If production function is concave, then demonstrate that profit function will also be concave Show that concavity of firm's production function implies concavity of its profit function.
(Hint: For a concave function, first order conditions gives the vector that maximizes the function)
Confusing point: How do I relate the concavity of production function to the profit function. And where do I begin this proof mathematically.

The "hint" is wrong and misleading.
It is wrong, because first-order conditions are sufficient for a maximum when the objective function is strictly concave, or at least strictly-quasi-concave, under some conditions. For example, a linear function is also concave (as well as convex), and in such a case, the first-order conditions are not sufficient for a maximum.
It is misleading, because concavity of a multivariate function depends on the signs of its Hessian matrix of second-order derivatives.  So the OP should compute the second derivatives of the profit function, with respect to capital and labor, since I guess the presumption here is that we maximize with respect to input quantities only, treating prices as exogenous constants.
Both of the above are standard material in many microeconomics books or "math for economics" books, so the OP should look them up there.
The proper "hint" would be that costs are linear in capital and labor...
...so the profit function has a non-linear part that relates to production function, and a linear part that relates to costs. And what happens to the second derivative when linearity is present ?


Central bank loss function $$
L_t = \gamma(\pi_t - \pi_t^\otimes)^2 + \hat{Y}_t^2
$$

Central banks loss function is given by the equation above. This loss is increasing and convex in the distance from the inflation target, i.e. the marginal loss increases at a faster rate than the increase in the distance from the target.
In Figure 1, we can visualise the preferences using indifference curves, where the ellipses have the center being the desired point (0, $\pi_t^\otimes$ = Target inflation point.) (saturation point - bliss point)
It is hard for me to see how the different ellipses are indifference curves, or convex from in the difference from target?. Any explanation would help a lot.

The intuitive way

Draw the function $L_t$, the graph below is build with $\gamma = 0.8$ and $\pi_t^\otimes = 1$

Convex in this context means that the function $L_t$ looks like a bowl, just one well defined minimum. And if you cut it with a horizontal plane, the result is an ellipse.

  The formal way

Just for convenience, define the variable $p_t = \pi_t - \pi_t^\otimes$ so the function becomes

$$
L_t(p_t, Y_t) = \gamma p_t^2 + Y_t^2 \tag{B1}
$$

First note that if $\gamma > 0$  then $L_t(p_t, Y_t) \ge 0$. To prove convexity you just need to show that for any pair $(p_{t,1},Y_{t,1})$ and $(p_{t,2},Y_{t,2})$ the condition

$$
L_t(t p_{t,1} + (1 - t)p_{t,2}, t Y_1 + (1 - t)Y_{t,2}) < tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \tag{B2}
$$

holds for $ 0 < t < 1$. This is actually fairly easy to show

\begin{eqnarray}
L_t(t p_{t,1} + (1 - t)p_{t,2}, t Y_1 + (1 - t)Y_{t,2}) &=& \gamma[t p_{t,1} + (1 - t)p_{t,2}]^2 + (t Y_1 + (1 - t)Y_{t,2})^2 \\
&\vdots & \\
&=& t [\gamma p_{t,1}^2 + Y_{t,1}^2] + (1 - t) [\gamma p_{t,2}^2 + Y_{t,2}^2] \\
&&\quad + t(1 - t) [\gamma(p_{t,1} - p_{t,2})^2 + (Y_{t,1} - Y_{t,2})^2] \\
&=& tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \\
&&\quad \underbrace{t(1 - t)L_t(p_{t,1} - p _{t_2}, Y_{t,1} - Y_{t,2})}_{> 0} \\
&< & tL_t(p_{t,1}, Y_t) + (1 -t)L_t(p_{t,2}, Y_{t,2}) \tag{B3}
\end{eqnarray}

So $L_t$ is convex. To show that indifference curves are ellipses, just fix $L_t$ to a constant value, let's say $A > 0$ then we have

$$
A = \gamma p_t^2+ Y_t^2 ~~\Rightarrow~~ \frac{p_t^2}{A/\gamma} + \frac{Y_t}{A} = 1 \tag{B4}
$$

which are just ellipses centered at $p_t = 0 = \pi_t - \pi_t^\otimes$ and $Y_t = 0$ with axis lengths $\sqrt{A/\gamma}$ and $\sqrt{A}$


Joint Dynamic Programming: Group Activity Here we have two agents who can spend their time doing some group activity ($h$) or staying at home ($l$). Each agent $i$ is trying to maximize their respective dynamic programming problem:

\begin{align}
& \max \sum^\infty_{t=0} \beta^t h^{\alpha_i}l_i^{1-\alpha_i} \\
& \text{s.t.} \quad h_i + l_i = \bar{T} \\
& h = \min\left\{h_i, h_{-i}\right\}
\end{align}

Is this problem well defined? That is, is there a solution/does the question make sense economically?
Is there an equilibrium where $h_i = h_{-i}$, and if so, when? Only when $\alpha_i = \alpha_{-i}$?

(Note that typical Inada conditions and non-negativity constraints apply.)

Edit: Okay, to make this an actual dynamic programming question (derp), I offer a modified version of this question as well for your perusal:

\begin{align}
& \max \sum^\infty_{t=0} \beta^t h_t^{\alpha_i}l_i^{1-\alpha_i}k_t\\
& \text{s.t.} \quad h_{i,t} + l_i = \bar{T} \\
& h_t = \min\left\{h_{i,t}, h_{-i, t}\right\} \\
& k_t = (1-\lambda + \frac{h_t}{\bar{T}})k_{t-1} \cdot
\end{align}

Where $k$ is some "capital" stock in the group activity that makes people enjoy the activity more, the more it is done. Obviously $\lambda \in (0,1)$

If I understand correctly the temporal dimension is merely a distraction as agents face a separate optimization problem every round with no intertemporal budget constraint. Hence
\begin{align}
& \max_{h_i} h^{\alpha_i}l_i^{1-\alpha_i} \\
& \text{s.t.} \quad h_i + l_i = \bar{T} \\
& h = \min\left\{h_i, h_{-i}\right\}
\end{align}
A continuous best response function exists for each agent:
$$
BR_i(h_{-i}) = \min\left(\alpha_i \cdot \bar{T}; h_{-i} \right)
$$
The equilibrium is
$$
h_1 = h_2 = \min_i \alpha_i \cdot \overline{T}.
$$


What determines the rate of interest on national debt for a country? From what I understand of Canada's rate of interest on the debt, after Pierre Trudeau, it was around 17.4%, after Mulroney, it was around 7%, and after Harper it was around 5.2%.

Finding these figures are somewhat difficult, but to the point; what is setting these rates? Does any of it have to do with a country's credit rating?

The first thing to recognise is that an interest rate is a price because the interest rate tells you how much the borrower has to pay in return for receiving a loan.

The borrower would obviously like to pay as little as possible for the money s/he wants to borrow. But how little s/he can get away with paying depends upon how willing people are to lend their money. Here are some examples of how willingness to lend affects interest rates.

If the borrower already borrowed a lot of money and looks to be in danger of defaulting (i.e. not being able to pay everyone back) then lenders will be less willing to give him/her their money for fear for losing it. The borrower will therefore have to agree to pay a higher interest rate in order to induce people to be willing to lend.
If times are hard (i.e. in a recession) and money is short then it again becomes more likely that the borrower will be unable to repay their loan. Again, the borrower must be prepared to pay a higher interest rate to compensate lenders for this risk, otherwise they will refuse to lend.
People have lots of choices about where to put their money: government bonds (from many different governments), the stock market, savings accounts, pensions, real estate investment, corporate bonds, commodities, etc. The better is the return they get on alternative assets, the less willing a lender will be to give you their money. You then have to be prepared to pay more to induce lenders to give up all those lucrative alternatives. So borrowers will pay higher interest rates when returns in general are higher.
Not only the returns on other assets, but also their volatility matters. In times of uncertainty, assets like stocks or commodities are likely to be more volatile so that the return investors get on them becomes more risky. Since lending to certain actors (e.g. the government) is relatively safe by comparison, people will be more willing to lend money to these actors during uncertain times. Thus, the interest rate will tend to be lower during times of volatility and higher when alternative assets are experiencing relatively smooth growth.
If inflation is high then money loses its value quickly. I am going to be reluctant to give someone my money for five years if, by the time I get it back, it is worthless—I'd rather spend the money now while it's still worth something. Thus, to induce people to lend, the borrower will have to pay higher interest when inflation is higher.

Note that this reasoning applies to borrowing more generally (i.e. also by consumers or firms) as well as to government borrowing.


Cointegration test in Stata I am doing a Engle-Granger test for cointegration and I am unsure about some commands.

"Cointegration and the ECM" (document) from learneconometric.com says I should use:

regress b f
predict ehat, residual
regress D.ehat L.ehat L.D.ehat, noconstant

However, "Time series" (document from Princeton Uni) says I should use:

regress b f
predict ehat, resid
dfuller ehat, lags(10)

So I am unsure about the last commands here. Should I use regress D.ehat L.ehat L.D.ehat, noconstant or dfuller ehat, lags(10) and what is the difference here? Also, how many "lags" should I include for the Dickey Fuller test?

Do the last one, the first one is just the same thing but you will not be using the in-built adf function. The second one does it better and you have a choice of including lagged differences to control for possible autocorrelation. If your data is monthly, give it a lag order of 12.


Why would an investor choose to invest in one index fund as opposed to another? Assuming no transaction costs or management fees, why would an investor choose to invest in a fund tracking the Nikkei 225 over say the S&P500.

If I was quite bullish on Japan, and I made the conscious decision to invest in the Nikkei tracking index, wouldn't this in effect be active management because I am using fundamental analysis to select my portfolio?

Assuming we take the academic consensus that passive > active management, would this decision make sense?

I can understand choosing a fund over another due to liquidity and fund characteristics, but not much else.

There's a lot to unpack here.

