Why does 'government consumption'in NK models not add to utility

You can include government consumption in the utility function of the representative consumer, but why? When we add exogenous government purchase shocks to a macro model, we enquire about the dynamics of the agent's consumption and labour supply change. This concept is clear and easy to measure.

Let's assume that you are including government spending in an additively separable utility function. For example, the representative consumer is happy because of the government is investing in coastal defences. But he is unhappy about the state's Olympic games project. It's all OK, but how are you going to defend these assumptions empirically speaking? I think you should introduce a few agents plus. How do you describe the preferences of agents about the different government spending projects (defence forces, park building, etc.)? It's a normative assumption, what makes this research Economics no more. This research becomes Political Science. Unless you find or do a reliable research on the preference of agents about government spending projects. But keeping in mind, political preferences are very far-fetched from the so-called rationality. There is always an emotional factor, which makes a spending project likeable or not. How to make these "soft" questions into "hard" science? It's a very hard question.

Anyway, what is the purpose of including include government consumption in the utility function? I can imagine an OLG model with a social choice problem. E.g. children want playground, adults park, and elders hospital.

To answer your sub-question, government purchase is wasteful by default in Macroeconomics, unless this purchase don't effect the production function in a positively. In any other case, the welfare of the economy can be suboptimal. If government buys tablets for every pupils in schools, it may makes the workforce more productive in the long run.

You can measure the effectiveness of government spending considering the fiscal multipliers indicator. It's a stylized fact that the effectiveness of government spending depends on the countries structural features (income, openness,  exchange rate regime, etc.).

See also: Enrique G. Mendoza, Carlos A. Vegh , Ethan Ilzetzki - How big are fiscal multipliers? New evidence from new data (01 October 2009)


How can median wages be significantly different in two neighboring countries with open borders?

The underlying reason are probably differences in labor productivity. Assuming decreasing marginal productivity of labor, the real question is why migration flows do not equate labor productivity and wages as would be predicted by simple GE models.

I can think of several factors:

Moving is costly
Czech people in Germany are not as productive as Germans in Germany, because they don't speak German.
People prefer to
live in their home country even if that means earning lower wages.

That being said, I guess that wages in the Czech Republic have been growing much faster than in Switzerland or Germany and the gap is indeed closing slowly.


Should the government regulate the price of bananas?

(source: dineshbakshi.com)

As you can see in the graph, price controls policies create excess demand, therefore you will end up with shortage or you will import more bananas.

I'm not a banana expert, but I assume bananas are pretty inelastic. Also, banana supply is inelastic in the short run (they have already planted bananas), therefore the excess demand will be less than the graph shows.

Whatever the case, price control policies generate deadweight loss and it depends on the elasticities who loses more (consumer vs supplier). This market is no longer efficient and that impacts the welfare negatively.


If a currency crashes, does that effect the value of rare or collector coins?

Yes, their value would collapse.

But not for the reason you might think.

It's not because they're the same currency, per se.

But rather because a major collapse in the US dollar's value would be part of a much larger economic collapse. Which means that disposable wealth would be rare, and the demand for indulgences such as collectors' coins would dry up. At the same time, you'd have lots of owners of such coins trying to liquidate their assets. So, there'd probably be both a big increase in supply, and a big decrease in demand, leading to plummeting prices.

Promoters of any number of yieldless "investments" such as bitcoin, gold, or collectables, will tell you that their investment would be a safe-haven in times of crisis. But at times of major crises, the correlations (the betas) tend towards one. Hedges stop being hedges.


How to get the effect of one dummy variable against many others?

If you only want to test whether the mean wages of South Dakotans is higher than for residents of other states, run:

Where South Dakota is a dummy variable like the one you've described earlier. I'm assuming you simply have a cross-section of wages at a single point in time, but if you have either repeat observations of individuals or a repeated cross-section, then you can include a t subscript for each of the variables and include time controls.

If  is positive and significant, it would be interpretable as the level or percent (if the model is run in log wages) increase in mean wages for a resident of South Dakota compared to the mean for the pool of residents of all other states.

If instead you wanted to perform pair-wise comparisons between South Dakotans and residents of an individual state, you could include a dummy for that state and then run a command like lincom in Stata to perform hypothesis tests between those states' coefficients.

If you wanted to compare South Dakota against every single other state, then make South Dakota the omitted category in your regression, and you could run the model you described of including dummies for all non-South Dakota states. The big issue with this approach is the massive multiple hypothesis testing issue you will encounter, which could be addressed through Bonferroni correction.


What is a supply side shock in macroeconomics?

In the framework of the ISLM model, a supply shock would be defined as any exogenous force that affects the ability of an economy to produce its natural level of output, this ability is represented by a production function that often relies on available resources and production technology. Therefore, any factors affecting the availability of resources or technology will be considered a supply shock (e.g. Drought and crops).

On the other hand, a demand shock acts on aggregate demand by influencing planned expenditure by economic actors or the amount of real money balances that economic actors desire to hold.

In short, if it affects available production inputs or production technology, it's a supply shock.

Notice that the GDP is just a measure of all goods and services produced in an economy for a given period of time. If you use the expenditure approach to GDP (Y = C + I + G + NX), a supply shock will not immediately act on this formula, rather it affects the formula Y=F(X) (where X is a vector representing production inputs). But note that the two formulas, meaning the two "Y"'s must equal each other, therefore, there will be changes in the first equation to match the second equation at some level of Y. However, you can't differentiate these changes from changes arising due to a demand shock unless you have the whole story. You can't just look at "C" for instance, and assume that any changes in "C" resulted from a supply shock.


What does commitment in adverse selection mean?

Think about an auction, where the designer is selling good and trying to sell it to the person that values it the most while collecting as much revenue as possible.

A direct mechanism means that the seller asks buyers for how much they value the good and based on that decides who gets the good and how much they pay. Suppose that the designer uses a second-price auction so it tells the buyers "tell me your valuation and I'll give the good to the person with the highest valuation, but this person will only pay the second-highest valuation.

If the seller has commitment power (for example if they can write down the rules in a binding contract, etc), this auction achieves exactly what the designer wanted to do, maximizes revenue, allocates the good to the person who values it the most, and it is incentive compatible. However, notice the temptation that the designer faces:

Suppose the highest valuation is $\$2,000$, and the second-highest valuation is $\$1,200$. If the designer lacks commitment power, the designer might change the rules after learning the valuations and charge a lot more than $\$1,200$ for the good. However, if the bidders know that the seller lacks commitment power they will not truly reveal their valuations and there is no guarantee anymore that the auction will obtain good results.

Now keep assuming that the seller has no commitment power, but consider the following mechanism that is NOT a direct mechanism. Suppose the seller runs an ascending price auction. That is, bidders yell out how much they want to pay and can outbid each other as much as they want. Whoever's bid is the highest gets the good and pays what their last bid. Now the seller does not know exactly the valuation of the buyers, and arguably is no longer tempted to change the rules in the middle of the auction.

In conclusion, if there is no commitment, a not-direct mechanism can out perform a direct mechanism.


In an article in the Times, Jon Moulton stated:

  "The Bank [of England] has had this [2%] inflation target for more than a
  decade yet it lacks any theoretical basis."

EDIT: Just in case you are interested, I have a theory about optimal inflation. If you wish money to be neutral - i.e. be a Veil over Barter then you would like money to behave as closely as possible to ordinary "stuff" that gets bought and sold. One property that the bulk of goods have is that they decay over time, i.e. they rot, rust, become obsolete etc etc. If you consider the average decay rate of "stuff" to be X% then inflation should be such that money loses its value at a rate of X% too. I'd say that the decay rate of "stuff" is generally greater than 2%, my feeling is that it is more like 5% or so.


Effect on GINI of eliminating poverty in China

Such calculation is extremely difficult, because all the factors involved in shaping the personal distribution of income: relative change in return to different skills/job (think e.g. on skill-biased technical change), demographical/educational composition change (e.g. many new educated young into the labour market), taxation/redustribution policies (if less poverty, less income support welfare needed?), variations in the functional distribution of income (ownership of financial assets?), etc. It also depends on whether you are measuring inequality at the individual or household level, and which equivalence scale you use in the latter. Not to mention that you need to somehow differentiate between urban and rural sectors to incorporate your change.

I can think of three options you can follow to get what you want:

Microsimulation: this requires to have a general equilibrium model where a population of (commonly) households are simulated, usually to analyse policy changes. For the case of China, an example is here. This is the best approach in terms of using economic theory, but it is very time-consuming.
Regression: here you estimate a regression to evaluate determinants of nation-wide inequality. Included in these determinants should be the rate of urban and rural poverty. Forecasting them (and other regressors) you can forecast future Gini. This method is a highly reduced-form approach and might be highly inaccurate or inconsistent (due to endogeneity, etc). Still, to get an idea, see this paper.
Percentile income growth: as suggested by ahorn, you could go atheoretical and simply predict income growth of each percentile. Then you can easily compute Gini. The plot of these percentile income growth are commonly known as "growth incidence curve". For an example on China see here (Figure 4).


Currency Devaluation - As a Means of Paying Off Debt

Say that at time $0$, $1$£ is worth $1$€. Indeed, like any good or service, a unit of £ can be seen as having a price in € (and vice versa).

At time $0$, Alice lends $10$£ to Peter. Assume that Peter can actually influence the exchange rate between £ and €.

At time $1$, Peter influences the value of $1$£ expressed in €, such that $1$£ is now worth $0.5$€. Peter then decides to reimburse Alice the amount of $10$£. The point is that this amount of $10$£ is only worth $5$ once converted back into €.

Alice, the lender, has lost $5$€ between time $0$ and time $1$. And for her, it was a bad operation given that she lives in a country wherein everything is priced in €.

For Peter, it may have been a good operation over the short term, especially if he does not need to import goods and services from €-based countries, indeed €-priced goods cost henceforth twice as much than at time $0$, since he would now need $2$£ to get $1$€, etc...

That being explained, I do not think that Alice is willing to lend Peter money anymore, at least not in a currency that Peter can influence... She has learnt the lesson and, in the future, she will always write debt contracts which amount will be explictly expressed in the currency that matters most to her, i.e. in €.


