EC 270 - 2026-09-15 - Lecture 02

EC 270 - Lecture 02 - 2026-09-15

Intermediate Microeconomics I · Logan McLeod · LH 1011

Week 2 · 11:30-12:48 · 77 min · Full transcript

Overview

This lecture started the six-week consumer theory block. Consumer theory assumes consumers choose the best bundle they both value and can afford. "Best" (preferences) is Chapter 3; today covered "can afford" (Chapter 2, the budget constraint). McLeod set up the budget constraint and budget set, first in $n$ goods and then in two. He graphed the constraint and read its intercepts and slope, with the slope interpreted as a market trade-off and an opportunity cost. He then used the model to see what happens when income or prices change, when the numeraire changes, and under taxes, subsidies and non-linear prices. He stopped at non-linear prices, so the negative-price (garbage) example on the slides was not covered.

Key concepts

Consumer theory: models how consumers make choices in the market (the grocery-store example: you don't buy T-bone steak for every meal because it's expensive). Choices depend on both what you like and what you can afford, so the model is built in steps.
Consumption choice set: every consumption option available to the consumer, e.g. all $n$ goods on the grocery shelves, each written $x_{i}$ .
Consumption bundle: $X = (x_{1}, x_{2}, \dots, x_{n})$ , a vector giving how many units of each good you consume.
Constraints on consumption: money (budget), time (less grocery shopping in a week with several midterms) and other resources (you can afford the steak but have no barbecue). The model only looks at the monetary constraint.
Prices: each commodity $x_{i}$ has a price $p_{i}$ , giving $(p_{1}, \dots, p_{n})$ .
(Disposable) income $m$ : the amount the consumer has to spend. Saving and borrowing are left out on purpose to keep it simple.
Affordable bundle: total expenditure $\leq$ income.
Non-negativity: quantities must be $\geq 0$ . A negative quantity would mean returning or selling goods, which isn't part of this model.
Budget constraint: the bundles that are just affordable, meaning they cost exactly $m$ and use up all your income.
Budget set: all affordable bundles, costing $\leq m$ . The budget constraint is the upper boundary of the budget set.
Two-good simplification: most of the term uses only $x_{1}$ and $x_{2}$ . Results from the two-good case generally carry over to $n$ goods.
Composite good: $x_{2}$ can stand for "everything else" besides $x_{1}$ (steak vs. everything else in the store). The slides add that $x_{2}$ can then be read as dollars spent on other goods, with $p_{2} = 1$ .
Three ways to express the theory: in English, mathematically and graphically. All three are used throughout the term.
Graphing in $x_{1}$ – $x_{2}$ space: $x_{1}$ goes on the horizontal axis and $x_{2}$ on the vertical. Only the first quadrant is used, since quantities are non-negative.
- Bundles on the line are just affordable.
- Bundles beyond the line (more of both goods) cost more than $m$ and are not affordable.
- Bundles below the line (less of both goods) cost less than $m$ and are affordable.
Horizontal intercept $m / p_{1}$ : all income spent on good 1, with $x_{2} = 0$ .
Vertical intercept $m / p_{2}$ : all income spent on good 2, with $x_{1} = 0$ .
Slope $- p_{1} / p_{2}$ : the minus sign means the line slopes down. The price ratio sets how steep or flat it is.
Economic meaning of the slope, part 1 (market substitution): the rate at which the market lets the consumer swap one good for the other. To get one more unit of $x_{1}$ , you sell $p_{1} / p_{2}$ units of $x_{2}$ at market prices.