It is increasingly the case, especially among passive strategies, that transaction costs and management fees are the primary dimension along which funds are differentiated.  Ignoring those costs seriously distorts the question.
If you invest dollars in the Nikkei that you might otherwise have invested in the S&P500 because you are "bullish on Japan" then you are speculating.  Whether you are engaging in "active management" comes down to how well-researched your position is and how frequently you change your holdings.
There's no "academic consensus" that passive > active.  In fact there is evidence that suggests the reason so-called "active management" has historically unperformed the market is that a lot of fund managers are reactionary. They follow the herd or bend to non-market influences such as pressure from terrified stakeholders, which leads to "buy high / sell low" behaviour. There is a middle-ground that is largely passive, but which actively looks for clear irrationality in the market and acts accordingly.
I would argue that "fundamental analysis" is "fundamentally different" when looking at an index rather than individual stocks, and therefore so too is "active management".  Remember that the fundamental tenet of portfolio theory is to optimize the tradeoff between risk and return, and so if we're ignoring fees then this balance is the fundamental quantity of concern with an index.  There is a sense in which you do not care whether the index holds Nissan; rather, you care whether the index is weighted too heavily in Japanese automakers.

Speaking personally, if I was passively riding S&P500 coverage I might contemplate jumping to the Nikkei if I saw geopolitical reasons to do so, e.g., America fully walls itself off from global trade while Japan signs key deals with Canada, UK and the EU and the markets have failed to price this reality in.  Unlikely to happen, but if it did, it would be a clear reason to move that doesn't require fundamental analysis at all.


why does income of executives level out in the long term after tax increases? Marginal Revolution University has the following question in its Elasticity unit:

In 1993, then President Clinton passed a law raising income taxes.
This tax hike was fully expected: He campaigned on it in 1992. What do
you expect happened to executive income in the first year of the tax
increases? What about in subsequent years? Hint: Top executives have a
lot of power over when they get paid for their work: They can ask for
bonuses a bit earlier, or they can cash out their stock options a bit
earlier. Literally, this isn’t their “labor supply,” it’s more like
their “income supply.”

Their answer:

In the first year income will decrease; in the long run it will remain
nearly the same

I don't understand this answer; it made sense to me that the income would increase in the short term, as:

the practices of assigning salaries and bonuses, etc, would be difficult to adjust in the short term (i.e. more inelastic), so they'd tend to stay the same, but also
in anticipation of a higher tax burden, CEOs would want to give themself higher bonuses before the tax hike goes into effect, so income in the first year would be high

And it would make sense to me that, once companies adjust for higher taxes, the income of CEOs would tend to decrease to compensate in the long term.
Could anyone point out where my reasoning is falling short?  I'm new to economics and haven't really developed my intution on the subject yet.  Thanks.

the practices of assigning salaries and bonuses, etc, would be difficult to adjust in the short term (i.e. more inelastic), so they'd tend to stay the same,

This might be true empirically (I am not an expert on CEO compensation), but if you are doing a test you have to follow assumptions in exercise (which will not always be realistic - undergraduate programs for example almost always assume linear demand and supply which is unrealistic save for some special cases).
In this case the exercise states:

Hint: Top executives have a lot of power over when they get paid for their work: They can ask for bonuses a bit earlier, or they can cash out their stock options a bit earlier.

This is basically teacher telling you throughout this exercise assume the salary can perfectly adjust even in short term.

in anticipation of a higher tax burden, CEOs would want to give themselves higher bonuses before the tax hike goes into effect, so income in the first year would be high

You need to read the set up of the problem very carefully. The set up says:

What do you expect happened to executive income in the first year of the tax increases?

So the question asks what happens to the income in the first year after the tax hike is passed. You are completely right that before they pass the tax hike the executives (within this problem) will ask for higher compensation before the tax hike, but note the problem asks you what happens in the first year after the tax hike already taken place. Well then the income will be lower as those executives within the assumptions of this exercise would rationally ask to be paid before the tax hike portion of money they would earn in the subsequent year. Hence their income in the in the first year of the tax increases will be lower.


Why is government debt level presented as ratio to GDP, not against government budget or revenue size? A country's public debt is often presented as ratio to GDP. For example, in Wikipedia, List of countries by public debt or National debt of the United States both use ratio or percentages of GDP.

Since government borrows the money in term of balancing budget and will have to repay using budget, shouldn't it make more sense to define the ratio against something related to the size of the budget, instead of the size of the whole economy? Someone pointed out to me that when seen this way, the debt ratio sounds more serious. For example, US public debt is ~18 trillion, which is slightly more than the annual GDP (this doesn't sound too bad), but actually roughly 6x government revenue in 2014 (~3 billion according to CIA World Factbook)

The ratio of government debt to GDP is a useful indicator because, broadly, the larger a country's GDP, the larger is the tax revenue the government could potentially raise to service the debt while maintaining other government expenditure at an acceptable level.  Thus a very high debt / GDP ratio would raise doubts about the government's ability to service the debt.

One reason why a larger GDP implies a larger potential tax revenue is that most of the transactions making up GDP from an income perspective, eg wages and salaries, company profits, could be taxed.  However, taxes could also be levied on items that are not part of GDP, eg capital values of property.  The more fundamental point, therefore, is that a larger GDP, again from an income perspective, tends to imply a larger aggregate ability to pay, out of that income, such taxes as may be levied by the government.

The government's ability to tax is also influenced by political factors and by the effectiveness of its tax collection, so the debt / GDP ratio can only be a crude indicator of ability to service debt.


The Dumping Argument and Trusts I have often heard the following argument against the antimonopoly/anticollusion law:

Whenever the entrepreneurs decide to enter into "illegal" collaboration, each one has an opportunity to gain profit through dumping. This will cause the price to fall to the desired free-market level.

Truth be told, I reckon this argument to be quite faulty. It works only when the entrepreneur aims at a short-term profit. Moreover, I cannot see how it works for trusts.
Yet, it got me thinking. Thinking of the last century, pretty much no collusions come to mind, just trusts: Standard Oil, Morgan's railroad monopoly, U. S. Steel (the US), the German Steel Trust (Germany), FIAT (Italy). Were there any decentralized monopolies? If not, did I underestimate the dumping argument? If not, what is the real reason behind it?

Edit: thanks to 1muflon1, who provided a very interesting link on collusions in the EU. It proves my ignorance in history (not knowing any cartels, that is). Besides, they pointed out that my terminology is a bit messed up. Sorry about that.

The argument is based on the famous prisoner dilemma result. For example, consider the following situation. We have firm A and firm B. If they both collude and set high price (high P) they both can get profit $\\\$7$, If A sets high price but B undercuts A (note undercutting is not the same as engaging in dumping see end of answer) then A will earn 0 profit because everyone will want to buy goods from B and B earns $\\\$10$ profit. If A chooses low P and B high P the opposite will happen. If they both choose low price they both earn $\\\$5$ profit. Moreover, we will assume that they cannot enter into a contract that would allow them to make commitment to high prices that would be enforceable in law.
The situation can be visualized in the matrix below:

                             Firm B
                             High P    Low P
     Firm A       High P     $7,$7     $0, $10
                  Low  P    $10,$0     $5, $5

What will happen in this case? Lets start looking at the problem from perspective of firm B:

what happens if firm A chooses high price?  You can easily see that the most rational response for firm B is to chose low price - thats how they maximize their profit as getting profit of $\\\$10$ is better than profit of $\\\$7$.
What happens if firm A chooses low price? In this case again it would be most rational for firm B to choose low price because its better to get at least the $\\\$5$ profit than zero.

Thus setting low price is strictly dominant strategy for firm B and if firm B is rational it should always set prices low. The mirror argument applies to firm A as well. Hence, due to this collusion cannot be sustained.
Few caveats:
The simple prisoner dilemma model captures the core of the argument but also its not entirely realistic as in real life firms can be argued to 'play' repeated game of competition and in repeated games conclusions can be sometimes sustained depending situation. For example, if firm play the above game repeatedly they can collude because repeated games allow for richer amount of strategies such as trigger strategies where firms can try to punish each other for deviating from the collusion equilibrium.
You can learn about these more complex situations in some industrial organization textbooks such as Industrial Organization: Markets and Strategies by Martin Peitz and Paul Belleflamme.
Nonetheless, in real life also most countries set very high fines and penalties for colluding firms as well as rewarding any firm that 'snitches' on other colluding  firms. Thanks to the fact that these penalties are very strict it is incredibly hard for firms to sustain any explicit collusion which is why they are rare but still occur sometimes - see some EU cases here (although tacit collusion - i.e. collusion where firms set high prices without actually making some secret agreement can still occur).

Furthermore, note selling for price lower than your competitor is not called dumpling.  Dumping means selling products for less than it cost to produce them not less than others. Dumping is a separate issue altogether.


Optimal pricing for Crypto-Currency Exchanges I have a general understanding of game theory, and want to try to apply it to crypto-currency exchanges, which are completely decentralised systems. I come from a finance background so I will define some key terms.

BID Price: Price at which the exchange is willing to buy an asset from you.

ASK Price: Price at which the exchange is willing to sell an asset from you.

ASK>BID, so as a trader, you will always pay the higher ASK price when buying and will get the lower BID price when selling.

The BID-ASK spread is ASK price minus BID price.

Consider the following autonomous Bitcoin exchanges with the following prices on a certain date:

As you can see each exchange quotes its own price and spread.

In a frictionless world, a trader can buy Bitcoins at the cheapest ASK price (in this case exchange B at 1477) and sell the coins on the exchange with the highest BID price (exchange A at a BID price of 1603). This arbitrage opportunity is also
found on the other exchanges as long as the ASK price paid by the trader is less than the BID price on the other exchange.

Now let's say I want to create my own exchange. If I want to eliminate all arbitrage opportunities, I would quote a BID price equal to the lowest BID (exchange B: 1475) and an ASK price equal to the highest ASK (exchange A: 1604). While this eliminates arbitrage on my exchange, it also makes me very uncompetitive given that my spread will equal 129. As such, no one will trade on my exchange.

Given that I want to keep my spread competitive, with a goal of minimising arbitrage opportunities. What is the best course of action to take for quoting Bitcoin prices?

This is a real case, and the figures in the table are actual Bitcoin\USD prices taken on a particular date for these anonymised exchanges.

You are missing the key element, transfer fees.  If the transfer fees are large enough, then the marginal cost to move between exchanges just needs to stay inside the gross bid-ask spread.  It can probably move outside that spread because there is an additional risk for one who moves between exchanges.  Not all exchanges move back and forth between real world bank accounts equally well, nor do all have equal trading depth.  Additionally, the time value of money between the exchanges is probably sufficient to eat up any arbitrage.  It may be worth giving up arbitrage profits if they are traded for the time value of money profits.  If the arbitrage is small and the cost is great, you can carry a wider bid-ask spread and get away with it as a market maker.

