Shouldn't corporate tax be calculated based on EBIT instead of EBT?

Indeed the debt tax shield is distorting. Any policy that will essentially distort prices to favor A over B will lead to too much A and too little B. So such differentials are (almost) always distorting (unless you actually want less B and more A due to externalities).
Here the tax deductibility makes debt cheaper compared to equity. We know this is distorting from the Modigliani-Miller theorem. It states that a company's capital structure is not a factor in its value, so it doesn't matter how much debt or equity they have. However, the theorem holds in perfect markets without taxes. As you learned, taxes destroy this result.
So what can we do?
We need to not influence the price (cost) comparison between debt and equity in any way. So this leaves two options:

Limit (remove) the tax deductibility of debt (as you propose).
Allow for the cost of equity to be equally tax deductible. This proposal is known as the allowance for corporate equity (ACE).

Both have been tried in different countries.
The issue with the first one is that you are asking firms to pay taxes on money they may not have. Suppose the tax rate is 10% and a firm has EBIT of 10\$ and interest of 10\$. The firm has no money left after paying its interest, but it now owes the Government 1\$ (10% of the 10$ EBIT). Where is the firm supposed to get this money? Borrowing could be difficult as they can barely repay their current debtor's interest. If the firm pays the full interest, it cannot pay the Government taxes and is now essentially in default. If the firm pays the Government first and does not pay interest to its creditors, it is also in default. We don't want that either. It's counterproductive, since one point of removing the tax shield was to reduce debt and default risk in the first place. So eliminating the tax shield would be impractical, but one could try to limit it.
The main issue with ACE is the potential revenue loss. The IMF estimates that if the allowance were given to all equity, corporate income tax revenue would fall by 5-12 %. (I am borrowing this excellent link from a great comment by sba222).
Side note: You propose taxing earning before interest and taxes, but note that all taxes are on earnings before taxes, so its just "EBI" that we're talking about.


Can we see the computer age in the productivity statistics?

It means that computers (or IT in general) contribute to the society (they make us better off and more productive at work), but this does not show up in GDP statistics. The reason is that many of these services are for free. As an example, the value of the Google Search is not reflected in a market price for running this search. The problem is related to the societal value of mothers raising their children at home. Other examples are environmental protection or caring for your parents. Since these services are not traded on markets, it does not enter GDP. Whether the Solow quote is actually true is however debatable. The value of the google search engine is to some extent reflected in the advertisement google is selling. And this enters GDP statistics. The relation with the productivity puzzle is whether productivity is actually falling or whether we need new concepts to measure productivity in the computer age.


Why min AC = min SRAC at the minima of AC curve?

Short-run implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$.
Given a value $q$ let us denote the cost minimizing value of $x$ by $x^*$, i.e.
$$
x^* = \arg\min_x SC(x,q).
$$
Cost minimization implies that in the long-run this will be the chosen value of $x$, thus
$$
C(q) = SC(x^*,q) = \min_x SC(x,q).
$$
Dividing by $q$ we get
$$
AC(q) = SAC(x^*,q) = \min_x SAC(x,q).
$$
The above holds for any $q$.


Are Cobb-Douglas preferences homothetic?

Note that the wikipedia article  is very specific:

  [...] defined a preference to be homothetic, if they CAN be represented by A utility function [...]

You chose a specific utility function to represent your Cobb-Douglas preferences. However there are infinitely many others. All monotonic transformations of your utility function represent the same preference. Take
$$
\hat{u}(x_1,x_2) = \left(u(x_1,x_2)\right)^{\frac{1}{a+b}} = x_1^{\frac{a}{a+b}} \cdot x_2^{\frac{b}{a+b}}.
$$
As $\hat{u}$ is a monotonic transformation of $u$, it represents the same preference. It is straightforward to check that $\hat{u}$ fullfils the condition set forth in the wiki article. So there is indeed such a utility function, that also represents the preference, hence the preference is homothetic.


In the case of externalities, does the initial allocation of rights affect the final outcome from the perspective of each party?

Yes you're correct.
Stigler's "Coase Theorem" merely asserts that if transaction costs are zero, then the initial allocation of rights will not affect the total size of the economic pie, but may affect the distribution of the pie.
Two examples:
Example 1. Profit > Damage.
A producer $X$ producing widgets earns \$3 in profits but causes \$1 of pollution damage to a neighbor $Y$.

If $X$ has the right to pollute, then the widgets will be produced anyway, and $Y$ will simply suffer \$1 of damage (while $X$ earns \$3 in profits).
If $X$ has no right to pollute, then $X$ has to compensate $Y$ \$1, but the widgets will be produced anyway, and $Y$ suffers no damage, while $X$'s profits are reduced to \$2.

In either case, the widgets are produced, the pollution damage is done, and the size of the economic pie is the same. In particular, the net addition to the economic pie is \$3 - \$1 = \$2.
However, the distribution of the pie differs: In Case 1, $X$ enjoys +\$3 while $Y$ suffers -\$1. In Case 2, $X$ enjoys only +\$2 while $Y$ is not affected (+\$0).
Example 2. Profit < Damage.
The profits remain \$3, but now the pollution damage is $\color{red}{\\\$4}$.

If the producer $X$ has the right to pollute, then the neighbor $Y$ will pay $X$ between \$3 and \$4 to not produce the widgets.
If $X$ has no right to pollute and must compensate for any damage, then $X$ will not produce the widgets.

In either case, the widgets are not produced, the pollution damage is not done, and the size of the economic pie is the same.
However, the distribution of the pie differs: In Case 1, $Y$ transfers to $X$ some amount between +\$3 and +\$4. In Case 2, no widgets are produced, no transfers are made, $Y$ suffers no damage, and $X$ fails to enjoy any profit.


The intuition is simply that there is an excess of savings on the market and consequently the market gives people a signal to save less. You can see the situation visualized below in a tikz picture of IS-LM model I drew shown below.
The $y$ axis measures interest rates both real and nominal and $x$ axis output. The IS curve gives all the combinations of interest rates and output at which investment equals to savings. Whereas the LM curve gives you all combinations of interest rates and output at which money supply equals money demand  (see Blanchard et al. Macroeconomics an European perspective for more detail behind IS-LM). Furthermore, the LM curve becomes flat at zero lower bound as, if nominal interest rates become negative, peoples preference for holding money as cash becomes infinite.  Furthermore, the $Y^*$ line here would represent the natural level of output economy can attain.
If there is too much savings in the market but too little worthwhile investment opportunities (i.e. there are excess savings) the IS curve will shift to the left. The intuition for that is that if there is low demand for saving and huge amount of saving then any equilibrium interest rate (which can be viewed as the price for saving) at which $I=S$ will be lower for a given level of output (where $S$ is saving and $I$ investment).  Since nominal interest rate cannot be negative it will stay at zero but the real interest rate will be given by the intersection of the IS curve with the natural level of output which will be below zero.
So the intuition is really just that people want to save too much while at the same time they do not want to invest enough so the market tells them to reduce their savings rate with negative interest rate.
The intuition is similar to why oil futures price became negative recently - too many people wanting to sell the oil while too little people being interested to buy it. With the oil price, there was also an additional problem that people did not have any space to store the oil and so they were willing to pay people to take it off them. Here the problem is not so much with storage space but the fact that people simply often have to save - for example, due to precautionary savings or other reasons and even though they could hold cash at nominal 0 rate the problem is that it will be eaten by inflation as from Fisher equation we know that nominal interest rate $i$ is approximately equal to the real interest rate $r$ plus inflation $\pi$ - which in turn implies that if the nominal interest rate is above real we get inflation $i \approx r+\pi \implies \pi \approx i-r$. So people might desperately supply savings even when the real interest rate is negative because they have no other choice (except for stopping to save altogether- which again might not be reasonable for many people who might need to save due to various reasons such as to form precautionary savings).


The paper is assuming that some form of "law of large numbers" (LLN) applies for the continuum. The expected value of capital for an individual agent is
$$\mathbb{E}[R\cdot 1_{R\geq R^*}]=\int_{R^*}^1 R\,dR\tag{1}$$
The LLN assumption says that when we have a mass-1 continuum of agents, their actual total will equal their expectation in (1).

What justifies this assumption? Imagine that rather than having a continuum of agents, we simply had a very large number $N$, with each of these agents still having $R$ drawn from the same uniform distribution on $[0,1]$, resulting in an expected value of capital given by (1). Then as $N\rightarrow \infty$, the standard law of large numbers implies that the sample mean of capital will approach the value in (1). If, as we take this limit, we normalize each agent to have a size of $1/N$, then the sample mean will equal the total amount of capital - so in the limit, the amount of capital will always be (1).

Here comes the weird and not-entirely-correct thing that economists do. We often say that having a mass-1 continuum of agents is like taking this $N\rightarrow \infty$ limit (after all, there is an infinite number of agents in the continuum...). We therefore infer that the total value of capital in the continuum will equal the expected value given in (1).

Like I said, this is not entirely correct or rigorous. Judd famously complained about it, pointing out that if we take a continuum of agents each with independent random values, then the resulting function will usually not even be Lebesgue integrable, so we can't even define the aggregate value in a rigorous way. Al-Najjar recently suggested that we replace the continuum assumption with a discrete construction that actually implements the $N\rightarrow \infty$ limit. There are some other suggestions out there too.

Most economists, though, just ignore these technical concerns and continue to assume that when you have a continuum with independently distributed values, you get the expected value when you integrate. I don't mind this, since it's just a convention, and they are right in spirit, since a continuum realistically is intended to model a very large number of agents. (This issue came up once with a coauthor, and after reading Judd and Al-Najjar and others for a few hours, we just threw our hands up and decided to ignore the problem! My personal, technically acceptable way to resolve it is to say that agents - rather than living on a continuum - live in the probability space itself, and define aggregation over agents as probability-weighted integration over this probability space. But this is pretty technical and obscure, and the basic point is that 95% of economists ignore these issues.)


