
No other combination of (x,y) exists that could increase utility!†
† At current income and market prices!
MUxMUy=pxpy
MUxMUy=pxpy
MUxpx=MUypy
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!
Any optimum in economics: no better alternatives exist under current constraints
No possible change in your consumption that would increase your utility
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.
Example: You can get utility from consuming Burgers (b) and Fries (f), according to the utility function:
u(b,f)=√bfMUb=0.5b−0.5f0.5MUf=0.5b0.5f−0.5
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.