Given the production function
\begin{eqnarray*} F(L) = 10 \sqrt{L} - 2L \end{eqnarray*}
marginal product is
\begin{eqnarray*} F'(L) = \frac{5}{ \sqrt{L}} - 2 \end{eqnarray*}
Since there is no cost for the family for using its own labor and it cannot hire from outside, family will maximize its production subject to the constraint that it has 4 units of labor.
\begin{eqnarray*} \max_{L\geq 0} && 10 \sqrt{L} - 2L \\ \text{s.t.} && L \leq 4 \end{eqnarray*}
Given that $F'(L)$ is falling in $L$ and is positive at $L = 4$ $(F'(4) = 0.5)$, family will use all 4 units in optimum. The corresponding output is $10\sqrt{4} - 2(4) = 12$.

With $10\%$ income tax, the objective of the firm will be to maximize total output net of tax which is equal to $0.9 F(L)$. Solution to this problem will stay the same as before because we are just multiplying the objective by a positive constant.

\begin{eqnarray*} \max_{L\geq 0} && 0.9(10 \sqrt{L} - 2L) \\ \text{s.t.} && L \leq 4 \end{eqnarray*}

Therefore, the equilibrium output is still $10\sqrt{4} - 2(4) = 12$ out of which 1.2 is paid as taxes and 10.8 is consumed.

With new technology the problem is
\begin{eqnarray*} \max_{L\geq 0} && 11 \sqrt{L} - L - 15\\ \text{s.t.} && L \leq 4 \end{eqnarray*}

The above problem again give $L = 4$ as the solution and the corresponding equilibrium output is $3$ which is way less than what the family had earlier. Therefore, it will not use this technology.


Distribution of loans by types of interest

http://www.oecd.org/economy/surveys/

To complete the answer, I recommend reading the full economic studies country by country: banks, finance, debt, economic sectors, taxation, economic risk etc. Read these studies before making décisions !

I sincerely believe that these studies will help you in your economic analysis and banking.

We must understand why an interest rate is high or low, why the volume of business loans is strong or weak.

Only this type of study can explain and analyze the data rates between companies, and compare from one country to another.

For example, credit rates can reach 15% in an African country because the inflation rate is high and the rate of the central bank is very high! Without economic studies, it is impossible to compare.

French/Français :

Pour compléter la réponse, je conseille de lire les études économiques complètes pays par pays : l'écosystème bancaire, les financements, la dette publique, les études par secteurs économiques, la fiscalité, les risques économiques etc. Lire ces études avant de prendre des décisions !

Je pense sincèrement que ces études vont vous aider dans votre analyse économique et bancaire.

Il faut comprendre pourquoi un taux d'intérêt est fort ou faible, comprendre pourquoi le volume des prêts professionnels est fort ou faible.

Ces études peuvent vous expliquer et aider à analyser les données concernant les taux entre les entreprises, et les comparer d'un pays à un autre avec une approche globale.


Change in Welfare from an Incentive based Supply Restriction

(This ended up being a long post, but I find the approach of the textbook a bit outdated).

It is interesting to examine the issue from a (simplistic) game-theoretic point of view.

Assume a fixed (no entry) large number $N$ of small identical producers, meaning that the actions of each one individually does not affect the market. The market demand curve and the market supply curve are common knowledge. For next period, the government wants to restrict production below the current market clearing. The government announces that it will pay an amount $T$ to any producer that respects the deal and restrict his production, on top of the revenue he will earn in the market.

We assume that each producer has to evaluate the following options: either he conforms with the government plan  or he doesn't. In combination with either "all others conform", or "all others do not" (here lies one aspect of simplification).

So each individual producer faces four possible future states:
1) All conform
2) He conforms while the rest don't
3) He doesn't conform while all others do
4) Nobody conforms.

The motive to not conform comes from the fact that if he alone does not conform, the market price won't be affected, because he is "small".

First, a diagram:

$P_c$ is the price that will prevail if all producers conform to the government plan and produce in total $Q_c$. Correspondingly, $P_d$ is the price that will prevail in the market if each producer attempts to exploit for his own benefit the price hike (each assuming that the others will conform to the government plan). Then the quantity supplied will end up being $Q_d$. Finally $P_L$ is the price which producers would normally want to produce $Q_c$, absent government interference. Note carefully that the long-term market clearing point is out of the picture. Nobody "cares" about it, now that the government jumped in and essentially, reshaped the market structure in terms of the market's prospects.

We now assume that the individual long-term supply curve is a zero-pure-profit, best-response function, so the payoffs for each possible state are expressed as deviation from this best-response function. This is a critical assumption.

Then, the game in normal form as experienced by each individual supplier is ($q$ is just $Q/N$)

\begin{array}{| r | r |r |}
  \hline
\hline
   &  \text{All others conform} & \text{All others deviate} \\
  \hline
\text {I conform}   & (P_c-P_L)q_c+T & (P_d-P_L)q_c+T\\
  \hline
\text{I deviate}   & 0 & (P_d-P_c)q_d \\
  \hline
\end{array}

Note the reason why the lower left corner has a payoff of $0$: if the producers produces $q_d$ and receives a unit price $q_c$, he is on his zero-pure-profit, best-response function -nothing is gained on top (so we do not ascribe any benefit to "be larger" in terms of revenues).

It should be clear from the above payoff matrix that "I conform" is already a pure dominant strategy without the need for any strictly positive government incentive $T$.

What I am (momentarily) saying, is that the Government needs only to announce the program, with a "token" incentive $T$, and everyone will conform to it. Essentially the government functions here as a "cartel creator", bringing all suppliers to cooperate in a decentralized manner in order to restrict quantity and increase price.

Well, it doesn't sound as a very convincing description of reality, does it?  "Token incentive programs" do not appear to work in the real world. So something must be critically mistaken in our assumptions.

And it most probably is: the assumption that a producer does not care about his scale of production. This neglects the fact that especially with small producers "costs" include their reward for their own labor and use of their own equipment. So a higher scale of operations does increase the income of the producer, even though it may not bring "pure profits" over and above the remuneration of the production factors.

So let's approach the construction of the payoff matrix in a different way. Assume that "all other costs of production" except the producer's own income (like materials, third-party services, additional laborers, etc) are constant per unit of output, denote it $b$. Then the producer income is $I = (p-b)q$.

The payoff matrix in terms of income becomes now

\begin{array}{| r | r |r |}
  \hline
\hline
   &  \text{All others conform} & \text{All others deviate} \\
  \hline
\text {I conform}   & (P_c-b)q_c+T & (P_d-b)q_c+T\\
  \hline
\text{I deviate}   & (P_c-b)q_d & (P_d-b)q_d \\
  \hline
\end{array}

We can reasonably argue that $P_d<b$ and so without the government incentive, Income in the right column will be negative. Note that here "I conform" is already dominant if all others deviate, because losses are smaller.

Here in order for "I conform" to become a dominant strategy we must have that

$$T > (P_c-b)(q_d-q_c)$$

This is essentially "the cost of avoiding treason". In order to deal with loss-aversion we could also argue that the government will guarantee "no-loss". Then the government incentive should be

$$ T > \max \{ (P_c-b)(q_d-q_c),\; (b-P_d)q_c\}$$


If Americans drive more and consume more oil, why are cars less efficient?

The price of petrol / gasoline is surely a major reason, offsetting the effect of longer distances.  In many European countries it is more than twice as high as in the US, mainly because of differences in tax rates, as shown here.

Given consumer rationality, the price of fuel might be expected to impact on fuel efficiency via choice between car models of different efficiency at the time of purchase.  But do people actually behave rationally in this situation?  Turrentine & Kurani (2007) found from interviews with households that consumers lacked the knowledge and understanding needed to estimate savings in fuel costs over time.  Although their sample was small (57), this finding seems plausible.  However, even if individual consumers do not behave rationally, it remains possible that the price of fuel is still for many an influencing factor, albeit imperfectly assessed, and that the aggregate effect of decisions by many consumers approximates to what might be expected given rational behaviour.

Allcott & Wozny (2014) tested the hypothesis that consumers, on average, are willing to pay \$1 more to buy a car with \$1 less in discounted future fuel costs.  They used panel data relating to purchases of different car models in the US over the period 1999-2008, relying on changes in gasoline prices over time (rather than differences between countries) to permit a test of the hypothesis.  Their sample was impressively large, including 86 million transactions.  Purchases were assumed to be informed by assumptions on future fuel costs. On their base specification, willingness to pay for \$1 less in discounted future fuel costs was estimated as \$0.76.  Alternative specifications with different assumptions on discount rates and the basis for forecasting future fuel costs yielded willingness to pay figures in the range \$0.42 to \$1.01.  Given the large sample, the standard errors of these results were small, at most \$0.06, even after clustering by model and age was allowed for.

The above strongly suggests that the price of fuel has influenced fuel efficiency in the US. I haven't seen a similarly rigorous study for Europe, or comparing the US and Europe, but it seems plausible that this finding would extrapolate, if only approximately, to a cross-sectional inter-country comparison.


What would be the consequences of Downsizing on the economy?

It would decrease consumption, lowering demand on many markets, meaning that prices would go down. If downsizing would rampant enough it's possible that economy would experience deflation, maybe even recession. As the result, maybe central banks would be decreasing key interest rates (i.e. make new debts cheaper), start buying bonds and lowering reserve requirements for banks in order to stimulate economy. While governments could increase government spendings, decrease tax burden and relax business regulations for the same reasons. Accumulation speed of government debts would be increasing (i.e. government would be borrowing more in order to finance their deficits caused by increased government spendings and decreased taxes). It would also make workforce more rigid, unemployed downsized people wouldn't be able to take some jobs. Like you can't take job of a burger-flipper at MacDonalds if you're downsized. Nations with downsized workforce would be able to offer lower prices while exporting their services abroad. Downsizing would be most widespread among countries with big service sector, like many deindustrialized and post-industrial countries. Imports of countries with downsized workforce would decrease, possibly leading to positive trade balance.


Can heterogeneous prices be consistent with competitive equilibrium?

