How can anyone possibly predict how the price of anything (stocks, Bitcoin, fiat, etc.) will change without "insider knowledge"? Other than the fact that all centralized (and thus "quick") exchanges require KYC/AML nonsense, making it impossible for me to have an account there, I've never dared to even try once to "gamble" or "speculate" on the Bitcoin price. Instead, I've used a decentralized exchange software (Bisq) to simply buy more Bitcoin for fiat money. Never once sold any Bitcoin.
But then I see all these videos of (apparently) intelligent people (but who have very obnoxious personalities) showing advanced visualizations of the Bitcoin (and altcoin) price with tons of "visual aids" that the software adds or, more commonly, they add themselves because they "notice patterns".
They are always very careful to mention that they are "not giving financial advice" and it's "for entertainment purposes only". But clearly, that's just some legal disclaimer, and then they go on to mention that you get a $680 bonus if you sign up to this or that exchange with their referral link, "only if you are an experienced trader". But what kind of "experienced trader" would be watching YouTube videos about trading advice, and not already have an account set up for trading? It seems to make no sense.
Either way, what I'm wondering is how it's possible for these people to predict how the price will go, at least often enough for them to be making huge profits. Which they appear to be doing. I'm not doubting that part. But how can they dare to trust that what they think is going to happen, is going to happen?
I've stared endless hours on Bitcoin price charts and really tried to follow news and watch videos and discussions and think deeply about this in the shower and in the bed, but I just can't make any kind of prediction other than:
I think that it will slowly climb to a million dollars USD per Bitcoin over the next few years, probably with some serious "dips" along the way.

That's not exactly valuable knowledge. It's not something that I uniquely believe, and it doesn't say anything whatsoever about the short term. It's useless for any kind of "trading strategy". I have no more fiat money to buy more Bitcoin, so I can just sit here and watch the price climb (and possibly go South by 30-80% at any moment). I do not understand how these people are able to see patterns and have access to information that I don't.
It seems like this German Bitcoin trader often is right about his predictions, because he will make a video saying: "It went eeeeeeeeeeeexactly as predicted, guys! Whaaaaat a time to be alive!"
When I see his dashboard with 32 BTC just in one account/trade, I feel like an ant compared to an elephant. He must have so much BTC to be able to have such a fortune in just one trading account to "waste" (potentially). I have too few Bitcoin to ever dare to "gamble" with them. Because that's what it is: gambling. It's basically closing my eyes and and hoping that the roulette wheel stops on the number/color which I bet on.
Speaking of gambling, I've never been a gambler in general either. I've never had a single debt, because it just feels wrong to me. And I don't dare to get one, because I realize how high the risk is for me losing that money. But at the same time, I'm in serious need of increasing my Bitcoin "fortune". So I really don't know what to do.
What is the secret behind these people? I doubt that they have gone out of their way to fake it all. I do believe that they are successful, but I don't understand what their "trick" is. Are their brains simply able to see patterns which mine isn't? Have they spent a huge part of their life reading thick books which I'm too tired to even open? I wish I could understand how they can be so confident about the near-future price changes when, to me, it's a complete mystery. I see no patterns whatsoever.

How can anyone possibly predict how the price of anything (stocks, Bitcoin, fiat, etc.) will change without “insider knowledge”?

You can do that by forecasting based on data.  Exact forecasting model will depend on exactly what you want to forecast but most economic variables can be forecasted with a degree of accuracy that is generally higher than just 50/50 random chance (see Elliott & Timmermann 2013 Handbook of Economic Forecasting).
Some series are more difficult to forecast,  for example it is very difficult to beat random walk when forecasting forex exchange rates (Kilian & Taylor 2003), but even in those cases it is not considered impossible only exceedingly difficult.
When it comes to bitcoin forecasting specifically you might want to inquire more about exact models used there at either Quantitative Finance or Bitcoin stack exchange, but just casually search of google scholar reveals plethora of models for forecasting bitcoin returns or trends in Bitcoin prices. For example, Xiaolei et al (2020) report relatively accurate forecasting of Bitcoin value trends with Light Gradient Boosting Machine.
In addition, you would likely want to have not just once forecasting model but multiple ones and analyze their forecasts together.


Net exports term in aggregate expenditure This might be a basic question, which I don't yet understand. The aggregate expenditure (i.e., in my understanding, the total amount of inflation factored (i.e. real) money spent in the economy by its participants) can be expressed as: $$Y = C+I+G+X-M$$

where $X$ is exports and $M$ is imports. Why isn't it $Y = C+I+G+M-X$? Aren't we spending in imports while having a cash flow by exports, i.e. effectivly reducing our expenditures at the end of the day?

We can go through the logic that GDP plus the imports (total goods) go to consumption, investment, government spending and exports. Then use GDP=expenditure and arrive to the answer. I don't want to go by that route but instead to focus on real money spent. We spend on imports and receive money on exports, i.e. opposite of the formula above.

Seems like what you are adding up is not the Gross Domestic Product, rather something like the Gross Domestic Expenditure.
$$
C+I+G+X-M = Y = C+I+G+M-X
$$
could only possibly hold if $M - X = 0$, that is if the trade balance was zero. If this is not true, e.g., if $M - X > 0$ than a country's gross expenditure is larger than its production, so counting these things will not both sum up to $Y$.


What was the Rogoff & Reinhart spreadsheet mistake? In class our professor said that the Rogoff and Reinhart research on growth and debt was discredited due to spreadsheet mistake. What was this mistake? Also did they perform their research in excel instead of R, stata or some other advanced program?

Reinhart and Rogoff (2010, RR) claimed that GDP growth is much lower when debt/GDP > 90%.
Herndon, Ash, and Pollin (2014, PDF, HAP) placed their errors into four categories:

Spreadsheet error: RR omitted Australia, Austria, Belgium, Canada and Denmark from the analysis (apparently a coding error somehow dropped the first five countries when sorted alphabetically).
One small transcription error: RR incorrectly transcribed a figure of "-7.6%" in one spreadsheet to "-7.9%" to another.
Inappropriate weighting: Example: New Zealand had only one year (1951) with debt/GDP > 90% and in 1951, its GDP growth was -7.6%. The UK had 19 years with debt/GDP > 90% and in those 19 years, its average GDP growth was 2.4%. The usual practice would be to count New Zealand 1951 as one country-year observation and those 19 UK years as 19 country-year observations. But instead, RR averaged the 19 UK years as one single observation that carried the same weight as that single New Zealand 1951 observation.
Selective exclusion of available data: Example. RR exclude Australia (1946–50), New Zealand (1946–49), and Canada (1946–50) without giving any explicit reason. But these country-year observations are precisely instances of high GDP growth and debt/GDP ratios.

Errors #1 and #2 are clear cut (and may be fairly characterized as "Excel spreadsheet errors").
Errors #3 and #4 are a little more subjective.
(Neither RR nor HAP state the program used. But as reported in e.g. NYMag, following repeated requests from Herndon, Reinhart emailed to Herndon an Excel file and it was in this file that Herndon discovered errors.)


What is the impact of inflation on Pakistan's exports? I want to know about the relationship between inflation and exports. What is the effect on exports when inflation increases or decreases?

There are some academic papers dealing with this question (but beyond the specific case of Pakistan)

Note first that inflation is inversely correlated with real exchange rates (as long as nominal exchange rates do not adjust instantaneously to prices).

The following numerical example from Gylafson (1999) illustrates the point.

Suppose the real exchange rate index $R$ is initially 100 and the inflation rate $p$ is 10% per year, so that $R$ decreases gradually to 100/1.1= 90.9 at the end of the year.
Suppose, moreover, that the nominal exchange rate adjusts fully to prices with a one-year lag, restoring $R$ to 100 at the beginning of next year. This means that the average value of R over the year is (100 + 90.9)/2=95.45.
Now suppose inflation increases to 20%, so that R falls to 100/1.2=83.3 at year's end. Then the average value of $R$ over the year is (100 + 83.3)/2=91.67.
Therefore, the real exchange rate is inversely related to inflation as long as the adjustment of the nominal exchange rate to prices is not instantaneous.

Then, currency overluation may hurt exports.

Gylafson (1999)'s paper is interesting because it relates exports and inlfation to growth. His main conclusion is that, in the period under review (1985-94), high inflation and an abundance of natural resources tended to be associated with low exports and slow growth.


Can printing money with simultaneous government intervention to increase supply prevent inflation? Just as above...If we print money and simultaneously introduce some sort of government intervention in order to increase supply/ production, then supply and demand would be 'balanced' and therefore price wouldn't go up. I must be missing something, can anyone tell me what it is?

Yes it can. There is nothing wrong in the above. The money market equilibrium is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing we get:
$$\ln P = \ln M + \ln V - \ln Y $$
Consequently, ceteris paribus, if $M$ increases by $1\%$ and $Y$ increases also by $1\%$ then the net effect on price level will be zero. The crucial assumption is of course that $V$ remains constant otherwise there could still by inflation/deflation.
However, this is not because this would balance supply and demand - or at least not as understood in econ jargon. When people talk about balancing supply and demand they just talk about them being in the equilibrium. Aggregate supply and demand can be in equilibrium even if inflation increases. Rather here it would be more appropriate to say that both supply and demand shift to the right by amount that makes leaves the price unchanged.


Concept of absolute advantage Can someone please explain the concept of absolute advantage with an example? I know it has to do with the fact that a nation should produce a commodity that is cheap relative to other commodities but I don't really understand the concept.

Say two countries have the same amount of resources. A has an absolute advantage relative to B if A has technology such that A produces a larger quantity of output using that given level of resources relative to B. So, lets say potatoes are inputs and mashed potatoes are outputs. A has an absolute advantage in mashed potato production if it can produce more pounds of mashed potatoes than B. Comparative advantage, on the other hand, deals with the opportunity cost. For example, the french fries foregone by each country when they decide to produce mashed potatoes.

It is worth noting that comparative advantage is the basis of trade. Not absolute advantage.


What would happen if the entire world switched to the 4-day work week? What would happen if the entire world switched to the 4-day work week?

Specifically would someone's quality of life drop significantly in a developed or a non-developed country?

Based on the OP's comment directed at @StevenLandsburg, it appears that what we are discussing here is "same $L$abor amount, more $H$eads". So if currently we have $H = L/40$ we will roughly have $H' = (1/0.8)\cdot(L/40) \Rightarrow H' = 1.25H$.