It sounds like you need decision theory and not game theory unless you are considering the behaviors of the other actors as part of the game.  You should start with de Finetti's Coherence Principle.  It is a weaker principle than the "no arbitrage" principle.  In its weakest form, it only states that odds are coherent if a bookie cannot take a sure loss in all states of nature through the clever combination of bets either by a single or by multiple actors.

What makes it weaker is that it never says that no arbitrage opportunity exists, but it would imply that if it did exist, then the market makers would capture all or nearly all the profits.  I would be surprised to find that the exchanges did not profit greatly from the arbitrage opportunities on each other's exchanges.  They are perfectly placed and have deep inventory that they could borrow from their customers.  They probably allow some to go by to get speculators into their markets.  They are profit maximizers.


Monetary Transmission in New Keynesian - Euler Equation I get that NK model basically consists of the three equations: Taylor rule, Dynamic IS curve and the NK Phillips curve.

I want to know the effect of nominal interest rate reduction on the transmission channels.

My lecturer says ''nominal rate goes down, real rate therefore rises, raising actual output relative to potential through the euler equation.  this then feeds into the philips curve and raises actual inflation.''

Q: How does the real rate rise when nominal rate goes down? And what is the Euler Equation here?

Real interest rate = Nominal interest rate - Expected inflation

Reducing the nominal interest rate by more than the decline in the expected inflation, with respect to the inflation target, leads the real interest rate to go down.
When the real interest rate goes down, the real exchange rate depreciates, both lead real monetary conditions to loosen. [Real monetary conditions = a1*RR_gap + (1*a1)*RER_gap].
Looser real monetary conditions lead to higher aggregate demand 'output gap' (higher domestic demand due to lower real interest rate and higher external demand due to the real exchange rate depreciation). [IS curve - Euler equation]
Higher aggregate demand puts more pressure on the real marginal costs, which in addition to higher inflation expectations that are directly affected by reducing the nominal interest rate, increase the inflation rate.


Studying Oligopolies I am a very Macro guy, but I need to dive a little in the world of Microeconomics. Specifically, I need to study oligopolies. From the general theory to empirics. Since I do not know the literature, I would like to ask if anybody knows a good starting point. Preferably a paper that explains the strategic interactions between firms operating in a oligopoly.

The basics are of course Cournot, Stackelberg and Bertrand competition, which you can find in any textbook.

If you are referring to needing references for research, then the paper you absolutely must know is Dixit and Stiglitz (1977) "Monopolistic Competition and Optimum Product Diversity". American Economic Review. With over 10k citations, the importance of this paper for research cannot be overstated. Especially for macroeconomics, which interests you, knowing this paper in and out is indispensible. It has been extensively used in international trade (e.g. Krugman's models), economic geography and monetary policy (most DSGE models incorporate it) to name a few fields.

While this paper is a specific form of oligopoly, namely monopolistic competition, it is very general and implicitly incorporates much of Cournot and Bertrand competition models. It is simultaneously relatively simple and tractable for use.

Most papers use a "light" version of Dixit and Stiglitz, however. If you search for explanations of this model online, you will find a lot of ressources that explain the different (light) versions step by step. I would actually read those before reading the paper.

A good book on the Dixit & Stiglitz model is also Brakman, S. and Heijdra B.J. (eds.) (2004): "The Monopolistic Competition Revolution in Retrospect." Because this model is so important, many interesting parts of Dixit and Stiglitz are also explained in Baldwin, R., Forslid, R., Martin, P., Ottaviano, G. and Robert-Nicoud, F. (2003). "Economic Geography and Public Policy."

You usually won't need to know any other model for applying oligopolies to research problems. An alternative is, however, the Salop (circle) model (Salop, Steven C. (1979), "Monopolistic competition with outside goods", The Bell Journal of Economics, 10). It may have an advantage depending on your setting, but is a bit outdated compared to Dixit and Stiglitz. In my experience Dixit and Stiglitz is preferred by the research community.


Growth in dividends will equal growth of earnings Let $Div_t$ be the dividend per share at date $t$.

$\displaystyle Div_t=\frac{\text{Earnings}_t}{\text{Shares Outstanding}_t}\times\text{Dividend Payout Rate}_t$ ------------------- (1)

My textbook says that if $\text{Shares Outstanding}_t$ is constant and $\text{Dividend Payout Rate}_t$ is constant, growth in dividends will equal growth of earnings.

However, differentiating (1), we get

$\displaystyle\frac{d}{dt}Div_t=\frac{\text{Dividend Payout Rate}_t}{\text{Shares Outstanding}_t}\frac{d}{dt}\text{Earnings}_t$

Clearly, $\displaystyle\frac{d}{dt}Div_t\ne\frac{d}{dt}\text{Earnings}_t$

Where have I gone wrong?

The textbook probably means growth rate and not absolute growth. Given your assumptions
$$
\frac{\frac{d}{dt}Div_t}{Div_t}=
\frac{\frac{d}{dt}\text{Earnings}_t}{\text{Earnings}_t}
$$
does hold.


Open access datasets for teaching IV regression I am looking for a dataset to show (to a group of engineers) how the instrumental variables technique is used in econometric practice.

I could always make up my own data, but I think it might be more interesting to everyone using real (not to complicated) data to replicate an actual study.

ps. Forgive me if this is off-topic for this site.

I can recommend this paper as an example:

The Colonial Origins of Comparative Development: An Empirical Investigation

Daron Acemoglu, Simon Johnson, and James A. Robinson

http://economics.mit.edu/faculty/acemoglu/data/ajr2001 (paper and data)

This example is famous not only thanks to the creative use of instrumental variables, but also because of the subsequent discussion about the validity of the instruments.

And relevant discussions:

Bazzi, Samuel, and Michael A. Clemens. “Blunt Instruments: Avoiding Common Pitfalls in Identifying the Causes of Economic Growth.” American Economic Journal: Macroeconomics 5, no. 2 (2013): 152–86.
Deaton, Angus. “Instruments, Randomization, and Learning about Development.” Journal of Economic Literature 48, no. 2 (2010): 424–55.

Secondly,

Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments

Angrist, Joshua D., and Alan B. Krueger

http://economics.mit.edu/files/18

  We discuss the mechanics of instrumental variables and the qualities
  that make for a good instrument, devoting particular attention to
  instruments derived from "natural experiments."

Examples follow.


The purpose of advertising for Pepsi What is the purpose of advertising for companies like Pepsi who advertise the same product and every person in the modern world knows about it? Why waste money?

Advertising is hard to understand if you take classroom economics models too literally.  I highly recommend diving into the literature on behavioural economics if you are truly interested. But for the purposes of your question, Pepsi runs advertising because it operates in a market in which consumer choice is primarily driven by two sources of cognitive bias: the anchoring and the availability heuristic.

On the one hand, new customers are born every day, and if Pepsi doesn't run advertising then those new customers will see a Coke ad first.  On the other, since existing customers don't actually make purchasing decisions based on a careful weighting of product merit, they need to be reminded that Pepsi exists and that there are warm, fuzzy feelings associated with the brand. This way, when they find themselves craving something to drink, Pepsi might be the brand that comes most readily to mind, because it's the last soft drink ad they saw.

This is just my opinion (built on some of the theory, but by no means do I claim it is the Complete Truth), but the anchoring heuristic in particular seems to explain brand loyalty in the cola market.  I grew up in the era of the "blind taste test" and my takeaway from those social experiments is that when the most significant product differentiation is along branding lines, consumers will tend to just stick with whatever product they encounter first, because absent labeling and the social utility that comes from a trite form of tribalism, the consumer is indifferent to the choice.


Cyclicality of Real Wages? Counter, pro or acyclical? Is the real wage counter or pro- cyclical? I've read in some textbooks that the real wage is mildly pro-cyclical, but I've also found some papers stating that it's acyclical.

So, which is it? Also, if the real wage is pro-cyclical, does that mean that there's some sort of price rigidity, i.e., the economy is not always on the Price-Setting curve?

Are Wages Procyclical?

Here's the entry from Table 1 in King and Rebelo (2000; Resuscitating Real Business Cycles). The variable of analysis is log-wages, detrended with HP filter. It has a std, relative std, first-order autocorrelation and correlation with output of 0.68, 0.38, 0.66, 0.12.

This leads the authors to conclude that

  The real wage is much less volatile than output.

You can see that for their long US period, 0.12 is a positive number, but pretty close to zero. This is why different authors take this number as either acyclical or weakly procyclical.

Price rigidity required for procyclical wages?

Assume that some sort of fluctuation in TFP "drives" the business cycle. Furthermore, if markets are competitive, the wage rate should be set by the marginal product:

$$ w = AF_L(K, L)$$

If $L$ stays constant over the business cycle (e.g. with inelastic labor supply), this setup will generate wages $w$ that vary with $A$, and hence any change in $A$ will result in a one-for-one change in $w$ and in $Y$: $Y$ and $w$ are perfectly correlated, while $L$ is fixed.

This is pretty much the prediction of the RBC model, and it is the opposite of what we observe in the data: $L$ is volatile, while $w$ is fixed. If you'd assume that $L$ responds strongly to changes in $A$, $w$ might be pretty constant over the business cycle. This is what we observe in reality.

And most of the labor-side of RBC research is to come up with ways that force $L$ to respond so much.

King and Rebelo (2000) should be a key paper of any introductory reading list for macroeconomics and I strongly recommend reading it.

The case of less competition

If there is literally zero competition, then firms will at all times only pay the household his outside-option value, making him indifferent between working and not working. If that outside option is constant over the business cycle (for which we have to some extent theoretical and empirical arguments), wages will be acyclical.

However, to the extend that there is no perfect monopsony, firms have to compete for workers and offer higher wages whenever workers are worth more. Hence, wages will be procyclical even if, to some extent, firms are not perfectly competitive on the labor side. The more competitive they are, the more pro-cyclical wages should be.

Alternatives and extensions to the neoclassical model

The above model was just a simple one to generate a case in which pro-cyclicality of wages can be generated without price stickiness. There is much more to the subject than an answer here can provide. Here's some more references:

Barro (1977): Long-term contracting, sticky prices, and monetary policy. *Wages are long-term contracts, hence if firms are less risk-averse than workers, firms will insure workers against wage changes over the business cycle and we can observe (optimally) constant wages
Beadry and DiNardo (1991) is a follow-up on that discussion
Haefke et al (2013, JME) is also a very recent follow-up
Bewley (1999): Why wages don't fall in a recession
Shimer and Hall (two separate papers, both in the same 2005, AER) are the standard references on whether the more modern and reasonable frictional DMP model can match labor (and wage) volatility over the business cycle.