As a starting point, you might take a look at Stigler's model of price dispersion. Quoting from a resource I found online,

  George Stigler’s 1961 “The Economics of Information” begins with:

  "One
  should hardly have to tell academicians that information is a valuable
  resource: knowledge is power. And yet it occupies a slum dwelling in
  the town of economics."

  Stigler documents the existence of price
  dispersion empirically, and proceeds to sketch various theories that
  might be applicable to price dispersion. Stigler is careful to
  distinguish price dispersion, which results from imperfect information
  on the part of consumers, from price discrimination, which involves
  distinct prices for different types of consumer preferences

  There are various reasons that price dispersion might arise. Clearly
  some consumers must be willing to pay higher prices than others, that
  is, distinct consumers have distinct reservation prices. (Note that a
  reserve price – the minimum bid in an auction – is a very different
  concept.) Differences in reservation prices can arise because of
  differences in knowledge – some consumers have access to price lists,
  others don’t – or search costs. If some consumers have lower search
  costs, then these consumers will search for lower prices.

You might also check out the Wikipedia articles on Price Dispersion and Search Theory.


Meaning of «intertemporal substitution effect dominates the income effect»

I suppose the right approach is to start with the statement: "Real wage is also a positive function of productivity".
Let's start with the income effect: If both goods, free time and consumption are normal goods (This is the standard case; normal good simply means: if you are richer you want to consume more of them), then the individual wants to increase both consumption and free time simply because he got richer and can afford it now. More free time translates into less working hours.
The substitution effect: As real wages increase the individual would miss one hour of higher salary after the wage increase. This makes free time relative to consumption more cost full than before. The substitution effect induces the individual to work more.
As long as the second (substituion) effect is greater than the first (income) effect labor demand will increase as productivity increases.

So far, the pretty standard interpretation for a change in relative prices. Regarding the intertemporal component of the substitution effect i found in Mankiw, Rotemberg and Summers 1982, p.8:

The intertemporal substitution hypothesis implies that labor supply should depend on the relative return from working in period t and from working in period t+1.This depends on the real wage in period t relative to the discounted real wage in period t+1.

I suggest this implies that as long as the productivity shock has a relative higher impact on real wages in $t$ than in $t+1$ the intertemporal substitution effect has a positive impact on labor supply in $t$. With all of this in mind you should be able to understand the proposition « labour will be a positive function of productivity, when  the intertemporal substitution effect dominates the income effect»


Why low labour productivity for Govt Services in UK and USA in 2000s?

The biggest issue (in the US national accounts, at least) is the fact that government production is recorded on a cost basis rather than a market value of output basis (due to the lack of market values for many government services).

From the NIPA Primer, pages 2-3:

  GDP is composed of goods and services that are produced
  for sale in the “market”—the generic term referring
  to the forum for economic transactions—and of
  nonmarket goods and services—those that are not sold in
  the market, such as the defense services provided by the
  federal government, the education services provided by
  local governments, the emergency housing or health care
  services provided by nonprofit institutions serving
  households (such as the Red Cross), and the housing services
  provided by and for persons who own and live in
  their home (referred to as “owner-occupants”).

  [...]

  In cases where there are no similar market transactions
  available to impute a value of the good or service being
  produced, the output of these services is valued by estimating
  the costs (such as employee compensation and
  purchases of materials and supplies) of producing the
  good or service.

As a result, any direct government production is recorded as though the workers add no value that they do not capture through their salaries. Given the technical difficulties (not to mention the political issues) that arise when one tries to come up with an alternative estimate of the value of government production, this is probably the most neutral approach, and the approach that is most accurate in differences, though it obviously gets the level wrong.


Solving a LRE model in King-Watson form using alternative algorithms

The two approaches specify the linear rational expectations problem somewhat differently, but one can reconcile the solutions they produce.  In addition to the difference in the equation system specification, KW use the "predetermined variables" concept while AM does not.

King and Watson (KW) address a model of the form

$$
  A E_t y_{t+1} = B y_t + C_0E_t x_t +C_1 x_{t+1}\\
  x_t=Q \delta_t +G \epsilon_t\\
  \delta_t=\rho \delta_{t-1}\\
  y_t=
  \begin{bmatrix}
    \Lambda_t\\k_t
  \end{bmatrix}
$$

the $k_t$ are predetermined variables.

The algorithm produces solutions of the form
$$
 y_t=\Pi s_t\\
  s_t=M s_{t-1} + \bar{G} \epsilon_t\\
  s_t=
  \begin{bmatrix}
    k_t\\ \delta_t
  \end{bmatrix}
$$

Anderson Moore (AM) cast the problem as

$$
  H_{-1} w_{t-1} + H_0 w_t + H_{+1} E_t{w_{t+1}} = \psi_{\epsilon} \epsilon_t + \psi_c
$$

and produce solutions of the form

$$
  w_t=R w_{t-1}+ \phi \psi_\epsilon \epsilon_t + (I-F)^{-1}\phi\psi_c
$$

where
$$  \phi=(H_0+H_{+1}R)^{-1}\,\, \text{and} \,\, F=-\phi H_{+1}
$$

In the AM setup, the $w_{t-1} $ are given as data and cannot be affected by the $\epsilon_t$ or
the  current or future values of $w_{t+i}\in, \{w_t,w_{t+1},\ldots\}$.
  In KW,  the ``predetermined'' variables at time t depend only on the already realized shocks.  So the
$\epsilon_t$ only influences future $k_{t+i} \in \{k_{t+1},k_{t+2},\ldots\}$  Consequently, to square AM and KW, the dating of some of the model variables
and their time subscripts  will differ.

Set

$$
 w_t=
  \begin{bmatrix}
\Lambda_{t}\\    k_{t+1}\\x_t\\  \delta_t
  \end{bmatrix}
$$

We will need to partition the KW $A=
\begin{bmatrix}
  A_1&A_2
\end{bmatrix}
$ and $B=\begin{bmatrix}
  B_1&B_2
\end{bmatrix}$
 matrices to accomodate  these differences.

Construct the AMA $H$ as

$$
H_{-1}=\begin{bmatrix}
  0&-B_1&0&0  \\
  0&0&0&-Q\\
  0&0&0&-\rho
\end{bmatrix}\\
H_{0}=\begin{bmatrix}
  -B_2&A_1&-C_0&0  \\
  0&0&I&0\\
  0&0&0&I
\end{bmatrix}\\
H_{+1}=\begin{bmatrix}
  A_2&0&-C_1&0  \\
  0&0&0&0\\
  0&0&0&0
\end{bmatrix}$$ and
$$\psi_c=0 \,\, \psi_\epsilon=
\begin{bmatrix}
0\\  0\\G\\0
\end{bmatrix}
$$

Then, we will find that
$$
  R=
  \begin{bmatrix}
    0&M_{\Lambda,k}&0&M_{\Lambda,\delta}\\
    0&M_{k,k}&0&M_{k,\delta}\\
    0&0&0&M_{x,\delta}\\
    0&0&0&M_{\delta,\delta}\\
  \end{bmatrix}
$$

So that for KW
$$
  M=
  \begin{bmatrix}
    M_{k,k}&M_{k,\delta}\\
    0&M_{\delta,\delta}\\
  \end{bmatrix}
$$

$$
\phi \psi_\epsilon=
\begin{bmatrix}
\bar{\bar{G}}\\
  \bar{G}
\end{bmatrix}
$$

The KW solution for $\Lambda_t$, $\Pi_\Lambda$ can be found by expressing the first few rows of R as a linear combination of the next few rows of R.

$$
  \begin{bmatrix}
      M_{\Lambda,k}&M_{\Lambda,\delta}
  \end{bmatrix}= \Pi_\Lambda
\begin{bmatrix}
      M_{k,k}&M_{k,\delta}\\
\end{bmatrix}
$$

See also
Gary Anderson. Solving linear rational expections models: A horse race. Computational Economics, 31:95–113, March 2008.


The link between US debt of $250 Trillion dollars and "Promise to pay" on US Dollar currency

I looked at that video only briefly. Trying to interpret what that interviewee said would be an opinion-based answer, which this site wants to avoid. I will instead try to answer the question without referring to his comments. In my opinion, I have serious doubts that the economist in the video could offer a convincing explanation of what he said.

Note: it is incorrect to refer to future entitlement spending as a “debt”. At most, it is a “contingent liability.” A debt is a financial claim, held by an owner. An inability to distinguish between debts and contingent liabilities is a signal that he is an ideologue, and not a reliable source. This also means that the 200+ trillion debt is a worthless figure, as it would need to be held in context against future tax revenues as a capitalised asset - which will also run into the hundreds of trillions of dollars.

(1) The linkage between “the promise to pay” on currencies large existing debts and contingent liabilities is presumably the threat of inflation or hyperinflation. In the case of a hyperinflation, a currency is often withdrawn from circulation, and so it cannot be used to pay for things. However, I cannot name a serious body of economists that treats hyperinflation in the US as a possibility under current policy settings.

(2) A promise to repay on currency is just a promise that it will be redeemed at par value by the US government. This means that banks can convert currency to deposits at the Fed (“reserves”) at par. Although this is just converting one US government liability for another, it does distinguish currency from other tokens that might circulate (foreign currency, gift cards, etc.). For an individual, since they do not bank at the Federal Reserve, it gives them a right to exchange currency at par for a bank deposit - or to settle tax payments.

From the perspective of Chartalism (or neo-Chartalism), the “promise to repay” derives its importance from the fact that currency will be accepted at par to settle future taxes. The prospect of the need to settle taxes is what gives currency its value.

Although settlement at par seems natural, it was not always the case. Pre-Federal Reserve, cheques (check in American spelling) drawn on other banks with weaker reputations would not settle at par. That is, you might need to write a cheque of \$110 to settle a debt of \$100 at another bank. The “promise to repay” (and payment system regulations) mean that these “internal exchange rates” disappear.


How does foreign trade work in a planned economy?

