Which countries have the comparative advantage in Ricardo's example? I am a complete beginner here trying to learn through Khan Academy. I found some data that correspond to Ricardo's explanation of comparative advantage. According to Wikipedia "England has a comparative advantage in producing cloth due to lower opportunity cost." Yet, when I work through it I get the opposite, where am I going wrong?

          Cloth Wine
England     100 120
Portugal    90  80

England

100 = 120
100/100 = 120/100
1 = 1.2

For 1 cloth there is an OC to England of 1.2 wine. For 1 wine there is an OC to England of 0.8333 cloth.

Portugal

90 = 80
90/90 = 80/90
1 = 0.8888

For 1 cloth there is an OC to Portugal of 0.8888 wine. For 1 wine there is an OC to Portugal of 1.125 cloth

England should produce wine because its OC is lower (0.8333 < 1.125).

Portugal should produce cloth because its OC is lower (0.8888 < 1.2).

The mistake that you are doing is that you are treating the numbers in the table below that I assume you took from wikipedia as quantities that can be produced but they are not

          Cloth Wine
England     100 120
Portugal    90  80

Wikipedia states that in the problem they present these are "hours of work necessary to produce one unit" of given goods. So its not that England can produce $100$ cloth and $120$ wine. It takes $100h$ of labor to produce $1$ cloth and $120$ hour to produce $1$ unit of wine. Same for Portugal. This is the original table from wikipedia from which I assume you took the numbers:

So in this example if England had access to only $100$ hours of labor it would be able to produce exactly $1$ cloth and only $5/6$ units of wine.  Hence actually the opprotunity cost of $1$ cloth is $0.833$ wine and for $1$ wine the opportunity cost is $1.2$ cloth- the exact opposite in terms of OC.

The same holds for the Portugal - you will find exact opposite result namely in Portugal the opportunity cost of $1$ cloth is $0.888$ wine and opportunity cost of $1$ wine is $1.125$ cloth.

Since the opportunity cost of producing cloth is lower in England it has comparative advantage in cloth production and since the opportunity cost of producing wine is lower in Portugal it has comparative advantage in wine making. Hence, also the final result will be opposite.

Consequently, in this case wikipedia is correct in stating that "England has a comparative advantage in producing cloth due to lower opportunity cost" given the numbers they assume. So England should produce cloth after all. You have to read problem very carefully - dont just assume that you know what the table presents.


Is this a valid or legitimate way of comparing countries on traits and characteristics? Country A has 16 million inhabitants and 2.000 MW of solar panel
generated power, also a GDP of $50.000 per capita

Country B has 160 million inhabitants and  2 MW of solar panel
generated power, also a GDP of $10.000 per capita

Would it be correct to say that country A has two thousand times as much solar
power because:

If country A was as populous as B, they would have 20.000 MW in solar power.
But if country A  was as poor as country A, they would have 4.000 MW solar power.  So, after adjusting for differences in population and
wealth, country A still has 2000 times the solar power that country B
has?

Is it correct to adjust for differences in population and
wealth in that manner, or not?

Follow up question: Is there a formal way, when comparing countries or
groups, to adjust for differences in various traits (such as
population size and GDP per capita?)

I have no idea whether there are formal rules, but you can emulate physical scientists and look at the units.

Dividing solar power by population is reasonable, as you get megawatts per capita. Those refer to real quantities.

However, you are dividing by GDP (in a roundabout fashion). The units of GDP are (currency unit)/(year). Since megawatts are a real quantity, and thus no currency value, the resulting ratio is (megawatts)(years)/(currency unit). This does not appear to correspond to anything.

You could try to fit a model that predicts solar capacity as a function of wealth and population, and the model calculations should implicitly correct the units.


Is it correct to say that wealth is "destroyed" when an asset bubble bursts? I was reading this article from the World Economic Forum, when I came across this passage:

  When the global equity and real estate markets hit their lowest points in March 2009, \$34.4 trillion of global wealth was destroyed. US households lost almost \$8 trillion in the stock market, on top of the \$6 trillion loss in the market value of their homes.

Is it fair to truth to say that the wealth was "destroyed"? I ask this because that type of phrasing gives the impression that the wealth was really there at one point, and that it wasn't merely an illusion.

I can think of a few reasons why this may be true:

The wealth really did exist elsewhere at first, but when resources were diverted from other assets/resources and invested into real estate – which turned out to be a bubble – those resources had been spent on assets that weren't productive (i.e. didn't give a positive return). The wealth was moved over from valuable assets to less valuable/worthless assets. Capital and labor was spent for nothing.
Wealth is never really expressed in intrinsic "real" terms, but is always a subject of valuation. Wealth is only worth what people value it to be, i.e. what resources they would give up for a transaction of that wealth.

So, is it correct to say that wealth can be destroyed if there was nothing substantial to back it up?

We need to distinguish two things here: the intrinsic value of the assets, and their fair value, or market value.

No matter how much the price of an asset changes on financial markets (say, stock exchanges), its intrincic value stays the same, all other things equals. Typically, you could use the cost of the asset to assess its intrinsic value: for a house, the price of its construction. For a share, the book value per share of the underlying firm. For bonds, the nominal (or face value) amount.

But assessing cost is sometimes tricky. The book value might incorporate things that are valued at fair value. For real estate, the price of the land is a large part of the total cost, and it changes all the time. For bonds, you could incorporate interests and discount them, but you need to choose a discount rate.

Thus we typically assess wealth using fair value. It's a concept precisely defined by IFRS (IFRS 13 - Fair value measurement). Basically, the fair value of an asset is the price at which you could sell it under normal market conditions ('at arm's length'). It is true 'value', as you could trade the asset against that amount of cash.

To come back to your question: wealth is measured at fair value, or market value. Therefore, the $34.4bn were indeed destroyed, or disappeared, even though the intrinsic value may have been constant. Your second suggestion is correct.


Interior Solution for profit maximisation problem A function $c: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$ is is said to be a cost function if

The value of function $c$ at $y = \textbf{0}$ is $0$: $c(\textbf{0}) = 0$
$c$ is continuous on the domain $\mathbb{R}^K_+$, strictly increasing and strictly convex on $\mathbb{R}^K_{+}$
For any $p \in \mathbb{R}^K_{++}$, the set $B(p) = \{y \in \mathbb{R}^K_+| c(y) \leq p \cdot y\}$ is compact, the interior of $B(p)$ is nonempty.