Assuming that there are $25$% more Heads to go to work, this is a redistribution of the same labor income from current earners to non-earners, possibly partly offset by the reduction of unemployment benefits and other transfers that are somehow related to non-earners, which could materialize as lower taxes.

Another interesting aspect here would be the effect on savings: if savings are non-linearly related to income, and then, reasonably expected to be a convex function of it (the higher the income, the higher proportion of it is saved), then such a redistribution would lower savings in favor of consumption (but only partially, since part of this +$25$% would come from households that had already some earners, thus tending to re-instate family income). In any case, lower savings would mean lower available funds for loans and investments. What would that do to capital accumulation, capital level and hence profit maximizing demand for labor?

Also, part of the non-earners that they will now go to paid work, may have been engaged in "non-market", "home production". What will happen with this aspect?

In other words, as comments suggested this needs a fully specified model, even if it is only a highly abstract one, and depending on the assumptions of the model, even within the "usual" variety of assumptions, conclusions will most probably differ.


What are 'Symmetric Agents' vs. 'Asymmetric Agents'? Here is a contextual example from Landry et. al. 2006 :

  We concentrate on an economy with n symmetric agents who
  derive utility from consuming a numeraire good, yi, a public good
  at level G, and (possibly) from their own contribution bi to the
  public good. Each agent faces a budget constraint yi  bi  w and
  derives ex post utility according to

  (1) $   Ui = u(yi) + \Theta h(G)  + \gamma f(bi)$

“Symmetric agents” generally means that all the agents have the same payoff function and the same information set. In this case, it’s the function you’ve specified. It’s often the case that, because symmetric agents will all respond the same way, they can be replaced with a single representative agent, in contrast to heterogenous agent models.


Is it possible for an equilibrium to be Trembling Hand Perfect for one player but not another in an asymmetric game? Is it possible for an equilibrium to be Trembling Hand Perfect for one player but not another (in an asymmetric game)?

Yes. It is possible for a strategy profile of one player to be trembling hand perfect and not the other.

Generally speaking, trembling hand perfect is used in the equilibrium sense. Certainly for an equilibrium to be called trembling hand perfect, the strategy profiles of all players must also be so.


How can I adjust the base year of this price index? I am given the following data for GDP deflator:

1997: 100.0

1998: 100.8

1999: 102.5

2000: 109.9

2001: 112.5

2002: 115.9

2003: 119.7

2004: 122.1

2005: 126.5

...and so on.

Currently, 1997 is the base year. However, I would like to rescale this index so that 2002 is the base year. Could I do this simply by solving 115.9-x=100 (x=15.9) and then subtracting 15.9 from each annual value? That is, would the following index be equivalent:

1997: 84.1

1998: 84.9

1999: 86.6

2000: 94.0

2001: 96.6

2002: 100.0

2003: 103.8

...and so on.

No, you want to re-scale proportionally, so you want to solve the equation $115.9x=100$, so that $x=100/115.9$ and then multiply each value by $x$. The value for 1997, for example, should be 86.3 (after rounding).


What is a number estimated as a Tornqvist Index? In the Solow growth model, to compute economic output $Y$, you need variable $A$ being total factor productivity. I found some data about my country's TFP but it's estimated as a Tornqvist Index. What does it mean? Can I simply use that number in the equation to calculate $Y$?

I've seen the Tornqvist used as many measures of total factor productivity but have not seen it in conjunction with the Solow growth model. My understanding of how the Solow growth models measures TFP is by having values for economic output, labout and capital, and assigning the disparity to TFP.


What statistical techniques can be used for measuring price elasticity I know that linear regression is used for calculating price elasticity. If my objective is to estimate the parameters of a causal relationship, can i use machine learning techniques like Random Forest or lets say ARIMA with regressors? Has anyone done this before? I tried searching in jstor.org but didn't find any.

There are typically two ways to go about estimating causal relationships in the data: by using exogenous variation in the "right-hand-side" variable, or by using structural estimation.

Unless there's an experiment or a quasi-experiment of some sort, the technique economist's apply is instrumental variable (IV) estimation. The idea is that to find the causal impact of x on y in a situation in which x is partly determined by y itself or by a third variable that also affects y, is to find a source of variation that only has a direct effect on x.

Machine learning techniques are not typically used to estimate causal relationships. Instead, they are often used for forecasting.

Mixing the two: machine learning and causal inference is a focus of current research, but apparently there are no solid, off-the-shelf methods to use. Intuitively, the issue is that in the example above, you will need all of x to produce the best forecast of y, but you want only the exogenous par of x to get a sense of the causal relationship between x and y.


How can Aggregate Supply = National Income I have researched two seemingly contradictory definitions for Aggregate Supply, and I don't know how to reconcile them:

The total level of income in an economy over a given period of time.

or

The total productive capacity of an economy, i.e the potential output when all factors of production are fully utilised.

How can these both be true?

Aggregate supply is the total supply of goods and services in an economy given the price level, over a period of time. So, according to this your first statement is correct.

The second statement emphasized on the potential output when all factors of production are fully utilized. It is a long term statement because only in the long run does an economy fully utilizes all the resources.

Here, second is a particular case of the general aggregate supply function definition.


Decreasing Costs, Increasing Returns to Scale, & C''(q) Given a profit-maximizing firm with production function $f(x_1,x_2)$, I understand that we can formulate a firm's cost function $C(q)$ by using the contingent demand functions $x_1^c$ and $x_2^c$. We can then use this cost function to see whether a firm has decreasing costs, namely if $C(tq)<tC(q)$. If the second derivative of the cost function, C''(q), is less than zero, does that also imply that this firm faces decreasing costs and thus increasing returns to scale?

Second derivative of cost function is actually the first derivative of marginal cost function. i.e.
$$
\frac{\partial^2C(q)}{\partial q^2} =\frac{\partial}{\partial q}\frac{\partial C(q)}{\partial q}=\frac{\partial}{\partial q}MC(q)
$$
Now if $\frac{\partial^2C(q)}{\partial q^2}<0$, this means that marginal cost is decreasing in output. If marginal cost is decreasing then that implies that firm's average cost is decreasing and hence it exhibits increasing returns to scale.


What is technology in the Ricardo model? I've been reading about the Ricardo and Heckscher-Ohlin models. I'm having trouble following what makes a worker in the former able to produce more or less of a given good.
So e.g. suppose a US worker can produce 10 computers a day where a Mexican worker can only produce 1. This is attributed to better computer-production technology in the US. But what is that technology? Is it better physical capital, in the form of computer-building machines? Is it better human capital? Is it ideas in some loose sense? Is it related to TFP?

this is attributed to better computer-production technology in the US. But what is that technology? Is it better physical capital, in the form of computer-building machines? Is it better human capital?

In a sense yes. For example, typical Cobb-Douglas function with capital labor and human capital would look like this:
$$Y = A K^{\alpha} L^{\beta} H^{1-\alpha-\beta} $$
Where A would be the technology that actually augments the production capabilities of the different factors ($K$ capital, $L$ labor and $H$ human capital).
You could have a technology that also augments just single factor, for example $ Y = K^{\alpha} (A_LL)^{\beta} H^{1-\alpha-\beta} $ or each factor could have its own separate technology. However, you should not confuse the technology with the stocks of capital, labor and human capital. This being said, you can consider $A$ as a factor that makes capital labor and human capital 'better'.
The technology here is considered in the loosest sense possible. For example, information technology such as skype or zoom allows people to give lecture online to large audiences making their stock of human capital more productive. Some idea about how to improve production process would also be considered. For example, if there are some teaching techniques that allow teacher to teach more efficiently that could be again considered a technology.
Most undergraduate textbook models will not explicitly include production functions in presentation of the Ricardian or Heckscher-Ohlin models, rather they will just assume that certain number of inputs just give you some outputs, but the production functions are included in more complex presentation of these models. See for example some graduate text such as Feenstra's Advanced International Trade.
Furthermore, the production function does not need to be necessary Cobb-Douglas, and it is not necessarily the most appropriate production function for the international trade models but it makes exposition of technology relatively easy so I used it as an example.

Is it related to TFP?

Yes it actually is the total factor productivity. The $A$ is also called the total factor productivity (see pp72 Bruda Wyplosz Macroeconomics). Also, the way how total factor productivity is measured is that people try to estimate the production function such as the one above by putting there all inputs in terms of capital, labor and so on  and using that to actually estimate what is the effect of technology.


How can economies grow if every account is perfectly balanced against others? In an economy with finite actors, if every account is perfectly balanced against another account, how can the economy grow when new money enters?  I am thinking of the following chain of events:

The Fed prints money
   + Fed: $+$ debit (imbalanced book)
A Bank recieves cash from one of the 12 regional Fed banks

Fed: $-$ credit (balanced book)
Bank: $+$ debit (imbalanced book)

Lady takes loan out from the Bank

Bank: $-$ credit (balanced book)
Lady: $+$ debit (imbalanced book)

Lady is payed by her company

Lady: $-$ credit (balanced book)
Company: $+$ debit (imbalanced book)

People buy good or service from company

Company: $-$ credit (balanced book)
People: $-$ debit (imbalanced book)

And from here I am too confused to continue $\dots$ I am not sure we can say credit and debit in this sense, I just mean to show there is someplace that is always unbalanced.  The conclusion I always end up with is that this chain is complete (all parties are balanced) when a new loan is taken out, .. but this makes another chain, which I feel just cannot be the answer!

The Accounting mistake is in step 1, obviously, since you performed a single-entry and not a double-entry:
The Central Bank books are not imbalanced: schematically speaking the CB credits the account "money creation" (or "Our future") and debits its own "Cash Available": this is the essence of "the printing privilege": the CB has the right to create money out of thin air.
Then, it credits its Cash Available account and debits the customer/borrower "Commercial Bank" -because Commercial Bank don't just get money from the Central Bank, they borrow it (or the Central Bank instead of lending money, it buys shareholder stock of the commercial bank or any other so-called "open market operations").

So all books balance. As regards growth in such a balanced-book situation, the key is in step 5, "People buy good or service from company":

What tells you that the firm has the capacity to serve the demand it faces? If demand is now increased because new money have flown into the economy (and initially people treat money as a store of value/purchasing power, not just as a medium of exchange in a static situation), the firm may have to increase its productive capacity to satisfy total demand.

If it cannot do that, the firm will see an opportunity to raise prices, hence new money will just cause inflation. But if the firm can increase its productive capacity, namely, if there is people unemployed and capital laying idle, then the firm will "call to arms" these currently unemployed factors of production, and it will put them to work and produce, thus increasing output and so creating growth.