Rosen's Diagonal Strict Concavity condition Consider a game with $n$ players, with strategy space $S \subset \mathbb{R}$, where $S$ is bounded set, and player's $i$ payoff function  $\pi_i:S^n \rightarrow \mathbb{R}$.
Rosen's condition (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965) for uniqueness of the Nash Equilibrium in n players game states that the  equlibrium will be unique when

payoff function $\pi_i(\textbf{s}) \; i \in N$  is concave in own strategy
There exists vector $\textbf{z}$ ($(\forall i \in N)(z_i \geq 0)\ \wedge (\exists i \in N) (z_i >0)$ such that function $\sigma(\mathbf{s}, \mathbf{z})=\sum_{i=1}^{n}z_i\pi_i({\textbf{s}})$ is diagonally strictly concave

$N$ denotes the set of players.

To define the concept of diagonal strict concavity, fist introduce 'pseudogradient' of function $\sigma$, defined with:
\begin{align}
g(\mathbf{s},\mathbf{z}) = \begin{pmatrix}
z_1\frac{\partial \pi_1(\mathbf{s})}{\partial s_1} \\
z_2\frac{\partial \pi_2(\mathbf{s})}{\partial s_2} \\
... \\
z_n\frac{\partial \pi_n(\mathbf{s})}{\partial s_n}%
\end{pmatrix}
\end{align}
Then, function $\sigma$ is said to be diagonally strictly dominant in $\mathbf{s} \in S$ for fixed $\mathbf{z} \geq 0$ if for every $\mathbf{s}^0, \mathbf{s}^1 \in S$ the following holds:
\begin{align}
(\mathbf{s}^1 - \mathbf{s}^0)'g(\mathbf{s}^{0}, \mathbf{z}) + (\mathbf{s}^0 - \mathbf{s}^1)'g(\mathbf{s}^{1}, \mathbf{z})>0
\end{align}

It is shown, in the paper I cite in the beginning, that a sufficient condition for $\sigma$ to be diagonally striclty concave is that matrix $\left[G(\mathbf{x}, \mathbf{z}) +G(\mathbf{x}, \mathbf{z})' \right]$ is negative defite for $\mathbf{s} \in S$, where $G(\mathbf{x}, \mathbf{z})$ is Jacobian of pseudogradient $g$ with respect to $\mathbf{s}$. I use ' to denote transpose of a matrix.
What is the intuition behind diagonal strict concavity condition?

So you want to find a maximum of $\sigma(s,z)$. If $\sigma$ is diagonally strictly concave you can do so by starting at any point and just following the gradient $g(s,z)$ until you find the maximum and no matter where you start, you will always end up at the same point (Start at the lower black points and follow the direction of the gradient (the direction of the steepest ascent).).

However, if $\sigma$ isn't diagonally strictly concave, you could end up at different maxima by starting at an arbitrary point and following the gradient (Follow the direction of the steepest ascent starting from the two lower black dots; you'll end up at two different points.).


The concept of expansion of an information structure I would like your help to understand the concept of expansion of an information structure in the incomplete information game at p.6-9 this paper.

Let me summarise the game as described in the paper.

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.

A basic game $G$ consists of

for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.
a full support prior $\psi\in \Delta(\Theta)$.

An information structure $S$ consists of

for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.

A decision rule of the incomplete information game $(G,S)$ is a mapping
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

Expansion:

Consider two information structures, $S^1\equiv (T^1, \pi^1)$ and $S^2\equiv (T^2, \pi^2)$. We say that $S^*\equiv (T^*, \pi^*)$ is a combination of $S^1$ and $S^2$ if

$T_i^*=T_i^1\times T_i^2$ $\forall i$.
$\pi^*:\Theta \rightarrow \Delta(T^1\times T^2)$ has $\pi^1$ and $\pi^2$ as marginals.

An information structure $S^*$ is an expansion of an information structure $S^1$ if there exists an information structure $S^2$ such that $S^*$ is a combination of $S^1$ and $S^2$.

My question:

The game, as it is described by the authors, seems to assume that, before receiving the signal $T_i$, each player $i$ knows nothing about what will be the realisation of the state. I call this as the baseline level of information assumed.

(For example, in other contexts, one may assume that the state is a vector of size $N\times 1$ and,  before receiving the signal $T_i$, each player $i$ knows the realisation of the $i$th component of such a vector. This would correspond to another kind of baseline level of information)
Let $\underline{S}$ denote the information structure that is totally uninformative, i.e., it does not add anything to the baseline level of information assumed (also called DEGENERATE at p.26 of the linked paper). In other words, $\underline{S}$ consists of

(a) for each player $i$, a finite set of signals $T_i$, where we write
$T\equiv T_1\times T_2\times ... \times T_N$.

(b) a signal distribution $\pi: \Theta \rightarrow \Delta(T)$ such that $\pi(\cdot|\theta)=\tilde{\pi}$ $\forall \theta \in \Theta$ for some $\tilde{\pi}\in \Delta(T)$. In other words, the conditional probability is equal to the unconditional one and our belief on the probability distribution of the state is not updated.

Notice that there are many ways to characterise the uninformative information structure (just by varying $T$ and $\tilde{\pi}$).
Let $\mathcal{S}$ denote the collection of all possible information structures. More precisely,

$$
\mathcal{S}\equiv \{S| T \text{ is a separable metric space}, \text{ $\pi:\Theta \rightarrow \Delta(T)$ is a probability measure on $(T,\mathcal{B}(T))$}\}
$$
where $\mathcal{B}(\cdot)$ denotes Borel sigma algebra.

Note that $\mathcal{S}$ contains all possible ways to characterise the uninformative information structure.
Question: Can we show that, for a given $\underline{S}$, each $S\in \mathcal{S}$ is an expansion of $\underline{S}$ (including $S=\underline{S}$)?

This seems to me to hold at the light of Theorem 1 combined with reading at p.26 of the paper "Now consider the case where the original information structure is degenerate (there is only one signal
which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria
correspond to joint distributions of actions and states that could arise under rational choice by a decision
maker with any information structure"

You have to be careful what you can say is that if $\underline{S}$ be given by $(T^1,\pi^1)$ and $S$ is any information structure, say given by $(T^2,\pi^2)$, then there exists an expansion $S^*$ of $\underline S$ given by $(T^1\times T^2, \pi^1\cdot \pi^2)$ where $\pi^1\cdot \pi^2$ is the product distribution of $\pi^1$ and  $\pi^2$. Note that I have constructed the expansion, thus shown its existence.

The information structures, $S$ and $S^*$ are different since their support is in different spaces, $T^2$ vs $T^1\times T^2$. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by $S$ and $S^*$. Alternatively, if $\succsim$ is the Blackwell partial order, then $S\succsim S^*$ and $S^*\succsim S$. Your intuition, however, is correct, any information structure will be "essentially" an expansion of $\underline S$.


What happens if the "control variables" are also endogenous? I work in Political Economy, and a lot of the models include "innocent" control variables such as population, inequality, colonial legacy, etc. so that the author can claim unbiasedness on their independent variable of interest.

But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?

If that's true, then what can we do? Leave those control variables out and they lead to omitted variable bias themselves. Include those in and they will contaminate everything in the model.

Example: A researcher wants to know if inequality leads to violence, and he controls for a few things:
\begin{equation}
Violence = Inequality + Growth + Development + \epsilon
\end{equation}
Seeing that Inequality is likely to be endogenous (because of the omitted variable Level of altruism), he will try to find a instrumental variable for Inequality. But aren't Growth and Development likely to be endogenous (i.e. correlated with Level of altruism) too?

This example may look silly, but my point is in Political Economy / Development work, there are so many factors at play (yet omitted) that I'm afraid many variables included on the LHS are endogenous. Yet often, the researcher only looks for an instrument for his pet independent variable only.

"But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?"

I don't want to emphasize this too much, but it's worth mentioning that this is not true in general. The following derivation will hopefully provide some understanding of the "contamination" you mention. As a simple counterexample, suppose that the data generating process is given by
$$
Y = X_1 \beta_1 + X_2 \beta_2 + Z \gamma + \varepsilon,
$$
where $Z$ is unobserved. Let $Cov(X_1,Z) = 0$, $Cov(X_2, Z) \neq 0$, and
$Cov(X_1,X_2) = 0$. Then, it
is clear that $X_2$ is "endogenous." But notice that because $Cov(X_1,Z) = 0$, our
estimate of $\beta_1$ will still be ok:
$$
\text{plim}\, \hat \beta_{1} = \beta_1 + \gamma \frac{Cov(X_1^*, Z)}{Var(X_1^*)} = \beta_1,
$$
where $X_1^* = M_2 X_1$ and $M_2 = [I - X_2(X_2'X_2)^{-1}X_2']$. Because $Cov(X_1,X_2) = 0$, $X_1^* = X_1$. So $Cov(X_1^*,Z)=0$.

"What can we do?"

One of the mains challenges of doing good econometrics is thinking of potential identification strategies. In the type of situation you describe, there is probably nothing you can do but to try to approach the problem a different way.


Prices set by firms are the same in a Salop circle. But why? Trying to calculate the prices charged by each firm, I don't understand why the solution argues that all firms set the same price. If I use the assumption that they are the same, I can figure out the value. I just don't understand why they should be the same.

For those unfamiliar with a Salop circle, Levy and Reitzes describe one, depicted below with evenly spaced firms (A through F). Consumers are uniformly distributed along a circle of unit circumference (say, on every point of the circle is a consumer, and the circle has a circumference of $1$) and served by one firm only. Firms only compete directly with their neighbors. Using profit maximization strategy and finding the market segements for each firm (through finding the indifferent consumers), the price set by firms is given by

$$
p_i=\frac{1}{4} (p_{i+1}+p_{i-1} ) + \frac{1}{2} (t/N^2+c),
$$
where $p_{i+1}$ ($p_{i-1}$) is the price set by the clockwise (counterclockwise) rival, $t$ represents the cost consumers experience per unit travel, $N$ is the number of firms in the market, and $c$ is the marginal production cost (this is the same for all firms).

But here is the part that I don't understand. From this, it is then argued that with a Bertrand-Nash equilibrium, firms cannot directly influence the prices of their rivals. And therefore the prices should be the same.