Economic meaning of the slope, part 2 (opportunity cost): the cost of the next best alternative forgone. With only two goods, the next best alternative is simply the other good. An extra unit of $x_{1}$ costs $p_{1} / p_{2}$ units of $x_{2}$ . The slides add that an extra unit of $x_{2}$ costs $p_{2} / p_{1}$ units of $x_{1}$ .
Change in income: the budget line shifts parallel, because the slope depends only on prices.
- An income increase moves it out, enlarges the budget set, loses no original choices, and cannot make the consumer worse off.
- An income decrease moves it in, shrinks the set, loses some choices, and typically makes the consumer worse off.
Increase in $p_{1}$ : the horizontal intercept moves toward the origin and the vertical intercept stays put. The line pivots inward around the vertical intercept and gets steeper. The budget set shrinks and the consumer is typically worse off.
Why the vertical intercept doesn't move when $p_{1}$ changes: at that point $x_{1} = 0$ , so nothing is spent on good 1 and its price doesn't matter. Only $p_{2}$ and $m$ matter there, and neither changed. (McLeod corrected a student here: it is $x_{1}$ that is zero, not $p_{1}$ .)
Decrease in $p_{1}$ : the line pivots outward around the vertical intercept and gets flatter. The budget set grows, no choices are lost, and the consumer can't be worse off.
Increase in $p_{2}$ (not on the slides): the vertical intercept moves down and the horizontal intercept stays put. The line pivots around the horizontal intercept and gets flatter. The conclusions mirror the $p_{1}$ case.
Numeraire ("unit of account"): you can peg one price, or income, to 1 by dividing the whole constraint by it. This writes the same budget line differently. Changing the numeraire changes neither the budget constraint nor the budget set.
- Numeraire price: the price the other price and income are measured against.
- Numeraire good: the good used to measure the value of the other goods.
Recurring theme: a piece of economic behaviour isn't described by one unique equation. A whole family of equations can describe the same thing, and this comes up "over and over again" this term.
Ad valorem tax: a tax on the value (price) of a good. At rate $t$ it raises the price from $p$ to $(1 + t) p$ . Example: HST is a 13% ad valorem tax.
Uniform ad valorem tax: both goods taxed at the same rate $t$ . Relative prices don't change, so the slope doesn't either. It works like taking income away, so the line shifts inward in parallel.
Quantity tax: a fixed amount per unit bought, e.g. the U.S. federal gasoline tax, about 15¢ per gallon at one point (McLeod thinks it's higher now). Price goes from $p$ to $p + t$ , which changes the slope: a tax on good 1 makes the line steeper.
Lump-sum tax: a fixed amount paid no matter what you buy. Like an income cut, it shifts the line inward in parallel.
Subsidy: a "negative tax" that lowers the consumer's price from $p$ to $p - t$ . Example: the federal per-vehicle subsidy for electric vehicle purchases.
Non-linear prices: the price depends on how much you buy.
- Bulk discount: chips are $4.99, or two for $7.50.
- Penalty for buying "too much": toilet paper is $2 for a 24-pack if you buy up to four, but the fifth jumps to $25.
- Either way, the slope changes at the quantity where the price changes, giving a kinked budget line. The interpretation is unchanged: bundles on the line cost exactly $m$ , and bundles on or under it make up the budget set.