What was the US monetary policy like in the 70s?

In the 70s U.S. Fed policy was in disarray and Fed was not following consistent monetary policy (at least not by present day standards). Generally monetary scholars characterize the Fed monetary policy in the 70s as "go-stop" policy. As argued by Goodfriend (2007):

At the heart of the disarray in monetary policy practice in the 1970s was the
tendency for a central bank like the Federal Reserve to pursue “go-stop” monetary
policy. Go-stop policy was a consequence of a central bank’s inclination to be
responsive to the shifting balance of public concerns between inflation and unemployment. The central bank would stimulate employment in the “go” phase of the
cycle until the public became concerned about rising inflation. Then aggressive
interest rate policy initiated the “stop” phase of the policy cycle to bring inflation
down, while unemployment rates moved higher with a lag. Public support for  interest rate increases evaporated once the unemployment rate began to rise, so it
was politically difficult to reverse a higher inflation rate.

Wage and price setters learned to take advantage of tight labor and product
markets in the “go” phase of the policy cycle to make increasingly inflationary
demands, which neutralized the monetary stimulus. As a result, central banks
became ever more expansionary in the pursuit of low unemployment. Lenders
demanded ever-higher inflation premia in bond rates which moved higher and
fluctuated widely. By pursuing low unemployment and fighting inflation only when
it became the predominant public concern, central banks then increased the
volatility of both inflation and output.

Other factors contributed to the disarray in monetary policy in the 1970s. In
the 1960s, the widespread belief in a long-run Phillips curve trade-off between
inflation and unemployment inclined central banks to allow inflation to drift
upward in the hope of achieving a permanently lower level of unemployment. Also,
the productivity growth slowdown of the 1970s caused central banks like the
Federal Reserve to overestimate noninflationary potential output (Orphanides,
2003). Oil price shocks that occurred in 1973 and 1979 – 80 worsened the inflation
problem, although neither of the oil price shocks produced the 3 percentage point
increase in U.S. inflation that occurred in the 18 months prior to Paul Volcker
becoming Fed chairman in August 1979.

Consequently, the Fed was pursuing generally quite erratic loose monetary policy with occasional abrupt tightening.


Is opportunity cost market prices of input goods or income from alternative investments?

The market value of the inputs are determined by the most lucrative possible investment opportunity. Yes, the concept of market value is a simplification. If your firm alone can make the investment then the market is not functioning in a competitive way and hence is unable to assign prices in an informative way. If other firms cannot make the same investment but they can at least make investments with similar returns then the market value of the inputs would still be informative.


Consider a production function

$$y = F(A,K,L)$$

where $A$ represent "technology" (in the broad sense including for example also organizational technology), $K$ is capital and $L$ is labor (again both broadly defined)

"Economies of scale" (a looser synonym of increasing returns to scale), is, in economics, always defined with respect to all arguments in the production function except technology.

We say that the production function exhibits increasing returns to scale if it is homogeneous of degree higher than one... but with respect to capital and labor only:

$$F(A, sK, sL) = s^{\delta}F(A, K, L), \;\; \delta > 1$$

Note that we have not multiplied $A$ by $s$. Note also that economies of scale is a static concept.

Let's turn to the "learning curve" concept: it uses the word "learning", presumably not by accident: but "learning" has to do with efficiency and not with scale.  And efficiency has to do with technology, which is the one aspect left aside when discussing economies of scale.

As a final note, I won't dispute the meaning for the "learning curve" that you report, but I have to say that it sounds strange to me: whenever I have encountered the concept, it was always in a dynamic setting, as a synonym of "learning by doing": as we spend more time with a production process, we become more efficient at implementing it (and even if scale of operations remains the same).

I understand that a falling average cost can of course come as a result of the above process, but the dynamic aspect is crucial, contrasted to a static one.

In fact, in my experience, in a static sense a greater scale of operations always eventually leads to increasing average cost, if we do not allow for time to pass, and so experience and efficiency to be gained, that can more than offset the inefficiencies that accumulate due to the complexity of operations at a larger scale.

The above do not refer to the "easy phase" of decreasing average cost at low initial levels of production due to the inescapable existence of fixed costs.


How to specify a Diff-In-Diff Regression with multiple time periods?

I am a bit confused because you said there are two treatments. I presume that one "treatment" group is the control group and the other is the treatment group. (I don't think you have a separate control group.) I am also confused because you said the treatment is identical until round 3 but then the treatment begins in round 2. I presume that there is no treatment in rounds 1 and 2, and the treatment group is treated differently thereafter in rounds 3, 4 and 5.

Your $Roundn$ dummy variables for $n=2,\ldots 5$ make sure that you handle the "round effects" (for the control group) properly in a nonparametric way. This looks fine to me. If you include only the $After$ dummy, it means that there is no trend within each of the "before" and "after" periods. You would not want that.

A single $round$ variable means that there is a linear trend in $Effort$ in the control group. You could try that, but I would wonder where the linearity belief comes from. Also, you lose only 3 more degrees of freedom by including the round dummies comparing to the linear trend model. That's not a big deal unless you have a really small sample. I would be happy with the full round dummies.

Your model assumes that the treatment effects (measured by diff-in-diff) are identical in all rounds 3, 4 and 5 (because you have only one interaction term). If you believe it is true, that's fine. If you believe otherwise, you can include three interaction terms $Treatment * Round3$, $Treatment * Round4$ and $Treatment * Round5$ instead of the single interaction term. If you want, you can test if those treatment effects are identical across rounds.


Ex-post vs Ex-ante Budget Balance in Auctions

These are widely used technical terms with a precise mathematical meaning:

Ex-ante budget balance means that the expected sum of all transfers is zero.

Ex-post budget balance means that the sum of all transfers is zero with probability one. Some authors require that the sum of transfers is always zero.


Can a government or central bank work to reduce inflation without hiking interest rates and causing the currency to appreciate?

Yes, but only if you also introduce capital controls. You should check out the Mundell–Fleming trilemma:

  The Impossible trinity (also known as the Trilemma) is a trilemma in
  international economics which states that it is impossible to have all
  three of the following at the same time:

  A fixed foreign exchange rate
  Free capital movement (absence of capital controls)
  An independent monetary policy

Wikipedia Impossible Trinity


This website is supposed to be technical questions and not opinion-based, so I will try to respond to two technical questions I can extract from the question.

The first question: is the loan “less expensive”? The obvious response is: how big is the loan? Note that if the loan is not too large, it would be similar to student loans, an existing programme. The analysis of the programme would overlap that of student loans. However, the loan programme being cheaper is an apples-to-oranges comparison: what happens when a person cannot pay back the loan? They are cut off from support, which is what the UBI provides.

There are other questions here about Universal Basic Incomes, which may help answer questions.

The second question I see is whether the loan rate could replace taxes. This seems implausible with the current size of government. The US Federal government has a current tax take that bounces around 20% of GDP. (Other countries will have a larger tax burden, but there are also state and local taxes in the US.) At a loan rate of 5%, that would require the loan principal to be 400% of GDP to match amounts. A debt level of 400% of GDP is well above private sector debt levels seen in practice, and would probably unsustainable (could not be paid back). Although my comparison is somewhat apples-to-oranges, it indicates that taxes are much larger flows than plausible interest payments.


No, term deposits aren’t affected, it’s a change reflecting how often transfers can be made from savings accounts, which changed last April.
Per their announcements page:

As announced on December 17, 2020, the Board's Statistical Release H.6, "Money Stock Measures," will recognize savings deposits as a type of transaction account, starting with the publication today. This recognition reflects the Board's action on April 24, 2020, to remove the regulatory distinction between transaction accounts and savings deposits by deleting the six-per-month transfer limit on savings deposits in Regulation D. This change means that savings deposits have had a similar regulatory definition and the same liquidity characteristics as the transaction accounts reported as "Other checkable deposits" on the H.6 statistical release since the change to Regulation D. Consequently, today's H.6 statistical release combines release items "Savings deposits" and "Other checkable deposits" retroactively back to May 2020 and includes the resulting sum, reported as "Other liquid deposits," in the M1 monetary aggregate. This action increases the M1 monetary aggregate significantly while leaving the M2 monetary aggregate unchanged.


Check for concavity, quasi-concavity with given value of function

First look at the shape of the indifference curves. (I assume that $f$ is a utility function here.) The better-sets are convex, so this is consistent with quasi-concavity of the function $f$.

Second, consider the convex combination $(2,2)=\frac{1}{2}(1,1)+\frac{1}{2}(3,3)$. You have $f(2,2)=2$, but $\frac{1}{2}f(1,1)+\frac{1}{2}f(3,3)=\frac{1}{2}1+\frac{1}{2}4=2.5>2$, which is inconsistent with the function being concave.


How do I construct the score process of a Markov model and verify that it is a Martingale?

The derivation of the score process is correct. To verify that the process is a Martingale, recall the definition. It becomes clear that if we substitute $W_{t+1}$ back into the equation
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t W_j \\
 \sum_{j=1}^t W_j (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Because $W_{t+1}$ are Normal with mean 0 and variance 1 (I'm assuming they're iid), then
$$
E[s_{t+1} \mid s_t ] = s_t + E \left [
\begin{matrix}
(1 - \beta_0) W_{t+1} \\
W_{t+1} (X_t - \alpha_0)
\end{matrix} \middle |
s_t
\right ]
= s_t
$$
and we are finished.


Is weak global economy always a "headwind"?