Suppose a firm has cost function $c(\cdot)$ and $p\in \mathbb{R}^K_{++}$ is the market price. What sort of topological assumption I need to add to guarantee the existence of an interior solution to the profit maximisation problem?
$$
\pi (p) = \max_{y\in\mathbb{R}^K_+} \left\{ p\cdot y - c(y) \right\}
$$
Remark 1: From assumption (2) and (3), the existence of solution is guaranteed. Due to strict convexity, uniqueness is also guaranteed. However, the solution is probably not inside the interior of $\mathbb{R}^K_{+}$. I could not find an example for this at the moment.
Remark 2: My prototype cost function to look at is $c(y) = y_1^2+y_2^2$. However, the cost is not necessarily to be additive separable.

Your assumption 3 is compatible with corner solutions of the kind $y_i=0$ for some $i$, and is not sufficient to avoid corner solutions for some prices small enough.
With production functions it is common to assume Inada's conditions to avoid corner solutions. With cost functions, such conditions are quite naturally expressed as
$$ \lim_{y_i\rightarrow 0} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=0 $$
$$ \lim_{y_i\rightarrow +\infty} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=+\infty. $$
In your discussion, you proposed the cost function $$c(y_1,y_2)=y_1^2+y_2^2+2y_1$$ to illustrate a corner solution. This function does not satisfy the first Inada conditions wrt $y_1$, hence the corner solution.
A general comment to conclude: it is not fully natural to forbid firm's exit from a given output market. In IO there is a quite dynamic literature on entry and exit. And no firm is producing all products available in the economy.


Use Brouwer's Fixed Point Theorem to Prove existence of equilibrium(a) with completely mixed strategies How can one use Brouwer's Fixed Point Theorem to prove that the following game F has a solution:

F is defined as N={L,R} Ai=(g,1-g) where g must be positive and smaller than 1, that is, each player plays a completely mixed strategy and has the following payoff matrix:

\begin{array}{|c|c|c|}
\hline
&LU&LD\\\hline
RU&0,0&6,3\\\hline
RD&3,6&0,0\\\hline
\end{array}

I reached the conclusion that one cannot use such theorem to prove there exists a solution to the game since the plays do not belong to a compact set. Can anyone use it?

Rather than deprive you of the joy of solving your own problem, I will show how one can also use Brouwer's theorem to show that the function $f:\mathbb{R} \to \mathbb{R}$
$$
f(x) = \frac{33+x}{16+x^4}
$$
has a fixed point, even though its range is not compact.

Note that $f(1) = 2$ and $f(2) = 35/32$. As $f$ is monotonically decreasing over $[1,2]$, this also means that $f([1,2]) \subset [1,2]$. Also, $[1,2]$ is compact and $f$ is continuous over this set. So $f: [1,2] \to [1,2]$ will have a fixed point by Brouwer's fixed point theorem.

Similar tricks are used in game theory and general equilibrium theory. By showing that the optimal decisions by some players cannot possibly be outside of some boundaries one can frequently create a compact space over which fixed point theorems are applicable.


Real yields and growth Found the following very interesting statement on bond prices,

"The yield on a risk-free government bond should equal the growth in nominal GDP since that represents the opportunity cost of holding a government bond both in terms of investment opportunities (real GDP) and the time value of money (inflation)"

http://globalfinancialdata.com/7-centuries-of-government-bond-yields/

Question) How does one square the circle of this statement and the "expectations hypothesis"?

Most significant factors in the price of a goverments bonds

Question)Is future growth or short term rates that determine longer rate? or are these two the same somehow?

The first statement is a result from an old economic model.

The two views can be reconciled by the following argument: the central bank is constrained by the structure of the economy as to its choice of the policy rate. If persistently too low, inflation rises, and if persistently too high, the economy risks falling into deflation.

In this case, the central bank allegedly needs to keep the real policy rate near the growth rate of real GDP. Arguably, not many people are convinced that is precisely the case, rather that there is a “natural rate if interest” (now preferably referred to as $r^*$) that the policy rate needs to revert to.

The argument is then that bond investors will be aware of this tendency, and thus bond yields will revert to the long-term average.

However, one would need to search for the modern literature on $r^*$ to get a better handle on preferred ways to estimate it. I am in the camp that $r^*$ does not exist, so I am not the person to offer such references.


Are there stable improvements of the Ramsey-Cass-Koopmans model? The Ramsey-Cass-Koopmans neoclassical model of growth is saddle path stable. In other words, it is stable upon perturbation from the steady state on a one-dimensional line, but is unstable towards a generic perturbation in the two-dimensional space of capital/labor ratio and consumption.

My question is whether there are (realistic, standard) neoclassical growth models that are stable upon perturbation (beyond the simplest Solow type models) ?

The economic essence of saddle-path stability, is that it requires conscious behavior and decision making from the part of economic agents, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, economic systems are unstable.

Contrast this with the basic Solow growth model: there, fixing arbitrarily the savings rate (i.e. depriving ourselves from making a decision on it period per period) makes the system properly stable. But once we want to optimize with respect to the savings rate, we find out that there is a single path that leads to long-term equilibrium.

Now, perusing the pages of a book like Barro and Sala-i-Martin's "Economic Growth", which contains many-many extensions of the basic RCK growth model, one will not find a model with intertemporally optimizing agents that is also mathematically stable.

ADDEDNDUM
Elaborating, in response to a thoughtful comment by the OP, my approach to the issue of "stability" of an economy is the following: The fact that economies tend to "correct course" is not an indication, let alone proof, that they posses some inherent stabilization mechanism independent of the purposeful action of the economic agents.
If this was the case, then indeed, we should have models that possessed the mathematical property of stability (I remind again that "saddle-path stable" although it uses the word "stable", is a mathematically unstable system).

On the other hand, economies do not plunge into chaos every day. The solution to the puzzle from my point of view is not that we "continuously fine tune the economy" -such dreams and visions have been tried and squashed (irrespective of what ideological/philosophical (dis) agreements one may have with socialism, the factual lesson to be learned from the failure of centrally planned economies is that an economy is such a complex system that no one can "plan and then fine-tune it", especially in our era).
My answer is that economies, after being hit by external shocks or internal institutional failures or technology upheavals, or.., tend to return to a dynamic path that tends to some fixed point by the aggregate consequences of decentralized purposeful action of economic agents that try not to fix the economy as a whole, but their own economic present and future. Economies survive not "because they inherently do", not because the government is "steering the boat with a certain and effective hand", but because individuals try hard every day to survive.

And the RCK models reflect just that: if one performs comparative statics on it, assuming an external shock that suddenly puts the economy out of the saddle-path, the prediction of the model as to what will happen in order for the economy to return to the (new) saddle path towards the (new) long-run equilibrium, is more or less what we observe in reality. But this happens not because the government makes it so, but because individual economic agents realize that they must do so in order not to eventually self-destruct.

I will indulge myself and I will link to a little exposition I crafted a year ago for a class of graduate students, to show them how the simplest RCK model can help them think about the economic crisis (and subsequent depression) in Greece -which was sudden enough to be described as a shock. The exposition can be found here.