This effect can certainly be written using the balanced-book Accounting Language, but I will leave that as an exercise.


Is the convexity of production sets necessary for the welfare theorems? I have read that the convexity of production sets (for instance non-increasing returns to scale) is not a necessary assumption for the first welfare theorem but it is for the second welfare theorem. Maybe I got things wrong and what the author actually tried to say is that convexity is not necessary for the efficiency of the competitive equilibrium but rather for the existence of the competitive equilibrium.

Could you please elaborate where the assumption of convex production sets is needed in the first and second welfare theorem?

Convexity of the production set is indeed not needed for the proof of the first welfare theorem but for the proof of the second welfare theorem. It is not a necessary condition though.

It is possible to interpret this as an existence issue. The first welfare theorem is about all competitive equilibria and holds trivially if there are none. The second welfare theorem, on the other hand, states that for a given Pareto efficient allocation, there is a price system and redistribution of endowments with respect to which it is a (quasi-)equilibrium.

The standard proof of the second welfare theorem uses a result of Minkowski on the separation of nonoverlapping convex sets by a hyperplane, but it is possible to prove a version of the second welfare theorem as a corollary to an existence result by a nice argument due to Maskin and Roberts. The argument is quite easy in the case of an exchange economy: Take a Pareto efficient allocation as the endowment distribution. If a competitive equilibrium exists from these endowments, everyone will end up with something at least as good as their endowment. Since the endowment distribution was Pareto efficient, nobody can end up with something better. So everyone must be indifferent between their demanded commodity bundle and their endowment, so they might as well just demand just their endowment. The argument generalizes to economies with production.


Indifference curves and equivalence classes Last week in class, my professor described indifference curves in the context of modelling consumer choice with multiple goods. He defined a partial-ordering operator $\prec$ and an equivalence operator $\sim$ to manipulate tuples that satisfied the budget constraint problem. So far, so good.

But then we went on to explore the behavior of indifference curves and I noticed that the $\sim$ was assumed to be reflexive, symmetric, and transitive. We learnt that indifference curves never intersect and that there is a completeness property that says that for all the tuples in our domain space belong to an indifference curve.

At this point, indifference curves looks like textbook equivalence classes to me, but I would like confirmation and perhaps some context from an economics perspective.

First of all, why not call them "equivalence classes" or openly saying that "indifference curves are equivalence classes". Second, does that mean that if I prove something about equivalence classes, then it automatically transfers to indifference curves in the context of this economic model?

Yes, indifference is an equivalent relation.

That was covered in other answers and comments. I'm here to point out that preference relations are not partial orders. A partial order has the property that

$$a\leq b, b\leq a \implies a=b$$

This not true precisely because of indifference. Rather, preference is typically assumed to be complete and transitive which means it is a total order on the quotient $X/\sim$


Perfect multicollinearity when estimating a gravity model everyone!

I am estimating a gravity model in order to analyze the impacts of tighter environmental regulations on international trade. More specifically, I am analyzing Brazil's trade flow.

My (linearized) model is as follows:

$ \ln(EXP) = \ln(GDP_O) +  \ln(GDP_D) +  \ln(POP_O) + \ln(POP_D) + \dots $

In other words, the exports from Brazil (origin) to a country D (destination) depend on both countries GDPs,Populations, Total Area and  another couple of variables.

My problem is: because Brazil is the origin for the exports to all other countries, GDP_O, POP_O and any other variables representing Brazilian data will be equal for all observations and as such there will be perfect multicollinearity.

How do I circumvent this? Every gravity model uses both the exporter and importer variables in order to explain the bilateral trade flow.

EDIT: See for instance the estimation carried out here:  https://drive.google.com/file/d/1DczfoFI_mkI8Tm4H0WcaVZTz9ZXLGb2K/view?usp=drivesdk

EDIT2: Specifically, here:

[Regression table] (https://i.stack.imgur.com/TQDA3.png)

I appreciate any help!

Kind regards,
Pedro!

My problem is: because Brazil is the origin for the exports to all other countries, GDP_O, POP_O and any other variables representing Brazilian data will be equal for all observations and as such there will be perfect multicollinearity.

This isn't perfect multicollinearity. It's a lack of time/space variation.  Multicollinearity occurs between features, not observations.  You would of course not have space variation in any given year, for the reason you note - but in a comment you said you ran a panel regression, so you should have that time variation.

  I want to regress exports as a function of, among other things, Brazil's GDP. As such, I want to see the effect of GDP on Brazil's exports and not the effect of GDP on exports to each country

Then, why are you using a gravity model for trade flows? Doing so spreads the effect size across a bunch of variables you seem to be uninterested in.  Total exports is just a number that relates to GDP through the accounting identity.  Since that identity is additive, a log-log form for your regression does not make sense.


Adding a non-binding constraint to the objective function I am dealing with a constrained optimization problem found in Tirole's Theory of corporate finance. My question is not related to the details of this model, but just to provide some context, we are solving for the optimal contract to secure outside financing in a setting with costly state verification. $y(\hat{R})$ is the probability of no audit when income $\hat{R}$ is reported. Audits cost $K$ to the lender. $w_0(\hat{R},R)$ is the reward for the borrower in case of no audit, and $w_1(\hat{R},R)$ in case of audit.

The objective function is:
$$\max\limits_{y(\cdot),w_0(\cdot,\cdot),w_1(\cdot,\cdot)}\left\{\int_0^\infty w(R)p(R)dR\right\}$$

It is subject to the incentive constraint for the borrower to report the true income $R$:
$$w(R)=\max\limits_{\hat{R}}\{y(\hat{R})w_0(\hat{R},R)+(1-y(\hat{R}))w_1(\hat{R},R)\}$$
and to the break-even constraint for lenders who provide $I-A$ to the entrapreneur.
$$\int_0^\infty[R-w(R)-[1-y(R)]K]p(R)dR\geq I-A$$

My question relates to the following.
The author argues that, since the second constraint binds at equilibrium, it can be added to the objective function. The problem is then equivalent to minimizing the expected audit costs:
$$K\left[\int_0^\infty[1-y(R)]p(R)dR\right]$$
subject to the two above constraints.
I don't understand why we could add the binding constraint to the objective function and find an equivalent problem subject to the same constraints.
Is there a general optimization principle that I missed along the way?
Thanks!

To illustrate what Tirole has done, let's consider a simpler environment.

Consider a utility maximisation problem over two goods, $x$ and $y$. The consumer has utility function $u(x,y) = f(x) + y$, where $f$ is strictly increasing and strictly concave. The consumer's problem is thus

$$ \begin{align}
\max_{x,y} &\quad f(x) + y \\
\text{s.t.} &\quad p_x x + p_y y \le m
\end{align} $$

Given the conditions on $f$, we know that the budget constraint must bind. That is to say that in any solution to the above maximisation problem, it must be the case that

$$ p_x x +p_y y = m $$

We can rearrange the above equality to

$$ y = \frac{m - p_x x}{p_y} $$

We can thus incorporate this fact into our objective function and use the expression derived above for $y$ to replace it. Thus, our optimisation problem now becomes

$$ \max_x \quad f(x) +  \frac{m - p_x x}{p_y}$$

which, in a way, is a simpler problem, given that I now no longer have to worry about the Kuhn-Tucker conditions. The second-order conditions are also now much easier to verify.

Tirole does something similar to this. He uses the fact that a constraint is binding to simplify the objective function by expressing the objective in terms of the binding constraint.


Econometrics - Need help interpreting results My absolute t-value is greater than t-critical value. This means that I can reject my null hypothesis which was that $\beta_1\leq 0$. Therefore, $\beta_1\gt 0$ and my alternative hypothesis is correct. However, my data has a negative slope. which tells me that $\beta_1\lt 0$. What am I missing here?

You seem to have $H_0: \beta_1 \le 0$ and $H_1: \beta_1 > 0$. In this case, you reject $H_0$ in favor of $H_1$ if your t-value (not the absolute t-value, but the t-value itself) is bigger than the one-tailed critical value. In your case, $\hat\beta_1 < 0$ so your t-value is negative and so you do not reject $H_0$ in favor of $H_1$.

Edit

Regarding significant* variables with the wrong signs (negative when expecting positive in your case), Jeffrey Wooldridge (Introductory Econometrics: A Modern Approach, 7e, end of Section 4-2f, p. 134) writes:

"A significant variable that has the unexpected sign and a practically large effect is much more troubling and difficult to resolve. One must usually think more about the model and the nature of the data to solve such problems. Often, a counterintuitive, significant estimate results from the omission of a key variable or from one of the important problems we will discuss in Chapters 9 and 15."

Note *: "Significance" means rejection of the null in favor of the two-sided alternative. "Significance" does not necessary imply the rejection  of the null in favor of a one-sided alternative.


Integral solution (or a simpler) to consumer surplus - What is wrong? Problem
Given demand $D(p)=A-ap$, and $A,a>0$ and a fixed price $0<p_1<A/a$ by some company.
Calculate the consumer surplus and its derivative with respect to $p$. What is this number?
My solution so far
I could not find a simple way to do it since quantity is not known, $q_0$. I calculated the consumer surplus as
$CS=\int_{p}^{A/a}D(p)dp=\int_{p}^{A/a}(A-ap)dp=\frac{1}{2a}(A-ap)^2=\frac{(A-ap)^2}{2a}$
and its derivative as
$\frac{\partial CS}{\partial p}=\left ( \frac{(A-ap)^2}{2a} \right )=2(A-ap) \left ( \frac{\partial }{\partial p} (A-ap) \right )\frac{1}{2a}=-(A-ap)\left ( \frac{\partial }{\partial p} p \right )=-A+ap$
Which I am almost certain is incorrect. I am not sure on how to approach this without a equalibrium or am I missing it?

Your calculation is correct. We can doublecheck your work with a graphical approach.
As shown in the figure below, $CS$ at some arbitrary and not necessarily equilibrium price $p_0$ is the gray-shaded area. If we take the non-integral approach, we get
\begin{equation}
CS=\frac12\left(\frac{A}a-p_0\right)D(p_0)=\frac12\left(\frac{A}a-p_0\right)(A-ap_0)=\frac{(A-ap_0)^2}{2a}.
\end{equation}
Differentiating with respect to $p_0$, we get
\begin{equation}
CS'=-A+ap_0.
\end{equation}


Are businesses and individuals buying clean / green / renewable energy contribute to slowing down the climate change? I'm not sure if an individual choosing a "green energy supplier" is making any difference.