Randomly, I came across a compelling counter-argument just today which I will share below. I'm not sure I buy into this theory whole-heartedly, but there are some valid points and it's worth reading through, even just for the sake of generating more discussion:

  America benefits when everybody is worried about problems abroad, and
  as a result of that they buy dollars. What does that do for America?
  Well, a stronger dollar keeps a lid on consumer prices for Americans
  and it keeps a lid on interest rates for Americans. Also if people are
  buying into dollars and dollar denominated assets, it helps sustain
  asset prices. That's a double win; the worse things get abroad the
  better things get in the United States. But it works the same way in
  reverse. When people have a more optimistic outlook on global growth,
  then they don't need the dollar as a safe haven. And now the dollar
  starts to fall. That means we now have upward pressure on consumer
  prices. That puts upward pressure on US interest rates. And that puts
  downward pressure on US assets, because there is no longer demand for
  US treasury bonds, for US stocks, for US real estate, because now
  people want to invest internationally. That's where the money will go
  because that's where investors will find growth, and that will in turn
  hurt the US economy. -- Peter Schiff


How would banking in UK change if minimum required capital to open own bank would be £100k instead of €5m?

The minimum required capital matters, but in the opposite direction: the more capital you have, the more likely you can challenge big banks. Because the costs of a new bank look like this:

(http://www2.deloitte.com/content/dam/Deloitte/pt/Documents/financial-services/dttl-fsi-uk-Banking-Disrupted-2014-06.pdf)

You need $15 mn just to keep it alive. Recall that, meanwhile, you are supposed to provide credit, so you'll have to borrow more money for this.

With 150K GBP in capital and fractional reserve rules, a new bank will be wiped out with losses. And it has no impact on the competition in banking, which is already quite tough.


Pretty much by definition, "necessary" jobs have to be filled. Wages would have to be bid up so that the positions get filled. This implies some form of relative price shock, or inflation. If robots can do those taks, great, but the plausibility of that outcome is left as an exercise to the reader.

The Universal Basic Income can be viewed as a rebranding of the Negative Income Tax idea that was floating around during the Nixon administration. (Milton Friedman was a supporter.) Studies were done at the time, and the problem with the scheme is the effect on income taxes.

My comments here are my interpretation of some of that analysis. One example is the article "The macroeconomics of a negative income tax" by Hyman Minsky (1969), which was republished in "Ending Poverty: Jobs, Not Welfare" (Minsky). Although Minsky is viewed as a heterodox economist, I believe this particular analysis ends up being similar to the consensus view at the time.

In order to hand everyone a non-negligible amount of cash each month, roughly the same amount of taxes will have to be raised. (One could debate about the exact amount of taxes required, but it would presumably be a significant proportion of the transfers, as it would be safe to assume that most of the transfers would be spent. Otherwise, there would be a major inflationary impact from the program.)

Unless the ultra-rich are willing to pay a lot more taxes, marginal tax rates would have to be high, starting at low levels of income. This creates a massive incentive to drop out of the formal economy, creating a death spiral for the tax system (or equivalently, an out-of-control inflation).

Therefore, the real issue is not the effect of artisanal markets, but rather the global interaction between all low-paying jobs and the tax system. For example, if the marginal tax rate for a minimum wage job is close to 50%, why take that job instead of just engaging in barter or under-the-table jobs, and using the UBI money to cover cash expenses?


Consumer Preference when Consumer only consumes $A$ or $B$

If Sally only prefers A or B but not both, this could be an example of Independent Goods. These are goods that are neither complements nor substitutes, and changes with one good generally do not affect the other.

However, with the information about Sally's preferences, it sounds to me like it could also be an example of Cross-Category Substitute Goods. These are goods that can substitute (imperfectly) for one another. Consumers may find one good more satisfying than the other based on their individual preferences and utility function.

  i.e. perhaps Sally wants a slice of pizza or Coke, and both cost \$1.
  She prefers the taste of Coke so she generally chooses it, but one day the price of Coke goes up to \$2. She
  may be satisfied to a similar degree purchasing pizza until the price
  of Coke comes back down. On a different day, she may want to purchase 2
  pizzas for the same amount as 1 Coke. There are many possible outcomes with
  cross-category substitutes.

It's a little unusual and subjective, because both goods in the example are in consumable food/drink categories, but one good satisifies hunger and the other good satisfies thirst. Thus, it's hard to think of one being a replacement for the other unless one thinks of them both being sweets or snack items. (One common example uses a chocolate chip cookie and a chocolate chip granola bar in place of Coke and pizza.)

If this isn't the exact term you're looking for, I hope this helps you on the right track at least.


What would be the "text-book" result of governement taking on massive debt immediately before a significant recession

As asked, this question is too vague and open-ended. But the following statements seem safe.

The effect of a fiscal stimulus is to raise growth and expected growth. All else equal, the central bank would be expected to raise interest rates, and inflation would be slightly higher, but still should remain near target (or else the central bank is going to have to explain why it is failing in its inflation-targeting).
A recession is a reduction in growth. Inflation normally falls, and the central bank cuts rates.

The effects are in different directions, hence the net of the scenario is ambiguous.


Mechanisms to incentivise social welfare

You say that losses are reduced by investment, so we can write the (private) loss as a function of own investment and others' investment: $l^i(c^i,\mathbf{c}^{-i})$ with $l^i_1(c_i,,\mathbf{c}^{-i})<0$.

Player $i$'s utility is $U^i(c^i,\mathbf{c}^{-i})=-f(l^i(c^i,\mathbf{c}^{-i}))-c^i$ and he invests until the first-order condition is satisfied:
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-1=0.$$
The social loss is
$$\sum_i\left[-f_i(l^i(c^i,\mathbf{c}^{-i}))-c_i\right]$$
so that the first-order condition for the optimal $c^i$ is
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})\right]-1=0.$$

This now looks like a standard externality model, where the external benefit of $i$'s investment is
$$b^i(\mathbf{c}^{-i})=-\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})>0.$$

If $l^i(\cdot)$ are publicly known then the problem can easily be solved as by imposing upon $i$ a standard Pigouvian subsidy of $b^i(\mathbf{c}^{-i})$ per unit of investment. This means that $i$'s payoff (including the subsidy) when $j\neq i$ choose $\mathbf{c}^{-i}$ is
$$\underbrace{-f(l^i(c^i,\mathbf{c}^{-i}))-c^i}_{U^i(c^i,\mathbf{c}^{-i})}+b^i(\mathbf{c}^{-i})c^i.$$
A rational $i$ will maximise this subsidy-augmented payoff; his first order condition is
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})\underbrace{-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,c^{-j})\right]}_{b^i(\mathbf{c}^{-i})}-1=0,$$
which is exactly the same as the first-order condition used to maximise social welfare. It is therefore a Nash equilibrium for each $i$ to choose the socially efficient level of investment given that everyone else does the same.

If the $l$s are heterogeneous and private information then things get more complicated because we have to get people to reveal their private information. Assume there is a benevolent central planner with the power to monitor agents' investments. One way to do this is then, indeed, to use a VCG mechanism. It would work like this: first, the central planner asks each agent to report their $l^i(\cdot)$. Write $\widetilde{l}^i$ for the reports. He then solves the first-order condition for the maximum of social welfare given the (possibly dishonest) reports. Thus, he maximises
$$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\sum_i\left[-f_i(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c_i\right].$$
Next the planner instructs all players to invest up to the optimal level. In return, he promises to pay to each $i$ a subsidy equal to
$$S^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)|c^i\right]-\max_{\mathbf{c^{-i}}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right].$$
Woah, what is that ugly thing!? The first set of square brackets is the welfare everyone other than $i$ gets. The second is the welfare everyone other than $i$ would get if $i$ were to drop out of the market all together. Thus, the difference between the two is $i$'s positive externality on everyone else.

Now let's look at $i$'s payoff:

$$U^i=S^i-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i$$

When evaluated at the equilibrium $\mathbf{c}^{-i}$ some terms cancel and we have

$$U^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]-\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right]$$

The second square bracket here does not depend on $c^i$ (remember, it is the surplus if $i$ were to drop out of the market) so $i$'s problem collapses to choosing $c^i$ to maximise $\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]$. Thus, $i$ has the right incentive to choose $c^i$ to maximise social welfare with respect to the reported $l$s.

What about the incentive to report $l$ honestly? The social planner is going to chose the optimal $c$s to maximise
$$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)$$
(this is just $\widetilde{W}$ with $i$'s utility added on and then subtracted again). $i$ would like to maximise
$$U^i=S^i+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$
which is
$$U^i=\underbrace{\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+z}_{S^i}+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$
where $z$ is the second term in $S^i$ (does not depend on $\widetilde{l}^i$). Thus we see that the social planner's choice will coincide with $i$'s optimum exactly when $l^i=\widetilde{l}^i$ so that $i$ has an incentive to report $l^i$ truthfully.


Can small countries use a foreign currency as their official currency?

Having your own currency means that you can conduct your own monetary policy. Adopting another country's currency means that you trade your ability to make decisions about your country's quantity of money, inflation target, etc. to a tested and solid mean of payment (the advantages you listed are good reasons).

You might consider European countries adopting the euro as an example of countries adopting a single currency. Ecuador adopting the U.S. dollar in 2000 might be a good place to look if you want to see the effect in a specific country.

There are quite a few countries in the world which use another country's currency as their official currency. This article lists them all (as of 2014):
http://qz.com/260980/meet-the-countries-that-dont-use-their-own-currency/


Macroeconomic effects of Profit Maximizing Federal Reserve

The Fed also conducts discount window borrowing. If pure profit maximising were the goal, they might only provide reserves through the discount window. They might even shake down weaker banks.

If confined to buying only Treasury securities, they would just need to maximise the real rate differential between currency and the nominal rate (plus, shrinking their balance sheet so that they no longer need to pay interest on reserves). The maximum level of the real policy rate that is sustainable depends upon your view of the natural rate of interest. If the belief is that the actual real policy rate cannot greatly depart from a natural rate, we would see little difference in policy.

