Utility is a number assigned to represent preferences; higher numbers indicate more preferable outcomes.
While the magnitude of utility isn't directly comparable (i.e., 4 utility isn't necessarily twice as good as 2 utility), comparisons are made in terms of trade-offs.
Path Dependence of Utility
Utility is path-dependent, meaning the utility derived from something depends on prior consumption and experiences.
Preferences at a given moment depend on prior consumption, experiences, and current state.
Example: Choosing cheesecake over ice cream, even if ice cream is a favorite, due to having ice cream recently.
Utility analyses must consider prior consumption; this is reflected in marginal utility.
Marginal Utility
Marginal utility is the additional utility gained from consuming one more unit of a good.
Example:
First pizza slice yields 5 utility.
Second: 4 utility.
Third: 3 utility.
To assess the happiness from a slice of pizza, you need to know how many slices have already been consumed.
Characteristics of Utility
Total utility increases with consumption, while marginal utility diminishes.
Diminishing returns: Each additional unit provides less additional happiness.
Example: $1,000 has different impacts on an average person versus Jeff Bezos.
Example: Calculus class enjoyment decreases with repetition.
More is better: Utility is strictly positive, assuming more of a good always provides some positive happiness, even if minimal.
Even an 81st slice of pizza provides some, albeit small, amount of happiness.
Choosing Between Two Goods
Every decision can be simplified into a binary choice between two options.
Examples: Work vs. not work, food vs. beverage, sleep vs. not sleep, apple pie vs. bacon.
Utility, indifference curve, and preference relation are different names for the same concept.
Indifference Curves
Indifference curves represent all combinations of two goods (good A and good B) that provide the same total utility.
Example: Combinations of pizza (good A) and ice cream (good B) yielding 24 utility.
Calculating Total Utility
Total utility is the sum of all marginal utilities.
Example:
First pizza: 5 utility.
Second pizza: 4 utility.
Total: 9 utility.
Identifying combinations of goods that yield a specific utility level.
Example: Three slices of pizza and three scoops of ice cream provide 24 utility (12 + 12).
Trade-offs and Compensation
When giving up a unit of one good, additional units of the other good are needed to maintain the same utility.
Example: If reducing pizza from three to two slices, additional ice cream is needed to compensate.
The third slice of pizza provides 3 utility; this loss must be compensated by additional ice cream.
To compensate for losing the third pizza slice (3 utility), two additional scoops of ice cream are needed (2 + 1 utility).
Continuous Data: Utility isn't limited to whole units; it can be measured for fractions of units.
Inferring Preferences
Economists build models based on the assumption that people are rational, and infer consumer behaviour based on their optimal decisions in the real world.
Utility as a Function
Utility is usually represented as a continuous function to allow for fractional units of goods.
The utility curve illustrates all combinations of good A and good B that provide the same level of utility.
Utility Curve Implications
Any point above the utility curve represents a better (higher utility) bundle of goods.
Optimization involves finding the best possible point, which lies on the highest attainable utility curve.
Marginal Rate of Substitution (MRS)
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while maintaining the same level of utility.
It represents how much of good A is needed to compensate for a change in good B.
The amount of compensation needed varies at different points on the curve.
Calculating MRS
If someone has four ice creams and gives up one scoop, the MRS shows how much additional pizza is needed to maintain the same utility.
The MRS changes depending on the position on the curve. Losing the fourth scoop of ice cream requires less pizza than losing the second scoop.
Since the second scoop of ice cream is more valuable (higher marginal utility), more pizza is needed to compensate.
MRS and Slope
The MRS is represented by the slope of the indifference curve between two points.
Calculus: As the two points become infinitely close, the line becomes a tangent to the curve.
The tangent line represents the slope of the curve at a specific point.
The slope changes at different points along the curve due to its curvature.
Contextualizing MRS
The definition of MRS depends on the context of the question: which good is being given up and gained.
Delta notation: Δy/Δx
Remember to read the question carefully so you are sure whether it is calculating the amount of good A needed if you give up good B, or vice versa.
MRS as Ratio of Marginal Utilities
Marginal Rate of Substitution (MRS) is the ratio of marginal utilities.
Formula: MRS=MU<em>icecream/MU</em>pizza
Context matters: the question should specify which good is being given up.
The formula can be inverted depending on the context: MRS=MU<em>pizza/MU</em>icecream
Example Calculation
If giving up one slice of pizza requires two ice creams to compensate, the MRS is either 1/2 or 2, depending on the numerator.
At three pizzas and three ice creams, the marginal rate of substitution is 1, meaning one pizza can be traded for one ice cream at that specific point.
MRS and Diminishing Returns
Giving up the last piece of pizza has a higher value because of diminishing returns.
At three pizza and three ice cream, the MRS is 1 only at that infinitesimally small point. As we move along the curve, the MRS changes.
Key Takeaways about MRS
Consider the starting point because marginal utilities are path dependent.
Consider how far you're traveling along the curve.
At three pizza and three ice cream, trading occurs at a 1:1 ratio, but this changes as quantities increase.
Budget Constraints and Utility
The first class covered building a model with mathematical relationships for budget. For preferences it gets a bit difficult.
Finding Optimal Consumption
The budget indicates affordability, while the utility curve represents preferences.
Any point on or below the budget curve is affordable.
Any point above the utility curve is better.
You should be spending your entire budget because having unspent budget is useless in this context.
Lens of Opportunity
The area between the budget line and the utility curve represents better and affordable bundles.
Economists refer to this region as the lens of opportunity.
The goal is find the highest affordable utility curve that lies on the affordable budget line.
Utility Curves
There are infinite utility curves, each representing a different level of utility.
Moving away from the origin in the graph indicates increased utility due to more consumption (more is better).
Optimizing Utility
Consumers aim to reach the highest possible utility curve within their budget.
The optimal choice occurs when there is one, single kissing point between the utility curve and budget line.
Utility per Dollar
The method for optimized decision making is when the utility curve touches the budget line exactly once.
That means at the optimal, with curved utility, the marginal rate of substitution must equal the slope of the budget line. MRS=MU<em>icecream/MU</em>pizza P<em>pizza/P</em>icecream
Economic Intuition
At the optimal point, if pizza provides twice as much happiness as ice cream, it must also be twice as expensive.
It is important when optimizing that you account for prices in your purchasing decision.
Disequilibrium
If this condition does not hold, the consumer is not optimizing and can rebalance consumption to achieve higher utility.
The equilibrium is reached when you will receive the same happiness with a dollar spent on either pizza or ice cream
The utility per dollar of pizza must equal the utility per dollar of ice cream
Just buy what makes you more happiness given how poor you are