This I already find strange: in the formula above, you would naturally argue that the prices set by firms $i+1$ and $i-1$ have an influence on that of $i$. And even if you were to assume they cannot influence each other's prices directly, how would you interpret the equation above? And how could you derive that the prices set by all firms are the same? Would it be derived mathematically from the formula above, or would it be argued by economic intuition in words?

tl;dr: Prices are identical because firms are identical.

Profits of firm $i$ depend on own price $p_i$ and on neighbors' prices $p_{i-1}$ and $p_{i+1}$. Prices are set simultaneously, therefore competition in this market is described by a (normal form) game. A strategy is a price, and your formula for $p_i$ is the best response function, showing firm $i$'s profit maximizing price, given its neighbors' prices. Since consumers are distributed uniformly along the circle and the firms are equidistant and have the same marginal costs, the game is symmetric. It is therefore natural to look for a symmetric Nash equilibrium as a solution of the game. To do this, you assume that your own price and your neighbors' prices are symmetric, i.e., identical, while at the same time your price is the best response to their prices. This gives you the common price of all $N$ firms in symmetric Nash equilibrium.


Calculus Based Microeconomics Book This is more of an informal question. I am studying microeconomics theory and have completed the intermediate microeconomics by Hal R.Varian. I really enjoyed the book and want to upgrade my skills. So please suggest me some good books that would serve the purpose. Thanks.

If you would like to go step above Varian's intermediate microeconomics and you like Varian as an author then his book Microeconomic Analysis is on graduate level.
However, I personally prefer more  Mas-Colell,  Whinston, and Green (MWG) Microeconomic theory. That is probably the most comprehensive microeconomics textbook there is on the market.


Deriving long-run cost functions from production function Suppose that I have a production function $(aK + bL)^3$ in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function without using Lagrange. I have obviously tried $MRTS=w/r$, where w is wage and r is rent for capital, to express L in terms of K or vice versa. This does not yield to anything as the variables K and L just get cancelled in the MRTS. I am utterly lost. How do I go about this?

Don't get stuck with math.

Look again at the production function. Perhaps draw out isoquant curves (for the same output, how could you vary $K$ and $L$). How are the isoquants for this production function different from those of standard Cobb-Douglas production function?

Hint: perfect substitute.


What is it called when there is no personal increase of wealth but GDP increases because of mutual sales? In a hypothetical situation, every person produces some product but instead of consuming it sells it to another person.
This situation would increase the GDP whereas self-consumption doesn't.
Say, person x cleans person y's car and vice versa, and both pays each other some amount Z. Or maybe person x cleans y's car, y cleans a's car, a cleans b's...and so on. Or let's say x sells eggs to y, y sells to x.
Either way, the persons involved got some product or service and is left with no extra money.
This situations increased the GDP, whereas self-consumption doesn't. Is it a flaw? If so, what's it called?

Yes this can be considered flaw of GDP since GDP’s intended purpose is to measure the value of output produced by a nation.  In fact in his report to U.S. Congress Kuznets (who is the person who developed the modern concept of GDP) mentions in a preface that national product (GDP) in his view should encompass all production at its hypothetical market value (not just a production actually sold at market):

If  all commodities produce and all personal services rendered
during the year are added  at  their market value, and  from  the resulting  total  we subtract  the  value  of  that  part  of  the  nation's stock  of  goods  which  was  expended  (both  as  raw  materials  and  as  capital  equipment)  in  producing  this  total,  then  the  remainder  constitutes  the  net  product   of  the  national  economy  during  the  year.  It  is  referred  to  as national  income  produced,  and  may  be  defined  briefly   as  that  part  of  the  economy's  end-product  which  is  attributable  to  the  efforts  of  the individuals  who  comprise a nation. [emphasis is mine]

Kuznets even explicitly mentions house production should be part of this income, but he concluded it’s omission is ‘unavoidable’ due to the fact that it is problematic to measure it:

Services of housewives and other  members of the family.
—The  volume  of services rendered by housewives  and other members  of  the household  toward the satisfaction  of wants must  be imposing indeed,
when totaled for  the  30 million families  comprising  the population  of  this country; and the item is thus large enough to affect materially any  estimate  of  national income. ... It  was considered  best to omit this large group of services from national income, especially
since  no  reliable  basis is  available  for  estimating  their  value. This  omission, unavoidable though it is, lowers the value of national income
measurements as indexes  of  the  nation's  productivity  in  conditions  of  recent  years  when  the  contraction  of  the  market  economy  was   accompanied by an expansion of activity within the family [emphasis is mine].

So already the original creator of GDP arguably considered this omission as a ‘flaw’ but an ‘unavoidable’ flaw.
In addition historically GDP was from its inception used also for welfare comparisons and economists generally agree that home production and self-consumption increases welfare (see England; 1998).
Consequently, it is valid from economic perspective to consider omission of non-market activity (such as home production) as a failure.
This being said  from purely accounting perspective this would not be flaw because GDP is defined as “the total monetary or market value of all the finished goods and services produced within a country's borders in a specific time period” (see here). So from pure accounting/statistical perspective it’s by definition not intended to measure non-market production, but as mentioned above this accounting/statistical definition was a result of expediency and not because it’s intended purpose was to omit non-market activity.
I am not aware of there being any special name within economics for this flaw. Generally, the above would be just called limitation of GDP, but limitation here is not some economic jargon.


Is the global Automobile Industry changing from an oligopoly to monopolistic competition? The current situation is the automobile industry is an oligopoly as there are few big firms and massive barriers to entry due to cost.

However, The industry seems like it is changing to a monopolist competition structure as new firms join the industry (Apple, Google). This is as a result of bigger differentiation’s in cars product (due to software)

The automotive industry is being disrupted by new trends such as electric vehicles and self-driving cars. These, still, represent a minor part of the market.

So the answer would be no, for the moment besides media attention, there weren't gigantic shifts in the industry as to have monopolies. Looking at the last years data it seems to remain an oligopoly with 4 very large manufacturers.

If you need further info this is usually not free - look for the automotive industry in business databases such as IBIS world, or get a report from a consulting firm such as this one.


Should I deflate a nominal variable when forecasting it? Say I have a structural equation that is forecasting Y. Y is a nominal financial variable that grows with time due to inflation. I have an economic determinant that is a rate and therefore does not increase with time. Is it wise or necessary to deflate Y so that the variables behave the same way? I do not particularly want Y deflated as this regression will be forecasted and this just introduces another hard to forecast variable.

In macro, the standard approach to forecast variables which are intrinsically nominal like GDP seems to be to separate the forecast. This is, you want to have a model for real GDP and for inflation. The final forecast for nominal GDP is the "combination" of both. This is for instance the standard practice of the Fed (of Atlanta) and the Bank of England (pages 39 to 41).

In your case of a financial variable, you could model the real (deflated) variable, and then, as inflation is given by the economy, just take the official inflation forecast of your country from the central bank. They are the real experts on this so is a safe thing to do in my opinion. Also, the market expectations of inflation, crucial in finance, are very much informed by central bank estimates. This approach gives you a cleaner estimation of your model too. As you say, your determinant is a rate, and thus inflation is just molesting in the left hand side.


Who enjoys the Net Domestic Product? Suppose the Net Domestic Product in a certain sector (e.g. agriculture) is 10 billion. Who enjoys this 10 billion?

Initially, I thought that this 10 billion is the gross income of the farmers. So some of it is paid as taxes while the rest remains with the farmers as net income.

But, I did the math with some real-life data, and found out that the net income of the farmers is much smaller.

So, where does the difference go to?

My conjecture is that the NDP include the income of other workers related to agriculture, e.g. truck drivers, traders and sellers in farmer markets. Is this correct? Or are they counted in a different sector?

In terms of attribution. Recall that Net Domestic Product is the total cost of goods produced minus capital depreciation. In terms of why farmers income is lower, than the NDP, you must subtract out all the costs the farmers have to pay to create that good. So fertalizer cost, reinvestment, wages, interest costs, supplies, insurance etc. etc. all need to be subtracted. Finally, the remainder is split between farmers (in terms of income) and shareholder of corporations that operate in agriculture who recieve a return on thier investment (retained earnings for corporation or dividend payments).

Finally, your farmer income dataset is not an accurate description of the rewards of thier labours. Many farmers are small business owner so, in addition to thier income, they are building equity by making the land more productive and creating a business.

In other words, you are almost looking at the gross margin of a farming business. The NDP is the Revenues (minus capital costs) to get to the farmers net income you have a a substnatial number of other costs to subtract (taxes, amortization, operating expense, interest expense etc.). Finally, the farmers net income is not nescairly the same as the FARM/ FARM BUSINESS' net income so the actual farmers income will be lower still.


After how many years will loan be paid with constant payment I'm solving the following problem:

  A young couple took a loan 100 000 USD. For the first 5 years they pay
  5000 USD yearly. After 5 years they pay 10 000 USD yearly. The
  interest rate is 5%.

  After how many years will the debt be fully paid?

After 5 years I got:

$78352,61655 = 5000 * \frac{1-(1+0,05)^{-5}}{0,05}$

Then I calculated how many years do they need to pay with constant payment of 10 000 USD:

$n = \frac{ln(\frac{78352,61655*0,05}{10000}+1)}{ln(1+0,05)}$

$n=6,78$ years

Are my steps correct?

Thanks!

Some of your equations are wrong and you need to account for the time value of money correctly in some places. It might be useful to write down a timeline and make sure to bring all payments to the same point in time (let's take time 0 as our reference)

Therefore, what you receive should have the same present value to what you pay (I'll use American conventions for punctuations):

$$100,000=\frac{5,000}{0.05}\left[1-\frac1{(1.05)^5}\right]+\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$ and solve for $n$.

So the first calculation was correct (though  your formula is not), since doing the first operation you get:

$$78,352.61665=\frac1{(1.05)^5}\left(\frac{10,000}{0.05}\right)\left[1-\frac{1}{(1.05)^n}\right]$$

However, hopefully you can see that you failed to account for the fact that the $10,000$ payments start coming at period 6 (so the annuity formula will  bring the payoffs to period 5 and you have to further discount them 5 more periods), further, you also did something funny with the signs when you solved for n. Instead I get that:

$$n=-\frac{\ln\left(1-\frac{(78,352.62)(0.05)(1.05)^6}{10,000}\right)}{\ln(1.05)}=14.2067$$

That means that they will finish paying the debt in period "$19.2067$", i.e. in period 20. That is after 5 years of paying $5,000$, 14 periods of paying $10,000$ and paying the remainder (less than 10,000) on period 20.