Buying bonds and keeping rates low works to lock in profits once, but the profits will dry up as bonds mature, and are replaced by low-yielding ones.


Yes, this appears to be correct. This intuition is fundamentally Keynesian.

Consider the AS-AD curve.  During the vertical portion, additional demand increases prices while supply remains roughly fixed. As a result, prices increase dramatically.

https://en.wikipedia.org/wiki/AD%E2%80%93AS_model


Would it be correct to use exchange rates of the biggest trading partners to the dollar as an explanatory variable?

When a variable is measured over a period of time, it is a flow variable.  When measured at a single point in time, it is a stock variable.  In your case, I think you are looking to explain the year/year change in credit, a flow.

Think about the case with the level of credit as the dependent variable.  Does one year GDP growth explain the overall level of credit in this economy?  Most likely not.

Try to be cautious here on how you interpret the regression coefficients.  For instance, if your coefficient on GDP is 1, you cannot argue that GDP growth is causing increased credit flows.  This is because this model likely suffers from endogeneity.

To make the case for causality, an important step is showing the reader that your model satisfies conditional independence.  That is, the variables on the right hand side of the regression are uncorrelated with the error term.

Otherwise, it's useful to disclose that an attempt at identification was not made, and that the coefficients are more are less statistical summaries of how the data move together.


You are right that you first have to find F3's best-response function. F1 and F2 take as given this reaction of F3 to whatever they produce.
Hence, you plug this best-response function into the incumbents' profit maximization problem. In that way, you take care of the fact that the incumbents anticipate F3's reaction, indirectly determining $q_3$.

You should also check if the resulting reaction quantity $q_3$ is non-negative! It can be optimal to deter the entry of F3 completely.


Are energy and capital complementary factors of production?

You say "energy and capital are complementary factors of production. So for instance if price of energy goes up, so will the price of capital.". I think you might have confused yourself because this kind of positive correlation in prices is not necessarily a property of complements (indeed, often one would expect the reverse to be true).

It is more helpful to think of complementarity as "if the price of energy goes up then the demand for capital will go down" (in much the same way that people will buy less butter if bread becomes more expensive).

The reason for this is not far from that proposed by andersan. Capital is often invested in machines (e.g., computers, robots, trucks, farm equipment) that require energy to operate.

Consider a factory producing cars. It can assemble the car in various ways (a) by having a team of workers (labour) do the work, (b) installing robots on an assembly line, or (c) some combination of the two. If the price of electricity doubles then having a bunch of robots on the factory floor suddenly becomes more expensive because the energy needed to run them became more costly. Thus, the guy who was just indifferent between having a robot and a worker will now strictly prefer to hire a worker and the demand for capital decreases.

Even if you have a business (e.g. long-distance transportation) where you cannot easily substitute capital for labour, there will buy some entrepreneur who is indifferent between buying some capital goods (e.g. planes) to start a business and not. An increase in the cost of energy (e.g. fuel) will reduce the prospective profitability of his firm and thus leave him strictly preferring not to invest.


Why is the price of capital ‘r’ ? (From Cost function)

Firstly, in your example the value of $r$ (as used by economists in this context) would be $1.03$, not $0.03$. Economists call this the "interest rate", but you might prefer to think of it as the "rate of return on capital".

Secondly, what we define as constituting one unit of capital is pretty arbitrary. Is a computer one unit of capital or ten units of capital? It doesn't make any difference, just like it doesn't matter whether you measure distance in meters or feet, so long as everyone knows what units you are using. Since this choice is arbitrary, economists usually make the simplest possible choice and measure units of capital in units of money (e.g., dollars), meaning that one unit of capital has a price of one dollar by definition: $P(K)=1$.

If we return to your formula with $r=1.03$ and $P(K)=1$:

$$K[P(K)+0.03P(K)]=P(K)K(1+0.03)=Kr,$$

we get exactly the cost you saw in your course.


Can monetary union work without fiscal union?

Yes, and that facilitates a comparison of economies & policies.

Note that the governmental participation you outline (tending to government's prominence in an economy) is only one approach of policy making. Another, opposite approach is neoliberalism, whereby a government's role in an economy is mostly supervisory and promotes market freedom.

It is the absence of fiscal union what allows a government to make the domestic economy more competitive by lowering taxes. And in a context of monetary union, the spectrum of fiscal differences becomes more evident, which streamlines an assessment of which economies or policies are more effective. In short, the combination of monetary union and lack of fiscal union is the "best of both worlds".

  for example by undertaking infrastructure projects. Does monetary union somehow limit the ability to use this lever too?

It certainly does because monetary union removes a degree of freedom: The members of a monetary union are subject to an additional constraint necessary to preserve stability of the currency they have in common.

But more important is that the constraint that ensues from a monetary union makes it more difficult for a government to go too far on policies that ultimately are detrimental or unsustainable.


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

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.


Snowdon and Vane "Modern Macroeconomics: Its Origins, Development And Current State" is probably the best book I have read which gives a clear exposition of all the major macroeoconomic schools from the classicals to the New Keynesians. It highlights what the different schools have contributed to modern macroeoconomics and compares them to the other schools. They even include interviews with economists such as Robert Lucas Jr. and Milton Friedman to mention a few. I guess this is exactly the book you are looking for and to be honest I think every single person studying economics should read this book. It is not an introductionary economics book but I think most people should be able to read and understand it.


A question regarding eurodollar futures rate

In theoretical literature LIBOR rate and central bank's rate (for example Fed's fund rate) are often used interchangeably (see many examples in    Freixas and  Rochet: Microeconomics of Banking).
The reason why they are used interchangeably is that there is quite a strong proportional relationship between them and LIBOR rate depends on the Fed's fund rate. Hence:
$$r_{\text{LIBOR}} = f(r_{\text{Fed}})$$
In fact they are approximately equal so that:
$$r_{\text{LIBOR}} \approx r_{\text{Fed}}$$.
You can see that this holds in real life pretty well from the graph below based on Fred data. Also as explained here the correlation between them is almost 1. Hence, for purposes of theoretical modeling we will often just simplify and say that $r_{\text{LIBOR}} = r_{\text{Fed}}$ even though in reality this holds only approximately (there is really no point in explicitly modeling $r_{\text{LIBOR}} = f(r_{\text{Fed}})$ - models are meant to be simplified version of reality).


Can't we measure the growth in the income levels of poor?

That's not specific to poor, any income increases with inflation.

The real problem is that the poor receive goods for free (charity, squatting, rent nonpayment) so their real income can never be measured.


This is more of a logic question than it is a statistics question. I would say that the first point to address would be to define "root cause." Does that mean it is a sufficient condition? A necessary condition? Line of evidence #1 (there exist poor countries with few child labourers) precludes the possibility that child labour is a necessary condition of poverty as clearly poverty can exist without it. #2 reinforces that conclusion (again, poverty existing without child labour, just at a different scale).

Regarding point of evidence #3, I don't see how that is relevant: whether or not some policy exists compelling parents to act in a certain way doesn't affect the relationship between child labour and poverty.

As you allude to above, correlation is not causality. In this case, it seems from the correlation you can make equally strong arguments going both directions: child labour causing poverty; poverty causing child labour. You can make a logical case for it built on case studies, but making a statistical case for causality would require digging much further into the data.


Why is there negative sign for income effect in slutsky equation?

Intuitively, if the price of a particular good in a basket rises, then the individual can no longer afford to buy the same overall basket of goods

If there were no substitution effect, then this would be likely to result in amounts of each of the goods being bought being reduced, even those whose price had not changed, since the overall effect would be similar to a real reduction in income. Since the impact is negative, the equation needs a negative sign for this income effect

In general there will be a negative income effect on all goods and a positive substitution effect on all goods apart from the one whose price has increased.  Which is dominant for a particular good may depend on the particular situation


What counts as transactions fee in Coase Theorem?

Transaction costs (or fees, though I have always heard it as costs) is anything that complicates or prevents a transaction. This includes the example you are suggesting, but also things like the costs of suing, or if many people own the river and it is hard for them to organize and strike a deal. Maybe the legal system is terrible and people break deals all the time with impunity, etc, etc.

What you want to say in such cases is that transaction costs are high. Only if something changes that make it harder to strike a deal, it would be appropriate to say that transaction costs increase.


To provide some input (which I do not necessarily endorse), the Tax Foundation has used its "Taxes and Growth" (TAG) model to estimate/simulate the effects on USA Economy of Thomas Piketty's proposal that the tax rate for very high incomes should rise to $80$%.
The results can be found on a report freely downloadable from here.

Their Key Findings are (quote)

  1) If ordinary income were taxed at the top rates of 80 and 55 percent, our model estimates that after the economy adjusts, total
  output (GDP) would be 3.5 percent lower, wage rates would drop 1.6
  percent, the capital stock would be 7.4 percent less, and there would
  be 2.1 million fewer jobs.

  2) If capital gains and dividends were taxed at the new tax rates along with ordinary income, the economic damage would be much worse.
  GDP would plunge 18.1 percent (a loss of $3 trillion dollars annually
  in terms of today’s GDP), the capital stock would be 42.3 percent
  smaller than otherwise, wages would be 14.6 percent lower, 4.9 million
  jobs would be lost, and despite the higher tax rates, government
  revenue would actually fall.

  3) Although Piketty’s proposed income tax increase may appear to target only upper-income taxpayers, all income groups would suffer
  from the economic fallout.

  4) Our model estimates that the after-tax incomes of the poor and middle class would drop about 3 percent if the higher rates do not
  apply to capital gains and dividends and about 17 percent if they do.

The reasons that I say that I do not necessarily endorse the above results is that the report provided, although produced by a "Michael Schuyler, PhD" is not a scientific paper -so we do not know what assumptions and structure does their "TAG" model incorporates.