The trade in itself was not that much different per se but it was restricted and overseen by the government.
For example,  USSR was net oil exporter and most of its oil was sold for US dollars at international market prices and then USSR used dollars to buy imports that government officials decided to import. For example, you can see overview of trade in USSR in Zoeter (1983). This is really not that much different from Russia's state companies selling oil and gas at international markets now.
One peculiarity of trade in many centrally planned economies was that it was not rare to trade in kind, as due to general economic mismanagement most centrally planed economies had very weak currencies so they either had to trade paying with hard currency (e.g. dollars), or trade was conducted in kind as the famous Pepsi -  Stolichnaya vodka/warship in kind trade (yes as bizarre as it might sound that is not typo they exchanged some warships for Pepsi as well).
From economic point of view there is no advantage to government completely overseeing/controlling trade to this extent (I don't think there are many serious economists who would even entertain that as a serious option). A disadvantage of such arrangement is that government faces an informational asymmetry when it comes to the preferences and demand of its population, so when government tries to guess how much goods should it import it will likely get it often wrong resulting in shortages. In fact foreign consumer goods were as a result not even widely avaiable to ordinary citizens (see this Investopedia explainer). As a result foreign imports were often avaiable primarily through special shops (in Russia 'Beryozka') where people had to pay in hard currency effectively excluding ordinary non-privileged people, or through black market.   However, it is hard to disentangle how much this was due to mismanaging trade and how much due to general economic mismanagement under central planning.


What is the proof for why border adjustments are exactly offset in theory by currency appreciation?

Here is a reference by Feldstein, Krugman that is often quoted when discussing the FX neutrality of border tax adjustments:
http://www.nber.org/chapters/c7211.pdf

Another useful reference that takes a somewhat skeptical view and lists more references is
http://voxeu.org/article/exchange-rate-implications-border-tax-adjustment-neutrality. All models developed or mentioned in these references are non-monetary models, i.e. changes in the exchange rates

I have not digested these references completely yet. However, my understanding of the basic argument is as follows:

In present value terms exports must always equals imports. Changes to taxation may disturb this equilibrium which may then lead to an adjustment of the exchange rate. However, in a world with inelastic labor supply as assumed by Feldstein/Krugman a VAT with border tax adjustments does not have any distortive effects as it changes neither the consumption/savings decision, nor the present value of imports or exports it will not have any real effects.

In principle already from this result one can conclude that border tax adjustments that limit any tax impact on domestic economic agents cannot change any consumption or production decisions as long as the economy is small, labor supply is inelastic and tax revenue is redistributed in a lump sump fashion.

However, a more rigorous argument is as follows:

With inelastic labor supply, households have to decide on their consumption and savings. The first order condition resulting from a standard intertemporal optimization model is (following Feldstein, Krugman)

${U_C}_1/(\delta{U_C}_2)=(1+r)p_1/p_2$

This condition shows that the consumption/savings decision changes only when relative prices or interest rates change.

Consider now various types of firms underlying Auerbach's border tax adjustments, where we again model the firm as in Feldstein, Krugman:

Exporting firms

$maxV=p_X X^1+(1-\tau)(p_D^1 D^1-w^1l^1-p_D^1k^2)+[p_X X^2+(1-\tau)(p_D^2 D^2-w^2l^2)]/(1+r)$

subject to

$Q^1(l^1)=X^1+D^1$ and $Q^2(l^2;k^2)=X^2+D^2$

Where D denotes domestic and X denotes export sales.
Labor market equilibrium is given by

$\bar L=l^1$  and  $\bar L =l^2$

Equilibrium conditions are in the first period

$p_D^1=p_X/(1-\tau)$; $w^1=p_X/(1-\tau)Q_l^1(\bar L)$

From this follows that in the first period and increase in $\tau$ leads to a proportional increase in $p_D^1$ and $w$. This means essentially that the exchange rate will adjust to neutralize the impact of $\tau$.

These conclusions can be shown to hold also in period 2, where there is one additional equilibrium condition $1=Q_k^2(\bar L,k^2)$.

Exporting firms importing an intermediate good

$$maxV=p_X X^1+(1-\tau)(p_D^1 D^1-w^1l^1-p_D^1k^2)-p_m^1m^1+\frac{p_X X^2+(1-\tau)(p_D^2 D^2-w^2l^2)-p_m^2m}{1+r}$$
subject to
$Q^1(l^1;m^1)=X^1+D^1$ and $Q^2(l^2;m^2;k^2)=X^2+D^2.$

Compared to 1. there is one additional equilibrium condition in period 1: $p_m^1/p_X=Q_m^1(l^1;m^1)$ and one in period 2 $p_m^2/p_X=Q_m^2(l^2;m^2;k^2)$.  However, also these addtional equilibrium conditions do not change the overall conclusion that changes in $\tau$ will only impact $p_D$ and $w$.

These conclusions also hold in a more general context, for instance, if there are importing firms producing nontradeable goods.


relation between the demand function and the marginal utility function

This is true for quasilinear utility function. Suppose
$$U = d + V(x),
$$
where $d$ is the numeraire. The budget constraint is given by $p_x x + d = I$. The utility is then
$$U = I - p_x x + V(x).$$
Maximizing this, we have the following FOC: $p_x = V'(x)$. That is, for a given $x$, my willingness to pay is $V'(x)$, the marginal utility.


Is there some definition about risk sharing?

Captive insurance is an insurance for purpose of insuring the owners of the said insurance. It’s a kind of self-insurance where multiple agents pool their capital together to insure themselves. So the term in that sentence refers to the pool of resources for insurance.

It’s called captive because the modern concept of this kind of insurance was first applied to a company that wanted to create insurance for its subsidiary mines that were referred to as captives and somehow the term catch on.


Does Okun's law work at all?

It was an empirical observation of the relationship between unemployment and an estimate of the "output gap" (the relative difference between actual GDP and potential GDP)

To the extent that the relative difference between actual GDP and potential GDP may be the same whether calculated using nominal GDP or real GDP, then the relationship should hold either in both cases or in neither. Otherwise the argument becomes one of saying that the output gap is affected by inflation


How do I begin to approach this dynamic discrete choice model?

$b_t$ is the decision variable, and the value function is formulated as a function of the state variable of the problem. In your case, the role of the state variable is played by the $p$ parameter, which may change in each period (updated). So your Bellman equation is

$$V(p_t) = \max_{b_t} \{u(b_t) + \beta E [V(p_{t+1})\mid t]\}$$

and I guess now you see where the updating of $p$ enters the picture.

On "Value function iteration", or "iteration on the Bellman equation", the book "Recursive Macroeconomic Theory" of Ljungqvist & Sargent contains clear and practical guidance.


There’s a large literature on this in both CS and Economics. For economics, a good starting point would be Vijay krishna’s Auction Theory book (2009 edition). It has a whole section on multi-object auctions.
For CS, check out some of the papers by Jason Hartline. He may have lecture notes as well. Naom Nisan and others have a book called algorithmic game theory where some chapters (if I remember correctly) deal with multi-object auctions.


Have social accounting matrices and input output models fallen out of fashion? if so, why?

National Statistical Institutes do still compile IO tables (see http://ec.europa.eu/eurostat/web/esa-supply-use-input-tables for EU versions, although these are 5-yearly as well).  They're generally more interested in producing the Supply and Use tables (which are then transformed into input-output tables) due to their usefulness in balancing the 3 measures of GDP.

There are also attempts by groups such as WIOD to produce world input-output databases.

Part of the reason for the long delay in producing IO tables following the reference year is simply the very large quantity of data required to compile them, plus the time taken to investigate and resolve issues highlighted by the confrontation of the many various data sources.

Social accounting matrices are also still produced, and frequently used to calibrate things like CGE models rather than being used directly via SAM multipliers.

Most of the work in input-output modeling I see these days relates to environmental modeling, but this is by no means exclusive. The International Input-Output Association (www.iioa.org) runs a publication called the Journal of Economic Systems Research which can point you at some current work.

The UN are also working on a handbook foot IO and extensions due some time this year, which aims to provide a range of options for countries with smaller or less well established statistical institutions to construct tables, as well as more established ones.


Shareholders equity does not include intangible assets per se. This is mainly bookkeeping not economics but generally in most double-entry bookkeeping systems all accounts will have two sides one is asset side and second is liability and equity. An aggregate of all these entries will end up in the balance sheet which will again have asset side and liabilities and stockholders equity side. The asset side of balance sheet tells you what firm owns and and liabilities and stockholders equity side where the sources to obtain those assets came from. An example of balance sheet would be:
Assets                              Liabilities & Stockholders equity
Cash              $100                     Stockholders Equity      $50
Intangible assets $100                     Bank Loan               $150

The formula for book value per common share is according to investopedia given by:
$$BV = \frac{\text{total equity} - \text{preferred equity}}{\text{Total Shares Outstanding}}$$
Note that neither the total equity or the book value in itself gives you any information what the amount of intangible assets is or they are included in calculation. The total equity can be calculated as Assets minus Liabilities but again you cant generally tell from total assets or liabilities what the intangible assets are.  In this case the total equity is $\\\$50$ but amount of intangible assets is $\\\$100$. The book value also does not give you any information or contain the intangible assets either.  To see what they are you would have to go back to balance sheet or some ratio that includes them.


Ok to normalize GDP change by population change?

I guess there might be some purpose to this but generally this is very different than $g-r$ and probably a bad idea.

It isn't the same people growing the population and growing GDP.  $\Delta GDP$  is the $\Delta$ (output of the existing population) + $\Delta$ (output of the new population). When you normalize by new population, you get

$\Delta$ (output per capita of the new population)
plus a nuisance term of

$\Delta$ (output of the existing population) / $\Delta$ population).

Unless incomes drive positive population changes this term is not going to behave much at all like $g-r$.


What about
\begin{array}{|c|c|c|}
\hline
&C&R\\\hline
M&0,0&0,0\\\hline
D&0,0&1,1\\\hline
\end{array}
?

This game has two pure equilibria, $(M,C)$ and $(D,R)$ but no mixed equilibra.

You are probably referring to Wilson's Oddness Theorem, which guarantess your claim for generic games. See
Oddness of equilibrium points.


Why is elasticity not constant on a straight line graph?

