This is a document I handed out to students on the first day of class. It slowly walks through what we were instructed to teach as part of the math review needed for Econ 101.
The useful bits are the explanations of functions, slopes, and expected values.
I teach tutorials in Economics 101. Yesterday an international student asked me to explain why “one of the American political parties” thinks cutting tax rates will increase fiscal revenues. I explained that their theory is that lowering taxes will encourage people to work, spend, and invest more; that this would spur economic growth; and that even a lower tax rate on this higher GDP could actually generate higher revenues.
“Oh, that’s interesting. Is it true?” was his reply.
I thought about it for a second. I briefly considered the implications of answering a politically-charged question. I worried a little that this was a student whose interest in economics was piqued, and that I ran the risk of blunting that interest. After contemplating whether I should answer whether it was theoretically possible or empirically verifiable, I gave him the most honest answer I could: “No.”
The consensus among economists is that although this “bigger cake” idea has some truth to it, lowering taxes does not increase the cake anywhere near enough to make a significant difference. Harald Uhlig thinks higher tax rates could increase American income tax revenue by 30%; the Congressional Budget Office thinks that although lowering tax rates would lead to more economic activity, the revenue generated by these new activities would only be a quarter of the revenue lost in the initial tax cut; and even Greg Mankiw, chairman of the Council of Economic Advisers to President Bush (the second), thinks this thesis is a bit silly.
| Latin phrases for economists | ||
|---|---|---|
| Phrase | Translation | |
| A priori | Pre-supposed; your “priors beliefs”. | |
| Ad infinitum | To infinity; and so on. | |
| Ad valorem | Per value; a 10% tax is an ad valorem tax, whereas a duty per gallon of gasoline is not. | |
| Ceteris paribus | All else equal e.g. “We would expect, ceteris paribus, that an increase in price would lower quantity demanded.” | |
| De facto | For all practical purposes, but not officially. | |
| De jure | By law. | |
| Ex ante | Before the event; in anticipation. | |
| Ex post | After the event; in retrospect. | |
| Per se | Literally; by itself. | |
| Prima facie | In the first instance; at first glance. | |
| QED | As has been asked to be shown; done. | |
| Ultra vires | Beyond their power, e.g. the court ruled that Congress were acting ultra vires. | |
| Greek letters for economists | |||
|---|---|---|---|
| Lower-case | Upper-case | Pronunciation | Economic meaning |
| \(\alpha\) | A | Alpha | Capital share of income |
| \(\beta\) | B | Beta | Regression coefficient (econometrics), or rate of time discounting (economic theory) |
| \(\gamma\) | \(\Gamma\) | Gamma | N/A |
| \(\delta\) | \(\Delta\) | Delta | Depreciation |
| \(\epsilon\) | E | Epsilon | Elasticity |
| \(\zeta\) | Z | Zeta | N/A |
| \(\eta\) | H | Eta | N/A |
| \(\theta\) | \(\Theta\) | Theta | Type, e.g. \(\theta_H\) might represent a “high type” |
| \(\iota\) | I | Iota | N/A |
| \(\kappa\) | K | Kappa | N/A |
| \(\lambda\) | \(\Lambda\) | Lambda | The Lagrange multiplier |
| \(\mu\) | M | Mu | Mean |
| \(\nu\) | N | Nu | N/A |
| \(\xi\) | \(\Xi\) | Xi | N/A |
| o | O | Omicron | N/A |
| \(\pi\) | \(\Pi\) | Pi | Inflation, or profit (lower-case) or the product of a series (upper-case) |
| \(\rho\) | R | Rho | The coefficient of autoregression |
| \(\sigma\) | \(\Sigma\) | Sigma | Standard deviation (lower-case) or the sum of series (upper-case) |
| \(\tau\) | T | Tau | Tax |
| \(\upsilon\) | \(\Upsilon\) | Upsilon | N/A |
| \(\phi\) | \(\Phi\) | Fy | N/A |
| \(\chi\) | X | Chi (rhymes with `guy’) | Used in statistics |
| \(\psi\) | \(\Psi\) | Psi (like `Si’) | N/A |
| \(\omega\) | \(\Omega\) | Omega | N/A |
Some Useful Latin and Greek (PDF)
Some Useful Latin and Greek (TeX)
Oftentimes I hear conservatives argue for freer markets on efficiency grounds. Typically this will take the form of a pseudo-counterfactual, along the lines of “Well if there were profits to be made in this market, why didn’t some firm come in and earn them?” People sometimes assume that the freer the market, the greater the efficiency.
On the other hand, liberals will often assert that economists have an uncritical, ideological lust for the free market.
Neither of these things are true. Here’s a simple example.
Let Sam (the seller) own an item that he values somewhere between $0 and $100. My example of this is an autographed portrait of Tom Hanks — nobody really knows how much it’s worth, but it’s definitely worth less than $100. Let’s call his valuation of the portrait \(v_s\). If the seller receives a bid greater than or equal to \(v_s\), he will sell it.
Barry (the buyer) comes along. He too values the portrait somewhere between $0 and $100. He values it at \(v_b\). Sam and Barry agree to simultaneously write down their offers (the lowest price Sam will accept, and how much Barry bids) and then to reveal their prices. If the bid is greater than the price, Barry buys the portrait.
To make the maths a little easier to grasp, let’s assume the valuations are uniform, continuous, and iid. This means that 1% of people value the portrait at $1 or less; 30% value is at $30 or less; 99.99% value it at $99.99 or less, etc.
Note that this is a free market situation. There is no government intervention or taxation. There is no large corporation acting anti-competitively. There are no externalities. It’s two guys negotiating on a price.
How much does Barry bid? To get the intuitive idea, let’s say his valuation is $60. If he bids $60, even if that bid is accepted he gets no “profit” since $60 was really the most he would pay anyway. However if he bids e.g. $55 although he decreases his chances (by 5%) of winning, he stands to make a $5 profit if he does win. Thus he may have an incentive to bid below his true valuation. You can skip this if you like, but denoting \(b\) as how much Barry should bid, here’s the mathsy bit:
\(\pi_B = \text{prob(}b>v_s) (v_s – b) + \text{prob}(b<v_s)(0) \\\pi_B = F(b)(v_s – b) \\
\pi_B = b(v_s – b) \\
\displaystyle \frac{\partial \pi_B}{\partial b} = v_s – 2b = 0 \\
b = \frac{1}{2} v_s\\\)
Thus the optimal bid for Barry is exactly half his valuation. What does this tell us?
In the original setting, trade should happen if Barry’s valuation is greater than Sam’s. By our assumption, that happens half the time. However Barry, behaving rationally, will lower his bid in the hope of maximising his expected profits. This means that trade will happen only one quarter of the time.
This implies that even in this very simple no-frills no-government setting, the free market will only work efficiently 50% of the time.
So no, the free market will not definitely realise all potential gains from trade and no, economists do not think that free markets will cure all of the world’s problems.
(This is based on a microeconomic theory question that I think was written by Lones Smith. It is an application of the Myerson-Satterthwaite theorem.)
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