Would a Weather and Climate Change class be helpful to peruse a career in environmental economics

It wouldn't be detrimental by any means, but I would regard it as non-essential.

non-essential, because you'll likely use quite a simplified (as opposed to simple) framework for thinking about interactions between the economy and the environment. See for example Golosov et al. (2011) https://www.nber.org/papers/w17348

and non-essential because the larger models with more complex interactions between the economy and the climate/weather systems, would normally be conducted alongside some climate scientists. The division of labour here is arguably preferable, given that climate scientists and economists bring their own respective strengths to the table. I believe this to be the case in some of the integrated assessment models (IAMs) such as RICE/DICE etc.

But overall, if you believe it to be beneficial to your cause; I don't think it would be detrimental. Certainly it would enrich any modelling you'd consider doing.


Why is VAR analysis all linear? Why not nonlinear?

First, it is important to note that there is a substantial literature both developing and using nonlinear VAR estimation (for example, see papers here, here, here, and here).

The reasons linear VARs are seen a lot, though, are similar to the reasons that least squares (in its varying forms) is seen a lot. The model is very transparent and analytically tractable. This is a benefit for both pedagogy (analysis can be done on linear VARs just using lag polynomials) and for interpretation of results. You also don't run in to the concavity and global vs. local solution issues you do with big nonlinear problems, which can make your results more credible in many circles.


What does "the world's strongest currency" title mean for investors?

The strong currency understanding in your question does not have any interpretation that would be an economic factor that should effect investment, or any other decision. It is merely units.

It answers the question of '1 US dollar buys least of which currency?'. It also answers the question of '1 Colombian peso buys least of which currency?', or any other benchmark currency of your choice.

The Kuwait government could say that they decided to replace KWD by KWD2 and 1 KWD would equal 1000 KWD2. Then everything that has a price of $x$ KWD would now have a price of $1000x$ KWD2. Now, your US dollar would buy a lot of KWD2, but your investment decision would not be effected by this change because there has been only a change in units, and no other intrinsic differences.


How does a refinance allow a mortgage to be repaid?

Say you buy a house for \$100. This is paid with:

A \$10 down payment (from your own cash). (This is your equity.)
A \$90 loan from a bank at 10% annual interest. (We call this a mortgage loan or more simply a mortgage.)

Notice you're paying a relatively high interest rate of 10% on your mortgage, perhaps because the bank is not very confident that you'll be able to repay the loan.

Say that overnight, the value of the house rises by \$50 to \$150. Now your equity has also risen by \$50, from \$10 to \$60. You can now refinance your mortgage and ask the bank to lower the interest rate on your loan, say to 5%. You are certainly happy to do this refinancing because you'll pay a lower interest rate. And the bank might be willing to oblige because it is now more confident that you'll repay the loan.


Differentiation of the value function in Burdett Mortensen (1998)

When you take the integral of a $\max_{\{\cdot\}}$ operator, I think you have to split the integral into two separate integrals with different supports on them.

Even if your value function is complicated and there is no differentiability, you only need continuity for an existence of a solution to solve the optimization problem.


If we define $\succeq$ on $X$ as the primitive, then the strict preference relation is derived as
$$ x\succ y ~\Leftrightarrow~ x\succeq y ~\text{ but not }~ y\succeq x.$$

Note that you can view this as a definition of $\succ$. Now we prove $y\succeq x$ iff not $x\succ y$:

$(\Rightarrow)$ If $y\succeq x$, by definition, $x\succ y$ is not true;

$(\Leftarrow)$ If not $x\succ y$, then there are only three possibilities: (i) $ x\succeq y$ and $y\succeq x$; (ii) not $x\succeq y$ but $y\succeq x$; and (iii) neither $x\succeq y$ nor $y\succeq x$. Case (iii) can be ruled out if we assume completeness of $\succeq$. Therefore, in both (i) and (ii), we have $y\succeq x$.

It's worth mentioning that Kreps (1990) introduces another approach to define preference relations on $X$. He takes $\succ$ as the primitive and then derives the weak preference relation as:
$$x\succeq y ~\Leftrightarrow~ \text{not }~ y\succ x.$$

In this case, there is no need to prove the equivalence since it is a definition.


First paper to introduce the "principal-agent problem"

The term principal-agent problem is due to Ross (1973) (per Stiglitz, 1989). Other early contributions to this literature include Mirrlees (1974, 1976) and Stiglitz 1974, 1975).


Marginal Rate of Substitution for perfect complements

I think this is sort of a trick question.
First, you are completely right MRS is undefined at the kink - this is trivial MRS is the slope of indifference curve, which in this case is L shaped, and derivatives are not defined on the kink.
So this leaves us with two other parts of L shaped function. The vertical part and the horizontal part.
Over the horizontal portion of the indifference curve $\alpha x_1 > \beta x_2$ the MRS is given as:
$$MRS= \frac{U'_{x_1}}{U'_{x_2}} = \frac{0}{\beta}= 0 $$
So here the MRS is clearly defined.
However, at the vertical portion where $\alpha x_1<\beta x_2$ we will have a problem since:
$$MRS = \frac{U'_{x_1}}{U'_{x_2}} = \frac{\alpha}{0} = \infty | x_1 \wedge x_2 \geq 0 $$
but here on account of division by $0$ some people still say that the MRS is not defined.
However, the tricky part here is that choice of which good goes on $x$-axis and which good goes on $y$-axis is arbitrary. Usually people would put $x_1$ on $x$-axis and $x_2$ on $y$-axis but it is completely fine in principle to put $x_1$ on $y$-axis and $x_2$ on $x$-axis.  In that case the result above would be exactly reversed.
If this is for an exam often people who draft exercises just make some simple answer key that might not cover all possible solutions and maybe the answer key included option where their role was reversed.


I was recently recommended 'Quantitative Techniques for Competition and Antitrust Analysis' by Peter Davis and Eliana Garcés. While it does not contain all the necessary info, it does refer to more specified books.


How did Portugal draw down their interest rates on pubic debt?

1.- The government imposed an austerity program and, in exchange, received assistance from the IMF and the EU.

2.- The ECB made it clear that it would do whatever as necessary to save the Euro, even buying sovereign bonds of Euro countries.

3.- A general improvement in financial conditions around 2013 throughout the world let to a decline in default premia.


Does $U(x,y) = x^2 + y^2 + 2xy$ represent transitive, monotonic preferences?

Any preference relation represented by an utility function is transtive.

Suppose $x \succsim y$ and  $y \succsim z$ and $\succsim$ is represented by $U$. Then $U(x) \geq U(y)  $ and $U(y) \geq U(z)$, so $U(x) \geq U(z)$  and then $x \succsim z$.


Is it valid to compare cryptocurrencies by using their smallest divisible unit?

No, the above calculations are not a valid way to compare cost of these curencies, for exactly the reason you state:  a cryptocurrency is just worth what someone is willing to pay for it.

They don't have intrinsic value. Cryptocurrencies are inherently wealth-destroyers, they are not wealth: they require resources to construct, and then just act as ponzi schemes thereafter. So their only value derives from the number and purchasing power of the people that can be fooled into buying them.


How to prove that a point is a maximum point

The first order condition results in a differential equation in $\beta(x_1)$,

$$\left[ x_{1} + \alpha x_{1} + \alpha \frac{N-2}{2} x_{1} - \beta(x_{1})\right] (N-1)  - (1 + \gamma) \beta'(x_{1})x_{1} = 0$$

$$\implies (1 + \gamma) x_{1}\beta'(x_{1}) = -(N-1)\beta(x_{1}) + (N-1)\left(1+a \frac{N}{2}\right)x_1$$

or generically $$g(x_1) y' = f_1(x_1)y + f_0(x_1)$$

Where $y \equiv \beta(x_1)$ and the other obvious mappings.

This has general solution

$$y= \beta(x_1) = C\cdot  \exp \{F\} + \exp\{F\}\int \exp\{-F\} \frac{f_0(x_1)}{g(x_1)} dx_1, \;\;\; F\equiv \int \frac{f_1(x_1)}{g(x_1)}dx_1 $$

and doing the work results in

$$ \beta(x_1) = \beta(z^*) = Cx_1^{-(N-1)/(1+\gamma)} + \frac{(N \alpha + 2)(N-1)}{2(N + \gamma)}x_1$$

I guess there exists a condition in the model that leads to $C=0$ so that we get the linear expression for the $\beta$ function

$$ \beta(z^*) = \frac{(N \alpha + 2)(N-1)}{2(N + \gamma)}x_1$$

Regarding whether $\Pi(z)$ has a global maximum at the stationary point $z^* = x_1$, working the 2nd derivative with respect to $z$, and evaluating it at $z^*=x_1$ is what one can do.

Then, if I haven't done any trivial mistake, it appears that a necessary condiiton for the second derivative to be negative at $z^* = x_1$ is that $\alpha(1+\gamma) <1$ and, given this, the condition for a global maximum at $z^*=x_1$is

$$N  > \frac{2+\gamma}{1-\alpha(1+\gamma)}$$

which appears to put a lower bound in the number of participants in the auction, which I guess is what $N$ measures.

But this is the 2nd time I had to guess (and a third guess is that parameters are positive), validating @denesp comment that even though the OP issue is mathematical, providing context and information on what the various symbols stand for, is needed if one is to have a good-quality question to work with. After all, this context is what would make this an economics question and not a math question.


Core Allocations are Pareto efficient allocations that must satisfy individual rationality i.e. these allocations must yield at least as much satisfaction to the individuals as their respective endowments. In the given question, core allocations are represented by line connecting $(e/2, e/2)$ to $(e,e)$.
Dashed line on the left graph is representing the set of efficient allocations and on the right graph is representing the Core allocations.