Why do many economists believe that money is neutral in the long run? If a central bank carried out monetary policy that consisted of multiplying each person's wealth by a common factor, I would find it plausible that all prices would get multiplied by the same factor and there would be approximately no real effects.

However, this is not how central banks carry out monetary policy. They print money and engage in market transactions that target certain prices. To me it does not seem plausible that this has no real effects, in the long run or the short run.

My question is, why do many economists believe that money is neutral in the long run? Do they take the quantity theory of money as proof that money is neutral in the long run? Is there particular empirical evidence, or influential papers that have convinced many economists that money is neutral in the long run?

There are multiple ways that a central bank can resort to apply its designated monetary policy. The most common one is called "open market operations (omo)", which is basically selling/buying government securities to/from private individuals or organizations. Through these operations, central bank affects the level of cash reserves in the banking system since it deposits/withdraws payments for these government securities to/from the banking system. When cash reserves in the banks go up -by central bank buying government securities-, these banks decrease their interest rate to deplete the extra added cash reserves, and the vice-versa.

As you can see, in this mechanism the only price that is changed "directly" by central bank is the price of government securities. Once the cash reserves in the banking system reaches its targeted level, the probability that the general price level in a specific market will be affected is equal for almost all markets. Thus the operations of central bank does not aim at specific markets, alterations in the interest rates affect the whole economy equally by average.

There are tons of literature, both theoretical and empirical, focusing on the neutrality of money. You can also search them via Google Scholars.


What are taxes for since law forbids printing of money? This question bothers me for a long time: why government keeps up with the taxes instead of with printing money? This would decrease its operating costs, simplify law and generally raise comfort of existence of all people. Why I don't hear nobody speaking about such idea?

"Why I don't hear nobody speaking about such idea?"

Because historical experience says it won't work.

By printing money instead of collecting taxes, what increases is the nominal disposable income. The "value of work" is certainly not increased. And the important question is, does this increased nominal income lead to higher consumption?

Consider a static society that produces food using labor $L_p$. This society wants also the government to offer some services, say law and order. So the government will also employ some of the available labor in the society, $L_g$ to offer these services. Now, policemen and judges must also eat to survive, so the food produced by the private sector must be enough also for them.

Say it is and part of it must be transferred to $L_g$ somehow. Initially, the government collects taxes for this purpose.
The private sector faces

$$P_c\cdot Q = P_c\cdot w_p\cdot L_p =P_c\cdot C_p + P_c\cdot T$$

where $P_c$ is the price level, $C_p$ is private-sector consumption in food units, $w_p$ is the wage per unit of labor in food units, and $T$ is taxes in food units The left-hand side is production the middle term is pre-tax nominal income of the private sector, the right-hand side is how this income is split between own consumption and taxes. The government looks at

$$P_c\cdot T=P_c\cdot w_g\cdot L_g $$

The government collects that tax, gives it to public servants as pay for their services and they in turn go to the private sector and purchase the food they need.

The economy uses money, and the current quantity of money printed is $M$, which is fully used to facilitate the transactions in a production-consumption cycle.

Then somebody comes up with the OP's idea: why collect taxes which nobody likes, and not print the money needed to pay the government employees?

So the government prints new money $M_n = P_c \cdot T$, stops collecting taxes and pays its employees with $M_n$.

Since taxes are not collected anymore, once production is finished and before consumption, the private sector has in its hands
a) Real quantity of food $C=C_p+T$
b) Money holdings $M = P_c \cdot C$

Government employees appear and show the new money quantity $M_n$ asking to buy the same quantity of food as before. The private sector complies. After the transaction, government employees have no money but they have food. On the other hand, the private sector has

a) Real food quantity $C - T=C_p$ as before
b) Increased money holdings equal to $M + M_n$

And the question is : how does the private sector experience its new situation? What does it mean for the private sector to have more money holdings than before? Well, the private sector accepted the new money brought by the government employees, so it must be the case that this new money is considered to bear the same value, the same credibility as the pre-existing money. This in turn means that all money bills are thought to have purchasing power. It follows that the private sector has in its hands more purchasing power than previously, $M+M_n$, but the same amount of output, $C - T$.

From here on, the standard story is that people will try to use $M+M_n$ to buy more food and consume more, but since available output is the same, $C-T$, this will increase prices (imagining a "bid war" to buy the output). This story is "standard", because it has been observed too many times around the world. Although governments have not eliminated taxes, they have on occasion printed additional money in an attempt to reduce the tax burden or increase government expenses without increasing taxes.

I will describe now two conceivable scenarios (alternatives or in sequel) that do not lead to increased prices:

Α) People do not attempt to increase their consumption with their newly increased purchasing power. They just stash away $M_n$.
Next year, the same cycle happens, the government prints an additional $M_n$ quantity of money to pay its employees, the private sector save this new money on top of its previous savings etc,: the price level stays the same, while the private sector has stashed $2M_n$ away. After $k$ years, the private sector will have saved $k\cdot M_n$ in paper money... To do what with it? Look at it and "feel rich" but never actually use it?

Well, historical experience tells us that people will tend to use this additional money, exactly because they accept that it carries purchasing power. It appears that people are not content to just look at stacks of money bills and rejoice. I mean, initially they do, but then they want to go out there and translate these stacks of money bills into increased consumption (the Scrooge McDuck type of person is, alas, a negligible minority).

B) People are not Scrooge McDucks and try to buy more than before with their increased money holdings. But sellers do not increase their prices, they even resist any attempts by the buyers to offer a higher amount of money,  and sell at the old prices until they run out of stock. Result: at the end of the day, people realize that their increased money holdings were not good enough to bring them higher consumption. But this will make them question and re-assess how valuable, after all, is this money. Which leads us to the exact same consequences with a price increase: all money will lose part of its carrying value in both cases.

People will enjoy the same consumption level as before, but they will feel happier now that they don't pay taxes. But since the government keeps printing new money year after year, prices will go to heaven. Again, historical experience says that people appear to dislike high inflation, probably because in the real world with millions of products the price system plays an important signalling role, which appears to be understood and be valued by the people. High inflation impairs this function, increasing the transactional costs of the people. So one should offset the gain in happiness from the feeling of not paying taxes, to the problems that will be created by high inflation... since recorded episodes of hyperinflation as well as high inflation were time and again ended by government action, it seems that people support such measures, which implies that even though they don't like paying taxes, they dislike more living in high inflation.

ADDENDUM
The OP seems particularly concerned about the tax collection costs, as he indicated in a comment to this answer, putting the matter "at the core" of its question. Here are estimates of the tax collection costs for many countries for three recent years, from OECD. OECD-30 mean value is at $1\%$ of revenue collected.

An excel file with the data series and the diagram can be downloaded here.

More info from OECD on the matter of tax collection efficiency, here.


why subsidy is given why not govt direct discount the prices? I have a great confusion about the fundamental of subsidy . Why government give subsidy when it can also discount the price but it doesn't do so ?

Supposing by "discount the price" you mean set a maximum allowed price that is lower than the current price.
At the current price supply is equal to demand, when the price drops, there will be less supply and more demand. Then there is more demand than supply and the government doesn't want this because some people don't get what they want. Subsidies solve this problem because the price will drop because a part of the profit isn't paid for by the consumer but by the government. Then supply and demand are equal and the price is lower.


Why is a higher inflation rate bad for international competitiveness? Why if there's a higher inflation rate in country A then its exports are less competitive and its trading partners prefer to buy from countries with lower inflation rates?

Competitiveness simply means how expensive goods of a country are when compared. To correctly compare we would need to transform everything into a(ny) single currency. To do this correctly we need two elements: a) the actual price of the good and b) its exchange rate. If a country has higher inflation, then nominal prices increase, the goods are more expensive, less desirable and therefore due to (a) competitiveness decreases.

However over time purchasing power partiy (called PPP) holds. If it holds (typically does in the long run) then due to higher inflation in a country, the currency will likely lose value (depreciate) and competitiveness will increase due to (b), in which case it will eventually cancel the aforementioned effect (a) in which case your confusion is completely justified, there should be no effect.

However since PPP does not always hold and especially not immediately, higher inflation reduces competitiveness.


Leveraged Return on Investment A general return on investment (ROI) formula is:

\begin{equation}
\text{ROI}=\frac{\text{Net Profit}}{\text{Total Investment}}.
\end{equation}

But if you think about an investment that requires some sort of down payment, thus it is deemed as leverated return on investment (LROI), what would be a sensible way to compute such return? Specifically, I am interested in having $\textit{down payment}$ in the denominator. Think of a construction project or home investment.

\begin{equation}
\text{LROI}=\frac{\text{?}}{\text{Down Payment}}.
\end{equation}

Any suggestions?

I am not sure I understand your question. In most projects, the bulk of the investment is a down payment, so the ROI is meant to capture precisely that. I.e. the average return that the money you put in the project will yield. This has in mind that the total investment is mostly done today and the returns come in the future.

If you restrict the denominator to only be the down payment, you seem to want to capture the average return of the money you initially put into the project (excluding the returns that are derived due to money put into the project after the down payment). The problem is how you attribute how much of the future profits are due to the initial down payment and how much due to the investments that are done in the middle of the project, etc.

This is why  I think  that the ROI is already capturing the notion you want. If the project will not require a lot of money after the down payment, then the denominators will not be so different anyway. However if the project might require a lot more money after the initial down payment, it is harder to claim that X amount of the net profits can be attributed exclusively to the initial down payment.

You are correct that the ROI does not provide any information about leverage, so maybe you want to complement it with some other ratio like (down payment)/(total investment) or another over-the-counter financial ratio. However, considering a measure like net profits/down payment will seem very deceiving to me.