My (potentially naive, uneducated) logic is as follows:

higher demand for green energy = less demand for other energy

Less demand, therefore lower price, those who are polluting the most now have fewer incentives to switch. Supply and demand, elementary market forces.

I would like to better understand if choosing green energy makes any difference?

Personal note: I'm in the camp of effective altruism. In my version of reality, giving £1 to a homeless is a selfish "feel good" act that does solve the underlying problem of education, employability, affordable housing, mental health support, resilience and life skills.

Your intuition is correct, actually switching from dirty good to clean good will also increase demand for clean good so price there might increase as well. However, this does not mean doing some voluntary change could not help as it also depends on how the quantity demanded changes with price (elasticity of demand). So switching is not necessarily unhelpful.

However, at the same time it is also only a partial solution. In order to fully  internalize the externality it’s necessary to apply Piguvian tax to the dirty good at a level that fully internalizes the externality to people consuming the good.


In a bank-based economy, how do banks determine how to split the economic pie with the businesses? In an economy with a bank-based financial system (little or no stock market activity allowed), I think I've got this much down:

The banks lend to businesses, who then make products and services. The businesses pay their suppliers according to the market rate. In other words, the businesses share their piece of the pie with suppliers according to how valuable they think the supplier is to them.

However, I'm not quite sure how the banks determine how much the businesses should get for their efforts. I am assuming that the banks don't compete among themselves to sell loans to the businesses. The dividing of the pie between the banks and businesses in this case seems rather arbitrary. I know the banks simply expect an interest on their loan money, but it seems like the fixing of the interest rate seems arbitrary. Is it based on perceived risk of the industry in which the business operates? How would this risk be calculated for emerging industries, where there is little or no information?

I would say that if you assume away competition among banks, then the interest rate would be equal to the rate of profit in the industry or firm considered. If there is only one bank in a position of monopoly as a supplier of loans, then it will set the highest possible rate of interest, and that is the rate of profit : firms can not pay higher interest without bankruptcy. Then of course you can argue that there is a decreasing marginal efficiency of "capital", therefore the rate of profit would depend on how much money firms borrow from our single bank, and things would be more complicated. But I think it might still be a good approximation for practical cases. And if there is no available experience on a new industry, then our bank has to guess about the future rate of profit.

If you assume perfect competition among banks, then the question is about the lowest possible interest rate, and that would depend on risk of default : banks have to gain some profits on most loans to compensate for the defaults. Again if you don't know about risk you have to estimate it.


How is productivity measured in public sector? How do they measure productivity in the public sector? Companies just say this is how much money we make per input but what about the public sector?

The BLS has a manual by Katharine G. Abraham (1998) on the subject.
https://www.bls.gov/lpc/iprprody.pdf
page 17 and page 21;  covers outputs and inputs respectively.

Measuring output:

“The specification and measurement of output is the single most troublesome problem
in computing productivity. It is a more serious problem for government service
than for private sector organizations. Some of the problems are endemic to a specific
government activity; others are more general in nature.”
…
“For most public sector output measures, physical quantities are used to calculate the
outputs.”

Measuring inputs:

“Two labor measures that are commonly used to measure private sector productivity are the number of hours and the number of persons. These two measures impart very different information and for this reason both are calculated. The preferred measure is the number of hours because this measure reflects the time worked to produce the outputs.

Hours are the preferred measure for government too, but governmental units do not usually collect these data. Instead, most governments collect and use the number of full-time-equivalent employees.”

The OECD has put out an article on the subject but at the country of Finland
http://www.oecd.org/sdd/na/2666079.pdf


Envelope Theorem in Keen and Slemrod (2017) This question pertains to the paper "Optimal Tax Administration" by Slemrod and Keen (2017). The IMF working paper is freely available on SSRN, however, it is not necessary to know the paper in order to understand the question.

I am confused on how to arrive at equation (5) in the working paper version.

Suppose there is a welfare function:

$W(t,\alpha)= wl - tz(t,\alpha) -c(e,\alpha) - \psi(l) + v(tz(t,\alpha) -a(\alpha)) $

Here, $v$ is an increasing and concave function, $\psi$ is an incresing and convex function, $t$ is the tax rate, $\alpha$ is tax enforcement, $c$ is evasion/compliance cost and $z$ is declared income, which depends on $t$ and $\alpha$.

Declared income is given by $z=wl(t,w) - e(t,\alpha)$, where $w$ is the wage rate, $l$ is hours worked and $e$ is concealed income. Both $l$ and $e$ are derived optimally.

The first order conditions for $l$ and $e$ are given by:

$(1-t)w - \psi'(l) = 0$

$t - c_e(e,\alpha) = 0$

Here $c_e$ denotes the derivative of $c$ with respect to $e$ and $\psi'$ is the derivative of $\psi$.

I am interested in $\frac{dW}{dt}$.

The author's invoke the envelope property to arrive at:

$\frac{dW}{dt}= -z + v'*(z + tz_t)$

Here $z_t$ is the derivative of $z$ with resprect to t and $v'$ is the derivative of $v$.

How  can I arrive at this expression?

At first glance, it appears the answer should be $\frac{dW}{dt}= -(z + tz_t) + v'*(z + tz_t)$, as $l$ and $e$ are chosen optimally and the envelope theorem cancels these terms out. Furthermore, one could expect both $z_t$ terms in the final derivative to drop out due to the envelope theorem. However, one and only one $z_t$ term remains in the authors' calculations.

The envelope theorem is just an application of first order conditions. In this case, it is better to simply use the first order conditions we know to derive the results, instead of using the theorem directly. I believe the authors invoke the term "envelope property" to concisely convey that they are using the previously found first order conditions.

Taking the total derivative we arrive at:

$\frac{dW}{dt}= wl_t -z - tz_t -c_ee' - \psi'l_t + v'*(z + tz_t)$

Inserting the expression for $z_t$ and grouping some terms we have:

$\frac{dW}{dt}= l_t[w(1-t)-\psi'] + e'[t-c_e] - z + v'*(z + tz_t)$

Note that the terms in square brackets are the same as the first order conditions for $l$ and $e$ and hence equal to zero. This leaves us with:

$\frac{dW}{dt}= - z + v'*(z + tz_t)$


factors of production (and their relative contribution) I am reading mankiw's intermediate economics text. what i find is that when he writes a production function for how much will be produced in an economy. he says it is dependent upon quantity of inputs (capital, labour) and technology (defined by the production function itself).

Now my question why he ignored raw materials, energy and land? When you need to produce more cars. you will need more energy, more raw material (steel or whatever) and more land (after a certain threshold).

Secondly, the idea of technology is kind of tricky. Like usually technology does not hang in thin air. It is usually embedded in the capital itself. pentium 3 as compared to pentium 4 computer, for instance. Also labour. every hour of labour is not equal. Highly skilled labor may be more productive.

So, how do you actually go about disentangling what is the contribution of "technology", capital and labour to output, in a real world data. Like some growth accounting exercises do. It may be very hard to say what is capital and what is technology. every dollar of investment in capital is not same, and it is not independent of technology.

People aren't ignoring things like raw material, energy, and land, they're abstracting them. They're saying, "We don't want to keep track of each of these separately, so we'll say that for our purposes, as an input to production, they're all really the same thing: capital." Whether that level of abstraction is appropriate for a specific problem can be questioned, and it is, and people come up with other models that refine the abstraction. For an example of this, which goes to your second question, rather than shovel all "technology" into either the production function itself or an investment which is reflected in capital, you can claim there is "labor augmenting technology", and that appears as a coefficient of the labor input- instead of y = f(K, L), you have y = f(K, AL).

When you talk about "disentangling" the parts of a production function, you have to be careful not to put something back into the problem that wasn't there when the abstraction was made, otherwise, you've changed your problem. To take your Pentium example, if you've made a model that includes a labor augmenting coefficient, you might want to put it there, otherwise, you might want to put that in capital, as an investment the firm has made. Although the production function represents "technology", the worker is still using the same basic production process, so it doesn't seem to belong there. But what you can't do is put it directly in labor, if you haven't allowed labor to be directly affected by technology.

The point is, however you claim it affects output, you have to make sure the abstractions you've made support it as an input.


Difference between CCAR and ICAAP stress tests What is the difference between ICAAP and CCAR?

Except that one is for USA and other is for Canada?

ICAAP stress tests are for internal capital or economic capital, which are requirements under Basel II pillar II.
CCAR stress testing is for regulatory capital, which is also under Basel II. US regulators provide specific scenarios and risk factor shocks.


1929 stock market crash and unemployment rate Thomas Sowell has repeatedly made the case that government intervention is largely responsible for the Great Depression. For example, he notes what happened with the unemployment rate in the months after the stock market crash of 1929. Based on Out of Work by Vedder and Gallaway (alternatively available through JSTOR), page 77, one sees that the unemployment rate went from 5% (Nov. 1929) to 9% (Dec. 1929) and down to 6.3% (June 1930). It is then that Hoover signs the Smoot-Hawley Tariff Act, which Sowell contends is the first cause of the Depression. He then notes that by December 1930, the unemployment rate had shot up to 14.4%. Furthermore, aside from minor fluctuations in early 1931, the unemployment rate dips just below that 14.4% figure in 1936, but that is still the unemployment rate in December 1939.

I'm wondering what alternative factors or explanations could explain the quick increase to the unemployment rate followed by a decrease that approaches the previous numbers and then the marked and sustained increase, or if such alternatives even exist.

One caveat to the author you refer to, Thomas Sowell is a staunchly conservative economist, so most arguments you hear from him will stick to the Econ 101 wisdom that government intervention is usually bad, tariffs are always bad (to be fair, it is hard to use them to correct externalities for global "public goods"), etc.

Although tariffs probably were not necessarily the right choice for a tightening economy, the general story is that our central bank in America back then was a lot less independent from politics and less knowledgeable of how to deal with liquidity crises. It ended up tightening monetary policy instead of loosening it to ease the economic contraction, exacerbating employment problems for a long time. Keep in mind as well that this period of unemployment over 10% lasted for over a decade or so; it's not quite feasible to tell what all the factors were that caused the high unemployment, especially without official data on unemployment for a lot of that period.


Unlimited supply of national currency? I was reading this article about Denmark central bank defending its currency peg:

http://www.bloomberg.com/news/2015-01-22/denmark-cuts-key-deposit-rate-to-minus-0-35-to-drive-down-krone.html

In it he said they had "plenty of Kroner to defend against hedge funds"

My understanding of the issue is since Euro is coming under pressure for launching QE program, a lot of funds are moving their money into countries like Denmark -- which cause their currency demand spiked -- and hence its price.