Firstly, the use of debt/GDP as the ratio is somewhat arbitrary. One could easily look at debt/government revenue, and it may be a more useful metric. However, the convention to use GDP for scaling these series. (One can argue that GDP represents the full income capacity of the economy for debt service.)

For countries that borrow in a foreign currency (or in a currency where they do not directly control the central bank), a high debt ratio indicates a greater risk of default. There is no gurantee that they can their hands on the foreign currency to repay debt. You need to look at other things as well, but it's one common metric.

If the country controls the currency it borrows in, it should be able to arrange its affairs to avoid default. (There is no guarantee, however.)

However, even if we ignore default risk, the greater the debt level, the greater the interest expense for the government. The exact mechanisms are debated by various schools of thought, but it is reasonably safe to argue that greater interest payments will squeeze out other spending if we want to keep inflation around a target level. There are two ways in which a greater debt load increases interest payments.

Even if the rate of interest is unchanged, a greater amount of debt implies more interest to pay. This effect holds by definition.
A greater debt-to-GDP ratio might increase interest rates. The magnitude of this effect is controversial. The example of Japan in recent decades (high debt ratios, low nominal yields) demonstrates that the effect is not very large, even if it exists.

There is a school of thought that argues that we need to "pay back" debt in some sense; this is often referred to as the governmental budget constraint. This view can be summarised as: greater debt now means greater future taxes, and people do not like taxes. (This is sometimes called Ricardian Equivalence.) The exact mechanism is controversial. (I have severe doubts about the mathematics behind it, for what that's worth.)

In the case where the central bank has bought a lot of government debt (quantitative easing), the expectation is that the central bank will eventually reverse that policy. In any event, the central bank needs to pay interest on resserves if it wishes to raise market interest rates to control inflation, and the consolidated government will face a similar interest cost as the case of bill issuance.

Finally, the money supply is related to debt levels; both are forms of government liabilities. In most modelling traditions, people only target a certain anount of money holdings (that earn no interest), and the rest is invested in bonds/bills. Therefore, we only expect money to grow in line with transaction needs, and this level may have little resemblance to total debt levels.


What is the relationship between indirect utility function and the expenditure function?

This classic picture from Mas-Colell, Whinston & Green should help you:


A question about intertemporal price discrimination

Airline seats are often priced lower when first released and the price can rise, especially as you get closer to flight time.
The effectiveness will depend on the market, pent-up demand, and thus substitution effects. There are not many substitutes for a given video game while for a flight there are many substitutes such as flying earlier or using another carrier. The intertemporal aspect comes in since people buying later were not able to plan ahead and so pay a premium versus people who did plan ahead or are able to fly later.
For something unique, a low price early on might be justifiable as generating publicity; however, I doubt extending that low price to all early buyers would be optimal.


A maximization problem with multiple goods and integrated markets

I think there is an approach problem in the constraint. (1) implies that:
\begin{align}
v_{n,t} - v_{n,t+1}=H_t - (\sum_{i=1}^{n-1} v_{it}-\sum_{i=1}^{n-1} v_{i,t+1})
\end{align}
As the number of fish caught that are not the $ n $ species increases, the number of fish that belong to the $ n $ species decreases. This, i understand, may arise from a production decision: company 1 decides to produce more fish of species not $ n $ and reduces its production of type fish $ n $. But what exogenous force can create this phenomenon? it would make sense if they are species that compete for the same resources in a limited space, but this would affect future non-current populations, and it would be necessary to see if it is reasonable that the effect has a linear form, it is more likely that it affects according to a logistics function. The other option is for one species to be the predator of another, but in this case it appears that it is not.
I suggest either directly assuming that there is no inter-species resource struggle or choosing a different function that determines the resource struggle among fish. If you decide that there is no fight between species, you can separate the population dynamics of each species and you could choose a function like this:
\begin{align}
P_{i,t} = P(v_{i,t} - v_{i,t+1},K) \\
\frac{\partial P_{t}}{\delta t}=\delta P_{ t}[1-\frac{P_{t,i}}{K}]  - (v_{i,t} - v_{i,t+1})
\end{align}
The differential equation $ \delta P_ {t} [1- \frac {P_ {t}}{K}] $ is based on studies in biology on the dynamics of fish population, you can see a simple model in this link https://scholar.harvard.edu/stavins/publications/problem-commons-still-unsettled-after-100-years. $ P_{i, t} $ is the population of the species $ i $ in the period $ t $, and $ K $ is the "Carrying Capacity", this is the maximum level of fish population that can exist. You would only have to adapt that differential equation to the discrete case.
As for the second constraint, why make the quantity produced of one species of fish decrease with increasing another? This is something that arises endogenously as the producer's decision: it is optimal for company 1 to produce more of the type 1 fish and less of the type 2. This second restriction is not necessary, I think. You can use a production function like this:
\begin{align}
f_{m}=f(v_{i}-v_{i,t+1},v_{-i}-v_{-i,t+1} ) \ \forall \ m \in M
\end{align}
Each company endogenously chooses which fish to produce. It is clear that with this objective function, companies that cannot sell some type of fish will simply not sell that fish. In addition, you can analyze interesting phenomena such as the presence of economies of scope or simply choose a simpler function to find equilibrium in the long term. I think these observations would solve the problem you are having.


How to estimate Robot density in a country?

Daron Acemoglu has written a fair bit about the impacts of automation on labour markets. Bearing in mind that mechanization and "robotization" are different in important ways (with that importance increasing as AI and machine learning become more sophisticated and ubiquitous) you could use his work as a starting point.  Certainly, you'd be remiss not to cite it if you're writing a paper.

In https://www.nber.org/papers/w24196, Acemoglu outlines the challenges to the question you're asking and offers a framework of approach.  In https://www.aeaweb.org/articles?id=10.1257/jep.33.2.3, the framework is applied, and there are links to the data which you can look at for insight into how to measure this stuff.


I will provide another hint, more "primitive" than the one offered by @Anoldmaninthesea, for those that are not still very familiar with the "big-Oh/little oh" notation and arithmetic.

What we are examining here is a sum, which, following the first obvious "hint" given, is to be decomposed in three separate sums.

Now, we are interested in what happens to the value of these sums as the number of summands goes to infinity... In such cases, essentially we are looking if "we have enough "$N$"'s for the infinite sum to go in value to something finite (given the a priori assumptions), and if yes, whether this finite limit is zero or not.

Consider the middle sum

$$\frac{1}{n} \sum\limits_{i=1}^{n}[-2x_iu_i(\hat{\beta_n}-\beta)\mathbf x_i'\mathbf x_i] = -2(\hat{\beta_n}-\beta)\frac{1}{n} \sum\limits_{i=1}^{n}(x_iu_i\mathbf x_i'\mathbf x_i)$$

$(\hat{\beta_n}-\beta)$ was taken out of the sum because it does not depend on the index $i$. It does depend on $n$, since it is an estimator function and not a specific estimate, but not on $i$.
The assumptions of the model tell us what happens to $(\hat{\beta_n}-\beta)$, if it remains unscaled, as $n\rightarrow \infty$. As for the summands, $(\mathbf x_i'\mathbf x_i)$ is a $k\times k$ matrix, that does not include itself any sum. So it is finite (the fact that the random variables involved have possibly infinite support does not bother us -the probability that they will take the value "infinity" is zero). So it is only the whole expression $(x_iu_i\mathbf x_i'\mathbf x_i)$ that multiplies in cardinality as $n$ increases - and by the assumptions of the model we know things about $x_i$ and $u_i$ and what happens to the average of their product. Etc.

An unrelated example to show what this "how many $n$'s" business means is to consider, for example, a random vector like

$$\frac{1}{\sqrt n}\left(\mathbf X'\mathbf X (\hat \beta_n - \beta)\right)$$

We "know" that this random vector converges to a multivariate distribution, i.e. it neither converges to a constant, nor does it diverge. Can we see this in our primitive way? We would want a finite asymptotic variance. The asymptotic mean will be zero so we need only to examine

$$V_n=E\left[\frac{1}{\sqrt n}\mathbf X'\mathbf X(\hat \beta_n - \beta)(\hat \beta_n - \beta)'\mathbf X'\mathbf X\frac{1}{\sqrt n}\right] = E\left[\left(\frac{1}{n}\mathbf X'\mathbf X\right)[(\hat \beta_n - \beta)(\hat \beta_n - \beta)']\left(\mathbf X' \mathbf X\right)\right]$$

What is the problem with the above expectation? The first term wll converge to something finite and non-zero (per initial assumptions), the second goes to zero, the third goes to infinity(because it includes sums whose cardinality will go to infinity). So the expectation is still of undetermined fate. But if we multiply and divide by $n$ (we can do that, we have not yet send $n$ to infinity) we obtain

$$V_n= E\left[\left(\frac{1}{n}\mathbf X'\mathbf X\right)[n\cdot(\hat \beta_n - \beta)(\hat \beta_n - \beta)']\left(\frac{1}{n}\mathbf X' \mathbf X\right)\right]$$

Now things begin to clear: the first and second terms will converge to expected values, so they can be taken out of the external expected value:

$$V_n\rightarrow \left(\lim_{n\rightarrow \infty} E\frac{1}{n}\mathbf X'\mathbf X\right)\lim_{n\rightarrow \infty} E\left[n\cdot(\hat \beta_n - \beta)(\hat \beta_n - \beta)'\right]\left(\lim_{n\rightarrow \infty}E\frac{1}{n}\mathbf X' \mathbf X\right)$$

The middle term inside the expectation  is the (multivariate) "square" of $\sqrt n (\hat \beta_n - \beta)$. We know that this last one converges to a zero-mean random vector, so the limit of the expected value of its "square" is its asymptotic variance, which is finite. So we have the product of three asymptotically finite expected values, and so the whole expression is finite, and so the variance of the expression we started with is finite, and moreover, non-zero (by the usual initial assumptions of the model).