Balance of Payments and Capital Inflows We know from the balance of payments identity that:
$$
BOP=CA+KA+FA=0
$$
 where $CA$ is current account, $FA$ is financial account, and $KA$
is capital account. The BOP is an identity that always balances. Although
in some cases, the intuition is quite clear how this happens, it is
not always transparent. Specifically, there are a few types of external
financing that take place: bond finance, FDI, portfolio investment
and government foreign bank borrowing. Note that these will show up
as a + in the Financial Account. Imagine then that these funds are
used to fund domestic purchases (building a damn, building an office
etc). These are technically inflows of external capital that are funding
domestic purchases, and hence not imports. I wonder then, how do these
activities balance out mechanically in the BOP? In other words, how
are capital inflows that go towards financing domestic consumption
balanced out in the BOP? Thanks!

Take a country like Uganda, a net importer. Let's say that the government borrows foreign capital to finance a dam construction. Using the funds, it pays for construction materials, most of which are probably imported; it pays for the design of the dam, the architects and specialists are foreigners; construction companies purchase fixed capital stock (trucks, cranes, etc), most of which are imported, and builders and workers receive wages and they spend most of their income on consumption of imported goods.
Hope this helps.


implied volatility, how to get parameters? I understand how to calculate volatility and how to calculate call or put price.
However I don't understand something about input parameters.

For an example.

enter link description here

At the moment I am writing this post, implied volatility for Appl with 190 Strike Call is 22,69%. My question is, based on what TERM? (Time). Because if you want to calculate real implied volatility you must need to know what period of time to include. Time to excerise doesn't tell you much, if you don't know what period you must watch. One month, half a year, one year, 2 years etc.. I assume this 22,69 implied volatility is based on days_to_expire_parameters/some_period. I am trying to figure what is this some_period. I

The implied volatility is defined as the volatility parameter that gives you the option price currently trading on the market. That is, you will observe market quotes for option prices or implied volatilities and you use the Black-Scholes formula to convert from one to the other. The implied volatility is not therefore something you computed based on stock price data, it's really a price determined by the market.

There are as many implied volatilities as there are options traded. In fact, options are frequently quoted in implied vols. You have one implied volatility for every strike price and for every maturity. This defines the "Volatility surface". Instead of the strike price you frequently see on one of the axes the "moneyness", that is the ratio of current price to the strike price.

Take a look at this graph.
https://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/23316/versions/2/screenshot.jpg

That means, for example, that the 3-month 1-moneyness implied vol is the volatility parameter that you need to input in the BS formula to obtain the observed market price of the 3-month 1-moneyness call option.


Recommended reading to understand American economic history In India Unbound, the author goes over India's economic history, right from independence to present day. I am interested in a similar reading for the United States, specifically conditions in the 20th century.

Any decent suggestions which comes close? I found 10 books but any one in particular which stands out and is a must read?

Currently this one stands out to me, other suggestions welcome. The Rise and Fall of American Growth: The U.S. Standard of Living since the Civil War

Found this to have more breadth on the overall US economy: The Rise and Fall of American Growth: The U.S. Standard of Living since the Civil War


Impact of working capital constraints onto output I'm looking for empirical evidence (optimally from natural experiments, as described below) of the impact of working capital constraints onto firm output, and the differentials of firm size.

I suppose that generally, stricter working capital constraints are bad for output, and typically these constraints are stronger for smaller firms (measured in output) and younger firms. However, I couldn't manage to find (a) paper(s) to establish that fact yet.

Where by working capital constraint I mean something along the lines of Kiyotaki&Moore (1997)

$$L, K: f(wL, rK) \leq \psi(F(K,L))$$

, for example, for some constant $\psi_0$,

$$wL + rK \leq \psi_0 \cdot F(K,L)$$

where $F(\cdot)$ is the production function, and $f$ and $\psi$ can be whatever.

I have, in other environments, seen that some people take bank liquidity to approximate $\psi$ and argue that for example the last recession was a natural experiment on a change of $\psi$.

So, one type of useful regression table would be (on the firm level)

dependent variable: output
independent variables:
Increase of financial constraint (here, decrease of $\psi_0$)
Dummy for firm size (or different size bins)
And/Or: Dummy for firm age (or different age bins)
additional controls

One example would be Chodorow-Reich (2014, QJE). The two downsides with that paper are that

He uses employment, not output as dependent variable (but that's tolerable)
He does not actually list the coefficients on the dummies for size. He just includes them as controls.

I can think of at least one paper where binding accounting rules are shown to have negative consequences on real investment:

Can Tight Accounting Oversight Distort Investment? Evidence from Mortgage Restructurings after the JOBS Act

  Abstract: I study the effect of accounting oversight on a bank’s level
  of troubled mortgage restructurings. If banks manage their accounting
  numbers (e.g. regulatory capital or earnings) in the wake of the
  financial crisis, then they may be hesitant to restructure troubled
  loans because such restructurings require the bank to mark the value
  of the loans down. I show that giving a bank more accounting
  discretion leads the bank to restructure more troubled mortgages. My
  identification strategy uses a provision of the Jumpstart Our Business
  Startups (JOBS) Act of 2012 that allows a subset of banks to
  deregister from the SEC. The first stage of the analysis shows that
  accounting oversight is of first-order importance in the
  deregistration decision; in particular, there is a sharp increase in
  the probability of deregistration from 7% to 30% for banks that can
  escape accounting oversight. By instrumenting for the decision to
  deregister with the bank’s newfound eligibility to deregister after
  the JOBS Act, I show that deregistration leads banks to slash their
  accounting and audit fees and make fewer provisions for loan losses. I
  next show that deregistration induces banks to double their investment
  in mortgage restructurings, which entail large upfront accounting
  expenses; this effect is driven by banks with low capital ratios, and
  estimates for non-mortgage troubled debt restructurings react
  similarly. These results suggest that banks manage their capital using
  real activities, and this is one reason for the low level of mortgage
  restructurings since the recession.