Theoretically, the responsiveness of quantity demanded is different at different prices. Take the equation for the elasticity of demand:
$$\epsilon_{D}=\frac{\Delta{Q}}{\Delta{P}}\frac{P}{Q}$$
The equation itself is non-constant as it depends on P and Q. Lets assume a linear demand curve with the simple form: $P(Q)=6-Q$. If we take an equal change in price, say $\Delta{P}=1$, and since the demand curve is linear in this case, we see it has a slope = $-1$ and thus the $\Delta{Q}=1$. If we have two situations where $P=5$ and $P=4$, then, to keep in accordance with our slope, lets say $Q=1$ and $Q=2$, respectively.
The two elasticities of demand for these two prices would be $\epsilon_{D}=5$ and $\epsilon_{D}=2$ for the higher and lower price respectively.

In short, the elasticity of demand is a function of the responsiveness of quantity demanded to changes in prices that depends on the current price/quantity combination.
At a higher point up the demand curve, the elasticity is higher, or demand more elastic, than at lower points.


Search and Matching model with 2 agents?

I am not sure if this is exactly what you are looking for, but this paper has model that sounds like what you want:
Shimer, Robert, 2006. “On-the-job search and strategic bargaining, ”European EconomicReview50, 811—830.
Have a look at this one too:
Shimer, R. (2007). Mismatch. American Economic Review, 97(4), 1074-1101.


What effect does changing the retirement age have on employment?

"(...) increasing the retirement age (...) possibly takes jobs from
  young people and stifles their career."

Since this is often surfaces as "the conventional wisdom" (not necessarily in economics circles), let's present a simple argument, that shows that there are bound to be distinct effects for people already in the labor force, compared to the people that have not yet started their working period in life.

To bring in the surface the core effect, assume a constant population, and a constant demand for workers (that does not mean necessarily constant output of course, since productivity and efficiency may rise).

Assume that in this society, people start working as they enter their 21st year, and retire when the enter their 61th year (40 years of work). An old person should retire today -but he doesn't: he will stay on for another year (say, because the government raised the mandatory retirement age).

Then, the young person that was about to enter the labor force in order to replace him, will have to stay out of it for one more year. He will now enter the labor force when entering his 22nd year and he will exit when he will enter his 62nd year. But this means that this person will work again for 40 years: Raising the retirement age, does not mean raising the numbers of years worked for the new/future workers. What happens is that their working life shifts as regards its beginning and end on their personal age-time, but not as regards its lifetime duration.

Does this "stifle their careers"? One could reasonably argue that starting one year later gives you one more year to accumulate human capital through education and training -and so when you enter the labor force, you will enter it better prepared than if the change in retirement age had not happened.

Casually observing western societies, this is what appears to be happening: people tend to acquire more and more education, and so also start working later in life. At the individual (or mass media) level, this may be negatively experienced/pictured as "high unemployment for young people" and "difficulty to get a job that makes the young people accumulating more education in their attempt to find a job". I presented above an alternative perspective on the matter.

But as regards those that were already workers when the change happened, and so have entered on the labor force in the beginning of their 21st year, will now work in total 41 years. If we envision a gradual rise in professional ranks and income, the change in retirement age will mean that they may have to stay one more year "where they were", and in that sense, increasing the retirement age may appear to "stifle their careers". But there is also Arrow's "Learning by doing": human capital increase due to on-the-job experience (a verified phenomenon even in a perfectly static environments). So this one more year will not be a total waste in terms of increases in income earning capacity -and it may pay off more for the younger workers, than for the older ones (since the former will gain it earlier in their working life). So the "career-stifling" argument gets a little weaker, even for the class of workers-already.

Obviously this is a very abstract approach to the matter... for example, what happens to societies where population is rising? How many people can a society afford to invest in -create human capital for young non-workers- at any given time?
...but I believe it reveals one aspect of the matter that we tend to overlook.


Optimisation and Theory of Comparative Advantage

Without more information on producers (such as their production functions) and consumers (their utility functions), we cannot compute the optimal levels of production and trade

The point of this simple numerical example is merely to illustrate the key and surprising insight behind the principle of comparative advantage: Even if the US is better than China at producing everything, both countries (including the US) can still benefit from trade.

The numbers were somewhat arbitrarily chosen (this is a point sharp students are often confused by and the teacher should have mentioned this). They could be tweaked and still successfully illustrate the principle of comparative advantage.


Why labor-augmenting technological progress increases employment and wages?

I found this excellent Economic Letter from the Federal Reserve Bank of San Francisco that speaks to this observation. It is from Carl R. Walsh (july 16,2004) (https://www.frbsf.org/economic-research/publications/economic-letter/2004/july/the-productivity-and-jobs-connection-the-long-and-the-short-run-of-it/  )

In this letter he notes that, newspaper articles blame strong productivity growth for a “jobless recovery,” as economic output grows yet employment does not.

While it appears that faster productivity growth allows businesses to increase production without increasing employment.

His answer lays in the difference between macroeconoimcs and microeconomics perspective on productivity and between the short-run and long-run effects of changes in productivity.

From a micro perspective, productivity growth and new technological innovations are constantly leading to structural changes in the economy, causing one industry to expand in terms of both production and employment, while other industries contract.   However, jobs in other industries also expand.

For example, the manufacturing sector productivity increase but there share of total employment declined while employment and output in areas such as the computer industry have grown rapidly.

These shifts in the economy cause jobs to disappear in some sectors while jobs are created in others.

In the short-run, if demand of products rise this will lead to expanding product. And if labor productivity is unchanged, then typically they need to hire more workers to do this.
But, if overall demand in the economy has not expanded, then an increase in labor productivity could lead to a fall in employment in the short run.

In the long run, there is an emphasis that an increase in labor productivity increases potential GDP. It does so directly by allowing more output to be produced with the same level of employment, but it also increases employment because it decreases the cost of labor to firms and promotes the creation of new industries.

The cost of labor is noted as not just wages and benefits but also cost relative to the output the workers are able to produce.

A rise in labor productivity lowers the cost of labor at a given level of wages and benefits. If higher productivity makes labor less costly, firms will find it profitable to expand employment.


Why is depreciation added to the GDP with the income approach?

In national income and products accounting (NIPA), the word gross (usually*) means including depreciation, while net means excluding depreciation. We have:

Gross domestic product (GDP) = Employee compensation + Taxes less subsidies on businesses + Gross operating surplus on businesses
Net operating surplus on businesses = Gross operating surplus on businesses - Depreciation

And hence:

Gross domestic product (GDP) = Employee compensation + Taxes less subsidies on businesses + Net operating surplus on businesses + Depreciation

You are exactly correct when you wonder why we include depreciation in National Income. Conceptually, what we are more interested in is Net domestic product (NDP), which excludes depreciation:

Net domestic product (NDP) = GDP - Depreciation = Employee compensation + Taxes less subsidies on businesses + Net operating surplus on businesses

However, depreciation is difficult to estimate. And so often, we simply look at GDP. This is why GDP is the more frequently quoted, even though conceptually we'd prefer to use NDP.

Example. In 2018, a country had 100 workers (labor) using 100 tractors (capital) to produce agricultural products. We find that in 2018, the country produced \$1M worth of agricultural products and paid out \$800,000 to the 100 workers.

Hence, in our NIPA accounts, we have:

Employee compensation = \$800,000.
So, Gross operating surplus on businesses = Total income - Employee compensation = \$1M - \$800,000 = \$200,000.

This \$200,000 goes to the owners of capital, i.e. the 100 tractors. The key is to understand that "gross operating surplus" is the residual (the leftover term) after any compensation paid out to employees and any taxes (less subsidies) paid. (For simplicity we assume there are no taxes/subsidies.)

And now we have:

GDP = Employee compensation + Gross operating surplus on businesses = \$800,000 + \$200,000 = \$1M.

We now have to estimate depreciation (i.e. wear and tear on the tractors), which in the real world is difficult, but in our made-up example here we'll just say is \$50,000. Then:

Net operating surplus = Gross operating surplus - Depreciation = \$200,000 - \$50,000 = \$150,000.
NDP = Employee compensation + Net operating surplus = GDP - Depreciation = \$950,000.

*Confusingly, there is also another sense in which the words gross and net are used in NIPA (and also more commonly in daily life) — example: Net lending = Lending - Borrowing. See 2.72 in the UN System of National Accounts (2008).


How to derive the demand function of $U(x_1,x_2)=x_1-x_2$

Some economics problems are deisgned to help us practice the basic tools (e.g., optimisation) of the trade. Others are designed to force us to think about the mechanics of a problem to help us understand the underlying economics better.

This problem falls into the latter category, so I recommend you give up on trying to usual calculus (which is premised on finding some sort of indifference curve tangent) and play around with the problem to figure out why this approach isn't working.

A consumer with $u=x_1-x_2$ is indifferent between the bundles $(4,3)$ and $(3,2)$. We can plot these and other indifference budles to draw an indifference curve:

Likewise, the consumer is indifferent between $(4,4)$ and $(5,5)$ and we can, in this fashion, get an indifference map:

From inspection of the utility function we know that the consumer is better off with more $x_1$ and less $x_2$, so utility increases in the direction indicated.

Now we can pick an endowment vector (e.g,. $(2,4)$) and a relative price (e.g. $-p_1/p_2=-1$) and plot the price line:

The consumer can trade from their endowment to any point along this line. By inspection, you should be able to figure out what the demand vector $\mathbf{x}=(x_1,x_2)$ will be.

Addendum: once you figure out what is going on here, your condition $$\frac{\partial u}{\partial x_1}=1+p_1/p_2$$
should make a bit more sense to you.


I don't understand your problem: negative marginals returns are a feasible situation. The fact that a rational optimizing firm won't let itself reach such a stage is another matter.


Terms of trade is a measure of relative prices, and prices only. There are no quantities involved. As a recent review paper puts it:

"with exceptions that I will note below, most trade economists have defined a country’s terms of trade as the price of its exports divided by the price of
its imports, $p_X/p_M$" (p.2).

Therefore, by definition, in a two country model, they are reciprocal. In a model with more countries, there can be differences, as some countries could import or export different goods.