But why did he say they had "plenty"? Apart from law, there's nothing physically stopping a country from keep printing its own currency right?

So I imagine its not hard at all for any country to defend a peg if its demand side spiked -- just keep printing currency to maintain the price?

Why does it seem that Denmark has limited supply of its own currency?

When a country, or perhaps a country's central bank, prints money it is increasing the money supply. When the amount of money available in an economy increases, the value of the currency will decrease. So think about the capital flight from the Euro to the Kroner that is causing the value of that currency to spike. Printing money can basically cause a 'reversal' of sorts. That is, it places downward pressure on the value of the currency.

So Denmark's banking authority might have some large stash of Kroner to meet the current demand. It can sell Kroner to counter-balance the new fluctuation in demand to keep its currency's value from spiking. Denmark might make this decision to avoid making its exports more expensive etc. However, it certainly does not have an unlimited supply of money. If it was willing to print an unending amount of money, Denmark's banking authority would actually  be manufacturing hyperinflation.

I've written this in a non-techinical fashion. Please let me know if you want more details.


MPC and MPS relationship I have a question from an old assignment that I cant wrap my head around.

Other things remaining the same, if households suddenly decide to save more at each level of income, then the consumption function will:

a) become steeper because the marginal propensity to consume (MPC) falls.
b) become less steep because the marginal propensity to save (MPS) rises.
c) make a parallel shift downward as autonomous consumption expenditure falls.
d) remain unchanged but the savings function will shift downward.

I don't understand why it can't be option B.
   Since

MPS+MPC = 1

and

C = C₀ + (MPC)Y (Consumption function = Auto. Consumption + MPC(Income))

Given this, if MPS increases - it should mean that MPC decreases and so the slope of the consumption function decreases. Hence, option B. The answer however is C and I don't understand why.

You are correct in stating that if MPS increases, then MPC decreases.  But the question doesn't actually say that MPS increases. What it says is that saving increases at each level of income. That's consistent with a scenario such as the following in which MPS does not change while autonomous consumption falls from $10$ to $5$:

$\qquad$Before: $S = -10 + 0.2Y$ and $C = 10 + 0.8Y$

$\qquad$After: $S = - 5 + 0.2Y$ and $C = 5 + 0.8Y$

In this scenario, consumption falls by $5$ at each level of income - a parallel shift of the consumption function as per option C.

Actually, the question is not as precisely worded as it could have been.  If C is given as the correct answer, the implication is that households decide to save the same amount more at each level of income.  That could have been made explicit.


Why would a natural monopoly lose money when MC=P, but a competitive firm would get positive accounting profit? I assume that the natural monopoly is required by law to have maximum price=MC. According to this video such regulation would require constant subsidies for monopoly, because otherwise it would go bankrupt:

https://www.coursera.org/learn/microeconomics-part2/lecture/VWpuW/3-1-6-natural-monopoly-regulation-though-marginal-cost-pricing

The monopoly losses money because P < ATC, while for a competitive firm P > ATC.  Somehow it seems that a competitive firm would have lower ATC than a monopoly. I don't understand why.

I found the answer in "Intermediate Microeconomics" by N. Gregory Mankiw.

  When a firm’s average-total-cost curve continually
  declines, the firm has what is called a natural monop-
  oly. In this case, when production is divided among
  more firms, each firm produces less, and average total
  cost rises. As a result, a single firm can produce any
  given amount at the smallest cost.

...

  By definition, natural
  monopolies have declining average total cost [due to economies of scale, I suppose]. As we first discussed in Chapter 13, when average total cost is declining, marginal cost is less than average total cost. This situation is illustrated in Figure 10, which shows a firm with a large fixed cost and then constant marginal cost thereafter. If regulators were to set price equal to marginal cost, that price must be less than the firm’s average total cost, and the firm would lose money. Instead of charging such a low price, the monopoly firm would just exit the industry.

From other book:

  If demand is large relative to the minimum efficient scale, a competitive market is likely to result. If it is small, a monopolistic industry structure is possible.

Of course there must be the point where MC=ATC, but if the demand will ever become big enough to go beyond this point, then it will probably mean that our natural monopoly isn't natural anymore, that it makes sense to (re)introduce competition because now we have diseconomies of scale. Although it can be exactly MC=ATC, but it will be just coincidence and as such is unlikely.


Economics of Health Textbook Recommendation Was wondering if anyone who has taught a class on health economics could give some background on which textbook they used for their class and why.

It seems like "Healthy Economics" by Bhattacharya, Hyde and Tu or "The Economics of Health and Health Care" by Folland, Goodman and Stano are the two most popular texts, but I haven't read either. Thanks!

The two main textbooks in any Health Economics class are: Michael F. Drummond - Methods for The Economic Evaluation of Health Care Programmes and Andrew Briggs - Decision Modelling for Health Economic Evaluation. These are must read books. Michael Drummond is a legend in the area of Health Economics. He is the author of two major textbooks and more than 600 scientific papers, has acted as a consultant to the World Health Organization and was Project Leader of a European Union Project on the Methodology of Economic Appraisal of Health Technology. He has also served on the Boards of Directors of the International Society of Technology Assessment in Health Care and the International Society for Pharmacoeconomics and Outcomes Research.


Proof of DCF model by mathematical induction I am confident with the concept of DCF. However, I wanted to check the following proof given that if investors hold a share in $ t $, sell it at $t+1$, receive dividend $\ D_{t+1} $ and the price at $t+1$, $\ P_{t+1}$. Plus, the current value of share is the present value of all future cash flows (dividends).

Note that:

$\ D_{t+j} $ stands for dividend at period $ t+j $
$\ P_{t+j} $ stands for price at period $ t+j $

Starting from:

1) $\ P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)} $

Induction Hypothesis:

2) $\ P_{t+j} = \sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s} + \frac{P_{t+j}}{(1+k)^s} $ with $j > 1 $

Induction Step:

3) $\ P_{t+j+1} = \sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s} + \frac{P_{t+j+1}}{(1+k)^{s+1}} $

Now, I was provided with the above proof which does not look right to me. My doubt is the following: why in 1 and 2 the price term on the right side is the present value of the price we are measuring and not the PV of the future price? The same is happening for the dividend terms in the summation. This seems inconsistent with 1 and counterintuitive.

Any help in assessing the rightness of the above proof or providing an entirely new one will be greatly appreciated! Also this is first post, happy to join the community!

I suspect your induction hypothesis should be an expression for $P_t$ not $P_{t+j}$ , and should be discounting $P_{t+j}$ not $s$ times but $j$ times i.e.

$$P_{t} = \left(\sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j}}{(1+k)^j}$$

for $j \ge 1$, which reduces to your initial statement $P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)}$ when $j=1$

Now consider your initial statement replacing $t$ by $t+j$ to give  $P_{t+j}= \frac{D_{t+j+1}}{(1+k)} + \frac{P_{t+j+1}}{(1+k)}$ and substitute this into your induction hypothesis to give

$$P_{t} = \left(\sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j+1}}{(1+k)^{j+1}}$$

which proves that the corrected induction hypothesis follows from your original statement


Short call in binomial option pricing model I am pretty new at this, so my apologies in advance if the question is too out of place.
I have been reading about portfolio replication models, and stumbled upon this example that I don't quite understand. It goes like this: there's a stock with current price $S_0 = 100$, and a call option on it with strike price $K = 100$. The up-state and down-state values of the underlying are $S_u=110$ and $S_d = 90$, respectively. Now the problem states

Let's assume we take a position $\Delta$ in shares and short one call option to create our portfolio, which is then comprised by two assets: the underlying and the call option.
If the price goes up, then the shares will be worth $110\Delta$ and we will loose 10. Therefore, the value of the portfolio is $110\Delta \color{red}{-10}$.
If, on the other hand, the price goes down, the call expires worthless and the value of the portfolio is $90\Delta$

My question is: why would I include this call option in my portfolio at all? Clearly I am not profiting from it in any sense, I either would loose money or not gain anything.

One reason why you might want to do this, and perhaps is the motivation for this example, is that it gives you a simple example for thinking about option pricing.

Under some assumptions, the value of any option can be replicated by continuously trading in the underlying asset. Here, you can think of the $\Delta$ shares that you have as your replicating portfolio, and the amount $\Delta$ is meant to be chosen so that the value of these shares is exactly the value (to you) of the call option you just sold.

You can then price the call option according to how much those $\Delta$ shares cost you. If you chose $\Delta$ appropriately, you break even, and this determines the "market price" for the call option you just sold. This example doesn't give you quite everything you need to calculate that yet, but I imagine whatever you're reading is building up to get you there.


Dot over a variable, Neoclassical growth model? Looking at the Neoclassical growth model, I have hard time understanding what dot over a variable mean? I know it is short for taking the derivative with respect to time, I don't understand the meaning behind it.

The second question concerns the Euler equation for consumption. What does it tell us? And is c(t)dot/c(t) the growth rate of consumption?

Maybe an example will help you. Imagine you start working at a company today and they offer you a salary of $50,000$ USD with a promise of raising it to $55,000$ USD next year. At the same date your boss is also hired with a salary of $100,000$ USD which will be raised to $110,000$ next year. Surely you will think it is unfair your boss will get a lot more next year than you will. But let's look at it from another perspective

  For you

$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{55,000 -50,000}{50,000} = 0.1$$

  For your boss

$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{110,000 -100,000}{100,000} = 0.1$$

Which is exactly the same number, people usually call this a relative increment, which in this case is $10\%$. So from this point of view it is the pretty fair. To make this a bit more technical call $S$ the salary,  $t$ today, and $\Delta t$ make it 1 year, so that the equations above becomes

$$
\frac{S(t + \Delta t) - S(t)}{S(t)} = r \Delta t
$$

where $r$ is the example above just means an increment of $10\%/{\rm yr}$. Now let's organize this a bit

$$
\frac{1}{S(t)} \frac{S(t + \Delta t) - S(t)}{\Delta t} = r
$$

And from this, you probably will recognize the notion of derivative. If you take $\Delta t$ a very small number you will get

$$
\frac{1}{S}\frac{{\rm d}S}{{\rm d}t} = r \equiv \frac{\dot{S}}{S}
$$

Note that this is just fancy notation to represent a fractional change of the quantity $S$. In other words: how much does the quantity $S$ change in a moment, compared with this current value.