Formulas & models

p_{1} x_{1} + p_{2} x_{2} + \dots + p_{n} x_{n} \leq m

A bundle is affordable when total spending is no more than income.

{(x_{1}, \dots, x_{n}) ∣ x_{1} \geq 0, \dots, x_{n} \geq 0, p_{1} x_{1} + \dots + p_{n} x_{n} = m}

The budget constraint: non-negative bundles that are just affordable.

B (p_{1}, \dots, p_{n}, m) = {(x_{1}, \dots, x_{n}) ∣ x_{1} \geq 0, \dots, x_{n} \geq 0, p_{1} x_{1} + \dots + p_{n} x_{n} \leq m}

The budget set: all non-negative affordable bundles.

p_{1} x_{1} + p_{2} x_{2} = m ⟹ x_{2} = \frac{m}{p_{2}} - \frac{p_{1}}{p_{2}} x_{1}

The two-good budget line, rearranged into standard form.

a x_{1} + c x_{2} = d : vertical intercept \frac{d}{c}, horizontal intercept \frac{d}{a}, slope - \frac{a}{c}

From the Math Appendix (Sept 10). For the budget line, $a = p_{1}$ , $c = p_{2}$ and $d = m$ . The transcript has McLeod saying the slope is " $A$ over $C$ "; the slides give $- a / c$ .

vertical intercept = \frac{m}{p_{2}}, horizontal intercept = \frac{m}{p_{1}}, slope = - \frac{p_{1}}{p_{2}}

The slope depends only on prices, so income changes give parallel shifts and price changes rotate the line.

p_{1} x_{1} + x_{2} \leq m

The composite-good special case, where $p_{2} = 1$ (from the slides).

\frac{p_{1}}{p_{2}} x_{1} + x_{2} = \frac{m}{p_{2}} or \frac{p_{1}}{m} x_{1} + \frac{p_{2}}{m} x_{2} = 1

The same budget line with good 2's price or income as the numeraire.

(1 + t) p_{1} x_{1} + (1 + t) p_{2} x_{2} = m ⟺ p_{1} x_{1} + p_{2} x_{2} = \frac{m}{1 + t}

A uniform ad valorem tax at rate $t$ acts like cutting income to $m / (1 + t)$ : a parallel inward shift.

m - \frac{m}{1 + t} = \frac{t}{1 + t} m

The equivalent income loss. A uniform ad valorem sales tax at rate $t$ is the same as an income tax at rate $\frac{t}{1 + t}$ . McLeod first said " $t$ over 2", then corrected himself to "over $1 + t$ ".

p \to p + t (quantity tax), p \to p - t (subsidy)

Per-unit taxes and subsidies change the price, and so the slope.

- \frac{p_{1}}{p_{2}} = {\begin{cases} - 2 & 0 \leq x_{1} \leq 20 \\ - 1 & x_{1} > 20 \end{cases}

The slope of the budget line under the quantity-discount example below.

Worked examples

Numeraire, with $p_{1} = $ 2$ , $p_{2} = $ 3$ , $m = $ 12$ :
- In dollars: $2 x_{1} + 3 x_{2} = 12$ , slope $- 2 / 3$ .
- In cents: $200 x_{1} + 300 x_{2} = 1200$ , slope $- 200 / 300 = - 2 / 3$ .
- With good 1 as numeraire (divide by $p_{1} = 2$ ): $x_{1} + \frac{3}{2} x_{2} = 6$ , slope $- 1 / (3 / 2) = - 2 / 3$ .
- Result: all three describe the same budget line and budget set, with the same ability to trade good 2 for good 1.
Quantity discount, with $m = 100$ , $p_{2} = 1$ , and $p_{1} = 2$ for the first 20 units of $x_{1}$ but $1$ for each unit after that:
- Vertical intercept: $100 / 1 = 100$ units of $x_{2}$ .
- Up to $x_{1} = 20$ the slope is $- 2$ . If $p_{1}$ stayed at 2 all the way, the line would hit the $x_{1}$ -axis at $100 / 2 = 50$ .
- The first 20 units of $x_{1}$ cost $2 \times 20 = $ 40$ , leaving $$ 60$ . (That puts the kink at $x_{1} = 20$ , $x_{2} = 60$ .)
- The remaining $$ 60$ buys $60$ more units at $1 each, so the slope is $- 1$ past the kink.
- Horizontal intercept = $20 + 60 = 80$ , not $100 / 1 = 100$ . That was the "tricky part".
Income or price change intuition (in-class Q&A):
- More income lets you afford more ("more T-bone steak dinners"), so the line shifts out.
- A higher $p_{1}$ means $m / p_{1}^{'}$ has a bigger denominator, so the horizontal intercept moves in and the line gets steeper.