Aggregating in a Continuum I am working with the following Economic model of labour and consumption decisions:

I have a population whose mass is normalized to one of consumers.
They derive utility from consumption $c$,supplying labour $l$ and a public good $y$
This is summarized by a differentiable utility function $u(c,l,y)$
Agents derive earnings $z$ from suppliying labour according to:
$$z=nl$$
where $n$ corresponds to a "skill level" such that: $n\in[0,\infty[$. Such skill level is distributed according to distribution of skills $F(n)$ with density $f(n)$.
$c_n$, $z_n=nl_n$ and $u_n$ are used to denote the consumption, earnings, and utility level of an individual with skill level $n$
the individuals maximize their utility function subject to a constraint:
$$c_n=z_n-T(z_n)$$

where $T(\cdot)$ is a differentiable "tax function"

 7.The public good is produced according to the sum of this tax function:

$$y=\int_0^{\infty}f(s)T(z_s)ds$$

We can use the equations above to express the maximization problem of the consumer as:

$$\max_{z_n}u\left(z_n-T(z_n), z_n/n, \int_0^{\infty}f(s)T(z_s)ds\right)$$

The first order conditions to this problem write (ommitting the arguments of the function $u$ for notation compactness):

$$u_{c}(\cdot)(1-T'(z_n))+u_l(\cdot)/n+u_y(\cdot)\cdot  \frac{\partial y}{\partial z_n}=0$$

I am interested in studying the last term $\frac{\partial y}{\partial z_n}$. It seems intuitive that $\frac{\partial y}{\partial z_n}=0$ since:

$$\frac{\partial y}{\partial z_n}=\int_0^{\infty}\frac{\partial f(s)T(z_s)}{\partial z_n}ds=0$$

However, I am not sure about the mathematical requirement necessary for the last equation to make sense. It seems somewhat contradictory to the intuition yo get when aggregating using sums, for instance a discrete analogous would be given by:

$$\sum_{s=0}^{\infty} T(z_s)$$

and it is clear that the derivative of the last expression with respect to $z_n$ would be equal to one.

Could you help me out here? Maybe some broad claryfication of integral aggregation would be required. Thanks!

To obtain $\frac{\partial y}{\partial z_n}=0$, you implicitly assume that each consumer takes its own skill level as given. And in such a case, in reality the consumer optimizes with respect to labor supplied only. Perhaps it would be better to optimaze with respect to $l$ given $n$.

Regarding aggregation the discrete analogous would not be what you write but

$$\sum_{s=0}^{\infty}p(s) T(z_s)$$

where $p(s)$ would be the proportion of the population having income $z_s$.

Note that $y = E[T(z_s)]$, the expected value of individual tax burden w.r.t skill level. The density of $s$ plays in the continuous case the role of "population proportions" in the discrete setting


Which regression technique is used for calculating price elasticity in practice Since we need to consider 'Endogeneity' between price and quantity while calculating price elasticity and since linear regression cannot handle the phenomenon of endogeneity if objective of the model is to find causality, instrumental variable regression is used.

Since theory backs the idea of using IV regression, does this happens in real world? I mean corporations and governments use IV regression for price elasticity or do they use linear regression despite it's shortcomings in it's inablity to handle endogeneity?

They use all types of methods.

Often, papers will use a very basic linear regression model and submit to a low prestige journal so one can justify ones grant and boost the H index as much as possible when meta analyses cite your article to clean up the mess.

It's up to the meta analysis to take into account that not all models have correct methodology and create a meta-analytical model to correct for all these inaccuracies.

Fortunately, there are sometimes dynamic lag models and the like to compare to and guess at an appropriate correction term for the others.

Here is an example for gas prices.

https://www.google.com/url?sa=t&source=web&rct=j&url=http://papers.tinbergen.nl/06106.pdf&ved=0ahUKEwia2Me-85LLAhUUzmMKHWvjAd0QFgghMAE&usg=AFQjCNHi6RH_xX7h9NLPcVRCFAMP0OtERQ&sig2=3ChFSPGwukN_WC4IlhitGA

Edit: A meta analytic model is a model to explain why studies obtained the results that they did. Different samples, methodologies, and so forth will produce different results, and a metamodel expresses this formally.

A dynamic lag model (also called a distributed lag model) is a model to predict the current value of a dependent variable based on past values of independent variables. For example, if a change in price causes a change in demand, you can model this by correlating price at time t-1 to demand at time t.


Labor demand and supply curves in terms of elasticities I would like to ask for your help to derive some expressions. Let the labor demand curve be described by:
$N^D = N^D(W/P, \bar{K}), \quad N^D_{W/P}= \frac{1}{F_{NN}}< 0, \quad N^D_K = -\frac{F_{NK}}{F_{NN}} > 0$
and the labor supply curve by:
$W/P^e = g(N^S)$
My question is, how can I express these two curves in terms of elasticities? I know that the result should be:
$\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} - \varepsilon_D \Big(\frac{dW}{W}-\frac{dP}{P} \Big) \\
\frac{dN^S}{N^S} = \varepsilon_s \Big( \frac{dW}{W}-\frac{dP^e}{P^e}\Big)$
where $\varepsilon_D=-F_{NN}/(NF_{NN})$ and $\varepsilon_S = g(N)/(Ng_N)$ denote the wage elasticities of labor demand and labor supply, respectively.
Thank you for your help and time.

This result is presented on page 14 of Foundations of Modern Macroeconomics. The key is to total differentiate both expressions.
Start with the supply of labor:
\begin{gather}
W/P^e = g(N^S) \\
d(W/P^e) = d\Big( g(N^S) \Big) \\
\frac{1}{P^e} dW - \frac{W}{(P^e)^2}dP^e = g_NdN^S \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{1}{g_N}\frac{W}{P^e}\frac{dP^e}{P^e} \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e}, \quad \text{where} \quad W/P^e = g(N^S) \\
dN^S = \frac{1}{g_N}\frac{g(N^S)}{W}dW - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e} \\
dN^S = \frac{g(N^S)}{g_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg] \\
\frac{dN^S}{N^S} = \frac{g(N^S)}{N^Sg_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg]
\end{gather}
To derive the expression for the demand of labor, one can follow a similar process (I will ommit the superscript $N^D$ for simplicity, but we know we are talking about labor demand). Recall that the FOC of the firm's problem is given by:
\begin{gather}
PF_N(N,\bar{K}) = W \\
F_N(N,\bar{K}) = \frac{W}{P} \\
d\Big( F_N(N,\bar{K}) \Big) = d\Big( W/P \Big) \\
F_{NN}dN + F_{NK}d\bar{K} = \frac{1}{P}dW - \frac{W}{P^2}dP \\
F_{NN}dN + F_{NK}d\bar{K} = F_N\frac{dW}{W} - F_N\frac{dP}{P} \\
F_{NN}dN = F_N \bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] - F_{NK}d\bar{K} \\
dN = -\frac{F_{NK}}{F_{NN}}d\bar{K} + \frac{F_N}{F_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = -\frac{F_{NK}}{N^DF_{NN}}d\bar{K} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg], \quad \text{where} \quad KF_{NK} = -NF_{NN}
\end{gather}


Competitive equilibrium in Leontief economies Consider an economy in which all consumers have, possibly different, Leontief utilities. Since preferences are not strictly convex, it is not guaranteed that a competitive equilibrium exists. I found some papers that discuss the computational problem of deciding whether a Leontief economy has a competitive equilibrium, but I am interested in general existence results:

A. What conditions on Leontief economies guarantee that a competitive equilibrium exists?

B. In particular, if the initial endowments are equal (each of $m$ agents receives a fraction $1/m$ of each good), is a competitive equilibrium guaranteed to exist?

