What's the Value of a High Quality Teacher?

Sucky teachers. We've all had them. But what's the value of a good teacher over a bad one? According to working paper from the National Bureau of Economic Research (NBER), a good teacher could be marginally worth over $400,000 per year over a bad one.

This paper in particular reminds me a lot of the discussions that were going on last year with the teacher's union. This paper puts a lot of empirical evidence behind those discussions to reveal the truth about pay structures for the market for teachers.

The abstract of the paper is below:

"Most analyses of teacher quality end without any assessment of the economic value of altered teacher quality. This paper combines information about teacher effectiveness with the economic impact of higher achievement. It begins with an overview of what is known about the relationship between teacher quality and student achievement. This provides the basis for consideration of the derived demand for teachers that comes from their impact on economic outcomes. Alternative valuation methods are based on the impact of increased achievement on individual earnings and on the impact of low teacher effectiveness on economic growth through aggregate achievement. A teacher one standard deviation above the mean effectiveness annually generates marginal gains of over $400,000 in present value of student future earnings with a class size of 20 and proportionately higher with larger class sizes. Alternatively, replacing the bottom 5-8 percent of teachers with average teachers could move the U.S. near the top of international math and science rankings with a present value of $100 trillion."

Merry Christmas

Econ Style

The GPI?

An interesting new way of measuring inflation... courtesy of my favorite folks over at Google. Link here.

The Real Reason Mr. Devine Insists That You Label Your Axes

He really just wants to prevent this from happening:

Zermelo's Theorem

In game theory, we separate games into two different categories: normal form games (think payoff matrices), and extended form games. Extended form games consist of players moving sequentially (one after the other). This changes the information given to each player, as the players that move later on will have knowledge about how the players before have moved. This is why extended form games are also called sequential games. Today I will focus on a special type of sequential games: chiefly those will only two players. Below is a sequential game with two players summarized as a tree diagram:

A                B

o----T-----o-----L-------o (3,4)

|                 |

|                 ------R------o (2,2)

-----D-----o-----L------o (1,2 )

                  |

                  --------R----o (4,3) 

Where payoffs are written in the form of (Payoff A, Payoff B). Each "o" is a node. 

In this game, A moves first, with a choice of either top (T) or down (D). B then follows, and moves with the choices Left (L) or Right (R), given the information about how A has moved. The correct way to analyze this game is to use backward induction. Basically, we start from the end of the tree. Given that A has chosen T, B will ALWAYS choose L because a payoff of 4 is greater than a payoff of 2. Given that A has chosen down, B will ALWAYS choose R because a payoff of 3 is greater than a payoff of 2. Given this information, we can basically simplify our tree diagram to the following: 

A                B

o----T-----o (3,4)

|                 

|                

-----D-----o (4,3) 

Now, given this choice, it is obvious that A will choose down (D), because a payoff of 4 is greater than a payoff of 3. Therefore, the Subgame Perfect Nash Equilibrium is (D, R). This is the process of backward induction: we start from the end of the tree and move backwards to find our equilibrium. Because the topic is rather extensive (no pun intended), if you'd like to find out more on this subject, please see the following lecture: Sequential Games Lecture.

Now, let's see a game that's a bit more complicated (credit to xkcd)... 

So as you can see, tic tac toe is just a two player sequential game. We can turn this diagram into a tree diagram and use backward induction to find out the Subgame Perfect Nash Equilibrium for this game (hint: the  equilibrium is a tie). But as I'm sure you'll find out very quickly, drawing the tree out becomes very tedious (which is why most analysis involves computers now). 

Now, I'd like to introduce the concept known as "Zermelo's Theorem." Basically, it states that in a given game with 2 players, perfect information, finite nodes, and 3 outcomes (1, 0, and -1 for Win, Tie, and Lose), one of the three things below will happen: 

1. Player 1 can force a win

2. Player 1 can force a tie

3. Player 2 can force a win

This may sound obvious and stupid at first, but think about the applications. When applied to chess, it states that there is a solution to chess. One player can force either a win or tie on the other players. We could draw a very large extended tree diagram of every possible move of chess and use backward induction to find the Subgame Perfect Nash Equilibrium. As I'm sure you've noticed, this is a considerable task due to the ridiculously large permutation of moves available. This is why the solution to chess has not yet been found (but just know that it does indeed exist!). 

A Note on Profit Maximization

In 101, you learn about the concept of allocative efficiency and the necessary condition of producing at the point where P=MC. Today, I'll show how this condition is derived. Recall that in a perfectly competitive market, producers and consumers alike are price takers: there are so many producers and consumers that a single one of either cannot alter the market price by a significant amount. In this case, market price (P) can be treated as a constant. As such, your profit function is as follows:

π = PQ - C(Q)

where π is profit, P is the market price, Q is the quantity produced by an INDIVIDUAL firm, and C(Q) is the total cost of the firm as a function of Q. Taking the first order condition (Maximize π, i.e. derivate with respect to Q and set the function equal to zero), we get:

P - MC = 0

where MC is the marginal cost (as an exercise, work out and see for yourself why the derivative of C(Q) is equal to MC).

From this step, it becomes obvious where the condition P = MC comes from. As seen, setting P = MC simply the profit maximizing condition in a scenario where P is a constant.

With this in mind, we can generalize the equation for cases where P is not a constant. When is this scenario possible? When P is NOT a constant, a firm has pricing power: that is, a producer does not need to take a price as given, but influences price as a function of the quantity that he produces. We denote this as P(Q).

With this in mind, we can write the firm's profit function as:

 π = P(Q)*Q - C(Q)

Noting that P(Q)*Q is equal to total revenue, I will now re-write the function as follows:

 π = R(Q) - C(Q)

where R(Q) is a total revenue as a function of quantity produced (Q). Taking the first order condition (maximize profit with respect to Q), we get:

MR - MC = 0

From this, we get the condition MR = MC, the condition a producer with pricing power uses to maximize profit. Note that this is not a contradiction to the P = MC condition; it is merely a generalization of it because in a perfectly competitive market, P =MR.

If the calculus is above you, fear not: you'll get there. I'm merely trying to show that all of these profit maximizing conditions are not merely theoretical, but mathematically derived.

Resuming Blog Activity (and Reagonomics)

Alright, so by popular demand (aka Mr. Devine) I'm starting this up again. If you are a part of his class and would like to contribute, shoot me an email at boyangz@umich.edu. I'm somewhat busy, so my contributions to this blog will be somewhat sporadic from here on out. My posts will begin to introduce some more advanced topics in college level Macro and Micro for students who want to learn about economics beyond the scope covered by the AP test. In my opinion, the AP exam is very theory heavy. I will begin to introduce some basic mathematical models and concepts that will be important in intermediate level economic theory. I will also try to post my comments regarding current economic news (QE2 ZOMG). Also, if some of you end up contributing, I would be happy to comment on those posts as well.

To close, I'd like to post a 100% inaccurate but 100% funny (in my opinion) take on Reagonomics. Enjoy!