Actually, I don't know which Wikipedia you are looking at, but in the English version, the first line states:

"Terms of trade (TOT) refers to the relative price of exports in terms of imports and is defined as the ratio of export prices to import prices."

For future reference, Wikipedia is an unreliable source. If you are in an educational institution, it is likely you have access to the New Palgrave Dictionary of Economics (the terms of trade article is here). If not, you can google "terms of trade pdf" and hopefully get the slides of a generous professor who is sharing his/her knowledge with the world.


Why do we need both the UMP and the EMP?

The solutions of UMP and EMP only coincide if certain conditions such as monotonicity of preferences is met. Hence they are different problems.

Duality is important because the Lagrangian multipliers frequently have an economic meaning. In the UMP the Lagrangian multiplier of the budget constraint is equal to the marginal utility of money.
In the EMP the Lagrangian multiplier of the budget constraint is equal to the marginal cost of utility.

In the EMP you can easily replace the consumer with a corporation deciding what inputs to use to create an output. In this case the Lagrangian multiplier would show the marginal cost of output.

In a similar way you could also alter the UMP to a problem where a corporation has to decide between some kinds of outputs and is trying to allocate resources in a way that will maximize profits. (E.g.: There are $m$ amount of man-hours and several kind of projects that can be taken on.)


The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?

The single most important feature that we care about a utility function is that if a bundle $X$ is preferred to a bundle $Y$ then $U(X)>U(Y)$ (with weak inequalities whenever appropriate). This is how we define a utility function. You can immediately see that if a function has that property, any monotonic increasing transformation of it will also have that property. However, if the transformation is decreasing, call it $V(X)=f(U(X))$ with $f'<0$, then $V(X)<V(Y)$ i.e. now $X$ gives less utility. Therefore, we see that $V$ cannot represent the same preferences.

What about the MRS invariance property? Well, once you understand that utility representations are not unique, (or unique up to monotonic increasing transformations), you can ask what is constant across all functions that represent the same preferences? The answer is: they have the same MRS.

You can clearly see that it is necessary for two functions to have the same MRS if they represent the same preferences, but it is not sufficient. As you pointed out, decreasing transformations also have the same MRS. Note however, that the interpretations of the MRS slightly changes when  considering monotonic decreasing transformations.


Adjustment to equilibrium in Cournot model with 2 firms

The trick is to draw the whole reaction function—including the part that coincides with the axis. Hopefully these figures make it clear:


There is a demand for debt and a supply of debt. The demand for debt is determined by the extent of profitable investment opportunities that are around. The supply of debt is essentially people's saving multiplied by a factor depending on how much risk financial institutions are willing to carry.

Suppose we live in a world where there are no clearly profitable investment opportunities, but many projects which return the same money you put in on average. So nobody would take on debt at positive interest rates. However, even a a zero interest rate, people would still want to save money, for example to insure against bad times, or for retirement. So there's an equilibrium in which the interest rate is zero, but people still borrow and lend.

Another possibility is that people just expect deflation, so a zero nominal interest rate still goes along with positive real returns on your investment.

It is not clear to me what you mean by sustainable, but clearly such situations can arise, albeit rarely. Japan has nominal rates close to zero for a long time now.


What will happen if the government not print money?

It causes unemployment to increase. Policy makers usually have to make a choice between inflation and unemployment. More money means inflation and less money means less investment, less job and more unemployment. If money becomes valuable then people will prefer to keep it in bank instead of creating jobs.


Key Performance Indicators are important elements of a company that can be used to asess a firm's current core competencies and business model. They are based on of key result areas. Here is my interpretation of the three indicators and some examples (There are many more):

Competency: A firm's business strategy and it's competitive advantage

Example: Retention, Value (Quality/Price), R&D Investment

Profitability: A overall indicator of market performance

Example: ROI, Gross Margin, EBITDA

Effectiveness: The implementation of a firm's strategy

Example: Market Share, Sales, New Market Segments

You may notice that the examples can be related to other indicators, because the indicators are a broad overview of the firm.


Not all components of the economy vary with the same intensity. As you can see in the figure below, consumption is about as volatile as output (GDP), but investment is much, much more volatile. You can also see the grey bars, which indicate NBER recession dates for the US, line up nicely with these dips in investment.


Why not just print money to combat deflation?

That's essentially what Quantitative Easing was supposed to be.

However, we've known for about 80 years that it doesn't work in times like these, and like the 1930s, when demand is flat because of a massive debt overhang. The demand for money just isn't there.

In these times, trying to simply expand the money supply is like pushing on a piece of string.

Japan's been battling with non-responsive flat demand for many years now (though driven largely by demographics). Monetary stimulus has failed. Fiscal stimulus has at least steadied the ship.


What is the average economic value of a human life?

It depends on the context, of course, but most often in policy analysis "the value of a life" has nothing (directly) to do with output, etc, but instead means the maximum amount that people would want the government to spend in order to save a randomly chosen life.

So in a country of 300,000,000, the question is:  What, to you, is the monetary equivalent of a 1/300,000,000 chance of death?  Because 1/300,000,000 is a very small number, we don't have to worry terribly much about willingness-to-pay versus willingness-to-accept.  (Theory tells us that for small changes, the two willingnesses are effectively equal.)

Returning to the question:  How much would you be willing to pay to avoid a 1/300,000,000 chance of death?  Now multiply that value by 300,000,000.  That's the value of one life, and arguably the amount we'd want our government to spend to preserve a randomly chosen life.

Obviously there are problems with heterogeneity (you and I might not answer the question identically).  But as a general rough rule, the estimates tend to come in somewhere under $10 million.


Application of Intermediate Value Theorem for General Equilibrium

A stronger hint: Write $p_2=1-p_1$, so that $Z(p)=Z(p_1,1-p_1)$. Use the conditions $Z_1(0,1)>0,Z_1(1,0)<0$ etc. and the intermediate value theorem to argue that there exists a $p_1^*\in(0,1)$ such that $Z_1(p_1^*,1-p_1^*)=0$.


Optimal taxing in case of negative externalities

I would have thought the social planner might want the value of $x_i$ to be such that $f'(x_i)=k(n-1)$ at which point the marginal gain to each individual offsets the marginal total cost to the others

while individual driver $i$ might want the value of $x_i$ to be such that $f'(x_i)=t$ at which point the marginal gain offsets the marginal tax

To me that suggests the tax rate should be $t=k(n-1)$


Is Marx considered an important classical economist?

Among economic historians, Marx is often considered the most important economist of the 19th century.  His attempts to provide a systematic explanation of the functioning of capitalism  was on a far grander scale than anyone who preceded him, and in that sense, he reset the bar very high for what would henceforth be considered comprehensive economic theory, even if parts of it were fundamentally flawed. Refer to Mark Blaug's classic text Economic Theory in Retrospect (Chapter 7) for a nice attempt to parse out Marx's economic theory from the rest of his ideas.  This is a standard and authoritative text in the history of economic theory.

Blaug says of Marx "Marx the economist is alive and relevant today in a way none of the writers we have thus far considered." Those considered were Smith, Malthus, Say, Ricardo, & Mill.

In terms of his understanding of the dynamic long run evolution of economic systems, the nature of technical change, business cycles, and volume of employment, his insights were often brilliant, transformative and unparalleled in any writers that preceded him, and remain deeply influential.  (I'm paraphrasing Blaug here)

[It's worth mentioning that Blaug did his Ph.D at Columbian under George Stigler, who was a key leader in the Chicago School. So Blaug was hardly a lightweight, nor a "pinko" ;) ]


Example production function with increasing returns to scale but diminishing marginal product

Decreasing marginal returns to a factor means that keeping the other factors fixed, the marginal output generated by this factor is decreasing. When looking at returns to scale, we change all outputs. Increasing a factor with decreasing marginal returns can have an indirect effect in increasing the marginal productivity of other factors. If we increase all factors at the same time, the indirect effects may outweigh the direct effect. The production function $F:\mathbb{R}_+^2$ given by
$$F(x,y)=(x+1)^{2/3}(y+1)^{2/3}$$
has decreasing marginal factor productivities everywhere but not decreasing returns to scale (it doesn't have increasing returns to scale either).


Why is fuel more expensive on the high way in Europe?

Because there is no indication in your question that you are a student or practitioner of economics, I am writing an answer for a lay audience. Let me know if you would like more technical detail.

A fairly general prediction from economic models of competition between firms is that the price that maximises their profit is higher the less sensitive is demand to that price (this sensitivity is measured by the "price-elasticity of demand"). Intuitively:

if demand is very sensitive to price then a cut in price will cause demand to increase a lot. The firm receives less for each unit it sells, but sells very many more units and is better-off overall.
if demand is not sensitive to price then the firm can increase price without causing its sales to fall very much and will therefore wish to do so.

There are lots of things that affect the sensitivity of demand. One example, as DornerA mentioned in a comment, is geographic location. If the nearest competing seller is very far away or inconveniently located then buyers will be reluctant to shop elsewhere, meaning their demand is likely to be less sensitive to price increases at locations that are geographically isolated in some way. This seems like a reasonable way to think of highway fuel stations.

A little thought should convince you that the same principle (that prices are higher when demand is less sensitive) holds for other cases where demand is price-insensitive too. For example:

why is food so expensive on trains/airplanes?
why are branded goods more expensive than generic ones?
why are the photographs from theme park rides so expensive?
etc.

Edit: A comment by AndrejaKo reminds me to add that the other very common reason why firms might increase their price is that they have higher costs. However, economics theory predicts that only unit costs (i.e. those that increase when you sell more units) matter for optimal pricing. Thus, for example, the fact that land near a highway is expensive would not seem like a very convincing explanation for high fuel prices (because the land costs the same regardless of how many units of fuel are supplied). But if, for some reason, supplying each litre of fuel was more expensive at highway fuel stations (because, for example, transporting the fuel there is more expensive) then we would indeed have an explanation for higher highway fuel prices.


What happens if the Central Bank does not absorb a BOP deficit/surplus?

If OA is a part of FA. The second equation follows from the first:
$CA + KA = FA \wedge OA \neq 0 \Rightarrow FA - OA = CA +KA -OA \neq CA +KA$.