Strict convexity of preferences is not needed in existence results for competitive equilibria. Leontief preferences are quite well-behaved. They are continuous, convex, and strongly monotonic.   If all endowments are strictly positive, the existence of a competitive equilibrium in an exchange economy (or a production economy satisfying standard conditions) exists by the first result of the original Arrow-Debreu paper.

Arrow-Debreu actually do not just require convexity, they make, as pointed out by denesp in a comment, the convexity assumption (III.c) on utility functions that $u(x)>u(x')$ and $0<t<1$ implies $u(tx+(1-t)x')>u(x')$. Plain convexity suffices for existence, but Leontief preferences do also satisfy condition (III.c).:  Assume $\min\{\alpha_i x_i\}>\min\{\alpha_i x_i'\}$. Then
$$\min\big\{\alpha_i (tx_i+(1-t)x_i')\big\}>\min\big\{\alpha_i tx_i\big\}+\min\big\{\alpha_i(1-t) x_i'\big\}$$ $$=t\min\{\alpha_i x_i\}+(1-t)\min\{\alpha_i x_i'\}>\min\{\alpha_i x_i'\}.$$


If the government builds a bridge, how do we know it's the best possible usage of real resources (i.e. steel, labor, etc) at the time? I would assume the "goal" of any economic activity is to make the largest amount of people the "happiest" -- i.e. Pareto Efficiency or sum-total 'utility.'
How do we know if this occurs? Either before, during, or after construction?

A short answer is that in case of provision of public goods in real-life we can never be completely sure if they are provided in optimal quantity due sheer difficulty of quantifying all costs and benefits, measurement problems when it comes to underlaying utility, due to uncertainties involved and many other factors.
However, this does not mean that no evaluation of public projects can be done. Most developed countries will have some dedicated analytic unit. In the Netherlands where I live that would be Centraal Planbureau [Bureau for Economic Policy Analysis] (CPB)  in the US it is Congressional Budget Office (CBO).
These units routinely engage in cost-benefit analyses of public projects and then make recommendations based on whether the projects net present value is positive. The net present value would be given as:
$$ NPV = B_0 - C_0 + \frac{B_1-C_1}{1+r} + \frac{B_2-C_2}{(1+r)^2} + ... + \frac{B_T-C_T}{(1+r)^T} $$
Where $B$ are benefits of the project, $C$ are costs and $r$ is discount rate or you can interpret it as a required rate of return.
The benefits and cost try to capture actually underlaying changes in utility. They will be usually estimated using hedonic pricing or various discrete choice models that can tell us something about peoples preferences and utilities based on observing their actions. For example, from the way how housing prices vary in cities with high congestion we can estimate how much people value time lost in traffic. The required discount rate is usually set to be equal to interest rate on government/municipality bonds or rate that government could obtain if it had invested the money into wealth fund. Of course, the way how these parameters are estimated or assumed is always a point of contention as this can't be done with infinite precision and complete certainty (this holds even in cases when we do ex-post cost-benefit analysis as most benefits and costs are virtually never directly observable and have to be estimated).
The above is, of course, just a short overview as providing an exhaustive review of how public cost benefit analysis is done is unfortunately beyond scope of SE. A good and highly cited example of public cost-benefit analysis is: Leape, J. (2006). The London congestion charge. Journal of Economic Perspectives, 20(4), 157-176.
Furthermore, just because project passes cost-benefit analysis it does not necessary means that the outcome is pareto efficient or even that it maximizes utility. However, it is reasonably good guard against completely frivolous spending.  This being said, advice given is not advice taken and such analyses are often ignored by politicians, as pointed out by Giskard public officials are not necessarily benevolent.


What are key economic mechanisms explaining long term damage during Great Recession can you tell me what could be key economic mechanisms that explain long-term damage of the Great Recession?

Big recessions produce mass unemployment and for long periods. People can get demotivated, and lose skills and training. As the economy evolves, new skills are needed and unemployed people might lose that. That is a long term damage. Think of all the youngsters in Spain, coming out of the university, ended up working in bars and other low skill jobs. That's not only demotivating but also bad for their human capital.

That's one example I can think of.


What is the relationship between economic growth and equality? Does economic growth create more or less equality?

More often than not economic growth will increase wealth and income inequality. This is because wealth is not uniformly distributed across the population, with a small percentage controlling a large share of the worlds wealth.
This small percentage of people tend to hold wealth in income-generating assets such as shares in top companies. This means that when the economy is during well and experience high economic growth, companies will be making a lot of profit. This profit is then paid out to their shareholders. This can increase inequality.
However, it's possible for government policies to break the link between economic growth and rising inequality. One way governments do this is with progressive taxes, where people with a higher income also face a higher tax rate. Another policy is inheritance tax rich limits the ability of rich parents passing on their wealth onto their children


What level are EU tariffs mostly? During the entire Brexit debate in the media I never heard anyone discuss levels of tariffs imposed on countries wishing to export there that had no special deal arranged. I then tried to look up tariffs myself and found this site which looks unbelievably complicated. I just wondered if anyone was familiar with this area and could summarise it - at least crudely.

EDIT: digging around the site some more, I think that cars generally have a 10% tariff form all of USA/Japan/India/China... see here for example but the language is so technical that I'm not sure I'm reading it correctly.

To summarise crudely: the tariffs are punitive, to give all producers within the EEA as much of an advantage as the trade negotiators could deliver. And because the EEA represents a wealthy group of around half a billion consumers (very many of whom are not individually wealthy; but collectively are wealthy), the EU has huge negotiating power on trade agreements: probably about even with China and the USA, and superior to everyone else. So countries outside the EEA tend to get a poor deal in general.

There is a wide range of tariffs set for goods and services for countries accredited to the WTO in their own right (the UK is not) - as you can see from your linked site. But each of the EEA's significant trading partners seeks a bilateral deal with it, because the pain is too high otherwise.

That site - the one you quite reasonably describe as "unbelievably complicated" - is as transparent and as clear a description of tariffs as I've seen anywhere. And yet it still has vastly over-simplified matters: there's a host of regulation that goes with the tariffs, to ensure that what's imported into the European Economic Area (EEA) is safe and fit for purpose.

Within individual trade publications, there was some discussion on levels of tariffs imposed on countries outside the EEA for that particular trade: so, for example, the Financial Times had quite a lot of coverage on the potentially devastating effect on the financial sector that provides so much of the UK's income; and various manufacturing industries talked about the existential threat that Brexit poses to them; but as you can see from your linked site, general-purpose publications didn't have a chance of getting into the detail across all goods and services.