How does high U.S. federal debt affect its citizens? My apologies if this is a silly question. My background isn't in economics, though wish I had some.
During the 2008-9 financial crisis, federal debt increased significantly and there was a small talk between my friends on how this much debt is unprecedented and the U.S. will collapse unless something is done to reduce it.
It turns out (at least from what I experienced), is that nothing really changed. Debt increased much more now and the economy seemed to be doing fine until COVID-19 erupted.
Recently a stimulus package has been released to alleviate the economy affected by COVID-19. More debt is added and I'm wondering how is the U.S. government just be able to print money without side-effects? (at least for now?)
My understanding back then was that an increase in government debt implies increase in tax but the tax actually got reduced when the Trump administration took charge.
Sources

Federal Debt: Total Public Debt as Percent of Gross Domestic Product
Federal Debt: Total Public Debt
Tax Cuts and Jobs Act of 2017

Rather than have a long comment thread, I will give a partial “answer” here.(Really a direction on how to ask a new question, or what existing questions to search for.)
The first thing to note is that this is a broad question, and hard to answer. The other thing is that answers can easily veer into opinions. Questions of this form are often closed.
The preferred format would be to ask about the linkage of debt levels to one topic of concern. The most common seem to be as follows (“it”=high debt levels).

Does it need to be paid back?
Does it imply a risk of default?
Will It cause GDP growth to slow?
Will it raise inflation?
Will it cause the currency to fall in value?
Will it cause interest rates to rise?

Each one of these is a somewhat complicated question, and it would be best to look at existing questions, and see whether they match your interests.
To what extent I can offer an answer, the safest general statement is that nothing has to happen immediately - Japan providing a data point (or even the US after World War II). Beyond that, you need to ask something more specific, as too general answers turns into opinions.


What are the most recent devopments with applying fractals to economics? I was researching fractals for my senior mathematics presentation and discovered that one of the most recent pioneers in that section of the field was able to apply fractal mathematics to the field of economics.  One of his first discoveries was finding a fractal pattern to fluctuations in cotton prices.
What are some other applications of fractals to economics?  (Obviously they may be too many or too few -but an answer collecting references and pointers to some of them is worthwhile, nevertheless).

Mandelbrot's original application to cotton prices, The Variation of Certain Speculative Prices, claimed that the price fluctuations followed a Lévy stable distribution with a stability parameter $\alpha$ below 2. ($\alpha=2$ corresponds to the normal distribution.)

To be honest, I'm not sure how this was an application of "fractals" where previous contributions were not. While the sample paths of the Lévy stable process we was investigating have fractal characteristics—for instance, a non-integer fractal dimension—so does good old Brownian motion, which has fractal dimension of 1.5. Indeed, the relationship between fractal dimension $D$ and the stability parameter $\alpha$ is $D=2-1/\alpha$, so that Lévy stable processes with $\alpha\in (1,2)$ actually have a $D$ closer to 1 than Brownian motion—in a sense, they're less fractal!

The contribution of Mandelbrot here was, I think, to illustrate the rich variety of possible stochastic processes that asset prices can follow. (Processes that happen to exhibit a variety of fractal features.) He proposed, for instance, both Lévy stable processes as in the cotton price paper and, later, fractional Brownian motion.

Since this isn't my field (and I'm relying on half-remembered undergraduate knowledge at the moment), I can't say too much about the cutting edge of research here. I do know that "tempered" Lévy stable distributions are still quite popular for modeling. (These distributions make tails enough thinner that all moments exist, whereas the usual Lévy stable distributions with parameter $\alpha$ have no moments $\alpha$ or higher. Not having a finite second moment is a big problem for financial applications, which is why this modification is necessary.) Variants of fractional Brownian motion remain common as well. This Wikipedia article, although not too well-written, discusses why such stochastic processes can be useful for financial modeling.

It's safe to say that Mandelbrot has a very, very active legacy in finance.


Sign of substitution and income effect of a price change I just want to confirm with my understanding.

It is correct to say that no matter price increase or price decrease, the substitution effect is always negative for both inferior goods and normal goods. Then for income effect, it is positive for inferior goods and negative for normal goods no matter price increase or decrease.

The income effect is negative for normal goods and positive for inferior goods. That is, you buy more normal goods when you are richer and less inferior goods.

In contrast, the substitution effect is negative when price increases and vice-versa. It always moves opposite to the price sign.


Printing Money vs. Issuing International Bonds I'm reading Niall Ferguson's The Ascent of Money and I had a question about the economic difference between printing money vs. selling bonds to foreign entities.

At one point in the book, the author says that much of Southern Civil War effort (in the United States) was funded by issuing Cotton-backed bonds. I.e the south issued bonds that could be converted to Cotton.

The author then points out that the turning point in the war was when the Union army captured New Orleans and setup a naval blockade, which prevented European investors from collecting their cotton.

This ruined the South's ability to sell bonds, and forced them to print money to pay for war expenses, which led runaway inflation.

My question is as follows: In this case, how is issuing bonds better than printing money? Wouldn't foreign bond purchases also lead to inflation, since there would be an influx of money into the south purchasing more goods? Why did the South even bother with issuing bonds, and why couldn't they just print money in the first place.

Bonds are backed by real resources. Money is backed by nothing.

Lets say I'm a British investor. The price of 1 kilo of cotton is 100 British pounds. If I buy a bond for 1 kilo of cotton redeemable in 1 year for 90 pounds then in one year when I redeem the bond and the 1 kilo of cotton is sent to me I can sell it for 100 pounds for a profit of 10 pounds or 10% (100 - 90). The number of pounds in the south has nothing to do with the value of the pound since no new pounds are being created. A southerner with payment in pounds can go to Britain and use the pounds and if anything the transfer of pounds from Britain to the South will cause deflation as the supply of pounds in Britain decreases since they are now in the South.

This is different from the South printing money which actually increases the money supply. If the south has 100K total confederate dollars in circulation which can be used to purchase 1000 arbitrary goods and services then each good can be purchased for 100 confederate dollars. If the south wants to purchase an additional 1000 arbitrary goods and tries to print an additional 100K this leads to inflation as now there are 200K total confederate dollars. The money supply in this instance has expanded whereas it has not in the bonds example.


Do claims in economics require proofs? My professor stated four axioms in class

Scarcity
Rationality (aka purposive behavior)
Stable Preferences
Equilibrium

I don't understand what these axioms are for. I am used to axioms with distinct mathematical properties (eg definition of a vector space) and these axioms don't have that.
I had an exam question that said

  True/False/Uncertain. The Law of Demand is a direct result of diminishing marginal utility.

and I was supposed to use the scarcity axiom. But what does it mean to show this claim is false? The principle of scarcity is just that "resources are scarce". That's not mathematical. How can I prove the above statement is false using it?

My Question:

How can I tell when a question requires a proof versus merely conceptual reasoning / explaining? What is the difference between the two?

These assumptions actually have distinct mathematical ground, but your professor likely decided to leave this out for (of course bad) pedagogic reasons.

scarcity: society has insufficient productive resources to fulfil all human wants and needs. Scarcity is the motive behind economics. If it is cheaper for everybody to get a given good from the ground than to produce it, then they are not economic goods, they don't have a market value and are not treated by introduction classes in economics. This can be stated as: for all actor, for all (studied) goods, the cost of acquiring the good is strictly positive.
Rationality: people adopt the best actions to achieve their goals. Usually, this is translated in maths by introducing a utility function from the possession set to a measuring set (usually $\mathbb{R}$), and stating people always act in order to maximize their utility.
Stable Preferences: Using the same utility function as above, this states that this function does not depend of time. This implies that if you prefer having a kilogram of soup than a kilogram of raw potatoes today, it will still be the case tomorrow.
Equilibrium: Here I have to guess, but given your example I suppose you're studying equilibrium on the market. I guess what your teacher meant is that you study the economy at a time when the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers.

All these definitions are not guaranteed to match with those implied by your teacher, but are consistent and widely taught in economics. When not otherwise specified, economical answers require referring to a consistent framework of axioms from which mathematics should be able to guide you to the solution you're trying to state. Different models can show different solution, but one model should give only one solution (which can be composite: "people buy more or less goods at the end of the month" is a valid solution, but "people buy more goods at the end of the month" and "people buy less goods at the end of the month" should not be obtained from the same data with the same model...).

However, economic practice does not require you to re-demonstrate any known result (mostly, the results which you know a name for can be used as starting points for your answers). If you're not confident with "explaining", you need to be able to think about a one-line mathematical justification for each explanation you give, and thus you will ensure your explanations give you the expected marks.


Is the CPI accurate? When regarding the CPI, why is the selection of G&S within it always favorable to what it has been for years? I ask because some economists and businessmen have seemed to find goods and services that have proved the CPI and inflation cohesion wrong. Are they fishing to deeply to find G&S that is to abstract? Or is that they are opening a larger topic of discussion that no one wants to believe?

Price measurement in most national and international agencies is based on the “statistical approach” to price indexes developed by Dutot (1738), Carli (1764), and Jevons (1865). The methodologies developed in these papers form the foundation of 98% of all consumer price indexes generated by government statistical agencies (Stoevska, 2008). Following this approach, statistical agencies are more likely to select a good for inclusion in the sample with a past high sales share. Because agencies typically choose products based on their historic sales shares, this explains why "the selection of G&S within the CPI is always favorable to what it has been for years".

Economic theory has largely rejected the "statistical approach" to price measurement in favor of the "economic approach," which asserts that all price indexes should be derived from consumer theory and correspond to the unit expenditure function. See Redding and Weinstein's paper "A Unified Approach to Estimating Demand and Welfare", they provide the argument and the above  references.


Consumer Theory question You plan to use the following specification for an empirical study:
$$e_i = \alpha_i + \sum_{j=1}^{n} \beta_{ij}p_i + \gamma_iy +\delta_i,
i=1,...,n$$
where $e_i$ is the consumer's expenditure on good $i$, $p_i$ is the price of good $i$, $y$ is the income and the $\alpha_i, \beta_{ij}, \gamma_i, \delta_i$ are parameters.

You want the specification to be consistent with standard consumer theory and three friends offer the following opinions:

1) “the equation will work for any values of the parameters;”

(2) “you need to impose the  $\beta_{ij}=0,\sum_{i=1}^{n}, \delta_{i}=1$";

(3)“you need to impose restrictions such as $\beta_ij = -\gamma_i\alpha_j, \gamma_i \geq 0,\sum_{i=1}^{n}, \gamma{i}=1,\delta_i=0 $

(a) Explain which of the friends gave sensible advice.
(b) Why should you really not listen to the advice of the other friends?

You could use the fact that $\sum_{i=1}^{n}e_i=y$ (adding-up property) to see which parameter values are compatible with the above specification and this identity.
Before proceeding, you should replace on the RHS $\sum_{i=1}^{n}\beta_{ij}p_i$ by $\sum_{j=1}^{n}\beta_{ij}p_j$.