Response to your EDIT:

$CA + KA \neq \frac{FA}{OA}$ follows from the first equation as well:

$CA + KA = FA \wedge OA \neq 1 \Rightarrow  \frac{FA}{OA}= \frac{CA+KA}{OA} \neq CA +KA$ in general. (I've checked your LateX expression it really says: "(FA \backslash OA)"). I don't understand why you would think that BOP has anything to do with division. For example Feenstra and Taylor say in their textbook:

$OSB=-Reserve FA=CA+KA+Nonreserve FA$, where OSB is your OA and their version of the BOP identity is $CA + KA + FA = 0$, with your FA as their -FA.


It seems you are confounding growth and growth rate.  The difference between the blue and purple curves, i. e. growth,  may be small when capital is small, but the ratio of the difference to  capital, i. e. the growth rate will indeed be larger the closer capital is to zero.


Is the economic behaviour of a competitive market as a whole the same as a monopoly market?

Intuitively, I would say no. The main power that a monopoly has is that the firm can change prices at will, whereas a competitive firm has no power over price.

The competitive market demand curve slopes downward simply because of the law of demand. The typical procedure is to take the price determined by the aggregated demand curve as given and set the individual firm's demand curve equal to that. This seems peculiar (to assume to know the market demand before you know the firm's demand), but given the assumptions of perfect competition  (particularly perfect knowledge), this does make some sense. If every firm has perfect knowledge of demand for their good, then they inherently know the market demand curve. The reason why this is important is because that is how you derive the price taking behavior of competitive firms.

Now that we have determined why price taking behavior exists, we can discuss whether competitive markets (as a whole) can behave like monopolies. The answer to this is that it would be nearly impossible for an aggregate competitive market to behave like a cartel (monopoly). The reason for this is because of the perfectly elastic nature of their demand. Suppose, somehow every firm agreed to increase their price to the exact same amount at the exact same time. Suppose further one firm had the ability to produce enough of the good to meet demand at a certain price. This firm (or any firms like it) would have a huge incentive to lower their price because by lowering their price (even by an infinitesimally small amount), they would instantly capture all of the demand for the good. If just one firm decides to cheat, then the whole market would end up right back at where they started at the original market price. For this reason, it is nearly impossible for an aggregated perfectly competitive market to act like a cartel.


Such government would be called net lender. However, note that for a country to be net lender it is not required that debt is zero, only that the amount it is lending is higher than its liabilities (debt). A country can chose to have some level of debt even if it is not necessary for it to have it because its revenues are higher than expenses. An example of net lender would be Norway (see here).


This Discussion Paper (PDF) suggests the use of political variables as an instrument for minimum wage.

See abstract (emphasis mine):

  Following the early 1980s apparent consensus, there has been a controversial debate in the literature over the direction of the minimum wage employment effect. Explanations to non-negative effects range from theoretical to empirical identification and data issues. An explanation, however, that has not been sufficiently explored is that a non-negative effect might be an upward biased estimate of a truly negative effect, resulting from the simultaneous determination of the minimum wage and employment. This paper estimates the employment effect of the minimum wage using a number of political variables – not previously used in the literature – as excluded exogenous instruments to control for the endogeneity of the minimum wage variable. The data used is an under-explored Brazilian monthly household survey from 1982 to 2000. Robust results indicate that an increase in the minimum wage has very small adverse effects on employment.

This article (PDF) (Which I like better anyway) uses the differences in the changes on minimum wages wrt to the federal minimum wages. It is worth noting that this paper has a pretty good reference list and useful introduction as well. I'd start here looking for further reading.

See abstract (emphasis mine):

  The magnitude of the impact of minimum wages on employment is a hotly debated topic in policy and academic circles. In this paper, we use cross-state differences in the impact of adjustments in federal minimum wages on effective minimum wages in each state - the maximum of federal and state minimum wages - to reassess this question and explain biases in past research. A rise in the federal minimum wage will have a larger impact on a state's effective minimum wage in states in which federal minimum wages are binding. Using CPS data for 1977-2007, we find notable wage impacts and large corresponding disemployment effects, yet only when we utilize the differential influences of federal minimum wages to instrument for state wage floors. State effective minimum wages are procyclical. Accounting for the endogenous determination of effective minimum wages at the state level turns out to be materially important for drawing accurate inferences about the impact of minimum wages on employment


What if price were a function of buyer's income?

This is just perfect price discrimination. Happens sometimes with niche drugs (where drug makers can know the ability of people to pay). In this scenario, consumer surplus is completely captured by suppliers.


What is the meaning of "labor input" in the context of incentive theories?

Since the authors state that the total labor input is:
$$\int t_i(k)dk$$
the meaning of the total labor input in this case would be that it is the sum of all attention $t_i$ allocated over those tasks $k$.
For example if we would assume that $t_i (k) = k$ then the labor supply across continuum given by $[0,1]$ would be equal to $\frac{1}{2}$ because $\int k dk = \frac{1}{2} k^2 + c$ and when you evaluate it between bounds $[0,1]$ the area would sum to $\frac{1}{2}$ (note that geometric interpretation of an integral is an area under the curve that you are integrating).
You can graph it by making assumption about the function, for example following my assumption of $t_i(k)=k$ the labor supply can be plotted (using tikz in LaTex) as:

Where the total labor supply is the area under the curve bounded by $[0,1]$. Of course you might want to impose some less simplistic assumption on the function $t_i (k)$.


A hypothetical question on neutrality of money

I think I figured out the answer using Excel. I assumed that there was a 100% inflation in goods market so that the prices of two goods doubled, while keeping the nominal wage fixed. The result was that an individual producer would still be making positive inflation adjusted profits. So I assume that in the scenario I described above, real GDP would still increase even though relative prices in goods market stays the same because that costs adjusted for inflation fall. When I also doubled the nominal wage so that the equilibrium in labor market is back to its initial level, profits were down back to the initial levels.

Here are the screenshots from Excel

A hypothetical firm and its cost curve.

Real price and real cost of the firm after a 100% inflation in the goods market.

Inflation adjusted profits of the firm after doubling the nominal wage from 120 dollars to 140 dollars. Profit levels are returned to their initial amounts and I presume that at this point firms would go back to produce the initial amount of real GDP.

I think what I got incorrect was that I was trying to compute real wages by dividing it to a ratio of nominal prices to CPI using the formula (W)/(P/P*) where P* representing CPI level. The correct usage should be (W/P*)/(P/P*) so that an equal increase in price of an individual good and CPI while holding the nominal wage fixed should result in a lower real wage, inducing higher profits than the initial level of prices and costs thus resulting in more labor demanded by the firm producing good A.

When I doubled the nominal wage and used the (W/P*)/(P/P*) formula, real wage went back to the initial level so that there was no profit opportunity so the firms reduced the extra real GDP supplied and therefore went back to their original levels of production.


Rearranging the steady state equation
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
we get
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\frac{a+1}{\sigma}.
$$
As $\alpha \in [0,1]$, the left hand side of the equation is increasing in $\overline{p}$ and the right hand side is decreasing. At least one of these is strictly monotonic because $\alpha$ cannot be $0$ and $1$ at the same time. Hence at most one solution is possible.


How is sending money out of a country draining its economy?

Whatever money earned in a country that is spent outside that country is taking away economic activity from that country, GDP can no longer be increased with that money.

Sending money to a country with a different currency usually implies that currency is bought. That will increase the value of the bought currency and decrease the value of the currency sold. A lower US dollar will make it easier for US exporters to sell their products abroad, which will lead to foreign importers selling their foreign currency in return for US dollars. That is how the money can come back at some point.

It can also come back if foreigners invest in the US. If the US has a lot of money from remittances flowing out and doesn't export as much as it imports, which has been the case for a long time, foreigners holding US stocks and bonds will be the result. You could argue this could lead to a dangerous dependency at some point, e.g., if you look at how much the US owes to the Chinese. Mexico in comparison is too small an economy to be such a threat and I agree with the comment above that losing a significant number of Mexican workers would be much more problematic to the US economy.

You should also take into account that the positive development effects of remittances in the receiving country should that country to become a more attractive partner for trade with the sending country, especially given the strong relationship that is built by the migrant community.


Take an individual's maximization problem:

$$\max_{x_i} = U(x_1,...,x_n) \quad \text{s.t.} \ \vec{p} \cdot \vec{x} = w$$

Where one of the goods $j$ is a riskless savings good (savings account, government bond, mattress, whatever).

You can then use the Slutsky decomposition to measure how a change in income will change the consumption of good $j$, which you can then adjust to be relative to income, $w$.

More relevant (though harder) would to set up the problem as a dynamic programming problem, where the consumer wishes to consumption smooth over their lifetime.


Effect of changing MC on monopolists' maximised profits

If MC increases, profits need to go down. Let $\pi (q(c))$ and $\pi' (q(c'))$ denote profits before and after the cost increase. Observe that producing $q(c')$ was feasible when the marginal cost was low and, therefore, it must be the case that

$$ \pi (q(c))=[p(q(c))-c]q(c)\geq[p(q(c'))-c]q(c').$$

In addition, we have that

$$[p(q(c'))-c]q(c')>[p(q(c'))-c']q(c')=\pi' (q(c')),$$

which gives us $\pi (q(c))>\pi' (q(c')).$


Multiple Forgone Job Opportunities and Economic Profit

We don't use all possible opportunity costs. We only use the cost of forgoing the next-best-alternative.


Why is discriminating monopoly better than competitive markets for producers?

You're looking for an answer while it's clear you are very early in your studies and will get to this stuff in due time.

With that said, here is some loose intuition about why a monopoly is better than competition for a single producer:

In a competitive market, every single consumer and every single producer has the same amount of "bargaining clout".  In particular, no single producer can set prices however it wants, because there is always another producer waiting in the wings to offer a better price.  The market-clearing price and quantity can be viewed as an equitable compromise between relatively equal powers (producers and consumers), and is "fair" in that sense.  There are nuances to this that you will learn, but in economic terms producers do not earn "profit" in a competitive market.  They break even.