Terminology, is elasticity used as "mean elasticity"? Given a demand $q$ and a price $p$ sutch that $q=q(p)$, the elasticity of demand is given by,
$\epsilon = \frac{p}{q}\frac{dq}{dp}$
which depends on the price.
But, when reading papers about estimation of elasticities (elasticity of oil or
elasticity of gasoline just to mention some examples), I typically find that there is one single value reported as elasticity but not clear how it is computed, is this just the plain mean elasticity?

I would not call it average elasticity rather its elasticity at an average price. For example take the first paper about elasticity of oil you cite (that is Cooper, J. C. (2003). Price elasticity of demand for crude oil: estimates for 23 countries. OPEC review, 27(1), 1-8.).
In that paper the Cooper estimates elasticity using the following model:
$$\ln D_t = \ln \alpha + \beta \ln P_t + \gamma \ln Y_t  + \delta \ln D_{t–1} + e_t$$
Where $\beta$ gives you the estimate of elasticity. However, $\beta$ is not necessary equal to $\bar{\epsilon}$ for any model specification (even though the above one would actually imply it -see the last paragraph), rather $\beta$ gives you point estimate elasticity at $\bar{P}$.
In fact generally OLS regression is constructed in such way that it intercepts point given by $\bar{y}$ and $\bar{x}$ - that is $\bar{y}-\hat{\alpha} -\hat{\beta} \bar{x} = 0$. Hence, more correct interpretation here would not be that $\beta$ gives you average elasticity (i.e $\bar{\epsilon}$) but rather that it gives you point estimate for elasticity at average price (i.e. $\epsilon_{\bar{P}}= (\bar{P}/Q)/(dQ/d\bar{P})$.
This being said note that using OLS model such as the one that is used in Cooper would actually imply that the elasticity is constant because its a linear model (linear in its parameters that is) where $\beta$ is assumed to be constant across all observations (even though you get perfect fit only at ($\bar{P},\bar{D}$). In a model with constant elasticity it would actually hold that $\epsilon_\bar{P} = \bar{\epsilon}$. Nonetheless, I would still caution against that interpretation. The reason for that is that in most cases it is commonly understood that the linear model is used as a simplification and not because people actually assume that elasticity of demand is constant. Mostly, people take this point estimates to be reasonable for small changes around mean price but not really claiming that they found a constant price elasticity of demand, and in most such models if you would look at errors they would get larger the further you get from mean estimate.


What are the best theoretical or empirical defenses of equilibrium analysis in economics? An Anarchist FAQ (see Wikipedia page here) focuses its opposition to mainstream economics by criticizing equilibrium analysis. The FAQ notes that,

[Equilibrium analysis] is essentially a static tool used to analyse a dynamic system. It assumes stability where none exists. [...] Thus the whole concept is an unreal rather than valid abstraction of reality. Sadly, the notions of “perfect competition” and (Walrasian) “general equilibrium” are part and parcel of neoclassical economics. It attempts to show, in the words of Paul Ormerod, “that under certain assumptions the free market system would lead to an allocation of a given set of resources which was in a very particular and restricted sense optimal from the point of view of every individual and company in the economy.” [The Death of Economics, p. 45] This was what Walrasian general equilibrium proved. However, the assumptions required prove to be somewhat unrealistic (to understate the point).

[...] Given that general equilibrium theory does not satisfactorily encompass such things as profit, money, growth, instability or even firms, how it can be considered as even an adequate representation of any real capitalist economy is hard to understand. Yet, sadly, this perspective has dominated economics for over 100 years. There is almost no discussion of how scarce means are organised to yield outputs, the whole emphasis is on exchanges of ready made goods. This is unsurprising, as this allows economics to abstract from such key concepts as power, class and hierarchy.

As for the argument about using unrealistic assumptions in economic models, I'm aware that Milton Friedman had written a defense of that practice in his Essays in Positive Economics. However, I am unaware of any work addressing the specific critiques of equilibrium analysis cited in An Anarchist FAQ, or defending equilibrium analysis in economics from a philosophical, or empirical perspective.

There is a paper fully dedicated to this topic by Backhouse (2004).

The concept of equilibrium is often misunderstood by non-scientists and non-economists.  Following Robinson (1956: 57):

The word equilibrium, in ordinary speech, describes a relation between
bodies in space. The scales of a balance are in equilibrium when the balance is at rest. . . . If we are continually throwing coppers at random into
either scale, the balance is continually wobbling and never reaches equilibrium; but, at any moment, there is a definite equilibrium position
which it would quickly reach if, from that moment, we left it alone.

Hence, the equilibrium analysis can be readily applied to dynamic systems. Equilibrium analysis can be viewed as sort of thought experiment where to organize your thinking passage of time is held constant. Now thinking about equilibrium analysis in the way above still has few issues.
Again Robinson (1956: 59) writes:

Nor can we apply the metaphor of a balance which is seeking or tending
towards a position of equilibrium though prevented from actually reaching it through constant disturbances. In economic affairs the fact that
disturbances are known to be liable to occur makes expectations about
the future uncertain and has an important effect on any conduct (which,
in fact, is all economic conduct) directed towards future results. . . . A
belief that a particular share is going to rise in price causes people to
offer to buy it and so raises its price. . . . This element of ‘thinking makes
it so’ creates a situation where a cunning guesser who can guess what the
other guessers are going to guess is able to make a fortune. There are
then no solid weights to give us analogy with a pair of scales in balance.
The metaphor of equilibrium is treacherous. . .

However, this issue was later addressed as discipline evolved Backhouse (2004):

There is, however, an alternative way to think of equilibrium, not as the
static equilibrium analogous to that of a pendulum, but as inter-temporal
equilibrium where expectations are fulfilled (Milgate 1979, 1987, Phelps
1987). Because this idea is now so strongly associated with the New
Classical Macroeconomics and related theories, it is worth noting that Erik
Lindahl and his Swedish colleagues used the notion to attack the MisesHakek notion of neutral money. If expectations were correctly anticipated,
any rate of inflation was consistent with monetary equilibrium. The door
was opened to discretionary monetary policy. (see Laidler 1999: 58–60).

Furthermore, Backhouse (2004) also argues:

... This is the notion that equilibrium is where (a) agents’ decisions are compatible with each other and (b) no agent has any reason to
change his or her behavior. The second part of this condition (which might
be thought to encompass the first) is often represented by saying that agents
are maximizing utility subject to the constraints they face.

Consequently, in economics there are already several possible notions of equilibria:

Equilibrium as the absence of endogenous tendencies for change.
Equilibrium as balance of forces (using a mechanical analogy).
Equilibrium as correct expectations.
Equilibrium as meaning that no agent has any reason to change his
or her behavior.