Flagged for exams

Slope of the budget constraint: "You might want to put a little star beside this concept of slope. We are going to revisit this quite a lot… understanding the slope is going to be very helpful as we move through the rest of the term." Know both what it is ( $- p_{1} / p_{2}$ ) and its two economic interpretations: the market rate of substitution and the opportunity cost of $x_{1}$ in units of $x_{2}$ .
Just affordable vs. affordable: "there is a distinction." The budget constraint means spending exactly $m$ ; the budget set means spending $\leq m$ .
Why an income change gives a parallel shift: McLeod asked for the reason. The slope is $- p_{1} / p_{2}$ , a function of prices only, so changing $m$ can't change it.
Why the vertical intercept doesn't move when $p_{1}$ rises: he pushed for the economic intuition, not just the graph. At the vertical intercept $x_{1} = 0$ , so you spend nothing on good 1. It's $x_{1}$ that is zero, not $p_{1}$ .
A change in $p_{2}$ is not on the slides but was worked through verbally: the line pivots around the horizontal intercept and a higher $p_{2}$ makes it flatter. "We can get there pretty easily." Be ready to do it by analogy.
Better or worse off: an income increase or price decrease "cannot make a consumer worse off", since no original choices are lost. An income decrease or price increase "may (typically will)" make them worse off.
Uniform ad valorem tax: the right reasoning is that the income side of the equation shrinks to $m / (1 + t)$ while relative prices stay the same, so the line shifts in parallel. A student reached the right answer via "prices increased" and was told they were "overthinking" it. Know the equivalence to an income tax at rate $\frac{t}{1 + t}$ .
Which taxes move which part of the line: ad valorem (uniform) and lump-sum taxes shift it in parallel. A quantity tax or subsidy on one good changes the slope.
Non-linear budget line intercept: find it by working out spending on the first block of units and then what's left. Don't just divide $m$ by the final price.
Numeraire: rescaling or renormalising the constraint does not change the budget set. "There's not a unique equation that describes that… We'll see that come up over and over again."
The course builds on itself: "by the time we get to chapter five, we're using the stuff that we're talking about in chapters two, three, and four." Falling behind makes it hard to "dig yourself out of that hole". This is the same scaffolding point from Lecture 1.
Academic misconduct warning: checking in on iClicker when you aren't actually in class is falsifying course records. That is academic misconduct.

Admin & deadlines

iClicker attendance started today (Week 2).
- Check-in window: from 10 minutes before to 10 minutes after the start of class.
- About 90 of 125 students had registered so far. This was the Section B lecture (he said Section A stopped at the same point), and the syllabus link for Section B is https://join.iclicker.com/PJIQ.
- McLeod said you need roughly "50 to 75% of the lectures" for the 5%. The syllabus grading table says best 16 of 21 lectures, about 76%. (The syllabus text says "16 (of 22)", so the syllabus itself is inconsistent.)
Attendance records: marked 1 (present) or 0 (absent) and transferred by hand to the MyLearningSpace gradebook. Final grades use the MyLearningSpace gradebook, so check it and email him ([email protected], with EC270 in the subject line) if you spot an error. Errors on his part won't be held against you.
Readings: skim each chapter before its lecture. Review the Math Appendix slides from Sept 10 on your own; he won't re-teach that math in class.
Next class, Thursday Sept 17: finish Chapter 2 (negative-price budget constraints are still left), then start Chapter 3: Preferences.
Assignment 1 (Chapters 2 and 3) is due September 27, 11:59pm, on Smartwork5.

Transcribed automatically from the lecture recording. Slides used: EC 270 - Ch. 02 - Budget Constraint (slides).pdf. Course info used: EC 270 - Math Appendix (slides).pdf, EC 270 - Rules of Derivatives.pdf, EC 270 - Syllabus - Fall 2026.pdf. Audio archived at /mnt/porsche/configs/lectures/archive/2026-09/EC 270 - 2026-09-15 - Lecture 02.opus.