Depending on your data source, the debt level will either be the face value of debt, or its market value. The usual choice is face value. The debt level is a stock.
Interest payments are an expense that will show up in the deficit, which is a flow variable.
If nothing else happens, then new debt = old debt + interest. More generally, (new debt) = (old debt) + (fiscal deficit) is the simplified relationship. (Note that this will be an approximation to real world data, as it is possible for some transactions to affect the debt level, but are not part of the definition of the fiscal deficit, and vice-versa.)


There is no "reasonable" choice here.

One can choose inches over centimeters, gallons of water over kilograms of water, etc. This depends on how you want measure the input quantities. No choice is wrong so long as it is consistent with your other modelling choices.


Why is the Supply of Labour More Elastic in a Single Province/State then it is in the Country as a Whole?

Because there is freedom of movement between provinces; and the language barrier is low. When international borders are involved, there is much less freedom of movement, and much greater potential for language barriers.

So it's easier for labour to move between provinces if incentivised to do so by a change in relative taxation rates; than it is to move between countries.


Is there any research, theory, or anything which shows how much of a market has to be in power of a few companies to be a force against free market?

The quote in the question isn't really rigorous about what a free market is, but it talks about monopolies and artificial scarcities, so I am interpreting the efficient outcome with price equal to marginal cost as being one necessary feature of what they understand as a free market.

Let's look at the cournot model of competition. There are $n$ firms, each with idiosyncratic marginal cost $c_i$. Each firm simultaneously chooses its output (supply) quantity, $q_i$. Given consumers' demand, this results in some market price $P(Q)$, where $Q=\sum_{i=1}^n q_i$ is the total industry output. We would expect the demand curve to be downward sloping: $P'(Q)<0$.

Thus, a firm's profits are
$$\pi_i(q_i,\mathbf{q}_{-i})=\left(P(Q)-c_i\right)q_i.$$
The first-order condition for maximisation of profit is
$$\frac{\partial\pi_i}{\partial q_i}=P(Q)-c_i+\frac{\partial P(Q)}{\partial q_i}=0.$$
Noting that $$\frac{\partial P(Q)}{\partial q_i}=\frac{\partial P(Q)}{\partial Q},$$
the first-order condition can be re-written as
$$\frac{P(Q)-c_i}{c_i}=-\frac{\partial P(Q)}{\partial Q}\frac{Q}{P(Q)}\frac{q_i}{Q},$$
$$\frac{P(Q)-c_i}{c_i}=-\frac{1}{\eta}m_i,$$
where $\eta$ is the price elasticity of demand and $m_i=q_i/Q$ is firm $i$'s market share.

The left-hand side measures a firm's price-cost margin, or its market power. We can compute the average market power in the industry by weighting each firm by its market share:

$$\text{avg market power}=\sum_i m_i \frac{P(Q)-c_i}{c_i}=-\frac{\sum_i m_i^2}{\eta}.$$

Note that $\sum_i m_i^2$ is the HHI index for the market. So this equation tells us that the average market power in an industry will be positively related to the HHI. For any positive HHI, there will be a positive mark-up and the market will deviate from the perfectly competitive outcome in which artificial scarcities do not exist. Put differently, only if there is an infinite number of firms, each controling an infinitely small share of the industry do we obtain the textbook perfect competition outcome.


Book Recommendation for Microeconometrics of Discrete Data

Personally, for choice analysis, I like

Discrete Choice Methods with Simulation by Ken Train (pdf)
Applied Choice Analysis: A Primer by Hensher and Greene (2nd edition, book)
Modeling Ordered Choices by Hensher and Greene (pdf)
Regression Models for Categorical Dependent Variables Using Stata by Long and Freese (3rd edition, book)

(1) is a fairly short, readable classic. (2) is much more introductory, with lots of intuition, though it devotes a lot of space to the nlogit software. (3) has the same features as (2), but for a more focused set of topics. (4) is also pretty introductory and focuses on using Stata. It is perhaps the least "economisty" of the four, but still quite good.

For count data, I like Cameron and Trivedi's count data book, followed by Winkelmann's.

The books you have listed are good, particularly Wooldridge, but they cover much more ground than just discrete data at the expense of depth.


Generalizing demand for perfect substitutes utility function

Let $b_i := \frac{a_i}{p_i}$, now choose the good that maximize $b_i$, let it be $b_j$ i.e. the $j$th good. Optimal choice is $x_j=m/p_j$, and $x_i = 0$ for $i\neq j$.


Does everyone prosper through development projects?

A lot of this comes down to risk. A developing country usually don't have very strong "institution" and are much more prone to various kinds of crisis (think military coup, hyperinflation, terrorist attack, nationalization*). An investor might also not be as familiar with the developing country in question* (and hence the risk).

As a result, investors require higher return (the "premium") to invest in these countries. You can see the risk premium by country here.

This is why developing credibility could be very helpful in terms of drawing foreign investment into a country.

* Added from comments below.


Analyzing Farm Bureau's claim that milk prices would double at the retail level if all the illegals were deported

It is worth noting that OP's original question before I edited it asked, "why would economists lie to us?" This already leaves a poor taste in my mouth; such a question is loaded enough as it is, only good for picking fights. The author stated at the bottom of the Progressive Dairy article is a lawyer, not an economist, and seems to be the basis for some of your figures in your analysis. The article from the Texas A&M AgriLife study cited by the Farm Bureau/NMPF did have economists writing on the topic, which I'll discuss briefly later.

Related to denesp's comment on cost of labor share, consider the firm's cost of milk production:

$$p_L q_L + \sum^n_{i = 1} p_iq_i = C$$

The cost $C$ is made up of the cost of labor (price of labor times quantity of labor) plus the cost of all other factors of production. The labor share of cost that was mentioned by OP would be:

$$\frac{p_L q_L}{C} \equiv \boxed{s_L = \frac{p_L q_L}{p_L q_L + \sum^n_{i = 1} p_iq_i}}$$

Suppose as in the hypothetical that the cost of labor doubles. OP argues that then,

  we can multiply the labor share of a dairy farm's expenses per hundredweight by 2

but this is clearly not true from a mathematical standpoint.

$$2s_L \neq \frac{(2p_L) q_L}{(2p_L) q_L + \sum^n_{i = 1} p_iq_i}$$

More than this, it also assumes there are no cross-price effects. That is, if the price of one good changes, it will not change the quantity demanded of other factors of production. If there is any sort of substitutability OR complementarity between labor and any other factor of production, this cannot hold. Empirically, low-skill/blue-collar labor is a complement to high skill labor and a substitute with capital.

All of this walks around a bigger issue. The article in question makes a claim about the effect of disemployment on doubling milk prices. It does not say that it will come about by doubling labor prices, so the argument posed above is just a strawman. That is, it may very well be that doubling labor price would not double the final price of milk, but that is not what the article is arguing.

That said, I don't particularly like the analysis done in the Texas A&M AgriLife study cited either. Ultimately, it has no inference/causal modelling and is just a survey with some summary statistics, with a lot of questionable assumptions.  There is like one price elasticity in that whole analysis as I could see, and its use was not compelling.

The lazy way to interpret this is to just say the methods of the study are bad and that its conclusions have no value and only serve to mislead the public. I don't think this is particularly true either. Private sector market analysis is definitely in some ways less stringent than publishing in, say, an economics journal however.

So what can we do?

The long, boring, rigorous way to estimate the impact of a large shock to the labor supply is to look at labor demand and labor supply simultaneously.

Consider an extremely simple set of linear estimations:

$$\ln(L_s) = \alpha_s \ln(w) + \vec{\beta}_s \vec{Z} + \epsilon_s$$
$$\ln(L_d) = \alpha_d \ln(w) + \vec{\beta}_d \vec{X} + \epsilon_l$$

Where $Z$ is a vector of control variables/supply shifters (e.g. non-labor income), and $X$ is the same but for demand. Notice that labor, $L$ and wage, $w$ are endogenously determined, which nudges us towards the use of IV estimation, rather than the typical OLS.

Labor Demand:

From here you'd have to think about what kind of functional form production should take: $Y = f(K, L, \cdots)$, where $K$ is capital, and you can put in some other factors if you want (legal labor vs illegal labor perhaps?), and derive conditional factor demands, which you can plug into the equation for cost which gets some $C(w, r, \cdots, Y)$, with $r$ representing the rental rate of capital.

Applying Shephard's Lemma, you can then derive the elasticities of substitution between two different factors of production, which are used to calculate the cost shares of of each production factor, and eventually, we can then find the own-price elasticity of the factors, used to help estimate $\alpha_d$.

Labor Supply:

Similar line of thought, but with a specification of utility and working with Frisch elasticities.

Plugging in the data (number of laborers, capital stock on farms, etc.) could prove difficult or impossible without existing data of course.

This is obviously a very rough sketch for an empirical framework for analyzing labor supply, but it is a foundation for analyzing all sorts of shocks to the labor supply. For a more detailed analysis, you can refer to Cahuc, Carcillo, and Zylberberg and their first few chapters on labor supply and labor demand.

Sometimes in the economic literature, undertaking a more detailed study like this requires a lot of funding and time, with in mind particularly the hassle of data. People want practical policy answers in the short run, particularly in the private sector. With limited resources, sometimes studies like the ones you've looked at provide "second-best results" and that is the analysis we have to work with. The productive, honest thing to do with studies that take on partisan issues is to think about how to feasibly improve them and ask specifically about how to look at the research being done on the topic.

There are plenty of voices accusing all manner of partisan research/analysis as misleading, and still plenty polemics out there who would waste their time writing polemics. It is not for this site.


Does non-binding price ceiling effect the market?

No there is no impact at all. A price ceiling of $10 means that the price cannot go above $10. Since the equilibrium price is already below $10 the creation of a price ceiling will not effect anything at all.

It is called an ineffective ceiling because it is precisely that, ineffective.


Why'd a company issue unpaid shares, to acquire or merge companies?

Why would a company issue unpaid shares?

Possible reasons are:

buyer's temporary lack of sufficient funds to perform the
acquisition, whence the issuer of shares is willing to grant credit for
the sake of "closing the sale";
fiscal planning (be it the buyer's and/or the issuer's planning) might be such
that delaying payment of shares will place him at a more convenient tax bracket
in the acquisition and/or disbursement fiscal year(s) (this is more evident in cash based accounting systems), and/or the delay of payment will allow
him to qualify for certain tax deductions;
issuer predicts the price of shares will fall (and possibly discourage prospects of merger/acquisition), so he hedges against
that risk by entering a forward contract with the buyer, with the incentive --to the buyer-- that transfer of shares will be effective immediately;
buyer is willing to commit to a higher price on unpaid shares so as
to anticipate upcoming legislation that will make it harder to
perform akin mergers/acquisitions.


Does the Tax Allowance Benefit Everyone (Regardless of Their Income)?

Not everyone benefits from the tax allowance, but most people with incomes above £11,000 do.  As you suspect, those with income higher than £11,000 only pay tax on their income above £11,000, and thus gain £11,000*tax rate.  However, the personal tax allowance is clawed back at incomes above £100,000 at a 50% rate, so those with incomes above £122,000 don't benefit from the allowance at all.  The official UK government site is here.

At the margin, this means that those with incomes less than £11,000 pay no tax on each additional dollar of income, while those with incomes between £100,000 and £122,000 pay an extra-high amount in tax for each additional dollar earned (the tax allowance doesn't change the tax rate on each additional dollar of income for incomes between £11,000 and £100,000, or above £122,000).  The entire picture is nicely summarized in this Telegraph article.

Here in Canada, we have an equivalent tax allowance, but without the clawback.


Are quantitative easing and helicopter cash really different tactics? And how does QE relate to Modern Monetary Theory?

What is QE:
QE is simply an asset purchase by central bank. As explained by Fed St. Louis QE is defined as:

large-scale asset purchases—in the hundreds of billions of dollars range—of, for example, mortgage-backed securities and Treasury securities.

Furthermore, under QE this is done with newly created reserves. Generally these assets are actually not government assets (i.e. bonds) but rather the assets of banks (although they can be government assets as well).
The idea of QE is to basically make an asset swap - a swap of central bank's  reserves for some assets of banks which make them reluctant to borrow more (for example the first QE Fed did focused heavily on mortgage based securities - which were at the heart of Great Recession - as it was argued that banks were reluctant to borrow more while holding them). Later QE's also included treasury bonds and other more safe long term debt. This is not printing money, but to the extent it encourages new borrowing it increases the money supply so the money printing analogy has some measure of merit in it.

Answer 1

Some sites describe 'helicopter cash' as an alternative to QE. That conflicts with the above understanding of QE as a way of raising funds, not of spending them. Under my understanding, one might do QE in order to finance a helicopter cash strategy.

Yes 'helicopter cash', or more correctly helicopter money is an alternative to QE. QE is an asset purchase program. Helicopter money just literary means giving away cash/deposits to people. The term originated from Milton Friedman's quip that in this case central bank might as well be literary dropping money out of helicopter. Helicopter money does not require any asset purchase from central bank. Central bank can as well just send every household a check or directly deposit the money into their account.

Answer 2

Some sites say the aim of QE is to lower interest rates, while other sites (and sometimes even the same sites!) say that QE is used when lowering interest rates has ceased to have any effect on economic activity. One of these must be incorrect. Under my understanding, the first is incorrect.

No both are correct and you are misunderstanding what they are saying. There are multiple interest rates in the economy. QE is used when Federal Reserve cannot bring down further its federal funds rate - when it hits zero lower bound. However, as explained in the same Fed St. Louis document I linked at the beginning, the aim of QE is to bring down interest rates on long term treasury securities and other financial instruments. Those are completely different interest rates than the one which makes QE necessary. Even if federal funds rate is 0 long term treasury rate or corporate bond rate might be positive.

Answer 3
No, open market operations and QE are different although the difference is subtle.  Investopedia has an excellent article on the difference between the two, here is short summary (but I recommend reading the full Investopedia article):

Open market operations are a tool the Fed can use to influence rate changes in the debt market across specified securities and maturities.
Quantitative easing is a holistic strategy that seeks to ease, or lower, borrowing rates to help stimulate growth in an economy.
Open market operations can be an important tool used in seeking to obtain the goals and objectives of quantitative easing.

Basically, open market operations is much narrower term, while QE is wider term and when central banks execute QE it might be partially done through open market operations.

Answer 4
QE is an asset purchase program, hence it will always involve a debt. However, central banks are part of the government. All profits including the profits from interest on government bonds are sent back to government (after cost of running the central bank is deducted see this previous Q and A about Fed profits - other central bank might have different institutional arrangement but the profits end up at the government).
Also it is technically possible for central bank to just 'scrap' the bonds after purchase, saying government never has to pay it back and giving up any claims to gov. revenue those bonds give central bank. Nonetheless, doing so would represent permanent increase in money supply and most central banks would probably think twice before going this route.

Does that mean that QE could not be regarded as a MMT-oriented activity?

I dont think this can be answered. There is no single formal modern monetary theory (MMT). MMT lives, for most part, in blogosphere not in actual research papers and with every new book on MMT new/different things are claimed about it so there is too much inconsistency to say what is consistent and what isnt with it. As far as I understand from what I have read about MMT central bank purchasing bonds from government would be in view of MMT just an accounting trick and equivalent to directly funding government spending as long as central bank commits not to cash those bonds, but keep all the caveats mentioned in preceding sentences in mind.
Edit in Response to Brian's answer
For example, this blog on MMT says:

It should be noted that QE is conceptually different from MMT, although both involve central bank purchases of government bonds. Under QE, the expectation is that the central bank will sell the government bonds it buys before they mature, so that the government will need to raise money (ultimately through taxes) to pay its debt to private holders of that debt. Effectively, central bank assessments of a temporary need for money-creation to address short-run economic circumstances have driven QE.

Hence the blog claims if assets are purchased without intention of them to be sold later then QE is consistent with MMT as I mentioned above in my answer.
However, an important caveat to the reader here is that does not automatically means that Brian's answer is incorrect (apart from the part arguing this answer made incorrect assertions) - MMT has no definitive agreed upon version of a theory as mentioned before. However, saying that this answer misrepresented view of MMT is itself not correct as this answer both cautioned that people dont even agree on what MMT is and as the quote above shows clearly under some MMT views QE is consistent with MMT as long as central bank does not intend to reverse the bond purchase.

Answer 5
QE and any other monetary policy differs from mere deficit spending (i.e. fiscal policy) tremendously. From macroeconomic perspective the goal of monetary policy is to stimulate economy through changes in the money supply, which in turn affect interest rates and prices in economy. Fiscal policy tries to stimulate economy directly through government spending.
Furthermore, fiscal policy might be conducted even when central bank does nothing and vice versa monetary policy can be conducted even when government does not makes any changes in its fiscal policy or even if government does not have any net debt. In general monetary policy and fiscal policy are macroeconomically different on fundamental level (but this answer is already way too long to go into any further details on these differences but you can consult any standard macro textbook - if you are layman Blanchard et al. Macroeconomics: an European Perspective is excellent source targeted at undergraduates so you wont get yourself lost. If you want something more involved and nuanced then Romer's Advanced Macroeconomics is the book to learn intermediate macro from).


Increased production costs associated with increased output produced? Why do firms have prices increase with the SRAS or Supply Curve?

this is for SRAS and AD, but I'm also going to apply this a bit fo microeconomics.. sorry for mixing, but my question applies in both ways, I believe

You are applying market equilibrium logic to the supply function. When aggregate demand shifts for some reason while the supply function is unchanged the equilibrium price will change.
If the short-run aggregate supply (SRAS) function is upward sloping then there are no economies of scale (in the short-run). This does not give you detailed information on what a firm is able to do long term.


Let's begin by fleshing out what the EViews code is saying.

The EViews code says that the variable CLOSE depends on its first difference and there is also an ARMA(3,3) error term included in the model. Supposedly, the ARMA error process is to whiten the residuals, which can be, and has been, argued against as a modelling strategy. It's also worth noting that there is no constant term in the equation. The reference to BACKCAST indicates how the MA process is initialized - the details of which can be found in the EViews documentation. ESTSMPL typically refers to the estimation sample.

Having understood the EViews code, the question is now: how does one estimate a linear regression model with ARMA errors using MS-Excel?

The easiest solution would be to find an existing add-in that provides this feature. An alternative method would be to write the code yourself using the MS-Excel programming language (VBA). Beware though, what makes this application tricky is the MA term; AR processes are easier to program and more common in certain fields, e.g. VAR models appear more frequently in economics than VARMA models. Another approach, if you are willing to delve into some code, would be to look at some R source code (either base code or from some time-series package) to get a sense of the task ahead if you do decide to program from scratch. Programming this from scratch seems like overkill, though, as this sort of estimation has been around for decades. Why bother if all you're interested in are the estimated parameters and not the underlying routines? It doesn't answer your question, but my advice would be to use R if EViews is not available to you. Or, seek out an MS-Excel add-in that will do the job for you.

A useful thread from the EViews forum is here: http://forums.eviews.com/viewtopic.php?f=7&t=465. Note: "Estimation, and forecasting of MA processes is complicated, especially when backcasting is used to obtain starting observations for the error terms."

Good luck!