We use the concepts above because they allows us to describe and analyze economy and economic relationship as well as to test theories we make about them. Following  Huw Dixon (1990: 356),

At its most general, we can say that ‘equilibrium’ is a method of solving
economic models. At a superficial level, an equilibrium is simply a
solution to a set of equations. ... However, there is more to it than that. Whilst economists rarely argue
over how to solve equations, they do argue over whether a particular
solution represents a ‘real’ equilibrium or not. What is at stake is the
economist’s view of economic agents and the market.

Next following further Backhouse (2004):

An economist who has argued the case for equilibrium analysis very forcefully is Robert Lucas (1980). He interprets equilibrium to mean individual
optimization and consistency of plans (market clearing though perfect
competition). The alternative is to use ‘free parameters’ – parameters that
are not based on individual optimizing behavior, such as those involved
in equations relating price changes to excess demand for commodities. His
reason is that we can never know whether free prarameters are reliable.
Equilibrium models, he claims, are reliable in that parameters are based on
microeconomic behavior which is observable and directly measurable in a
way in which free parameters are not. When economists had no means of
handling more complicated equilibrium models, it was necessary that they
work with free-parameter models in order to explain dynamics. Such work
was immensely productive.

... The substantive points in his argument are that
microeconomic behavior can be established by experimentation; that this
behavior can be described in terms of optimization; and (implicitly) that
aggregation is possible without the introduction of what he calls free
parameters. They define Lucas’s view of how economies work at the macroeconomic level. These are the points on which discussion should centre, not
the issue of equilibrium. This was nicely expressed by McCallum (1986:
412):

Whether or not one works with a model in which “all markets clear” is a
matter of convention, but whether this clearing pertains to auction-type
markets, or to ones with nominal contracts (perhaps implicit) that have
allocational effects, is not.
Another way to express Lucas’s support for equilibrium models and his
hostility toward free-parameter models is that he believes that it provides
explanations of why parameters are what they are. Free parameters are
unexplained, however strongly they are supported by data.

Hence to sum it up.

Equilibrium is far more wider and complex concept that commonly perceived.
It is a way how to put structure on raw data and how to understand mechanisms that generate the data we observe
It is useful concept even for empiricists as it helps us test different explanations for economic phenomena.

The arguments above respond to proper criticism of concept equilibrium in the philosophy of economics. There quoted passages from that anarchist website have far more problems and can be dismissed more easily.

[Equilibrium analysis] is essentially a static tool used to analyse a dynamic system. It assumes stability where none exists. [...] Thus the whole concept is an unreal rather than valid abstraction of reality.

This is incorrect prima facie.

You can use static tools to analyze dynamics systems if you do it at a given point in time.

For example, the whole concept of equilibrium is omnipresent in physics (see the famous Feynman Lectures on Physics). Physical systems such our solar system, earths crust etc are often analyzed using concept of static equilibrium (here is full lecture on this topics). Now I am fully aware Economics is not Physics, the point is that general statement made there is prima facie false assertion.
Static models can and are routinely applied to systems that in real life are ultimately dynamic (since time always marches forward all systems are). This is because we often on purpose want to analyze problem holding the passage of time constant as that makes problem more tractable. This is done in any field of science and there is no reason why it should not be applicable to economic problems. It would  be like arguing English can be used in one field but not another without providing any additional reasons.

The concept of equilibrium directly extend to dynamic analysis. As one can see in any modern micro or macro textbooks (e.g. see MWG Microeconomics Theory or Advanced Macroeconomics by Romer) economics often analyzes equilibria in dynamic fashion or explores dynamic properties of various equilibria.

However, the assumptions required prove to be somewhat unrealistic (to understate the point).

This was already addressed by the essay of Friedman which you cite. Models are not meant to be realistic if they would be they would cease to be models in the first place. Models simplify reality because it is beyond the capability of any human mind to comprehend whole reality at the same time. Good models will simplify reality in some aspects while be true to reality in others. One can criticize any model for being unrealistic and realistic in wrong places but arguing that 'realism' of a model per se, without any qualifications,  is relevant criterion to judge it is false.

There is almost no discussion of how scarce means are organised to yield outputs, the whole emphasis is on exchanges of ready made goods.

The above is literary not truth, as the discussion of how scarce means are organized to yield output is virtually omnipresent in economics (see sources cited above).
The criticism above is not exclusive, one could criticize sentence after sentence mentioned in that excerpt you posted in your question. Even if you would like to criticize equilibrium analysis most of that excerpt is not carefully constructed criticism but rather a discussion of some straw-man version of equilibrium/economic analysis.


Economy with constant amount of money A thought experiment: (Based on my understanding of economics)
Suppose we are guaranteed that no natural calamity/war/pandemic happens for one year (from year $0$ to year $1$) in the world. Then the scientific knowledge ($K_{year}$) of the world is $K_1 \geq K_0$. Hence we are reducing the scarcity of resources which would lead to a higher quality of life (or more precisely cheaper goods while the income remains the same). But my practical experience says that things actually get expensive with each year (although it is also noteworthy that income also rises) because the government prints extra money to account for an increase in $K_i$.
My problem is that although logically both idea looks the same (i.e. good getting cheaper with a constant level of income  $=$ income growth outpaces prize increase ), I find the latter method mathematically more complicated. Given that in all branches of science, we describe things in the least complicated way, then why is that economics is done in a more complicated way as shown above. From my perspective, the amount of money in an economy should remain constant (apart from replacing the old torn bills with a brand new one) and things should get progressively cheaper until we reach a technological utopia where everything becomes free.
A possible explanation to my question: By not increasing the money flow, we are not able to reward the individuals who brought about the technological advance as the things get cheaper uniformly for everyone, whereas if we print the extra money, then that extra money goes to person responsible for the change).
I would like to hear your comments/criticism on the accuracy of my understanding.
P.S. I am using money in a broad sense which would also include things like property etc.

There are several problems with the idea:

My problem is that although logically both idea looks the same (i.e. good getting cheaper with a constant level of income = income growth outpaces prize increase ), I find the latter method mathematically more complicated.

Here you are confusing economics as a science with economic policy. Yes in a science we prefer more parsimonious explanation to one that is less parsimonious (assuming they both have equal explanatory power) but that does not mean we should prefer more parsimonious policy. That is equivalent of saying that simpler medical treatment should be always preferred. Just because a more parsimonious biological theory/explanation is more efficient and hence preferred does not mean we might as well prefer herbal tea compared to complex antibiotics.
Its not that there are two theories one which says that inflation must occur as some sort of natural process when technology expands and second theory that says that technology can expand freely. Here you are talking about policy choice about what we choose the money supply to be along some  path of technological progress.