By contrast, in a monopoly market, the single producer has (nearly) all of the "bargaining clout" because they control the means of production.  Therefore, they have tremendous freedom to set the price or quantity in such a way that they earn a positive "profit" because there is no other supplier who is able to undercut them on price.  This profit for the producer comes at the expense of the benefits consumers would otherwise enjoy in a competitive market (e.g., other things purchased with the price savings).

Again, these are kind of hand-wavy statements, but illustrate the general idea.


Why is the Dollar Index strong after the Fed dropped the Fed Fund Rate?

It is always very hard to precisely identify the causes of a particular index variation. Here, note that the U.S. Dollar Index (DXY) is a measure of the value of the US dollar relative to a basket of foreign currencies. So, the DXY Index goes up when the U.S. dollar gains value when compared to other currencies. Given that the economic slowdown is global, the US dollar could strengthen relative to other currencies


Where can I find County level income data for college graduates?

Try looking at the Current Population Survey. Using a tool like DataFerret you can find education (PEEDUCA) and income (HUFAMINC), along with the county code. That gives you the micro data, finding median income by education level can then be done in your statistics package of choice.


Is there any reliable data on what % of the UK population currently earn less than £17,300 annually?

https://www.gov.uk/government/statistics/percentile-points-from-1-to-99-for-total-income-before-and-after-tax/

The ONS has freely available data on the percentile points of the income distribution as estimated from the Survey of Personal Incomes. The data is collected on a yearly basis, and is currently designated as "Percentile points from 1 to 99 for total income before and after tax".

You should be able to calculate the answer to your question approximately with this information and some linear interpolation. If you really want the exact number, you probably need the actual micro-data, which I'm guessing you will need to submit a request for, unless the ONS anonymized it.

You should check whether these still need to be deflated to get real income.


I really think a great place to start in general is Olivier Blanchard's "Macroeconomics". However, this is an economics undergraduate text and may not be optimal for a mathematician, especially given the structure you expect.

More to your taste may be the following standard texts:

"Introduction to modern economic growth" by Daron Acemoglu.

"Recursive Macroeconomic Theory" by Lars Ljungqvist and Thomas J. Sargent

"Recursive Methods in Economic Dynamics" by Nancy L. Stokey, Robert E. Lucas and Edward C. Prescott

"Foundations of international macroeconomics" by Maurice Obstfeld and Kenneth Rogoff. A bit old by now, but a classic and a good place to start for international economic topics. Alternatively, try:

"Open Economy Macroeconomics" by Martín Uribe and Stephanie Schmitt-Grohé.


Is Malthusian theory of population growth being realized?

The annual growth rate of the global population has been in decline since about 1967 (five decades ago).

The absolute annual growth peaked in 1987 (three decades ago).

Malthusians claim that:

population growth is geometric or higher; and
food production growth is arithmetic or lower.

If either of those do not hold, then Malthusian theory does not hold. And it turns out that neither of them hold:

Population is not growing geometrically (not even arithmetically); and
food production turned out to be capable of greater than arithmetic leaps in growth: food production per capita grew by 45% between 1961 and 2013 (source: UN Food & Agriculture Organization FAOSTAT Food Production Indices for the world, element code 434, domain code QI, Area code 5000, Item code 2051, Agriculture (PIN);)

Some relevant factors: cheap, reliable, ubiquitous contraceptives; education and emancipation, particularly for women; cheap and plentiful crop fertilisers; mechanisation of farming; and selective crop breeding, by taking the old practice of eating the best of each crop and sowing the worst, and doing the exact opposite.

Here are two charts, using the global historic data from the US Census organisation for 1800-1950, and the UN data for 1950-2014.


Can You Help Me Understand this Definition of Economic Efficiency

Suppose the economy is in a state where one person could be made better off without making anyone worse off. Most people would agree this is not a good state to be in, i.e. we can do better.
That means a "pareto improvement" is possible.

So what we would want is to in fact make that person better off (as no one else gets harmed and one person benefits). In trying to improve upon that state, we change it. We keep doing this until it is no longer possible to make anyone better off without making someone worse off. I.e. we stop improving when it's no longer possible to improve.

So a state where pareto improvements are possible is not efficient since we can do better. Hence, states where no pareto improvements are possible (we can't do better) are the best possible ones. The best possible states are called efficient.

Hence, efficiency means a pareto improvement is no longer possible. In other words: it is not possible to make someome better off without making someone else worse off.


How are recessions even possible if its impossible for banks to lose money?

Recessions are defined as fall in output (usually measured as a consecutive two quarters of negative GDP growth) not loss of money. Output (GDP) might very well fall even when firms are on average more profitable and people earn more money.
For example, in the 2020Q2 the median wage in US actually increased according to U.S. Bureau of Labor Statistics, yet US experiences unprecedented $32.9\%$ GDP decline as reported by Bureau of Economic Analysis.
How can that be? Well it is because the lowest paying people are more likely to get unemployed. In the same way, most of the time you will see the least profitable firms stop producing during recessions not the most profitable ones. Hence even when output of economy declines those who remain employed and economic active might actually earn more so the idea that recession is caused by firms or banks loosing/retaining money per se is simply fundamentally wrong. Its not the loss of money, but loss of output that matters.
Generally speaking recession occur when either aggregate demand or aggregate supply (or both) fall, which in turn leads to loss of output. This is because when aggregate demand falls this means people want to consume less goods and services and hence firms will also supply less of them, and when aggregate supply falls the production is affected directly (figure below shows some graphical examples).

Aggregate demand changes for example in response to changing incomes, wealth, inflation expectation or interest rates and aggregate supply in response to technology shocks, changes in firm costs etc. These are just an examples,  you can find more exhaustive discussion in macro textbooks for example Blanchard et al. Macroeconomics: an European Perspective is an excellent resource.
Most recessions do not even originate within the banking system. Rather banking system can be thought of  as an economic highway through which macroeconomic shocks propagate to the rest of the economy. For example, it is generally accepted that Great Recession was caused by collapse of housing bubble not with banks themselves. Rather the banks played role in propagation of the recession. Furthermore,  banks can also exacerbate recessions due to the role they play in money creation as fall of money supply can make recession worse while expansion of money supply can help alleviate them but exploring this is beyond scope of this answer as your question is about how recessions occur (these issues are discussed in most macroeconomic textbooks you can again have a look at the above mentioned Blanchard's textbook).


Algorithm to find Bayesian Nash Equilibria

A brute-force algorithm might not be the right way to go. Sometimes it is not even feasible for finding Nash equilibria with perfect information. This is because even if players and types are finite, BNE's are a profile of (possibly mixed) strategies that maximize the expected payoff. If the game is sequential, this expectation may depend on the own strategies and the strategies of other players. So the space over which you are looking for best responses is quite large. Moreover, the equilibrium is a fixed point for which, in general, we don't know a lot. For example:

Do you know if the equilibrium exists? is it unique? can you define a contraction operator to update players' strategies that will take you to the equilibrium?

Coding will only work if equilibrium exists, and it will work nicely if your game defines a contraction. In contrast, if it is not unique, you will have to be more careful about the algorithm you use to find/approximate a fixed point.

A more useful approach would be to make assumptions (informed guesses) on how the equilibrium will look like and then verify that such an equilibrium exists. Some common assumptions are: "Symmetry": similar players playing similar strategies in equilibrium, "naive strategies": the equilibrium strategies are relatively simple (for example maximizing instantaneous payoff even though the game is dynamic), etc.

This is no simple task, but often you learn more this way. Keep in mind that often you have a multiplicity of equilibria and BNE must be refined to perfect BNE or sequential BNE or other refinements depending on the application.


What are the economic impacts of different professions?

Enrico Moretti (U.C. Berkeley) works on a related question: The multiplier effect. He finds that high tech industries have the largest multiplier. For each new high-tech job in a city, five additional jobs are created outside high-tech in that city over the next 10 years.

Moretti E. Local multipliers. American Economic Review 2010;100:1-7.
Moretti E. Local labor markets. In: Ashenfelter O, Card D E, editors. Handbook of Labor Economics. Amsterdam, The Netherlands: Elsevier; 2011.
Moretti, E. and P. Thulin, Local multipliers and human capital in the United States and Sweden, Ind Corp Change (2013) 22 (1): 339-362.

A quote from his book, The New Geography of Jobs:

  With only a fraction of the jobs, the innovation sector generates a
  disproportionate number of additional local jobs and therefore
  profoundly shapes the local economy. A healthy traded sector benefits
  the local economy directly, as it generates well-paid jobs, and
  indirectly as it creates additional jobs in the non-traded sector.
  What is truly remarkable is that this indirect effect o the local
  economy is much larger than the direct effect. My research, based on
  an analysis of 11 million American workers in 320 metropolitan areas,
  shows that for each new high-tech job in a metropolitan area, five
  additional local jobs are created outside of high tech in the long
  run.

  [And] it gets even more interesting. These five jobs benefit a diverse
  set of workers. Two of the jobs created by the multiplier effect are
  professional jobs—doctors and lawyers—while the other three benefit
  workers in nonprofessional occupations—waiters and store clerks. Take
  Apple, for example. It employs 12,000 workers in Cupertino. Through
  the multiplier effect, however, the company generates more than 60,000
  additional service jobs in the entire metropolitan area, of which
  36,000 are unskilled and 24,000 are skilled. Incredibly, this means
  that the main effect of Apple on the region’s employment is on jobs
  outside of high tech.

There is also a Wikipedia page on the Local multiplier effect.


Government spending/borrowing difference/ national debt increase disparity

There are many ways of calculating public sector finances.

One approach is similar to corporate accounting and the UK Treasury publishes Whole of Government Accounts.

Those are different in construction to the Public Sector Finances statistics published by the Office for National Statistics, who follow international National Accounts methodology

In respect of your question of what happens when these bonds mature and are in effect rolled over, does not lead to a difference in treatment: in both cases the repayment is not treated as government expenditure but of finance, and if the new bonds cover the same amount as the old bonds then in effect nothing changes and Government debt stays the same.

The translation from expenditure and revenue to borrowing and changes in debt is not simple.  The ONS produced an explanatory article. Separately the ONS published this diagram as part of a bulletin last year which illustrates some of the calculations they did for the 2015/16 financial year: