what is the Relationship between Zero nominal interest rates and liquidity traps

The paragraph that you included likely assumes that nominal rates won't go below zero, and therefore that $r_n$>=0 at the zero lower bound. This has been challenged by unconventional monetary and exchange rate policies (i.e. the EU, Switzerland). This likely leads to a few assumptions under classical economic frameworks:

-The Money Demand curve is not horizontally asymptotic at $r_n$=0

-If it is assumed that there is rate mobility below a zero nominal rate (it can go lower and lower with market manipulation), $\frac{d\left(r_n\right)}{d\left(Q_{md}\right)}$≠0 where $r_n$<0. Logically, a -0.41% rate affects consumers differently than does a -0.68% rate (remember that this is effectively the risk free rate; risk premiums will yield positive rates that can be interpreted using classical liquidity preference theory)

This tool has been used for different reasons. Switzerland isn't using negative nominal rates to escape a liquidity trap, but to keep the Franc from appreciating and sending their trade deficit through the roof. The EU on the other hand has resorted to this unconventional monetary policy to try to stimulate growth and avoid a deflationary spiral. They hope to incentivize banks to lend money at the negative rate plus a risk premium rather than be charged to hold money as reserves by the ECB. Consequences exist for these policies, however. Because consumers would rather hold money as cash than lose it in a negative yielding savings account, the money supply of the banking system weakens, and rates can rise back up. Due to its only recent implementation, negative interest rates are likely still considered experimental at best.

Anyways, the paragraph argues that an above zero inflation target allows more policy flexibility, as $r_r$=$r_n$ when inflation equals zero. This effectively gives borrowers cheaper "real" lending while lenders still gain a rate on their assets above holding money as cash (and therefore no bank runs or contracted money supply). Again, this paragraph is either assuming that the ZLB equals zero, or that negative nominal rates are not sustainable. Mankiw would probably argue that a higher inflation target (which leads to generally higher nominal rates in equilibrium), would've given the EU a better ability to manipulate aggregate demand with a "target" lower bound of 0% nominal interest.


CES production function application problem

Your y2 needs to be the output associated produced with those inputs. Given that you have a CES production function and you use World Bank Country data as inputs, you need output of the countries in that specific year. Typically we use GDP as a measure of output at country level. You can get it a.o. from the World Bank.


Why doesn't QE cause the federal funds rate to continue decreasing?

At the time of writing this answer, the Federal Reserve pays 0.1% on reserves: link to Fed webpage.

Banks have no incentive to lend Fed Funds at rates below 0.1%, as they can keep the excess reserves and earn a higher rate of interest.

Interest rates are largely set by price signals made by central banks, not quantities.


I will toot my own horn and cite - Phillips and Wrase. (2006) "Is Schumpeterian ‘Creative Destruction’ a Plausible Source of Endogenous Real Business Cycle Shocks?," Journal of Economic Dynamics and Control, vol. 30 no. 11 pp. 1885-1913.

We found it difficult to match the volatility using Schumpterian mechanisms alone.  The business cycle asymmetries from our model were also exactly backward.  That is, a sudden discovery leads to a jump in output which slowly peters out.  While recessions tend to suddenly jump downward and then slowly recover.

The mechanism in our model was that an increase in productivity due to the discovery, leads to a reallocation of resources away from innovation toward production of output and this slows down the innovation process.


Why does the Laffer curve for consumption have no peak?

This is not a general result. As the auhtors describe in page 10 of their paper, the result obtains only when government lump-sum transfers $s$ do not vary with tax revenues, and instead government spending $g$ does. Then consumption taxes do not affect labor supply, for any level of the tax rate. The relation becomes monotonic, and the Laffer curve with respect to the consumption tax rate becomes also monotonic. This is something that the authors are not very clear in the main text, but it can be detected in the various proofs at the end: they do not argue that the Laffer curve with respect to all three tax rates examined (consumption, labor, capital), becomes monotonic. Only its dimension with respect to the consumption tax.

And more over, on page 9, when government spending is fixed and it is lump-sum transfers that vary with tax revenues (their Proposition 2) they are not clear on what happens, with respect to consumption tax rate. They do say that here the usual Laffer curve emerges, but they comment on a scenario with $\tau^c$ fixed.

Mathematically, the result hinges on the different feasibility constraint that applies in the two cases (eq. $13$ for the fixed government spending case, eq. $14$ for the fixed-transfers case).

In the model, the difference between government spending and transfers, is that the first provides directly utility, while the second increases disposable income in a lump-sum fashion.

Consumption taxes affect utility: a higher consumption tax rate $\tau^c$ will decrease consumption and hence utility of the same income level. When this is offset by higher utility-generating government spending, while transfers remain unaffected, the incentives related to work/income generation ("activity") remain unaffected, pre-tax income remains the same, tax revenues increase: no extremum point in the Laffer curve. Essentially, this affect how much utility the individuals obtain from private consumption, and how much from government spending.

If alongside the increase in the consumption tax rate, we increase transfers keeping government spending fixed, this creates a substitution effect, a disincentive for labor income: it is better to sit back and collect, rather than work and collect. The individual can increase his utility indirectly, by decreasing the disutility from work. This creates conflicting forces: eventually, for a sufficiently high consumption tax rate (and a correspondingly higher level of transfers) he will decrease his labor supply and hence his labor income more than the increased transfers, thus lowering the consumption tax-base.

The above are of course static effects, "inside" each time period, since the Laffer curve is a static concept.


Do all mixed strategy Nash Equilibria satisfy Trembling Hand perfection?

No. Trembling hand perfection would be an additional consideration.

Consider this case:

Two agents engage in a two stage game. In stage one, player one selects Rock, Paper or Scissors. In stage two, player two selects Rock, Paper or Scissors. They receive payouts depending on whether or not they match. They both have identical payout matrices.

Payoffs (Match, Unmatched)

Rock (1,0)
Paper (1,0)
Scissors (1,-1)

In a mixed strategy nash equillibrium consider player one. He can choose any mixed strategy such that

p(Rock)+p(Paper)+p(Scissors)=1

and they would all be equally rational. Since even if he chooses scissors, a rational player two would match him. However, taking into account trembling hand perfection, the subset of strategies where

p(Scissors)=0

dominate the previous strategy set. This is the case because we are concerned that player two has a chance of making a mistake if we choose scissors and not matching us. Therefore, mixed strategy and trembling hand mixed strategy give us different sets of strategies which fulfill our Nash equilibrium criteria.


How does one interpret the usda wholesale market price reports?

MANGOES: MARKET STEADY. flats 1 layer [[cargo type]] HT [[Haitii]] Francis [[variety]] 9s [[9 fruits in 1 layer flat]] 11.00 [[price for 9 fruits]] 10s [[10 fruits in 1 layer flat]] 12.00 [[price for 10 fruits]] MX [[Mexico]] Keitt [[Variety]] 7s [[count]] 5.00-6.00 [[price]] few 6.50 [[a few at a different price]] ...etc... 25 lb cartons [[cargo type]] DR [[dominican republic]] Banilejos [[variety]] 36.00 [[price]]


If you're into growth economics the Nonlinear Dynamics in Equilibrium Models (Eds. Stachurski, Venditti, Yano) collection might be an interesting read.

Hommes' book Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems seems to be nice fit as well.


$\newcommand{\fone}{\color{red}{f_1(q)}}$
$\newcommand{\ftwo}{\color{blue}{f_2(q)}}$

For the sake of simplicity, call

$$
f(q) = \frac{w}{k}q +  \frac{rk}{q} = rk\left(\underbrace{\frac{1}{q}}_{\fone} + \underbrace{\frac{w}{rk^2}q}_{\ftwo} \right) = rk (\fone + \ftwo)
$$

where I have factored $rk$ out of the expression. Now you want to understand each term separately:

  $\fone = 1/q$

This term is drops as $q$ increases, and diverges when $q$ is small.

  $\ftwo = \alpha q$, with $\alpha = w/rk^2$

This is a linear term with slope $\alpha$: it is small for small $q$ and large for large $q$.

  Combined

In this particular case the function one of the terms grows while the other shrinks. So in extreme cases only one matter. The question is where is the point in which one becomes more relevant than the other.

If you notice above I always use the expressions small and large, but these are relative words. You can actually find a value $q^*$ at which these two terms are equal, and this defines in which each term dominates.

So, if $q < q^*$ this is what I mean by small $q$ and therefore $\fone$ dominates. If, on the other hand $q > q^*$ $f$ will be dominated by $\ftwo$. To find $q^*$ we make

\begin{eqnarray}
\fone &=& \ftwo \\
\frac{1}{q} &=& \alpha q \\
q^* &=& \alpha^{-1/2}
\end{eqnarray}

With this in mind, below there's a graph for $\alpha = 1$


Tax falls out of the equation.

Based on your model, tax can still be in the equation, but it is hidden inside of your $C$ and $G$.

First, as you mentioned, if consumption is not constant, then adjusting the tax rate will change $C$ (lower $T$ higher $C$).

But the tax rate is also imbued in your $G$- the government must have some budget. In a balanced budget, $G=T$, which can be a model constraint.

We may not impose $G=T$, but suppose revenue must be a tenth of government spending; $$G = \frac{T}{10}$$

Thus, lower taxes must lower government spending (thus lowering $Y$), or imply higher taxes in the future (See Ricardian equivalence)

  giving individuals massive tax cuts should influence them to consume and/or save more money

This is true, but we must consider the effects on $G$ and a looming future $\tau$ due to massive tax cuts now.


You may want to take a look at Evolutionary Game Theory. Instead of using the regular equilibrium concept developed by Nash, it uses the concept of an Evolutionarily Stable Strategy (ESS).
The idea isn't that players are rational and are choosing strategies given what others are doing to maximize payoff. The idea is: each mutation of that species is "programed" to adopt a particular action (instinct), and when different mutations of the gene interact, it will generate probabilities of survival. Thus, an equilibrium strategy is an ESS, which are the types of actions instinctive to a certain gene type, which will tend to survive and procreate, passing forth the gene.
Game Theory is a field that has had intensive collaboration between Biology and Economics. I am sure you will find some interesting stuff.


Effects of shocking budget or indifference curves on optimum point?

I'm not sure what you mean by "shocking the preferences". But it's very easy to find an example in which the optimal solution is still at the corner after a (small) positive income shock.

Suppose $u(x,y)=\sqrt x+y$, $p_x=p_y=1$, and income is $m=0.1$. Performing utility maximization subject to budget constraint, we get
$$
x^*=m,\qquad y^*=0.
$$
Now give income a positive shock: $m'=m+\delta$. As long as $\delta\le0.15$, the optimal solution is still going to be a corner solution: spend all the income on  $x$ and zero on $y$.


How to determine demand without the price of a good?

Consider that, if two goods are perfect substitutes, their marginal utilities are constant with respect to each other. Hence with two goods, you will consume only one or the other*.

If $\frac{a}{p_x} > \frac{b}{p_y}$, then $x^*(M) = \frac{M}{p_x}; y^* = 0 $

otherwise if $\frac{a}{p_x} < \frac{b}{p_y}$, then $x^* = 0; y^*(M) = \frac{M}{p_y} $

*(assume their prices are not the inverse of their marginal utilities, in which case you are indifferent).


What is the concrete meaning of Britain saying they export Banking Services?

Exports of banking services  means that banks in the UK charge companies and people abroad for banking services, which might include for money transfer, foreign exchange and other broking, wealth and asset management, merger and acquisition services, and other things that banks do. It may include their interest rate margins between taking deposits and making loans, as an implicit fee for financial intermediation

It does not include actual foreign investment, such as direct investment, portfolio investment or financial investment: those do not count as exports of services in the Balance of Payments though income from them do count as primary income in the Balance of Payments current account

Nor does it take into account that many of the banks in London are in fact predominately foreign owned or that many of their staff are foreign.  If they operate from London then they are treated as British banks for these purposes, though their profits may then count as an outward flow to their owners in the Balance of Payments

In 2017 UK exports of financial services (excluding insurance and pension services) were worth about £61 billion; imports of financial services were almost £11 billion


Possible "debt-to-..." ratio indicators for debt-to-economic growth relationship?

Median household income comes to mind. This can be a useful denominator in capital-heavy economies where labour's share of income is relatively small, as it frames debt and deficit in terms of the burden faced by the most fundamental unit of concern in public policy.

In Canada, where I'm from, the OECD finds that labour share has dropped from close to 70% in 1990 to just over 60% in 2007, while nominal GDP nearly tripled in the same period. National debt, meanwhile, increased roughly 50%, and so while debt-to-GDP would have fallen over this period, the debt-to-household-income would have risen. This was a period of heavy investment in Canada's resource extraction industries, with new technologies like SAG-D coming into their own and kicking off a wave of multi-billion dollar injections of capital.  I don't have time to dig up the stats for the province of Alberta for this period, but I would expect these trends to be even more extreme there, as this is where the majority of these investments occurred.


This is a "there can be no adequate answer" answer, for periods prior to "imperialism as a world system."  (Personally, I'd pick 1880s here, ymmv.)

Gross domestic product as a conceptual tool contains a number of assumptions about how economies function which are unsustainable prior to the generalisation of capitalism in the Marxian sense.  While it might be possible to produce homologues for GDP in societies prior to "production" in the capitalist sense, these link poorly with the assumptions you wish to use GDP for.  In societies without a generalised measure of value, production occurs in traditional manners.  The state, particularly prior to Louis XIV era innovations, lacks the capacity to mobilise the economy to particular ends.  This is to say, there is no "gross" productivity, and there is no capacity of a state to mobilise a percentage of productivity.  The state acts as an agent amongst others in traditional relations of production to move portions of portions of social product to its ends.

Earlier the ability to talk about the state comes into question.

Apart from these issues of concepts of measure, there are issues with measurements themselves.  Data series are incredibly fragmentary and difficult to interpret.  Consider, for example, Measuringworth.com's data series on English wage/price series.  The wages and prices originate in monastic records which were not typical of the relations between agricultural labourers and land controllers.  Even the data points from "private" markets in labour here are suspect, 'with' or 'without' beer?  'With' or 'without' crofts being destroyed regularly?  The customary nature of pre-capitalist economies makes data sets less and less meaningful numerically because of the number of externalities.  Even if data points rigorously record the externalities ('with' beer, but 'with' destruction of waste communities of free peasants, and 'with' regular rapine due to civil war), other data points won't have adequate records.

The alternate way forward would be situational and context interpretive single case studies.  To look at institutional stressor indicators due to war.  Did dynastic families fail to reproduce, or were they subject to heightened baronial outrage; were peasant revolts common, what grains were eaten, etc.

We can't apply a stable category, due to categorical assumptions about capacity to mobilise economies and what is worthful being changeable.  We can't gain equivalent data points, due to the complex structures of worth in the premodern.


Accounting for machines in the labor theory of value

"I have read that the Labour Theory of Value holds that the value of a good or service is determined by the total amount of labour involved in its production."
That would be Adam Smith's version of the LTV, which according to him only holds in pre-capitalist societies.
Karl Marx's version is thus:
the value of a good or service is determined by the total amount of socially-necessary labor involved in its production.
Socially necessary labor is basically the average amount of labor a society (industry, trading network or whatever) requires to produce a good.
It takes far less labor time on average to produce randomly mixed chemicals than to produce penicillin from chemical precursors.


Who paid the difference on a warrant if i am winnnig

A warrant is very similar to a call option (see "Comparison with call options" section here: https://en.wikipedia.org/wiki/Warrant_(finance)#Comparison_with_call_options)

So we can suppose we're dealing with a call option: If $P_t$ is the price of the underlying asset at time t and $K$ is the strike price, then the value of the call option is $V_t=\max(P_t-K,0)$ (if the option has not been exercised by time $t$). This value function indicates that if I own the call option, I will not exercise my right to buy the underlying asset at price $K$ unless the asset is worth more than $K$ at time $t$. By exercising my right at time $t$, I gain $P_t-K$, and the issuer of the option, by selling me an asset worth $P_t$ at a lower price $K$, loses exactly the amount I gain.

Your question seems to be getting at the issuer's intent: Why would you issue a warrant where you would have to pay money on average?

A warrant can be used to:

Make bets, or
Provide monetary compensation without giving up cash flow immediately.

1) As you've suggested, the bank may be hoping to never pay out a warrant. But if everyone else also believes that the value of the warrant is 0, then why would the warrant exist in the first place? Maybe the bank has information that others don't, so some people will buy the warrants from the bank, believing that the warrants have positive value less than the price they paid for them. This would be a use of warrants for betting, both for the bank and the people who purchase the warrants.

2) Companies that have promising future cash flow prospects but have poor short-term cash flow would often like to compensate their workers highly to attract talent, but cannot afford to do so immediately with money (e.g. tech startups). Stock warrants will go up in value with the company's value, so such non-cash compensation is often acceptable as a (partial) substitute for cash salary.


Are unemployment benefits considered to be an investment that may lead to a multiplier effect?

With monetary offset by the central bank the fiscal multiplier is approximately zero for any government spending.

See eg Why the Fiscal Multiplier is Roughly Zero.

Basically, if any government spending pushes up inflation (or a similar measure targeted by the central bank) the central bank will 'print less money', eg by increasing interest rates.  If government austerity were to suppress total spending and thus inflation, the central bank 'prints more money', eg by lowering interest rates or doing more QE.

We can see that happening eg with the American fiscal cliff---the US federal deficit was cut in half, and pundits forecast the end-of-the-world, but if anything job growth accelerated.  Similarly, we can already see that Trumps promised extra spending just leads to the FED raising rates.  Inflation expectations both from the Feds forecasts and implied by spreads of inflation indexed treasury bonds over convention treasury bonds both stay relatively flat at 2%.

Under a gold standard or a central bank regime that targets things other than inflation or aggregate spending, government expenditures can have a non-zero fiscal multiplier.


Is Dominique Strauss-Khan really "the best economist of France"?

Jean Tirole is a French economist, with a degree from a French school, working at a French school, and as of the June 2015 the tenth most cited living economist as measured by Repec with 13,649 citations. That's about the same citation count as an entire top 100 ranked department. Everyone else in the top 10 works in an American institution and was born in the USA (8) or Russia (1). Dominique Strauss-Khan doesn't even make the top 5 percent list.


Does Bangladesh and India's economy face similar economic obstacles?

A good starting point is to look at the World Development Indicators (WDI) of the World Bank. You may compare Bangladesh and India's economy over hundred indicators.

For instance, you will find that the percentage of the population living below the national poverty lines was around 31.5% in Bangladesh (in 2010) versus 21.9% in India (in 2011). The 2014 annual report documents that the unemployment rate was quite low in both countries between 2008 and 2012, around 4-5%.


Why should marginal cost be equal to the price an item is sold at?

You should keep in mind that the definition of profit in economics is not the same as in accounting. In particular, economists always deduct opportunity cost from accounting profits, and the price = MC formula has to be interpreted in this way.

For example, let's say you are a self-employed web-developer and you may make a profit in an accounting sense. Economists will subtract from this accounting profit the opportunity cost of not working elsewhere. Looking at it this way, the profit of a self-employed web-developer is just a wage he pays to himself.

The same is true for larger firms. Those firms usually pay out accounting profits to their shareholders. Economists would interpret this as the cost of using the equity capital provided by owners.


Are all constant-sum games games of pure conflict, or only zero-sum games?

For most purposes, there is no difference between (two-player) zero-sum and constant-sum games. One usually assumes in game theory that players maximize expected utility. Payoff functions are von Neumann-Morgenstern utility functions. One does not change the preferences induced by such a payoff function if one adds or subtracts a constant. One can, therefore, transform every constant-sum game into a zero-sum game without changing preferences and behavior.


Is human psychology really reflected into inflation?

Economics is, first and foremost, a discipline that tries to understand human behavior, in particular human behavior regarding scarce resources and choices over them. You are correct in doubting a "we kind of get used to it" explanation if you want an economic explanation. While people may give a lot of reasons to why they make prices behave the way they do, empirically they behave as acting upon the laws below.

Inflation has been plaguing humanity since it developed coinage (barter and money-less economies can't have inflation) and many theories have been developed to explain it. Causes at a fundamental level have changed as economic theories came and went, but there's a consensus among most modern economists.

Without going too much into details, money itself is a good, and as such it's influenced by basic laws of supply and demand. If people desire more money to spend on goods, its value will go up (and the price of things in term of money will go down); if people desire less money, its value will go down (and the price of things in terms of money will go up). In the long term, inflation is a purely monetary phenomenon: more/less money around means more/less supply and more/less goods around to spend money on mean more/less money demand. In the short term, other effects dominate, such as:

If people expect prices to go up, for whatever reasons, they'll mark up prices in a self-fulfilling prophecy. That's why part of a central bank's job is aligning expectations.
Certain prices, like wages, don't update frequently (especially if they ought to go down to reach equilibrium). This is one of the main reasons most central banks strive to reach a ~2% inflation rate: instead of getting wages down by increasing unemployment, they lower wages to equilibrium across the board.
There can be a spiral were firms raise prices and workers demand higher wages to pay for those prices, and firms need to rise prices to pay for wages...
Around Christmas, people buy more. Central banks know this and put more money on the market to account for this extra demand.
What constitutes money is a hard question to answer, but in general it depends on how easy it is to buy goods with it. More money that no one spends won't imply more supply. This means you can print all the money you want and have no extra inflation... if you don't spend it.
If there's a systemic effect, like higher gas prices or exchange rates, a lot of prices will go up to align with lower purchasing power for a country. This is why central bankers tend to focus on "core" inflation instead of actual inflation.

You can see that it has a quite complex dynamics, but I hope the above will give you an intuition behind how to think about inflation.


I would point you firstly to work pioneered by the late Gary Becker on applying the principles of economic optimisation to non-market behaviour.

Becker's insight was that the kinds of trade-offs people face when trading goods and services also affect many other kinds of decisions, including things such as criminal behaviour or marriage and family life. For example, Becker modelled criminals as deciding what crimes to commit based on an evaluation of the costs (if caught) against the benefits flowing from the crime's proceeds (whether material or psychological). A list of Becker's papers can be found on Repec.

Becker's style of applying economics to such topics has been influential and there are now huge literatures applying economics to social behaviour of every variety—to the extent that it is hard to even know where to begin summarising such literatures.

At the time of Becker's death, many leading economists wrote articles summarising his impact on the field of economics.

Becker's contribution was recognised with the award of the 1992 Nobel Prize. The citation read

  for having extended the domain of microeconomic analysis to a wide range of human behavior and interaction, including nonmarket behavior.

The Nobel press release is here and his Nobel lecture "The Economic Way of Looking at Behavior" was published in the Journal of Political Economy.


Why is bartering uncommon in modern countries?

The main likely reasons why barter is not more common are:

The inconvenience of having to find another party who both offers what you want and wants what you offer.
Even if such a party can be found, the possible complexity of negotiating a "fair" transaction (eg I'll do your electrical job if you'll clean my windows monthly for the next 3 months).
I don't want to encourage it, but for those seeking to evade tax, the comparative ease in many circumstances of doing so via a normal monetary transaction, but with payment in cash.

A further point is that most countries with VAT have a significant "VAT threshold", that is, a minimum value of annual sales at which businesses are required to register for VAT and charge VAT (where applicable) on their sales.  In the UK, for example, the threshold is currently £85,000. So an individual could provide services not exceeding that amount in any year and would not have to charge VAT.  If two such individuals were to enter into a barter arrangement, there would be no VAT saving. The Netherlands is atypical in that its VAT threshold is unusually low: 1,345 Euros (see here, table on p 2).


A first price standard and reverse auction are formally equivalent to each other, and the same method can be used to solve both:

First Price Auction

In a first price auction, $n$ bidders choose their bid, $b_i$, as a function of their value $v_i$ (distributed according to $F$. They seek to maximise their expected payoff:
$$[v_i-b_i(v_i)]\Pr(b_i\geq\max_j b_j).$$
If we look for a symmetric equilibrium in which bidders with higher values bid more then the probability that I have the highest bid is just the probability I have the highest value:
$$\Pr(b_i\geq\max_j b_j)=F(v_i)^{n-1}.$$

A useful trick is to think of a bidder as choosing what kind of value to "pretend to have". If we call their choice $\widetilde{v}$ then a bidder's probablem is to
$$\max_{\widetilde{v}}[v_i-b(\widetilde{v})]F(\widetilde{v})^{n-1}.$$

Differentiation yields the first-order condition:
$$(n-1)F'(\widetilde{v})F(\widetilde{v})^{n-2}[v_i-b(\widetilde{v})]-F(\widetilde{v})^{n-1}b'(\widetilde{v})=0.$$

In equilibrium, we know that a bidder should not want to pretend to be anyone other than his true type, so this first-order condition must hold at $\widetilde{v}=v_i$:
$$(n-1)F'(v_i)F(v_i)^{n-2}[v_i-b(v_i)]-F(v_i)^{n-1}b'(v_i)=0.$$
This is a differential equation that can be solved for the equilibrium bid function, $b(v_i)$. In, particular, if values are uniformly distributed on $[0,1]$, we have $F(v_i)=v_i$ and the equation becomes
$$(n-1)v_i^{n-2}[v_i-b(v_i)]-v_i^{n-1}b'(v_i)=0,$$
which can be simplified to
$$(n-1)\frac{1}{v_i}[v_i-b(v_i)]=b'(v_i).$$

The solution to this differential equation is
$$b(v_i)=v_i\frac{n-1}{n},$$
which is what you have in your question.

Reverse Auction

In a reverse auction, $n$ bidders choose their bid, $b_i$, as a function of their cost $c_i$ (distributed according to $F$. They seek to maximise their expected payoff:
$$[b_i(c_i)-c_i]\Pr(b_i\leq\min_j b_j).$$

If we look for a symmetric equilibrium in which bidders with higher costs bid more then the probability that I have the lowest bid is just the probability I have the lowest cost:
$$\Pr(b_i\leq\min_j b_j)=[1-F(c_i)]^{n-1}.$$

A useful trick is to think of a bidder as choosing what kind of cost to "pretend to have". If we call their choice $\widetilde{c}$ then a bidder's probblem is to
$$\max_{\widetilde{c}}[b(\widetilde{c})-c_i][1-F(\widetilde{c})]^{n-1}.$$

Differentiation yields the first-order condition:
$$-(n-1)F'(\widetilde{c})[1-F(\widetilde{c})]^{n-2}[b(\widetilde{c})-c_i]+[1-F(\widetilde{c})]^{n-1}b'(\widetilde{c})=0.$$

In equilibrium, we know that a bidder should not want to pretend to be anyone other than his true type, so this first-order condition must hold at $\widetilde{c}=c_i$:
$$-(n-1)F'(c_i)[1-F(c_i)]^{n-2}[b(c_i)-c_i]+[1-F(c_i)]^{n-1}b'(c_i)=0.$$
This is a differential equation that can be solved for the equilibrium bid function, $b(c_i)$. In, particular, if costs are uniformly distributed on $[0,1]$, we have $F(c_i)=c_i$ and the differential equation can be simplified to
$$-(n-1)[1-c_i]^{n-2}[b(c_i)-c_i]+[1-c_i]^{n-1}b'(c_i)=0.$$
$$(n-1)\frac{1}{1-c_i}[b(c_i)-c_i]=b'(c_i).$$
The solution to this differential equation is
$$b(c_i)=\frac{1+c_i(n-1)}{n}.$$

Comparing the two

The two solutions,
$$b(v_i)=v_i\frac{n-1}{n},\qquad b(c_i)=\frac{1+c_i(n-1)}{n}$$
are equivalent in the sense that a bidder in the first price auction shades (bids below his true valuation) by exactly the same amount as a bidder in a procurement auction bids above his true cost. You can confirm this by, for example, plotting these two bid functions for some value of $n$. Here's an example with $n=2$:


Can there be a sample such that SRF and PRF do not intersect?

It is theoretically possible, but extremely unlikely if the random sample is large.

To see that it is possible, consider the very simple case shown below in which the population consists of just four points in the X-Y plane which happen to lie at the corners of a parallelogram.

Assuming linear regression lines, the population regression function (PRF) will be as represented by the continuous line.  If a sample regression function (SRF) is fitted to a sample of two points and the points happen to be A and B, then the function will be as represented by the dashed line which is parallel to the PRF.

Note however that even in this case, the probability that a random sample of two points will produce a SRF parallel to the PRF is only one third.  There are six possible such samples, two of which (AB,CD) yield SRF's parallel to the PRF, and four of which (AC,AD,BC,BD) do not.  In a realistic case in which both the population and the sample are much larger, and the points do not lie in such a simple geometrical pattern, the probability of a SRF parallel to the PRF will be very small indeed.


This website (in Russian) states that, in 1825,
you could sell 1 Imperial Ruble and get 0.72 US dollars. I don't know Russian so I cannot tell the source of that data.

Those 0.72 US dollars in 1825 are worth US$15.49 in 2016, according to this website.

Therefore, 1 Imperial Ruble of 1825 was equivalent to US$15.59 in 2016.

Additionally, this website has information of prices of different goods and wages for Russia since the 16th century.


Categorisation of Harmonised System codes for economic analysis

If you are looking for classification like Agri-food trade presentation that covers products other than agricultural, you should check out the Combined Nomenclature (CN) of the European Union and its Explanatory Notes.

It is established on the basis of the Harmonized System and it covers all the products that can be subject to customs and tariffs in the EU. Seek Annex I to the the basic CN Regulation (Council Regulation (EEC) No 2658/87 on the tariff and statistical nomenclature and on the Common Customs Tariff). Be sure to use the latest version, though, as it is updated on a yearly basis. Here's a link to 2018 version, but if you want to be accurate, you should also include amendments (there were five of those since publication of 2018 verion). Perhaps the most readable is of course Explanatory Notes to the CN (so something like this).

If you are interested in jurisdictions other than EU, please find links to cusoms nomenclature under Harmonized System's wikipedia page, you've posted. I'm sure all of them will have the same problems as Harmonized System. It is simply impossible to explicitly list all the products in the World and such listings will always be half open with use of phrases like 'not elsewhere specified'.

Hope that helps.


Think of your function as

$$Y=[A_1Q^\sigma+ A_2X^\sigma]^{1/\sigma}$$

where $Q=L^\alpha K^\beta$ (I removed unnecessary bits like $\gamma$ and exponent of $A_i$). This is a standard CES, where the optimal use of $Q$ and $X$ depend on $\frac{A_1}{A_2}$ and $\sigma$.

For a relatively low substitution ($\sigma<0$), an increase in $A_1$ (holding factor prices fixed) means firms make use of that extra increase in productivity of factor $Q$ by using less of it and more of $X$. Since $A_1$ corresponds to a Hick neutral change regarding $Q$, you will use the bundle $Q$ with the same initial $K/L$ ratio than before the change in $A_1$. Therefore, you will reduce both $K$ and $L$. In consequence, an increase in $A_1$ could be thought as labour-saving technical change with respect to $X$, and in absolute value (although notice that $K/L$ remains unchanged).

The reverse is true if $0<\sigma \leq 1$, namely an increase in $A_1$ leads to more use of $Q$ relative to $X$, thereby expanding the use of $L$ and $K$.

Meanwhile, in the inbetween case ($\sigma=0$), the function collapses to a Codd-Douglas:

$$ Y = \left(L^\alpha K^\beta\right)^{\phi} X^{1-\phi} $$

where $\phi = \frac{A_1}{A_1 + A_2}$. Here a change in $A_1$ does not affect the K/L ratio, albeit it does affect the optimal ratio with respect to $X$. But commonly technical change is though as a change in some $A$ rather than in the elasticities of factors, so this case does not nest well with the above. I think this might be because the original function is not properly normalised. This is, the terms $A_1$ and $A_2$ do not add up to one, which is necessary given your original assumption of constant returns to scale.


Mathematical open problems that (when answered) might unlock MAJOR mathematical (micro)economics/finance/econometrics discoveries

People have argued that if $P \neq NP$ then efficient markets are impossible and certain equalibria may not exist. However, they may hold approximately, so I'm not sure if this qualifies. Additionally, if it turns out that $P=NP$ then certain economic optimization problems (e.g. in logistics) become easily solvable. On the other hand, if $P \neq NP$ then it might imply that encryption is very safe and therefore certain forms of economic arrangements will still be possible in the face of technological advance.


To answer your question I will assume that

$\bf{A1}$ firm 2 always "trust" in the output reported by firm 1, and

$\bf{A2}$ firm 1 always "keep its word" if announces an output.

Without $\bf{A1}$ and $\bf{A2}$ I think you need more information to solve the game.

Based on how the question is written, at the first period firm 1 has the opportunity to whether announce or not its output. So, lets say that firm 1 must to chose between $\{A, NA\}$ ($A$ for announce and $NA$ for not announce) at $t=1$.

Then, if firm 1 chose $A$ the resulting game will be the same as point i) of your question. In the same way, if firm 1 chose $NA$ the resulting game is the point ii) of your question.

Finally, you only need to obtain the utility of firm 1 for both cases to know what is the best action for firm 1 at $t=1$. The subgame perfect equilibrium must to include the action of firm 1 at $t=1$ and the quantities of both firms at $t=2$.

PS: In point ii) you have a mistake, the Cournot game is symmetric so there is impossible to get different quantities in a Nash equilibrium. (sorry for my bad english)


Confounding versus endogenous variables. What is their relative hierarchical position?

The question is not totally clear, but I will attempt to give you some guidance.

To answer your first questions, confounding variables are not a type of endogenous variable. We do not observe nor are we interested in the confounding variables, which means they are not endogenous variables in our model.

You later give the correct definition of an endogenous variable "An explanatory variable in a multiple regression model that is correlated with the error term". Hence, there can be several reasons why a variable would be endogenous, but they all have to do with correlation with the error term.

An endogenous variable is one, which cannot have a causal interpretation in a regression model. When something leads to a variable being endogenous, we call this a source of endogeneity.

As you mention, an important source of endogeneity are omitted variables. If we omit a variable, which affects the outcome variable, then the omitted variable is captured by the error term in our model. If the omitted variable is correlated with an explanatory variable, then that explanatory variable becomes endogenous, because it is now correlated with the error term.

Other important sources of endogeneity are reverse causality and simultaneity, as you mention. For reverse causality, we have that the explanatory variabe is supposed to influence the outcome variable, but if the outcome variable also affects the explanatory variable, then in our regression model, this latter channel is necessarily captured by the error term. This is because such a channel determines the relationship between the outcome and explanatory variable, but we cannot capture it in any variable. Putting the explanatory variable on the left side instead and using the outcome variable as an explanatory variable does not help either, because we then have the same problem. Then the explanatory variable is correlated with the error term, which makes the explanatory variable endogenous.

So overlapping features are that all sources of endogeneity stem from a correlation of an explanatory variable with the error term. Distinctions are the source of this correlation and consequently how to deal with it. For example, omitted variable bias may be dealt with by observing and adding the omitted variable to the regression. This strategy does not work when trying to address reverse causality.


How can valuable retail real-estate stay vacant for an extended period of time?

Before applying irrationality to explain the phenomenon, consider the following simplistic model of a rational owner: assume there is the high rent $R^h$ and the low rent $R^{\ell}$. The owner faces the prospect of earning either

$$V(h) = \sum_{t=1}^{T_1} \beta^{t-1}R^h + \sum_{t=T_1+1}^{T_2} \beta^{t-1}\cdot 0+ \sum_{t=T_2+1}^{T_3} \beta^{t-1}R^h +...$$

or

$$V(\ell) = \sum_{t=1}^{\infty} \beta^{t-1}R^{\ell} = \frac{1}{1-\beta}R^{\ell}$$

Favor the "low-rent" scenario by ignoring all terms in $V(h)$ except the first sum (this reflects that the property may go unleased for "many years", so remote-future income gets fully discounted). Based on these considerations alone, if we observe the "high rent" scenario, it must be the case that

$$V(h) > V(\ell) \implies \frac{1-\beta^{T_1}}{1-\beta}R^{h} > \frac{1}{1-\beta}R^{\ell} \implies R^{h} > \frac{1}{1-\beta^{T_1}} R^{\ell}$$

which can be re-arranged to give

$$\beta < \left(\frac {R^h-R^{\ell}}{R^h}\right)^{1/T_1}$$

The right hand side can be estimated by real world data: obtain average magnitudes for the high and the low rent, as well as the average time a business under the higher rent pays it prior to fold (the $T_1$ parameter). These will lead to an estimation of the upper bound for $\beta$ in order for the above inequality to hold, and justify the "high rent" scenario observed.
Then one can examine this upper bound and contemplate whether it appears "reasonable".
As a numerical example, assume $R^h = 2R^{\ell}$ and $T_1=2$ (say years). Then it must be the case that

$$\beta < (1/2)^{1/2} \approx 0.71$$

"Too low" (i.e. too high discounting) one would tend to say. But don't look at benchmark values of $\beta$ that are used in macro-models. Try to think what the discount factor attempts to capture here.  And in any case, a person with very high discounting, is not "irrational" -this is how he experiences the world.

And then, there are other considerations: accepting low rents tends to drive the asset values down. Also, It may be the case that this is incremental income for land owners, above some comfortable level from other sources, meaning declining utility of wealth may kick in. Further, constant occupation physically wears the asset down faster. Etc.

My point is, "people are just jerks" (i.e. arguing for irrationality) can explain everything - and so it really explains nothing.


There are two cases to consider:

1) Company B buys the shares of company A, but keeps A as a separate entity, a $100$% subsidiary. The individual Balance Sheets here will be

COMPANY B
\begin{array} {| r | r | r |}
  \hline
\hline
  \text {ASSETS} & \text {USD} & \text{LIABILITIES} & \text{USD}\\
  \hline
\text{Goodwill} & 10 & \text{Equity} & 10\\
\text{Participations} & 50 & \text {Debt} & 50\\
\hline
\text {TOTAL ASSETS} & 60 & \text{TOTAL LIABILITIES} & 60\\
  \hline
\end{array}

The overall value has not changed, only the composition of the Assets. Nothing changes in the Balance Sheet of Company A - it was not a party in the transaction, its shares was the object of the transaction

COMPANY A
\begin{array} {| r | r | r |}
  \hline
\hline
  \text {ASSETS} & \text {USD} & \text{LIABILITIES} & \text{USD}\\
  \hline
\text{Assets} & 100 & \text{Equity} & 50\\
\text{} &  & \text {Debt} & 50\\
\hline
\text {TOTAL ASSETS} & 100 & \text{TOTAL LIABILITIES} & 100\\
  \hline
\end{array}

but what is interesting here is the Consolidated Accounts, the two companies viewed as one from a financial point of view, even though for legal and other purposes they are separate entities. Here we have

GROUP BALANCE SHEET

\begin{array} {| r | r | r |}
  \hline
\hline
  \text {ASSETS} & \text {USD} & \text{LIABILITIES} & \text{USD}\\
  \hline
\text{Assets} & 100 & \text{Equity} & 0\\
 &  & \text {Debt} & 100\\
\hline
\text {TOTAL ASSETS} & 100 & \text{TOTAL LIABILITIES} & 100\\
  \hline
\end{array}

How come? First, Debts of both companies are owed to Third Parties, so their amount doesn't change. So Group Debts = 100. Second, The Assets held by mother company B are "internal" - the shares of company A (face value and premium), so they are ignored, "written off" in the consolidated accounts. What Assets remain are those held by company A. So Group Assets = 100. We are then led to find that Group Equity is zero... which is correct, because if we write off Assets we have to write off an equal amount from the other side, of Equity.

Can we arrive at zero by some other road of reasoning?
Well, the two Equity figures add up to \$60. Hmmm, and how much did you pay Third Parties to leave company A? \$60. What Equity was injected into the Group, exited as payment to Third Parties for the buyout. This is why Group Equity ended up zero.

2) Company B bought the shares of company A, but then merged with it. In this case we don't have individual and Consolidated Statements, only one Balance Sheet, which is the same as the Consolidated BS in the previous case.

Leveraged Buyout indeed.


Estimating Equilibrium Price for Advertising using Simultaneous Equations Model

You're definately on the right track. What you are doing is estimating the inverse demand and inverse supply in your market. The variables Visits Print and Prod can be used as "shifters" i.e. instruments for the quantity in the equation in which they are not present. Visits can be used as an instrument for quantity, when estimating the inverse supply equation and Print and Prod can be used as instruments for quantity when estimationg the inverse demand equation.

A well known example of how this works is Grady's article on Fulton fish market. You can find STATA code that replicates that article here: https://www.stata.com/data/s4poe/chap11.do .

The data sets for the stata code are here: http://www.principlesofeconometrics.com/poe4/data/stata/ .

Of course you should test the validity of the assumptions behind the use of IV in your SEM with the Hausman, Sargan-Wu and Stock-Yogo tests.


I think hot weather alters firms' desire to sell at some given price because hot weather increases the demand for ice cream. (Mankiw's book)

Yes, you are wrong. As the quoted part explains,

In this case, economists say there has been an increase in “quantity supplied” but no change in “supply.” Supply refers to the position of the supply curve, whereas the quantity supplied refers to the amount producers wish to sell.

Are you willing to sell your current computer for say \$1 000? What is the lowest price that you are willing to sell for? (Just a ballpark estimate will do.)
If you came up with a number, you did that without knowing anything about the prospective buyers of your computer.
Of course it is very nice to be able to sell for more, if the customers are - for some reason - willing to paying more, but the ability to do this is determined by the equilibrium. The supply curve merely describes the vendors' preferences. What is the lowest price at which they are willing to sell / what quantity are they willing to sell if the price turns out to be X.


What's the economic argument (if any) for taxing dividends as ordinary income?

Dividends have to be taxed as income, as otherwise people can incorporate, have their wages taxed as revenue for the corporation, and then pay themselves a dividend. (Canada adjusts taxes on dividends in an attempt to make these two approaches equivalent for taxes purposes.)
Capital gains are at a lower rate for reasons that are debatable. However, in order for a gain to qualify as a capital gain, the principle is that there has to be a possibility of a loss. (To what extent that principle is achieved can be debated, since there is an incentive to exploit this lower tax rate.) In the case of share buybacks, even though the firm is purchasing shares, there is no guarantee that the sellers are selling at a profit.

I am a consultant in Canada, and one of the first things to decide is whether to form a corporation or not. The logic varies upon the tax code in each jurisdiction. The following article discusses the situation in the United States: link to small business advice website.


Why is market power presumably high when goods are fungible and demand relatively inelastic?

"Fungibility", when referring to goods, essentially means homogeneity -all units of the good are exactly the same from all aspects that matter to consumers (including "exterior" aspects, like location etc), and so one can substitute for another.

If one knows that in a market essentially only one homogeneous good is traded, then one knows that he must look for market-power sources elsewhere and not in product differentiation (since the latter is non-existent), like for example in barriers to entry, cost structure etc.

If "fungibility" is narrowed down to the inherent characteristics of the good, then geographical location or other such aspects, may become possible sources of product-differentiation, and so of market power.

Fungibility is a property coming from the supply side, so by construction is unrelated to the elasticity of demand (which depends on preferences). Therefore it is a useful piece of information on its own.


The National Center for Education Statistics , Digest of Education Statistics does have post-secondary education (university and community college) enrollment in the US per state from 1995 to 2017. There are some tables that has the data going back to 1970 but it skips some years so you may want to perform a  some sort of missing data imputation method (i.e. linear interpolation) to impute the middle years,  if you need all the years for your analysis.
However, their website does require some skillful navigation to say the least.
Hopefully,  my instructions will help.  Once you go to this website: https://nces.ed.gov/programs/digest/
On the left pane of your window go to the List of tables and figures section.  There is a  dropdown box below, select a year.
It goes to 1995.  Once you have selected a year go to chapter 3  Postsecondary Education, then select degree-granting institutions, then enrollment.   At this point you will need to find the table that has the state data.
From what I understand of the data that you are looking for, I saw it is in table 195 for most years (1995- 2006) , then its table 190 in 2007,  2008- its table 203  , 2009 - table 219 , going forward for most years the table number that has the state data your looking for seems to change but I am pretty sure its there in chapter 3.
For your research project a recommendation is to pay attention to how the NCES Digest of Education Statistics changes how define their state level enrollment data.   They also have more education data by state if you are curious to see what they have that could support your work.
If you run into trouble with these tables I would recommend to call their office.  The staff list is located here https://nces.ed.gov/ncestaff/stafflist.asp
I hope this helps.


Consider the Slutsky equation,
$$
\frac{\partial x}{\partial p} = \frac{\partial x^c}{\partial p} -
\frac{\partial x}{\partial I} x.
$$

A giffen good is the case where the income effect $\frac{\partial x}{\partial I} x$ is negative and large (in magnitude) enough so
that $\frac{\partial x}{\partial p} > 0$.

From Wikipedia:

  There are three necessary preconditions for this situation to arise:
  (1) the good in question must be an inferior good,
  (2) there must be a lack of close substitute goods, and
  (3) the good must constitute a substantial percentage of the buyer's income, but not such a substantial percentage of the buyer's income that none of the associated normal goods are consumed.

(Sufficient when (1) also adds that the good is so inferior that the income effect is greater than the substitution effect.)

A Giffen good does not generate utility directly through its price. Contrast this to a Veblen good where there the user actually derives utility directly from the price.

See this link for some info about empirical evidence.


Logit and probit regressions do not help with endogeneity at all. What they do is bound the predicted value (aka conditional expectation) of your response variable between 0 and 1. As such they are appropriate for performing regression analysis for binary outcomes like:

Whether someone wins or loses a sports game
Whether someone is going to be employed or unemployed

To understand this, consider the univariate regression model

$$y_i = \beta x_i + u_i$$

Say $y_i = 1$ if player $i$ wins a tennis match and $y_i$ = 0 if a player loses a tennis match, and you are trying to work out whether they will win based on the difference between their world ranking and that of their opponent, $x_i$.

If you perform ordinary linear regression you might get an estimate of $\beta$ which is, say, $0.05$. Perhaps most of your observations for $x_i$ (i.e. the difference in world rankings) are between $0$ and $20$, which means that your predicted value of $y_i$ (i.e. $\beta x_i$) is between $0$ and $1$. You can then easily say whether you are predicting a win or a loss for your player depending on whether that predicted value is greater than or less than $0.5$. However there may be certain situations where the difference in world rankings is $>20$, in which case you will get a prediction >1.

In one sense this is unproblematic because based on your model this player is clearly the forecast winner. However you may prefer not to have unbounded predicted values, in which case a probit or logit transformation or model is appropriate.

Taking probit as an example, you specify that the probability that a player wins, conditional on the difference in rankings is not $\beta x_i$ as before. Instead you state

$$p(y_i = 1 | x_i) = \Phi(\beta x_i)$$

where $\Phi$ is the cumulative distribution function (CDF) of the normal distribution. As you can see in the normal CDF diagram here, no matter how high or low the value of $\beta x_i$ is, the variable on the y-axis never exceeds $1$!

In addition, if the distribution of your error term $u_i$ is logistically distributed and you use logit, or normally distributed and you use probit, your estimation technique will be more efficient (i.e. you will have smaller standard errors) than if you performed the simple linear regression. This point is more complex though so I don't provide a proof.


Can resource be considered scarce if its supply is limited, while quantity of the resource is practically unlimited?

Economists view everything people want, strive for, or can’t achieve effortlessly as scarce (you can read this nice article at EconLib). Under such broad definition virtually everything is scarce as anything requires at least some minuscule amounts of effort.

However, if you want it is possible to narrow this definition down and be more specific as in most models a scarcity can be checked and defined by looking whether at zero price there is still excess demand or not - if there is equilibrium then price must be positive, and we would say that this indicates scarcity.

Breathable atmosphere on land and outside extremely polluted city under such definition would not be considered scarce because supply is so high that at 0 price all demand is completely satisfied, but if you need some breathable atmosphere for scuba diving or in those cities where they sell oxygen tanks, or in space breathable atmosphere would be scarce because supply there is not large enough so that the equilibrium price is 0 and all demand is satisfied (there is no excess demand).

Now we can try to apply this to your example of Dyson sphere. First, I doubt this sphere will just pop out into existence and it will require no maintenance. There will be very likely exorbitant fixed and maintenance costs of such structure. Since the energy output will depend on how well the sphere is maintained I doubt it is even feasible to say equilibrium price of energy could ever be driven to zero as the supply depends on marginal costs which won’t be zero in that case.

Second, let’s say that as a thought experiment that marginal costs of operating the sphere are 0, it would still not be enough. Sun gives out a lot of energy approximately (rounding up) $4 (10^{26})$ joules per second or wats. Now that is a large amount of energy but to build a Dyson sphere or Dyson Swarm (a more realistic version of the concept) you have to be at least type II civilization on Kardashev scale and such civilization would also have insatiable lust for more and more energy. Before I became economist I was studying astrophysics. We did not discussed Dyson sphere or swarm much but when it was brought up during break/after class discussions etc. it was always in context of being necessary to satisfy the needs of type II civilization which will probably want to run some ancestor simulations, or for building kugelblitz black holes that could serve as engines for starships or do other stuff that requires unfathomable amounts of energy. So even with output of whole start being collected such civilization would most likely still have such exorbitant demands for energy that it would still be expensive and price would be nowhere near zero, and there would be huge excess demand even at zero price.

This being said assuming that such civilization would only need small fraction of the energy, and Dyson sphere/swarm would have no marginal costs, so everyone’s demand is satisfied (at zero price there is no excess demand) then you could argue energy ceases to be scarce. Also you can use the general definitions and this example to generalize this for any good/service.


Plotting money market equilibrium for 2020

The graph is almost correct given the assumptions about what is happening but not quite. The proper graph should look more like this picture I made in LaTex using tikz:

Where $^*$ denote the new state of things given your assumptions.
You are right that money demand shifts to the left and money supply to the right, but your graph missed an important fact that money demand at zero lower bound (ZLB) becomes flat. This is one of the reasons why monetary policy looses its potency around ZLB (see Blanchard et al. Macroeconomics a Eurpean Perspective pp. 96).
Correction of some misconceptions:
In your post you also have some misconceptions let me correct them:

This is the tricky part, as money demand is not something we can directly observe in practice, though I tried my best to be objective/conservative.

We actually can observe money demand at a given point in time (only the whole demand function is not directly observable). Furthermore, we actually even can, using econometric methods, properly estimate the money demand function.  See Jawadi & Sousa (2013) or Narayan, Narayan,  & Mishra (2009), for examples of that. However, of course this cannot be done in real time but only retroactively as we collect enough data about given time period, so to do that for period covering entire COVID19 crisis this won't be possible until few years down the road.

helicopter money offsets deflationary forces

central banks are definitely trying to expand money supply right now but not via helicopter money. Helicopter money means central bank is directly sending money to households. This has not yet happen to my best knowledge (at least not in US, EU or UK), although there are economists calling for central banks to do so (like Gali 2020 in his VOX$^{EU}$CEPR post).


Correlation between salary level and housing prices in a town

The graph below plots average house prices against average incomes for various US cities. The relationship is strong in the cross-section, especially if you ignore Honolulu.

Richard Florida, Housing and the Crisis, Part IV

Is it also true dynamically, that is, are changes in income associated with changes in house prices (not just levels)? Van Nieuwerburgh and Weill (2010) say yes:

  We set up and solve a spatial, dynamic equilibrium model of the
  housing market based on two main assumptions: households with
  heterogenous abilities flow in and out metropolitan areas in response
  to local wage shocks, and the housing supply cannot adjust instantly
  because of regulatory constraints. In our equilibrium, house prices
  compensate for cross-sectional productivity differences. We increase
  productivity dispersion in the calibrated model in order to match the
  30-year increase in cross-sectional wage dispersion that we document
  based on metropolitan-level data. We show that the model
  quantitatively matches the observed 30-year increase in dispersion of
  house prices across US metropolitan areas. It is consistent with
  several other features of the cross-sectional distribution of house
  prices and wages.

Why Has House Price Dispersion Gone Up?


How to find the maximum profit in a graph?

Short answer:
Shift the profit line parallel downward until it only touches the loss function in only one point. That's the point where the maximum gap occurs.
Reason:
The maximum occurs where Marginal Cost=Marginal Revenue.
You can see this from basic profit maximization:
$\max Profit = \max (Revenue - Cost) $
We solve by taking first derivatives, call them $D$, and setting to zero.
Hence $D Revenue - D Cost =0$. Note that what we mean by Marginal Revenue and Marginal Costs are just first derivatives of Revenue and Cost, respectively.
So clearly Marginal Cost = Marginal Revenue.
Graphically this means the slope of the cost function equals the slope of the revenue function at the maximum profit point. This is because the first derivative gives the slope of a function.
So shift the revenue function parallel downward toward costs until it only touches on one point. They have the same slopes at that point. This is because a) revenues here are linear (a straight line) and have the same slope everywhere and b) a straight line that touches a function in one point only (called a tangent) has the same slope as the function at that point.
So that tangent point is your profit maximization point.
Similarly, if you shift revenue upward to the other tangent point with costs, you will find the maximum loss point.


Why does the gold standard require countries to keep exchange rates fixed?

This is required just due to the simple definition of what gold standard is. For a monetary arrangement to qualify as gold standard money has to be convertible into gold. Hence central bank or some other government body must be always prepared to convert notes into gold. That is part of both the dictionary and economic definition of how it works.
To see why this requires fixed exchange rate lets say that this gold conversion is set in US such that $\\\$1$ converts into $1$ gram of gold. Now suppose EU would set its conversion rate to $1e = 2g$ of gold. If the exchange rate would be anything else than $E=\frac{\\\$}{e}=2/1$ the gold standard would eventually have to collapse in long run.
For example if exchange rate $E$ would be $1$ (i.e. $1EUR=1USD$) then everyone would take their dollars convert them to euros demand gold from European Central Bank to the point when either it would left without any gold to support the gold standard or it would change the exchange rate. If the exchange rate would be higher than $E=2$ opposite would happen. Consequently, under gold standard maintaining fixed exchange rate is necessary otherwise it cannot survive due to the fact that under gold standard notes must be convertible into gold.


Is M&A valuation a good proxy for company fair value?

The thing to always remember is that the valuation of any asset always comes back to the price that someone is willing to pay for it (i.e. if you can’t hope to sell it for that price, it’s not worth that much).
When you describe using the “value of a friendly M&A deal” as a proxy for fair value, I assume you mean price paid for the very company you are trying to value. (Analysts valuing a potential acquisition target won’t know ex-ante what the ultimate deal price will be so they instead will look at precedent transactions for comparable companies to get a range for what type of multiples buyers are paying).
Assuming you know the price paid in the friendly M&A deal, yes that is a very strong data point for what that business is worth. That being said, there are some caveats. For instance, say the buyer was a “strategic” (meaning an often bigger company in a similar industry as opposed to a financial buyer like a private equity firm). There may be some synergies only available to that buyer that raise the price they are willing to pay for it. For instance, say the buyer can close plants, eliminate duplicative sales roles by moving the acquired company’s products onto their sales platform, etc. Those cost savings only available to the strategic buyer make it a more valuable acquisition target than a pure financial player.
Alternatively, the context of the acquisition matters. Say the acquirer bought the business in 2008 (during the financial market collapse) vs in a booming period or in a low vs high interest rate environment. Make sure the M&A transaction you are comparing to is recent because the market (and the company itself of course) can change over time.
Lastly, was the acquisition “market tested”, meaning there were multiple bidders? A friendly transaction where the acquirer was an insider can sometimes result in shady below-market prices paid. Check out lawsuits relating to Delaware shareholder appraisal rights.
The price paid in a friendly M&A deal for the company you are trying to value is an incredibly strong data point, but I would caution against saying it’s better than a hostile price. The acquirer, context, timing and whether the acquisition was market-tested are all important factors in assessing the fair value (or intrinsic value) of a business.
And of course, when looking at any single acquisition, the acquirer might just have been wrong and overpaid—and you could never reasonably expect another buyer to pay that much going forward.


Does a Robinson Crusoe economy have a substitution effect and an income effect?

I don't see why equilibrium $MPL(\ell)$  would be fixed.

I think Robinson's income is measured by what he can get using his total time $t = l + \ell$.

Assume that given the original $A$ and total time $t$ his optimal choice was $(c_1,l_1)$. If the technology improves to $A'>A$ he can reach $(c_1,l_1)$
 (or $U(c_1,l_1)$ if you take the Hicksian approach)
 by only using total time $t' < t$.

Let us denote the optimal consumption bundle given technology $A'$ and time $t'$ by $(c_2,l_2)$. This is the intermediate step separating the income and substitution effects. The difference between $l_2$ and $l_1$ would be the substition effect. The 'income' in the two situations is considered equal as $(c_1,l_1)$  (or $U(c_1,l_1)$ if you take the Hicksian approach)  is barely feasible in both cases. Any difference in leisure consumption is due to the different transformation rate between leisure and consumption.

Let us denote the optimal consumption bundle given technology $A'$ and time $t$ by $(c_3,l_3)$. The difference between $l_3$ and $l_2$ is the income effect. The technology is identical in both cases, so any difference in consumption is due to the difference in 'income'.

Perhaps your instructors are trying to say something about the backward bending labor supply curve. It is difficult to tell from the available information.


The statement

  and now there is less income than product (because less output was sold, which leads to lower wages)

misses that the workers are paid to work a certain number of hours in production, and that production happened, so current wages are unaffected.

Meanwhile, the firm’s income is unaffected by the inventory build, since it is a form of investment, and thus the wages used to produce the inventory are effectively capitalised on the balance sheet.

Incomes might only fall in subsequent periods, as production is lowered in response to the unplanned growth in inventories.

As such, there is no real mystery to be explained.


Can central bank interest rate hikes lead to a price gain in equities?

perhaps flight to quality? ie reduced expectations of euro equity profits cause equity reallocation out of euro into chf.

and increased eur bond yields perhaps cause bonds reallocation out of chf into eur because of a) higher , ie more attractive coupons , b) eurchf xchange rate perhaps expected to strengthen


How can we get negative **real** interest rate?

There is a market for inflation-linked bonds. The quoted yield on the bonds is the equivalent of a real yield, with the inflation rate corresponding to expected inflation.

The real yields On US inflation-linked bonds are currently negative. Link to FRED data.

There is no mystery to this.

Nominal yields are effectively pinned to the expected path of the nominal policy rate.
Inflation rates follow developments in the economy.
There is no law of nature that says that the nominal yield has to be above the rate of inflation.


What economics aggregate(s) could suggest an imminent change in exchange rates?

If they expected the peg to break then it is clear that the franc would gain in value. Since the currency has been held artifically cheap by the central bank, i.e. would have gained value if the central bank had not intervened, then the moment the central bank stops intervening (abandons the peg) the currency will appreciate.

Why would it have gone up otherwise, i.e. what are the factors driving exchange rates? We call these "fundamentals". Basically exchange rates are determined by supply and demand for a currency, just like for anything else. So basically anything influencing supply and demand for currencies would be cause for a change in exchange rates. Further expectations of changes in fundamentals cause them as well. These include the trade balance (typically governs the long run behavior of exchange rates in reality) and the difference between domestic and foreign interest rates (typically govern the short run behavior in reality).

Now one might ask, why did they expect the peg to be abandoned? That depends on the specifics about Switzerland. I'm not an expert there, so I'm sorry if I cannot give a very concrete answer. However pegs are typically unstable and usually abandoned eventually. This is because the longer the central bank keeps a currency at the "wrong" price, the more pressure from the market there is to correct it which means the central bank must fight harder and harder to keep it and eventually gives up or decides its not worth it anymore.


Textbook claims that price level accompanied by decrease in interest rates?

Presumably the context is deriving the AD curve from the IS-LM model. Start from money market equilibrium $$M/P=L(i,Y).$$
If $P$ falls then the supply of real money balances rises. With $M$ and $Y$ fixed the nominal interest rate $i$ must fall in order to induce people to hold the greater supply of real money balances. This is because the alternative to holding money is holding it in bonds which pay interest. To get people to hold more money (which pays no interest) we need the opportunity cost of holding money to be lower, so yes your interpretation is correct.

With sticky inflation expectations (recall the Fisher equation $i=r+\pi^e$) the change in the nominal interest rate will map one to one to changes in the real interest rate. This means that the LM curve shifts down at any given level of $Y$ in $(r,Y)$ space. This produces a new equilibrium in the IS-LM model with a lower real interest rate and a higher income (since the lower real interest rate has stimulated investment). Thus a negative relationship between $P$ and $Y$: the AD curve.


Are stock trade commissions evidence of anti-competitive practice?

The money goes to banks/brokers. In fact most of the revenue in the financial sector comes from fees like that.

This is to compensate the bank/broker from taking the risk of carrying the opposite position on their books for a while, and the labor put into actually trading for you. A part of it is probably also profit. Since there are large entry costs, one would expect the financial sector to earn  profits.


Since we are suspecting a corner-solution, it is better to write the problem explicitly with its constraint. Even better, use the Fritz John (FJ) conditions rather than the Karush-Kuhn-Tucker (KKT) ones. We will mention the differences as we go along.

$$\max_{\alpha} \int u[w+\alpha(z-1)] dF(z),\;\; \text{s.t.}\;\; w-\alpha  \geq 0$$

The lagrangean under the Firtz John formulation is

$$L_{FJ} = \lambda_0\int u[w+\alpha(z-1)] dF(z)\; + \;  \lambda_1(w-\alpha)$$

The new element is the multiplier on the objective function, $\lambda_0$. Without loss of generality we can specify

$$\lambda_0 \in \{0,1\},\;\; \lambda_0 + \lambda_1 \neq 0$$

What is that we gain here, compared to the much more widely known and used KKT-conditions?
If a solution necessitates that $\lambda_0 =1$, we obtain the KKT conditions with the constraint qualification satisfied. If a solution necessitates that $\lambda_0 =0$, it reflects, among other special cases, the case where the constraint qualification fails to hold.

  (A standard example is the case where the feasible set for $\alpha$
  has been reduced to a single point due to the constraints imposed.
  Then we will find that the only solution dictates that $\lambda_0=0$,
  which has an intuitive explanation: if $\alpha$ can take one and only
  one value due to the constraints, then the objective function "plays
  no role" in the determination of $\alpha$ and so it gets a zero
  multiplier).

Back to our problem. The first order condition is

$$\frac {\partial L_{FJ}}{\partial \alpha} = \lambda_0\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) - \lambda_1 \leq 0$$

(note the "lower or equal to zero" which is the case when optimizing under inequality constraints, rather than just "equal").

First, we establish that $\alpha^* >0$. Due to $u'>0$ and the (strict) convexity of the utility function,$u''>0$, and the assumption that $E(z) >1$ we have (using Jensen's Inequality)

$$E[u(w+\alpha(z-1))] > u(w+\alpha(E(z)-1))] > u(w + 0\cdot(E(z)-1))=u(w)$$

We turn now to examine cases.  Since the two multipliers cannot be both zero, and $\lambda_0$ takes only two values, there are three possible combinations.

Examine the case $\lambda_0 =1$.
Then $\lambda_1$ can in principle be zero or positive. Examine the case where $\lambda_1 =0$, i.e. the constraint is not binding which implies that $\alpha^* < w$.  With this candidate pair of multipliers, $\{\lambda_0=1,\lambda_1=0\}$, the first order condition would become

$$\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) \leq 0 \Rightarrow E(zu') - E(u') \leq 0$$

Since $u'>0 \Rightarrow E(u') > 0$. Also, since $E(z)>1$ we have that

$$E(u')< E(u')E(z) \Rightarrow E(zu') < E(u')E(z) \Rightarrow \text{Cov}(z, u')<0$$

But this cannot hold,  because, since $\alpha^*>0$ and $u''>0$, we have that $u'$ will be strictly increasing in $z$. So the covariance of $z$ and $u'$ cannot be negative. But then the pair of multiplier values $\{\lambda_0=1,\lambda_1=0\}$ cannot be a solution, and this happens due to the assumption $u''>0$.

We are left with the cases
$\{\lambda_0=1,\lambda_1>0\}$, or $\{\lambda_0=0,\lambda_1>0\}$. In both these cases, $\lambda_1 >0$ i.e. the constraint is binding , i.e. we will have $\alpha^* = w$. QED.

  References

The Fritz John conditions have been stated in
"F. JOHN. Extremum problems with inequalities as side conditions. In “Studies and Essays, Courant Anniversary Volume” (K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker, eds.), pp. 187-204. Wiley (Interscience), New York, 1948"

and have been generalized in

"Mangasarian, O. L., & Fromovitz, S. (1967). The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. Journal of Mathematical Analysis and Applications, 17(1), 37-47.


Manias, Panics, and Crashes

by Charles Kindleberger

(Full book pdf)

An easier reading compared to Reinhart and Rogoff. It informally explains how crises occur and presents some famous examples.


What is 2nd round of rationalizability for a 1st price auction?

Elimination of (weakly) dominated actions will not get you far. In fact, all bids strictly between 0 and the value are undominated. Hence, a buyer's optimal bid in a first-price auction can't be determined without knowing the bidder's belief (unless the buyer's value is 0).
Claim: For any type $x>0$, all bids $c \in (0,x)$ are undominated.

$c$ is not dominated by any bid $b>c$: $b$ yields a strictly smaller
than $c$ if all other bidders bid $b_j<c$.
$c$ is not dominated by any bid $b<c$: $b$ yields a strictly smaller utility than $c$ if all other bidders bid $b_j=c$.

You can show that, for all types $x>0$, bidding zero is dominated by any $b\in (0,\frac{N-1}{N} x)$, though.
Suppose $p$ is the probability that all other $(N-1)$ bidders bid zero. Then, bidding zero yields utility $\frac{p}{N}x$: winning by a randomization because everyone bids the same and then winning the lottery and paying nothing.
However, when bidding $b$ from the interval above, your exepected payoff is at least
$$\geq p(x-b) > p (x - \frac{N-1}{N} x) > \frac{p}{N}x.$$


Is a lack of "rules" dangerous for cryptocurrency

ELI5: What actually makes makes currency worth anything? : explainlikeimfive

One reason why fiat money has value, is that the government insists that people pay taxes, and the only payment of taxes they will accept is in their own currency. So, everyone has to earn or acquire enough in that currency to pay their taxes.

You asked

If one day individuals decide to not put any value behind XYZ-Therum Crypto (using a generic name), won't that cause a collapse?

Yes is your answer. As at July 29 2020, no governments allow citizens to pay taxes with cryptocurrency, unlike fiat currency. Thus "If nobody is actually using them, that means the platform has no adoption, which means the tokens aren't worth anything and it's all just a dangerous speculative pricing game..."

For more detail, read ELI5:Crypto currencies how are they actually worth anything? : explainlikeimfive.

TL;DR: BTC's intrinsic value is represented by fractional participation in the 'illegal economy.'
You're getting a ton of 'BTC is fiat!', which feels true, but this isn't really true. Its the same situation for the USD, as much as people like to scream "THE DOLLAR IS JUST FIAT!!" over at places like zerohedge (don't go there - that website is trash).
Let's start with the USD (but the same is true about the yen, euro, ruble, etc). The thing about the US dollar is that there is an unequivocal end demand for US dollars, imbuing it with an intrinsic worth: if you want any level of participation in the US economy, you need dollars to buy it (to get access). Think of it this way - the US dollar is the vehicle by which you gain access to the US economy.
"But but but you can just convert....." No you can't 'just convert'. What would you say if your employer told you tomorrow that they were going to pay you in rubles, euros, yen or even gold? You wouldn't say anything - you'd quit. Why? Because you have to convert to USD to do anything within the bounds of the US economy. Paying your rent, going to Kroger, buying a car, getting a computer, hiring a maid, paying your taxes all require USD. In the case of your employer paying you in an alternative currency, instead of quitting, you might demand a raise to cover transaction costs, taxes and hassle as you convert back to USD.....so you can participate in the US economy. As such, there is a floating but measurable demand for USD. It is not fiat in the way that that it is typically described.
In slightly more technical terms: the 'intrinsic value' of the USD is a 'right of participation' within the US economy and all ROPs have an intrinsic value. We can debate the value of the ROP, but we are both gonna agree its there. I come on here all the time and talk about how the size of your money supply needs to be linked to the size of the economy - this is the manifestation of that link and an explanation of precisely why it has to be linked. SIDENOTE: John Nash (Nash Equilibrium & A Beautiful Mind) was working on 'solving' the precise math on this when he died, but he admittedly did not think it was a solvable problem.
Back to BitCoin: BTC's intrinsic value is tied to the size of the 'BTC economy' in that the value of a single bitcoin represents a 'right to participate' in the BTC economy. So what do people use BTC for? Well, let's be honest: mostly illegal shit. Here's a really interesting thought piece by a guy who tried to value BTC by measuring against an estimate of the illegal economies that BTC helps to facilitate. We can argue the particulars in this piece all day long, but I really like the economic/financial construction behind the theory - the guy is definitely approaching it the right way.
Now there are some obvious problems here, with the most obvious two being: 1) you are 'long' illegal activity which is very risky from an investment standpoint (I don't mean BTC holders 'go to jail', I mean you have no reasonable recourse if shit goes against you, such as criminal prosecution of the criminals that comprise the BTC economy); and 2) governments want to maintain their currency monopolies and will be fighting you forever (and again you have no recourse).
So how does BTC become really valuable? If BTC is allowed to edge in on the 'legit' economy, then BTC becomes very valuable b/c it becomes a viable alternative to traditional currencies (eg, you now get global access to local economies.)
However, let's be super fucking honest: the last thing the US gov't is going to allow is for a competitor to the USD, so when you buy BTC keep in mind that the government will be fighting you. And yes, they can stop the growth of BTC by restricting access by preventing any company that does business in the US from accepting BTC in any, way shape or form. Additionally they could prevent any transaction that might have come from BTC via existing money laundering laws. That alone could (close to) zero BTC overnight. If you think that could never happen, you're living in fantasy land.
In summary: BTC is granting you fractional participation in an illegal economy - that's its intrinsic value. However, I would not call it a currency (or an asset or a value store) because you do not have any of the legal protections that are inherent with currencies, assets or value stores. This is why its speculative as fuck.
Source: current quant hedge fund manager with a career on wall st. People like me get drunk together and talk about this shit all the time.


What is the "gross national saving" for?

The first point is that this is a flow statistic: it is the annual excess of income over consumption, in contrast to level statistics such as gross national debt which are largely accumulations of past flows.

As what gross national saving is for: it funds investment, which affects future income and thus future consumption.  Indeed, one of the tautologies of macroeconomics is that Savings = Investment.

A low savings/GDP figure may indicate excessive consumption or insufficient  income.  It may lead to either inadequate domestic investment or the need to borrow from foreigners to fund domestic investment.

A high savings/GDP figure may indicate insufficient consumption or excessive income.  It may lead to either unproductive domestic investment or the need to lend to foreigners to recycle the funds.

There is no optimal figure for savings. Comparative figures for different countries and global trends over time are published, for example from the World Bank.


The labour rate index is actually synonymous with a more popular term, the labour cost index.

In the same way that the consumer price index measures inflation of consumer prices, the labour cost index measures the cost of purchasing labour. For example, 12 dollars could buy you more labour 30 years ago than it can now. Inflation indexes like these can be used to normalize historical data. That way we can compare changes and trends across years without worrying about labour cost being inflated.

https://nso.gov.mt/en/nso/Sources_and_Methods/Unit_C2/Labour_Market_Statistics/Pages/Labour-Cost-Index.aspx

https://www150.statcan.gc.ca/n1/en/pub/71-586-x/71-586-x2001001-eng.pdf?st=mrJ1RXhQ


How to derive cubic cost function from a problem of constrained optimization?

Given a Cobb-Douglas production function your cost function will be cubic exactly if $\alpha + \beta = \frac{1}{3}$, and in this case you will have $a = b = c = 0$. There is no going around this. If you want to have the kind of cost function you describe you will have to slightly alter your Cobb-Douglas function, e.g.
$$
f(k,l) = (k-1)^{\alpha} \cdot (l-2)^{\frac{1}{3} - \alpha}.
$$


Relationship between expected utility and independence axiom

If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom  of independence.

For example in prospect theory, consumers display loss aversion, which  means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected utility form, but the utility function is still continuous. This preferences violate the independence axiom.

Similarly, you could have preferences that do not admit the expected utility form, and satisfy the independence axiom, but not continuity.


EU-Canada Comprehensive Economic and Trade Agreement?

I did not have either the time or the courage to plough through the text. The last consolidated text I could find stems from 2014 and has since been amended with an annex. Changes made are unclear to me.

The full text can be found here.

Excerpts that might be useful in this case are article 1.2 and 1.3 with respect to the scope of the application. Other excerpts might be the articles 17.2 and 17.3.

AS for the relevance of articles 1.2 and 1.3:
CETA applies to both the citizens and the territory of the agreeing countries. So, assuming USA corporations can have, and often do have, offices in Canada these companies can through these offices take advantage of the CETA agreement, assuming they keep to the regulations and requirements guaranteed under CETA. This means not everything from TTIP will be directly transposable. There are other treaties that cannot be overruled by CETA.

But in a way yes - a corporation holding office in the Canadian territories could in essence access the European markets under the same conditions. At least that is what I understood. I get back to the question when I'm through the 1598 pages of text and annexes (the latter being the greater part, but not less important).

Vague answer I know. I'm looking forward to a more in-depth conclusion because what you ask is something that a lot of people here in Europe are worrying about.


Why doesn't land value go to 0 when tax rate goes to 100%?

The correct answer is that it does not. It goes to zero as tax rate increases to infinity.

The reason for that is that the rent derived from land (assuming away any necessary land maintenance) can be thought of as a windfall gain. As a result all incidence of burden from such tax must fall on the holder of the land. Already If the tax is equal or above 100% the whole rent is confiscated and hence the holder of land would be willing to sell it away for nothing, although in the model you shown 100% tax rate would not reduce the property value to zero but to:

$$y=\frac{ra}{r+1}=\frac{0.01}{1.01}\approx0.0099>0$$


I'm just gonna try doing it. The problem is:

$$\text{min } C = rK + wL \\
\text{s.t. } q = K + \ln(L) $$

The implied constraint, of course, is $L \in [1, \infty)$ (non-negativity constraint, as $q > 0$, and $K = 0$ is, of course, possible).

We can try solving this using Lagrange multipliers. Define the Lagrangian:

$$\mathcal{L} = (rK + wL) + \lambda(q - K - \ln(L))$$

The first order conditions are:

$$\mathcal{L}_{K} = r - \lambda = 0$$
$$\mathcal{L}_{L} = w - \lambda(\frac{1}{L}) = 0$$
$$\mathcal{L}_{\lambda} = q - K - \ln(L) = 0$$

Using some algebra, we get $r = wL$ and $L = e^{q-K}$. Substituting into the original cost equation:

$$C = K(we^{q-K}) + we^{q-K} = we^{q-K}(K + 1) $$

I don’t think this can be simplified much further.

That seems to be the same answer that you got (I just expressed it in terms of wages and capital). So not sure – maybe I’m making a mistake.


Is over-valuation of a start up genuinely detrimental to the start up's future?

I believe this answer is relevant:

Why a new startup should ask a big investment company to value them as low as possible?

In brief, if founders accepted the investor offering at the highest valuation, this investor would be their last one.

But if a startup is looking for money alone, it is a bad startup.

The access to reputable investors (who valuate realistically) gives more opportunities. It includes experienced members of the board, better brand to attract candidate employees, and the track record of closing investment rounds with famous investors.


You ask:

  Is there any economic value contributed by humanities faculty?

A response from Neo-classical economics:

If there are for-profit companies hiring graduates from these faculties, then yes, they do produce value. The sum of their wages is their contribution in terms of value-added to GDP.

Same applies to economists. If they are hired by companies, then their production is valuable. I see no particular difference between economists and any other degree, and I do not see why you make such an a-priori distinction, other than an implicit view of economists as superior (widespread in the field, btw).

Now, the argument about the value of graduates that is allocated to non-profit organisations, including the state, universities, NGOs, and etc, goes exactly in the same way. Consider Greenpeace. If donors of Greenpeace consider it's activity as valuable, and Greenpeace wants to hire economists and artists, then their production has value, because some people benefit from Greenpeace activities. As such, there is no reason why you would not value the contribution of all these careers. Again, the argument applies to both economists and non-economists.

The only situation where you could argue that there is no value produced is when a degree is producing graduates which are then hired to run those same degrees, in a perpetual circle. I am afraid that economics is just as likely to suffer from this than any other discipline.


Updated International patent protection index from Park (2008)

Apparently not. A 2015 article by the author uses his index to evaluate economic growth across country and time, stopping in 2005, same as the index in the original paper. And yet, many of the other variables are in principle available for more recent years.

The first best here would be for you to contact the author. He might be in the process of updating his index. Actually, if you look into his working papers, you can see he is doing some work on patents. Even more, there he offers some (more updated) data on copyright and Trademark. You might contact him to know better what this is.

The second best is that you expand the index yourself. For example, you can use the data just mentioned. Alternatively, you can use the subcomponents of this or this index (e.g. property rights or investor protection). There are more indexes here (e.g. intellectual property protection), and here.

One way to do this update is to take the original Park's series, and see which one of the above match its better (higher correlation, for instance). From here you obtain your "proxy". Then, simply extend Park's index with the newest data on this proxy. This is simply an exercise of "concatenation".


Why does BC Hydro want us to use less power?

BC Hydro is a a public body: that means it can run for the wider public good, rather than purely as a profit-maximiser.

Pretty much everywhere in the world, electricity markets are dysfunctional. They typically have high negative externalities.

Historically, electricity prices haven't included all the costs of the electricity, which means that electricity has been over-consumed, relative to economically-efficient levels.

Energy-efficiency is, typically, the cheapest (when measured as a whole-system cost) way to provide additional energy services.

Furthermore, there is some evidence that energy-efficiency measures have a lower uptake than would be efficient, due to consumers having insufficient information, and due to them having irrationally high discount rates for future energy-bill savings.

So the ads can help to correct these imbalances.


Consumer Theory (Finding demand functions)

Given the data:

Utility function: $u(x, y) = \sqrt{x} + y$
Income: $I > 0$
Prices: $p_X > 0$ and $p_Y = 1$

Utility maximization problem is

\begin{eqnarray*} \max\limits_{x,y} & \ \ \sqrt{x} + y \\ \text{s.t.} & \ \ p_Xx+ y = I \\ \text{and} & \ \ x\geq 0, \ \ y\geq 0 \end{eqnarray*}
We can substitute $y = I - p_Xx$ in the objective and convert it into a single variable optimization problem:
\begin{eqnarray*} \max\limits_{x} & \ \ \sqrt{x} + I- p_Xx \\ \text{s.t.} & 0 \leq x \leq \frac{I}{p_X} \end{eqnarray*}
Differentiating the objective with respect to $x$ yields:
$\frac{1}{2\sqrt{x}} - p_X$
which is a decreasing function of $x$. The interpretation of the derivative is that it is the net marginal benefit curve of consuming X. As long as the value of the derivative is positive, it pays to spend more on X. Therefore, the utility maximizing choice will be to spend all the money on X if the net marginal benefit curve is still positive at $x = \frac{I}{p_X}$ i.e. $\frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X>0$ holds, and the equilibrium choice will satisfy the property $\frac{1}{2\sqrt{x}} - p_X = 0$ otherwise.
So, the demand function for X is:
$\begin{eqnarray*} x(p_X, p_Y = 1, I) = \begin{cases} \frac{1}{4p_X^2} & \text{if } \frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X\leq 0 \\ \frac{I}{p_X} & \text{if } \frac{1}{2}\sqrt{\frac{p_X}{I}} - p_X>0  \end{cases} \end{eqnarray*}$


How to calculate payback period of the project?

This is more finance than economics but here it is:

The payback period is pretty much how much time you need to recover the initial investment.

Each month, you are saving 10000 - 500 = 9500.  Then you simply just divide 50000 by 9500. This comes out to be around 5.

I made some simplifying assumptions: Discount rate is 0. This is usually not true, so you would probably have to discount the cash flows. Note: This will probably involve monthly compounding. Check out the formula for annuity.

Also, just FYI: payback method often results in wrong decisions. When you have a choice, always use NPV.


The easiest consistent explanation is using multiplication: The Q2 estimated loss of $231$ is $\frac{231}{150}=1.54$ times the Q1 estimated loss of $150$.  This is a combination of

The exposure being $1.5$ times the previous figure
The loss given default being about $0.933$ times the previous figure
The probability of default being $1.1$ times the previous figure

and this is consistent with $1.5\times 0.933 \times 1.1 \approx 1.54$ - it does not matter which order you do the multiplication

It is not that far away from saying $50\% -6.7\% + 10\%$ is about $54\%$, except that that is not precise (they actually add up to $53.3\%$ and other examples could be worse) but this would be talking about percentage changes in percentages, which is often confusing

If you looked at the effect on the old estimated loss of $150$ of each change if the others had not changed, you would have got $+75$ (not $+76$), $-10$, $+15$ which add up to $+80$ rather than the actual $+81$, and other examples could be worse


Examples of Covariance Stationary Time Series

Let $\epsilon_1, \epsilon_2, ... ,\epsilon_t,...$ denote i.i.d. stochastic variables, each having a standard deviation of $\sigma$.

Let
$$
x_t = \sum_{i = 1}^t \epsilon_i.
$$
Then
$$
cov(x_t,x_{t+h}) =
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right).
$$
Because of the i.i.d. property
$$
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right)
=
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
$$
and furthermore because of independence
$$
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
=
\sum_{i = 1}^t \sigma +
\sum_{i = 1}^t\sum_{j \neq i} 0
=
t \cdot \sigma.
$$
Now we have shown that for this time series
$$
cov(x_t,x_{t+h}) =
t \cdot \sigma
$$
depends on $t$ hence it is not covariance stationary.

For a covariance stationary time series, let
$$
z_t = \epsilon_{t-1} + \epsilon_{t}.
$$
By performing the same analyis, you can conlude that for $t > 1$,
$$
cov(z_t,z_{t+h}) =
\left\{
\begin{array}{ll}
\sigma & \text{ if } h = 1 \\
0 & \text{ if } h > 1.
\end{array}
\right.
$$
Since this depends on $h$, but not on $t$, the time series $z_t$ will be covariance stationary (after $t = 1$).