Good economic indicators for presidential performance I'm trying to put together a good list of economic indicators that press and economists look to when evaluating a president's "performance". These indicators should be ones that are updated at the least monthly.

What I have so far (with notes where necessary):

Unemployment rate
Debt
Deficit
GDP growth rate
Median household income (is this updated monthly?)
Median home value (is this updated monthly?)
Gallon of gasoline
DJIA
S&P 500
NASDAQ

Am I missing any and are the ones I flagged updated enough to be relevant?

Thanks

Get rid of the gasoline and the home value. Home value can only be used as an indicator of net worth and gasoline is dependant in most cases on too many outside factors. An important one you missed is the trade deficit, and I would add us bond interest rates too. Because they are a good indicator of how investors think about the us. I would also add the average household income, cause the difference between the average and the median is also interesting.


Are there any examples for $u(x_1, x_2) = \max\{x_1, x_2\}$ in real word? I know how its graph looks like, and it's like when you want to choose between 2 inferior goods you choose the cheaper one so you can have more, but is there another examples?

One way to think about this utility function is "I can only use one of these two goods, and I will use the one that I have more of"

Some examples:

Choosing between two sets of equally-valued but non-matching dishware. Once you have one set, the other one is useless to you.
If your budget constraint is such that you can only have one kind of platform of some kind, then "items to be used with that platform" works. As commented above, Android apps and iPhone apps, or multi-platform video games, or DVDs vs Blu-Rays.
If you were an opportunistic politician deciding whether to pander to group A or group B, knowing that whichever group you pick the other will hate you, then "number of voters in group A" and "number of voters in group B" would go into your utility function as max(A, B).


Research Design: Indifference curves and budget lines I have a basic Idea on how to construct indifference curves such that we must use two goods and then ask for pairs of bundles which are preference indifferent.

When doing applied research on estimating a consumers demand for a given bundle of goods, (which is probably done by survey as far as I'm concerned) what are the main questions we are seeking to answer in order that the data will work out nicely when using such a framework? How should such questions be worded on a survey?

In order to derive a preference indifference curve between different quantities of two goods, one possible questioning protocol would consist in:

First choosing any arbitrary combination of quantities for the bundle of goods (e.g. 20 units of A and 5 units of B);
Then choosing any arbitrary quantity of one of the goods (e.g. 15 units of good A);
Asking the consumer whether he (or she) is willing to exchange the first bundle of goods (20 of A and 5 of B) by a bundle of goods comprising 15 units of A (or any other quantity as specified in the previous paragraph) and 10 units of B (or any other quantity provided that is greater than 5);
If the consumer chooses the first bundle then the previous question should be asked again provided that that the previous 10 units of B be replaced by a greater quantity (e.g. 15 units of B); otherwise, if the consumer chooses the second bundle then the question of the previous paragraph should be asked again provided that that the previous 10 units of B be replaced by a smaller quantity (e.g. 8 units of B);
When the consumer considers that he (or she) is indifferent between two bundles, it means that the two bundles are indifferent and therefore the corresponding points should lie over the same indifference curve.
Following the previous procedure for a significant set of other combinations of quantities (bundles) would allow to draw the corresponding indifference curve(s).


Is this an an endogeneity/simultaneity problem? I would like to know if the logic in these two situations is correct.
Situation 1: Let's say we have a continuous dependent variable, $y_1$, that then has a causal impact on an unobserved variable, $\rho$. This unobserved variable then has a causal impact on a variable, $y_2$, which has a causal impact on $y_1$. We want to estimate $ \frac{\partial y_1}{\partial y_2}$. So, we have a set of structural equations as follows:
$$y_1 = f(y_2, \mathbf{x_1})+ e_1$$
$$\rho = f(y_1, \mathbf{x_2})+ e_2$$
$$y_2 = f(\rho, \mathbf{x_3})+ e_3$$
where the $\mathbf{x_i}$ terms are exogenous and the $e_i$ terms are errors. By substituting the second equation into the third, we can see that we would have simultaneity and our estimates of the impact of $y_2$ on $y_1$ would be biased if we could not control for $\rho$ or some proxy for it in our estimate of the structural equation for $y_1$.
I am particularly unsure about this last part in italics. Could we use the proxy for $\rho$ in the estimate of the structural equation for $y_1$, or would we have to do 2SLS, with the proxy for $\rho$ being included in the first stage but excluded in the second?
Situation 2: Let's say we have a percentage dependent variable, $s_1$. Let's say the complement of $s_1$ is made up of two other percentages, $s_2$ and $s_3$. Furthermore, let's say that $s_3$ has a causal impact on an unobserved factor, $\rho$, and that $\rho$ has a causal impact on $y_2$, which has a causal impact on $s_1$. We want to estimate $\frac{\partial s_1}{\partial y_2}$. Thus, we have the following structural equations:
$$s_1 = f(y_2, \mathbf{x_1})+ e_1$$
$$\rho = f(s_3, \mathbf{x_2})+ e_2$$
$$y_2 = f(\rho, \mathbf{x_3})+ e_3$$
Let's now say that $s_2$ is more or less constant across observations. Thus, there is generally an inverse relation between $s_1$ and $s_3$. This implies that we can rewrite $s_3$ in the second structural equations in terms of $s_1$:
$$\rho = f(1 - (s_1 + \bar{s}_2), \mathbf{x_2})+ e_2$$
Then, just as in situation 1, by substituting this equation into the structural equation for $y_2$, we can see there would be simultaneity and our estimates of the impact of $y_2$ on $y_1$ would be biased (once again, I am unsure about whether controlling for $\rho$ or a proxy for it in our estimate of the structural equation for $y_1$ would solve this).

Let us consider Situation 1.
Let us assume that $\rho$ is observed. If it does not work when $\rho$ is observed, there is no reason why it (using a proxy of $\rho$ as instrument) should work when $\rho$ is not observed.
$\rho$ is endogenous so we can't just include it as a regressor in an equation. Thus, let us consider IV estimation.
Obvious instruments for the first equation are $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$. Can we use $\rho$ as an extra instrument? It depends on whether $\rho$ is relevant and whether $\rho$ is exogenous in the first equation (i.e., the $y_1$ equation). First, $\rho$ is relevant as it is correlated with $y_2$ (unless the last equation is degenerate). Next, is it exogenous?
To check it, let's go maths and consider the following simple model (without intercepts and exogenous regressors, for simplicity):
$$y_1=\alpha y_2 + e_1,\;\; \rho = \beta y_1+e_2,\;\; y_2=\gamma \rho + e_3.$$
Write them in matrix form:
$$
\begin{pmatrix}
1 & 0 & -\alpha\\
-\beta & 1 & 0\\
0 & -\gamma & 1
\end{pmatrix}
\begin{pmatrix}
y_1\\ \rho\\ y_2
\end{pmatrix}
=
\begin{pmatrix}
e_1\\ e_2\\ e_3
\end{pmatrix}
.
$$
Using Cramer's rule, we get
$$\rho = (\beta e_1 + e_2 + \alpha \beta e_3 ) / (1-\alpha\beta\gamma).$$
$\rho$ is correlated with $e_1$ unless $\beta=0$, so $\rho$ can't be used as an instrument.
Edit
What happens if $y_1$ is regressed on $y_2$ and $\rho$? Then will $\alpha$ be consistently estimated? For simplicity, suppose that $e_1, e_2, e_3$ are $iid$ standard normal. Then the OLS estimator vector converges in probability to
$$\begin{bmatrix}
 \alpha\\ 0
\end{bmatrix}
+
\begin{bmatrix}
 E(y_2^2) & E(y_2 \rho)\\
 E(y_2\rho) & E(\rho^2)
\end{bmatrix}
^{-1}
\begin{bmatrix}
 E(y_2 e_1)\\ E(\rho e_1)
\end{bmatrix}
.$$
Letting $e=(e_1,e_2,e_3)'$, write $\rho = e'g$ and $y_2 = e'h$ for some $3\times 1$ nonrandom vectors $g$ and $h$. Then the second term is
$$
\begin{pmatrix}
 h'h & h'g\\ g'h & g'g
\end{pmatrix}
^{-1}
\begin{pmatrix}
h_1\\ g_1
\end{pmatrix}
.
$$
The first element of the above is the determinant inverse times $g'gh_1 - g'h g_1 = g'(gh_1 - hg_1)$. Let us calculate it. We have already obtained $g$. We now need $h$:
$$y_2 = (-\beta\gamma e_1 - \gamma e_2 + e_3) / (1-\alpha\beta\gamma).$$
Let us ignore the determinant part (common). We can work out with $g=(\beta, 1, \alpha\beta)'$ and $h=(-\beta\gamma, -\gamma, 1)'$, ignoring the determinant. Then
$$ gh_1 - hg_1 = [0, 0, -\beta (1+\alpha \beta\gamma) ]' \text{ times a constant}.$$
Thus,
$$ g'(gh_1 - hg_1) = -\alpha \beta^2 (1+\alpha\beta\gamma) \text{ times another constant}.$$
The OLS regression of $y_1$ on $y_2$ and $\rho$ gives an inconsistent estimator in general, but gives a consistent estimator of $\alpha$ if $\alpha=0$ or $\beta=0$. Especially, if $\alpha=0$, then the OLS estimator is consistent. That is, we can regress $y_1$ on $y_2$ and $\rho$ if we want to test $H_0: \alpha=0$. This is unexpected. Not sure if my algebra is correct.


Direct revelation mechanism's sets of strategies and types Mas-Colell, Whinston and Green in Microeconomic Theory describe the direct revelation mechanism as it follows:

The employed notation is the following:

$θ_i$: Player i's type

$Θ_i$: Set of types for player i

$S_i$ Set of strategies for player i

f: Social choice function

g: Mechanism outcome function

X: Set of collective choices

If $S_i$ is a set of strategies and $Θ_i$ is a set of types, how can $S_i = Θ_i$ ?

In mechanism design you are free to choose the rules of the game. The designer can determine $(S, g)$, i.e., what players can do and what happens when players played some strategy profile $s \in S := \times S_i$.

In a direct mechanism, players are simply asked to report their type. Hence, every player $i$ must have a strategy that corresponds to "I am type $\theta_i$" for all $\theta_i \in \Theta_i$ such that we can, without loss of generality, set $S_i = \Theta_i$ and each strategy corresponds to a type report.

Together with the revelation principle, direct mechanisms are very powerfull, see my post here.