In most cases the quantity $r$ is not a constant, but depends on time. Such in your second expression, but do not loose track of the meaning there: It tells you how the relative change of quantity $c$ is affected by other factors. For example $r$ will make it grow, but $n$ will make it decrease (because it is substracting)


Why does lowering stock rate of returns increase stock prices? (i) I don't understand point 1 beneath. Why does the Federal Reserve's buying assets lower risk premiums?

(ii) I don't understand point 3 beneath. Why does lower stock rate of returns increase stock prices? Won't lower stock rate of returns lower stock demand? Won't some investors buy something else other than stocks?

Why the stock market is going up... and it's not because the Coronavirus is going away : investing

  I think I know why the stock market is up... and it's not because the Coronavirus is going away.

  It requires some macroeconomic and finance knowledge, but maybe, just maybe:

  Fed actions (buying assets) are lowering risk premiums
  Fed actions (lowering rates) are lowering asset yields overall
  Both of these actions lower stock yields, which increases stock prices
  Lower stock yields = longer duration for stocks (dividend discount model)
  Longer duration + lower stock yields = less overall sensitivity to the lower expected earnings for next 12-24 months due to Coronavirus
  Plus... a maybe.... maybe long-term inflation expectations due to printing money = higher earnings (in pure dollar terms) for cash flows 5-30 years from now

  Run those things through a discounted cash-flow model, and you get higher stock prices. Near-term earnings become less important to the overall picture.

In the part 1 there must be some omission. Just by itself a random Fed purchase would not reduce risk premiums. However, a targeted Fed purchase can reduce risk premium as non-trivial part of risk in any firm stems from liquidity risk. Through purchasing assets of illiquid firms Fed can provide them with liquidity that they need and thereby reducing the risk that they fail just due to illiquidity. Also as Fizz points out in his comment this is just one example, there are several other channels that can also lead to risk reduction as well.

The intuition behind this is that the stock prices represent present value of the future expected income streams you get from the stock and at lower interest rate the present value of this income stream becomes higher. To see why this makes sense consider practical example where the stock always pays dividend 10\$ in perpetuity. Then in order to actually calculate whats the present value of getting the stream of 10\$ at 10% you can use the Gordon formula which is just based on calculating the value of infinite sum. In simple Gordon stock pricing formula given as $P=\frac{D}{r-g}$ where $D$ is dividend and $r$ rate of interest and $g$ rate of growth which we can for sake of simplicity ignore and assume its zero so we get $P = \frac{D}{r}$.
At the 10% the infinite sum of 10\$ will be worth 100\$ because
although your are getting the dividend in perpetuity due to time
value of money money in future is less valuable than money now and
becomes less value the more you look into the future and hence this
infinite sum in present value terms becomes smaller and smaller
which allows us to actually calculate finite value even for an infinite stream of money. However, now if the interest rate drops to 1% the present value of those dividend streams becomes higher and the stock will now cost 1000\$ as now the future is discounted less heavily. To put it simply, at lower interest rate the money you receive in future have higher value today.


Rational preferences/individual decision-making theory I am taking advanced micro course this semester. In one of the problems we need to determine whether the preference relation is rational (i.e. complete and transitive). Since we have not really discussed the set of two-dimensional real vectors $X=R^2$ (non-negative), I am wondering whether these preference relations are indeed rational, and if yes, how can it be shown
(i) The relation ≽ defined by $(x_1,x_2)$ ≽ $(y_1,y_2)$ if and only if $x_1≥y_1$ and $x_2≥y_2$
(ii) The relation ≽ defined by $(x_1,x_2)$ ≽ $(y_1,y_2)$ if and only if $x_1≥y_1$
(iii) The relation ≽ defined by $(x_1,x_2)$ ≽ $(y_1,y_2)$ if and only if $\min\{x_1,x_2\} ≥ \min\{y_1,y_2\}$
(iv) The relation ≽ defined by $(x_1,x_2)$ ≽ $(y_1,y_2)$ if and only if $x_1>y_1$ or $x_1=y_1$ and $x_2≥y_2$

(i) Is not complete. For instance, (10,5) is not $\succeq$ (9,6), because $10>9$, but $5<6$. However, (9,6) is also not $\succeq$ (10,5) for the same reason. Hence, there exists a pair of bundles $A,B$ such that neither $A\succeq B$ nor $B \succeq A$. Thus, it is not rational.
(ii) and (iii) are both rational. You can either see this by showing transitivity and completeness directly or you exploit a central result in microeconomics: A preference $\succeq$ can be represented by a utility function $u$ only if it is rational.
For (ii), you can define $u(x_1,x_2)=x_1$ such that $x_2$ doesn't matter for comparisons as the preference states. For (iii), you can define Leontief preferences, $u(x_1,x_2)=\min(x_1,x_2)$.
(iv) is a funny case, because it shows why the result we exploited above is just "only if" and not "if and only if." Lexicographic preferences are rational, but they are not representable by a standard utility function.
For any two bundles either the first dimension differs, or the first one is equal and the second one differs or both are equal. So for any two bundles we have either $\succeq$ or $\preceq$ or both, i.e., the preferences are complete. They are also transitive because $X\succeq Y$ implies that either $x_1>y_1$ or $x_1=y_1$ and $x_2\geq y_2$
and $Y\succeq Z$ implies that either $y_1>z_1$ or $y_1=z_1$ and $y_2\geq z_2$. Therefore, either $x_1>z_1$ or $x_1=z_1$ and $x_2\geq z_2$, implying $X\succeq Z$.


Best way to measure inflation in real time? I'm looking to measure inflation in real time preferably minute to minute. What would be different feasible approaches to this? I know that the CPI measures inflation quite accurately but this is only on a month to month basis with the Bureau of Labour releasing their findings with a few month lag. The metric should be able to reflect a sudden inflationary spike like in the case of hyperinflation for example. Would much appreciate your help and thoughts!

The highest frequency real time data that I am aware of the Billion Prices Project, which is daily. Link to website. These are prices on the internet, which can be mapped to the US CPI relatively well.

Otherwise, consumer prices in bricks and mortar stores are Generally not set “minute by minute,” nor is there a way to sample them at that frequency.

The closest you can get is countries in hyperinflation, where prices end up being indexed to a foreign currency. That is, the price paid is a fixed price in a foreign currency, multiplied by the exchange rate. This mechanism results in rapid changes, even during the time to drink a coffee (anecdote from the Weimar hyperinflation). Foreign exchange rates are the usual way of estimating a hyperinflation, since there is no good way to survey prices.


Clarification on amortization of constant payments I have found this formula on wikipedia :
$$P=C_k(1+i)^ {(n-k+1)}$$

which describe the constant payment that has to be paid every year ($C_k$ is the part of the initial loan that is extinguished with the $k$ payment) in order to pay off a loan after a period of $ n $ years , all of them at the constant annual interest rate $ i $ .

My problem is that $ 1 + i $ is elevated to $ n-k+1 $ while I expected $k$ because it's the numbers of years passed when the $ k $ payment is done . It seems that the loan is paid inside out , namely starting to pay the interests at the time $n$ on the first year and so on…Am i right?

Let $L$ be the loan amount, then $L=P\frac{1-(1+i)^{-n}}{i}$

The balance at time $t$ is defined as the present value of the remaining payments,

$B_t=P\frac{1-(1+i)^{-(n-t)}}{i}$

By using the following equations

$I_k+C_k=P$ and $I_k=iB_{k-1}$ where $I_k$ is the interest paid at time $k$ we derive the following equation

$C_k=\frac{P}{(1+i)^{n-k+1}}$ and after a simple rearrangement we arrive at the formula you presented.


How do I calculate the impact of an independent variable on a dependent var. when the independent var. changes from 0 to some positive value? I will try to explain my question using two production functions here.

Let

$Y$ = Yield of a certain crop (tons/hectare)

Assume yield (output) is a function of two inputs, $Y = f(N,I)$, where

$N$ = Nitrogen fertilizer (input) (kg/hectare)

$I$ = Irrigation (mm/hectare)

Case 1: Assume I have data on 400 different plots. Each observation includes yield (output) and N (input). There is no irrigation in this case, $I = 0$.

$Y_1$ = $\alpha_0$ + $\alpha_1$ * $N_1$ + $\alpha_2$*$N^2$ + $e$

Case 2: Irrigation (say up to 100 mm/hectare) is introduced in all plots, therefore, the value of $I$ ranges from 0-100. Irrigation increases yields in most plots. However, for a large number of plots, irrigation alone does not increase yields and additional N is required to increase yields. I have data on both inputs and the corresponding yields (output).

$Y_1$ = $\alpha_0$ + $\alpha_1$ * $N_1$ + $\alpha_2$*$N^2$ + $\alpha_3$ * $I_1$ + $\alpha_4$ * $I^2$+ $\alpha_5$ * $N*I$ + $e$

What I want to measure is:
(1) What is the impact of irrigation on yields?
(2) Given that irrigation increases yields, how much additional Nitrogen (on average) is necessary to get to these yield increases?

a) From what I understand, I can't use a diff-in-diff type estimation here because the impacts of irrigation are heterogeneous, i.e., I can't simply add an irrigation dummy to the regression.

I would appreciate any help. Thank you so much!

Just thinking out loud here, not saying this is correct others may have a better idea. I'm going to roll with this assumption that you are using best econometric practices so that you trust your estimates and want to take action based on them.

(1) The partial effect of going from $I=0$ to $I=1$ would be
$E(Y_1|I=1,N=\overline{N})-E(Y_1|I=0,N=\overline{N})$

$=\hat{\alpha_3}+\hat{\alpha_4}+\hat{\alpha_5}\overline{N}$

When irrigation is already positive, the marginal effect of irrigation on yields just be

$\frac{\partial E(Y_1|N,I)}{\partial I}=\hat{\alpha_3}+2\hat{\alpha_4}I+\hat{\alpha_5}N$

(2) So you are curious what level of Nitrogen makes it so that the marginal effect of nitrogen is the same as the partial effect of irrigation? I think you could get at this by finding the $I^*$ and $N^*$ such that

$\frac{\partial E(Y_1|N,I)}{\partial I}=\frac{\partial E(Y_1|N,I)}{\partial N}$?

We've already found $\frac{\partial E(Y_1|N,I)}{\partial I}$ above, so we need to calculate
 $\frac{\partial E(Y_1|N,I)}{\partial N}$

$=\hat{\alpha_1}+2\hat{\alpha_2}N+\hat{\alpha_5}I$

From here $\frac{\partial E(Y_1|N,I)}{\partial I}|_{N=N^*,I=I^*}=\frac{\partial E(Y_1|N,I)}{\partial N}|_{N=N^*, I=I^*}$

$\hat{\alpha_3}+2\hat{\alpha_4}I^*+\hat{\alpha_5}N^*=\hat{\alpha_1}+2\hat{\alpha_2}N^*+\hat{\alpha_5}I^*$

$\rightarrow (\hat{\alpha_5}-2\hat{\alpha_2})N^*=(\hat{\alpha_1}-\hat{\alpha_3})+(\hat{\alpha_5}-+2\hat{\alpha_4})I^*$

$\rightarrow N^*=\frac{(\hat{\alpha_1}-\hat{\alpha_3})+(\hat{\alpha_5}-+2\hat{\alpha_4})I^*}{(\hat{\alpha_5}-2\hat{\alpha_2})}$

I bet the optimal irrigation and fertilizer decision would equate the marginal product of fertilizer per dollar spent on fertilizer with the marginal product of irrigation per dollar spent on irrigation which would be

$\frac{\hat{\alpha_3}+\hat{\alpha_4}+\hat{\alpha_5}N}{P_{irrigation}}=\frac{\hat{\alpha_1}+2\hat{\alpha_2}N+\hat{\alpha_5}I}{P_{fertilizer}}$


Deriving support in bidding strategy I'm considering some question, and I'm not sure what it asks me to do:

  Consider a two-bidder auction with two types of players, high type and
  low type ($v_h>v_l$). The probability of a low type is $0.3$. The high
  type plays a continuous mixed strategy with support
  $s\in[\underline{S},\overline{S}]$. For some bid $s$ the probability
  of winning is given by the
  cdf$$\frac{3}{4}\bigg(\frac{S-v_l}{v_h-S}\bigg)$$

  What is the value of the upper bound of the support $s$?

I guess the derivation should be roughly as follows. First, it is easy to verify that the chance of high valuation player to win against another high value player is

$$\frac{S-v_l}{v_h-S},$$

Then we have something like this

$$\big(\frac{3}{4}\bigg(\frac{S-v_l}{v_h-S}\bigg)+0.3\big)(v_h-S)=\frac{S-v_l}{v_h-S}$$

Which doesn't quite give the right answer. I suspect I need to weight some arguments by probabilities.

If it helps, the answer is $$\overline{S}=\frac{7v_h+3v_l}{10}$$

I agree with the comments that suggest that the question is almost indecipherable. However, it should be easy, given the cdf, to derive the bounds. Why? Well we know for a cdf $F$, we must have $F(\underline{S}) = 0$ and $F(\overline{S})=1$. We can use the latter to work backwards from the upper bound i.e.

We guess that $F(s)$ is of the form

$$F(s) = k\bigg(\frac{S-v_l}{v_h-S}\bigg)$$

We must have $F(\overline{S}) = 1$ and using your solution, we have

$$\begin{split}F(\overline{S}) = F\bigg(\frac{7v_h+3v_l}{10}\bigg) = k\bigg(\frac{\big(\frac{7v_h+3v_l}{10}\big)-v_l}{v_h-\big(\frac{7v_h+3v_l}{10}\big)}\bigg) &= 1\\
k\bigg(\frac{7v_h+3v_l}{10}\bigg)-kv_l &= v_h-\bigg(\frac{7v_h+3v_l}{10}\bigg)\\
\frac{7kv_h - 7kv_l}{10} &= \frac{3v_h-3v_l}{10}\\
7kv_h - 7kv_l &= 3v_h-3v_l\\
k &= \frac{3}{7}\end{split}$$

Thus, I suspect that your mixed strategy for the high type is given by the cdf $$F(s) = \frac{3}{7}\bigg(\frac{S-v_l}{v_h-S}\bigg)$$


Game theory for showing interest and availability when dating The more cynical (or perhaps realistic) of us would argue that there's a fair amount of game theory going on in the dating game.

For example a classic ploy is 'Don't ring for three days, because you don't want to come across too keen'.  Of course this rule is now widely considered obsolete (as would be natural in any game).

A fairly straight forward hypothesis I would put forward is:

Everybody in the dating game is looking to get hitched with their highest possible value match.
By conveying unconditional interest in the other partner, you let them know that they can safetly 'get you' and so it's safe to pursue higher value mates knowing that they can score at least as highly as you.
Therefore, the smarter strategy is to not let your dating interest know how available you are for them, so that they are incentivised to make commitments to you.

Now I know, this seems like a deeply cynical and flawed way of thinking. I'm not looking for a social criticism of this thinking. Rather I'm interested if there have been any studies carried out, where by the attractiveness of someone is assessed according to the availability they've communicated, etc.

Sending costly signals may work, at least when the recipient is less attractive than the sender.
There's also a nice popular science book by Paul Oyer called Everything I Ever Needed to Know About Economics I Learned from Online Dating that covers some of this ground, including the paper linked above.
Another theoretical paper suggests that costly signals that are worthless to the recipient work nicely, because the cost signals to the recipient that the donor has resources and values her highly, but by being worthless, it screens out "gold-diggers" that merely want the gift. Perhaps setting a pile of cash on fire would do the trick?


Alternatives to Pigouvian tax Two common drawbacks of Pigouvian subsidy mentioned in the literature are related to monetisation and measurement of social cost (Baumol) and reciprocity of social cost (Coase).

What alternatives to Pigouvian taxes are proposed in the literature? Have any of such alternative measures been implemented in practice?

The most obvious answer is Coasian bargaining. What Coase showed in his famous "The Problem of Social Cost" is that if there are no transaction costs and if utility is transferable then it suffices to allocate property rights—i.e. to give one party the right either to engage in the externality-causing activity or to prohibit it. The two parties will then engage in bargaining with the result that the socially efficient level of the activity is undertaken. The idea is that if an activity has private value $v$, but imposes external social cost $c$ on others then

if the private individual has the right to participate in the activity then others would collectively be willing to pay up to $c$ to persuade him not to. This offer will be accepted only if $c>v$ so the activity takes place only if it is optimal.
if others have the right to prohibit the activity then the private individual would pay up to $v$ for them not to do so. This offer will be accepted only if $v>C$ so again the activity takes place only when it is optimal.

This example assumes a negative externality, but the same approach works in the case of a positive externality. For example, if the private benefit is $v$ (which may be negative if the activity is very costly) but there is an external benefit of $u$ then third parties would collectively be willing to pay up to $u$ to encourage the private individual to engage in the activity. Thus, the activity takes place only if $v+u>0$---i.e. exactly when it is efficient.

This solution is an important element of carbon trading schemes, which are one of the main ways that countries are attempting to tackle the problem of anthropogenic global warming.

The Coasian solution has the attractive feature that it is relatively decentralised (there is no need for a central planner to accurately determine the size of the externality). Although this solution appears to work very well, it has a couple of important drawbacks:

The zero transaction cost assumption is strong. This is particularly true when an activity imposes a small externality on a large number of people so that there is potentially the need for a large number of bilateral payments.
If the externality falls on a large number of individuals then paying a subsity creates a public good problem: each individual could try to free ride and hope that a large enough subsidy is paid by others.

The Coasian solution therefore works best when either

the externality falls mostly on a single 'large' agent who can therefore engage in Coasian bargaining without the fear of free riding and only incurring transaction costs once.

or

agents are able to use contracts, their government, or some other device to act collectively as if there were the single large agent in 1.


Troubleshooting Utility Maximization with the Lagrange Method I'm trying to solve a utility maximization problem through the Lagrange method.
The utility function is something like $u(x,y)=x+B(y-a)$.

However, I'm running into problems as x and y do not come up in the First Order Conditions. I was wondering if anyone had an intuitive explanation as for why, and how to get around it?

You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions.

Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x_1,x_2,...,x_n)=a_1x_1+...+a_nx_n$ will be constant at $MRS_{i,j}=\frac{a_i}{a_j}$

A consumer will buy none of a good $x_i$ whenever $\frac{u_i}{p_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$.

For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes.


What would happen if the U.S. "refinanced" its national debt? Interest rates are at historical lows. I believe they went negative in parts of Europe recently. Since interest on the national debt is one of the biggest, if not the biggest, expense item of the U.S. federal budget, would it make sense to "refinance" the national debt somehow by borrowing new money at these lower rates to payoff the old debt at the higher rates?

The Treasury continuously issues new debt, and retires matured issues. (This is called “rolling over debt.”)
The Treasury has no right to pay debt off early (“refinance”).
The closest thing that can be done is to repurchase debt. The Federal Reserve - whose common equity is owned by the Treasury - is repurchasing debt during “quantitative easing” operations.


maximizing leontief-type utility function How do I maximize the utility function:
$ U(x,y)= max(ax,ay)+min(x,y) $ , where $ 0<a<1 $ with respect to prices $ p_{x}, p_{y} $ respectively and income $ m $.

I know leontief-type utility functions are solved by graphing and not by using Lagrangians, buy how do I graph this function?. ( I know there would be kinks on the graph, but how do I find those kinks)

You can re write $u(x, y)$ as
\begin{eqnarray*} u(x, y) = \begin{cases} ax + y & \text{ if } x > y\\ ay + x & \text{ if } x \leq y\end{cases} \end{eqnarray*}

When one plot the indifference curves, this is how they will look:

Here $\mu_1 > \mu_2 > \mu_3$.


Was there a drop in liquidity in the corporate bond market during the Global Financial Crisis? It is often said that the increase in the perceived default risk led to a fall in the demand for the corporate bonds. This led to the fall in their prices and hence, their interest rates rose significantly. In fact, the risk and liquidity premium rose 360 basis points from 1.85% before the crisis to 5.45% afterwards. However, can we also attribute this fall in the demand for corporate bonds to a decrease in liquidity of these assets. It seems to make sense to say that as a result of there being less buyers, the market for these corporate bonds is thus less liquid. This decreased liquidity further diminishes the attractiveness of these assets, causing the demand for them to fall even more. I am asking if such a vicious cycle exists: Increased perceived default risk, leading to decrease in number of buyers, leading to decreased liquidity, leading to further decreases in the number of buyers, and so on?

"...if such a...cycle exists..."---i.e. in economic language, whether liquidity and demand are endogenous to each other. Yes, of course they are. Liquidity (or illiquidity) of any asset can be viewed as an equilibrium outcome where agents confirm each other's beliefs---similar to the liquidity/illiquidity of a bank. As such it has a self-fulfilling aspect. An asset is liquid because people believe it is and therefore are willing to trade/make markets for it. This in turn leads to demand for the asset, which makes it liquid.
There are many instances during the 2007-2008 Financial Crisis that illustrate this. In the US, the Fed took positions to restore confidence in illiquid markets. The Fed eventually took no loss on those positions. Of course, the Fed also took similar actions to prevent/stop runs on financial institutions.
There are also similar examples during the current Covid-19 episode. In March 2020, liquidity of corporate bond market returned as soon as the Fed announced its corporate bond purchase program---long before it was actually implemented. When the program was eventually implemented (which the Fed must, to signal its credibility), only a small portion of planned purchase was made. Market's belief in liquidity was maintained as long as the Fed is believed to be credible. (See also here.)


Find the optimal labor supply when labor utility is linear I have been working the afternoon on an intertemporal exercise where I'm blocking on something very basic. Have been looking on previous posts but didn't find a similar question.

We have the utility function of the household : $ U(C_1,C_2,n_2)= \log(c_1) + \beta \log(c_2) - \beta n_2 $

We have an allocation $ \bar y $ for period 1 and none for period 2, the household can only work in period two.

Here is the budget constraint : $ c_1 p_1 + c_2 p_2 = p_1 ( \bar y - x ) + qx + wn_2 + \pi  $

And the production function of the firm : $ f(k) = k^\alpha n_2^{1-\alpha} $

What I did until now is replacing $ c_2 $ in the utility function to have it depending on $c_1$,$ n_2$ and $ x $ only. Then I derive it for each variable and I can get the optimal condition for consumption :
$ \frac{c_2}{c_1} = \beta \frac{p_1}{p_2} $, the other derivatives gives me : $ w = p_2 c_2 $ and $ q = p_1 $ .
By replacing $c_2$ in the budget constraint, I can determine $ c_1$ and then $c_2$, and x is given by the market clearing condition $ \bar y = c_1 + x $.

But I can't find anyway to determine $n_2$, it's usually in a logarithm so we can find it in our First Order Conditions...

Edit :
$ x $ is the amount of capital given to the firm by the household, so we sould have $ x=k $ on capital market

$ q $ is the price of capital per unit

$ \pi $ is the profit of the firm , i.e. $ \pi(k,n_2) = p_2 k^\alpha n_2^{1-\alpha} - kq - wn_2 $

Edit 2 :
Rereading my question, I realize I should precise :
The household can only consume the firm production in period 2, which is why he contributes to the capital and labor of the firm, earning wages and profits.
Therefore, we should also have $ f(k,n_2) = c_2 $ I believe

Edit 3 :
My profit maximisation gives me :
$ \frac{\alpha}{1-\alpha} n_2 w = k q $

Edit 4 after @X recommendation :
So I do the lagrangian and derivatives, which gets me :

$ \frac{1}{c_1} = \lambda p_1 $

$ \frac{\beta}{c_2} = \lambda p_2 $

$ \beta = \lambda w $

$ p_1 = q $

After computing, I find : $ c_1 = \frac{p_1 \bar y + w n_2 - \pi}{p_1(1+\beta)} $ and $ x = k = \bar y - c_1 = \frac{\beta p_1 \bar y - w n_2 + \pi}{p_1(1+\beta)} $

But still having trouble regarding my original issue, $ n_2 $ : I did , according to your advices :
$ \frac{\alpha}{1-\alpha} n_2 w = k q $

$ \frac{\alpha}{1-\alpha} n_2 \frac{\beta}{\lambda} = \frac{\beta p_1 \bar y - w n_2 + \pi}{p_1(1+\beta)} p_1 $

$ \frac{\alpha}{1-\alpha} n_2 \beta p_1 c_1 = \frac{\beta p_1 \bar y - w n_2 + \pi}{(1+\beta)} $

$ \frac{\alpha}{1-\alpha} n_2 \frac{\beta}{1+\beta}(p_1 \bar y + w n_2 - \pi) = \frac{\beta p_1 \bar y - w n_2 + \pi}{(1+\beta)} $

$ \frac{\alpha}{1-\alpha} n_2 \beta(p_1 \bar y + w n_2 - \pi) = \beta p_1 \bar y - w n_2 + \pi $

$  n_2 \beta(p_1 \bar y + w n_2 - \pi) = \frac{1-\alpha}{\alpha} \frac{\beta p_1 \bar y - w n_2 + \pi}{\beta(p_1 \bar y + w n_2 - \pi)} $

Still pretty stupid as results ... I also used the central planner way and found $ n_2 = 1 - \alpha $ , which is weird..
I'd still like to go at the end of this way if someone has an idea.

Any idea ?
Thanks.

I guess you mean that the condition $-\beta +\lambda w =0$ does not include labor. But you have the profit maximizing wage relation that includes labor and wage, and so combining, you can obtain an expression that includes the marginal product of labor, and the multiplier lambda, which can be expressed in terms of the other decision variables.

Ok, I see you got lost in running around your equations.

Profits are zero since the firm is a price taker and the production function is homogeneous of degree one (constant returns to scale).

$$ \frac{\alpha}{1-\alpha}n_2 w = k q \implies n_2\frac{\beta}{\lambda} = \frac{1-\alpha}{\alpha} kq$$

Since $\lambda = 1/c_1p_1$, $q=p_1$ and $k = \bar y -c_1$we get

$$n_2\beta c_1p_1 = \frac{1-\alpha}{\alpha} (\bar y -c_1)p_1$$

$$\implies n_2 = \frac{1-\alpha}{\alpha\beta } \left(\frac{\bar y}{c_1} -1\right)$$.

Now, go back to the budget constraint

$$c_1 p_1 + c_2 p_2 = p_1 ( \bar y - x ) + qx + wn_2 + \pi$$

Inserting the various optimal conditions, you should arrive at

$$c_1 = \frac {\bar y}{1+\alpha \beta}$$

which leads to $n_2 = 1-\alpha$.


Independence of latent price and market microstructure noise When examining how Market Microstructure works and affects price formation, there is talked about the:

"Assumed independence of of latent price and microstructure noise"

From "On the Correlation Structure of Microstrucure Noise: A Financial Economic Approach", Diebold and Strasser, 2013.

I am struggling to understand what this means exactly?

For example, when examining a lot of financial trades and quotes, and looking for a underlying latent price - what does it mean to assume independence of the noise? And what do you assume the noise is then specifically? "Wrong" trades? Computer-errors?

I guess you're asking three inter-relatated though conceptually separate questions:

What is the latent or efficient price?
What is market microstructure noise (as opposed to non-market-microstructure noise)?
What does it mean to assume that the efficient price process and the microstructure noise process are independent?

I won't attempt to provide a full examination of the issues raised by these questions: entire articles could (and have been) written about them. I'll only attempt a summary review.

The efficient price process is, by assumption, unobserved. Nevertheless, under the maintained assumptions of frictionless markets, perfect and symmetric information, and rationality of all market participants, one can apply some basic theorems to infer some statistical properties of the efficient price process. Most importantly, the efficient price process must generally be a semimartingale with respect to some filtration. In a continuous-time setting, it is also straightforward to show that the continuous (i.e., non-jump) component of the efficient price process must be local Brownian motion with local drift and variance components.

If the observed price process isn't a semimartingale, it's either because there's an additional component (usually, but not always, called market microstructure noise) or because the maintained assumptions (rationality, perfect and symmetric information, etc) aren't valid to begin with. Let's pursue the first reason.
To make market microstructure noise more than just a catch-all (and tautological) component, it's important to relate any observed or known deviations from the semimartingale property to actual properties of the ways in which trading is conducted. I'll mention three such phenomena (there are several more, of course):

Trading is known to produce not just one price, but two prices: the bid and the ask price. The bid-ask spread is in general non-zero even in a competitive market, for reasons well-explored in the literature of the past 25 years. (In econometric practice, one frequently calculates the average of bid and ask prices and tries to model this average as some kind of martingale but, strictly speaking, that's not always valid.)
Because of inventory and liquidity effects, there's often predictability of price movements at the very highest sampling frequencies. E.g., if prices are observed every second, it may be possible to predict price movements out to a few seconds. This predictability is at the heart of what algorithmic trading rules are all about.
There may also be some mean-reversion of prices and volatilities at the very highest sampling frequencies due to the so-called "bid ask bounce": Empirically, it's been found that if a given trade happens at the bid, the very next trade is more likely to happen at the ask than at the bid, and vice versa. Because there's a spread between bid and ask prices, there's a bit of extra volatility that arises at the very highest sampling frequencies. This usually shows up in so-called volatility signature plots, which plot the intraday sampling freqency $\nu$ against the associated realized volatility $\sigma_\nu$: At the very highest sampling frequencies, $\sigma_\nu$ is frequently an increasing function of $\nu$.

Note that these "noise components" are generated as a consequence of the fact that trading takes place in real places among real people. As such, these noise components are not akin to those generated by computer programming errors. I.e., flash crashes in certain stocks are not an example of market microstructure noise.
To assume, then, that the latent or efficient price process and the market microstructure noise process are mutually independent really means that they're generated by two underlying processes which don't interact with each other. E.g., if new information enters the market, it should affect the value of the semimartingale but not meaningfully alter the noise process. (The interaction between the two processes is usually assumed to be additive, but I suppose one could generate a statistical theory that works with multiplicative rather than with additive noise.)

Naturally, it is not possible to verify directly that the independence assumption is correct, as one of the two underlying processes isn't observed. However, if it were the case that some observed departures from the semimartingale property are such that they cannot be explained usefully/credibly as being related to various market imperfections (such as the bid-ask bounce), one would have reason to doubt the empirical usefulness of the entire approach to modelling observed prices.


Serial correlation and clustered correlation in FE and FD estimators I'm running a regression on panel data, one time with Fixed Effects and one time with First Differences. The estimators are really different (the FE estimator is statistically significant and the FD isn't). While I'm running the regressions with the Clustered Standard Errors, it affects a lot the S.D of the FD estimator but not the S.D of the FE estimator. When I'm running Wooldridge test for serial correlation I get that there is no serial correlation. My questions:

Is it possible to be within clustered correlation for the FD model but not for the FE?
If there is no serial correlation, is it possible to be a within clustered correlation?

Yes. If the idiosyncratic error is iid, xtreg (FE) with no options (ordinary se) is valid, but the ordinary se for FD, reg d.(y x), is invalid because the differenced error is serially correlated.
Yes, it is. It's cluster-robust, which means OK  regardless of the presence of within-group correlation.