From my perspective, the amount of money in an economy should remain constant (apart from replacing the old torn bills with a brand new one) and things should get progressively cheaper until we reach a technological utopia where everything becomes free.

There are several problems with this:

First, what ultimately matters is not what the price tag is but how much real resources you have to spend for goods and services. If due to inflation apple costs $\\\$10,000$ but you earn $\\\$1,000,000,000$ per hour then in real terms the apple in such situation is much cheaper then in situation where apple costs $\\\$0.10$ but your hourly wage is $\\\$0.01$.
You are ignoring well documented nominal rigidity especially when it comes to wages (this is so notorious you can see this mentioned in any textbook - for example Blanchard et al. Macroeconomics an European Perspective). For the sake of brevity I wont go into details but in the presence of nominal rigidity in short run deflation creates excess unemployment and depresses economic activity and inflation has the opposite effect. This is because in short run when aggregates prices fall firm might be still locked to higher nominal wages as those might be slower to adjust and this forces firms to unnecessarily lay off some workers. Due to this most economists advocate that over business cycle there should be steady slow (about $2\%$) inflation in order to help 'grease the wheels' of the economy.

A possible explanation to my question: By not increasing the money flow, we are not able to reward the individuals who brought about the technological advance as the things get cheaper uniformly for everyone, whereas if we print the extra money, then that extra money goes to person responsible for the change).

This is also incorrect. Even if money supply would remain constant that does not mean that there is any lack of money to reward people for their additional work or effort. In long run prices & wages would just adjust downward so even if people would earn less in nominal terms they could actually buy much more as before (an opposite situation as the one shown with apple before). What matters ultimately is not how much money you have but the amount of goods and services you can consume.
Economy actually existed and functioned well before money was invented at all. If there would be no nominal rigidity and we would live in a world where money is even in short run completely neutral then deflation would not matter at all.


Criticism of "Modern Political Economics" by Varoufakis, Halevi, Theocarakis The book "Modern Political Economics" is quite critical about "neoclassical economics", the basic claim being, if I understand correctly, that the models which "neoclassical economics" (armed with mathematical insights starting from John F. Nash Jr, Gerard Debreu and Kenneth Arrow) investigate, are not simplifications of some realistic models, but that they (the models) are just wrong (because of what the authors call "Inherent Error").

Is there a serious criticism of the views such as the ones exposed in this book?

added after a comment asking for citations:

maybe not a successful choice, but nevertheless (the following two citations are short descriptions of chapter 3 and chapter 8 of the book):

"Chapter 3 The odd couple: The struggle to square a theory of value with
a theory of growth

The odd couple of the title are value and growth. From the very start, political economics found it difficult to square the two; to create models or accounts of how the exchange value of things was determined in a growing economy. The chapter begins at the beginning, with the French Physiocrats, before moving to Adam Smith and David Ricardo’s attempts to tackle this conundrum. The Inherent Error makes its first fonnal appearance in these works, before it returns again and again in the following chapters. The essence of the Inherent Error is the impossibility of telling a credible story about how values and prices are formed in complex (multi-sector) economies that grow through time."

"Chapter 8 A fatal triumph: 2008’s ancestry in the stirrings of the Cold War

During the Second World War, economic policy was in the hands of the New Dealers, who ran the economy on a trial and error basis and in the light of the accumulated experience of trying, not with great success, to kick-start the ailing US economy during the traumatic 1930s. Meanwhile, a group of scientists (mostly of Central European origin) were manning the agencies, laboratories and divisions of the civilian and military authorities whose job it was to solve practical problems (e.g. logistics, planning of transportation systems, price setting) by means of advanced mathematical methods. However, after the war ended, and the Cold War began to take hold, both the New Dealers and the Scientists lost out in the struggle for the hearts and minds of academic economics. The winners of that ‘game’ were a small group of Formalists, with John F. Nash, Jr, Gerard Debreu and Kenneth Arrow at the helm. The chapter tells the story of that triumph, which gave neoclassical economics a whole new
push, by focusing on the person that the book portrays as the era’s most tragic figure: John von Neumann. His ‘fate’, the chapter argues, was an omen for the type of economics that would prove instrumental in the run up to the Crash o f2008."

and yet another citation:

"The rest, as they say, is history. Debreu and Arrow emerged from Nash’s seminar and in a few short months applied what they had heard to hammer out their own existence proof in the context of a multiple-sector model economy: one that effectively brought back from the dead Walras’ (fully neoclassical) idea of a General Equilibrium complete with determinate prices for everything, including labour input and capital goods (recall Chapter 6). This they accomplished by procuring General Equilibrium’s ultimate mathematical proof; an existence proof that showed under which conditions a set of relative prices exists such that all
markets (including that for labour) are in equilibrium.
This existence proof was to mark a new neoclassical turn in political economics; a turn that altered the discipline’s course and returned neoclassical obscurantism to the throne from which it had been removed by the combined forces of the fall of 1929, the analysis of John Maynard Keynes, the engineering brilliance of von Neumann and, last but not least, the experiences of economic policy during the New Deal and the Second World War."

Neoclassical economics or rational analysis should always be a starting point for an analysis of how a given system or model works.

However, that is all it is - a starting point. You cannot use a model that is solely rational to predict or infer much about how the real world works, because humans are not rational. That is why rational economics are critized, but do not match up with observed behavior.

Therefore, any real critique has to show how rational economics do match up with observed behavior.


Should I stay or should I quit? Is there an economic theory of quitting an activity? A theory that weighs the investment costs put into something and the opportunity costs of pursuing it.

I am aware of Fershtman and Gneezy Quitting in Tournaments paper that investigates the decision of quitting in the middle of a race or of a tournament. But, I am looking for a more general model of quitting something such as playing an instrument, a sport or even a job. Dropping-out from (grad)-school may obey the same investment-opportunity cost trade-off.

Update: I am looking for some explicit references modeling the exiting or quitting process, even if it concerns some decisions in particular without any generalization.

Edit: Here is why I picked @Ubiquitous' answer.

Check out sunk cost. Although this might seem completely counterintuitive, the investment you have made should be irrelevant to your decision to stay or quit, if this investment has already been made and cannot be recovered.

What matters is whether the benefits you are yet to incur outweigh the costs (that you are yet to incur). So the decision should be a forward-looking one.

The above reference mentions that in the real-world, people often do not behave in this way. According to sunk cost theory, this is irrational, but this is only one theory.


Computational Bayesian Statistics and Economics What is there that links computational Bayesian statistics and Economics(preferably macro)?
I'm looking for a primer or gentle introduction on the matter. It doesn't need to go into much detail.

Any help would be appreciated.

Not aware if there exists a "gentle" introduction, but you might try Zellner's "Bayesian Analysis in Econometrics and Statistics" (1997) or Koop's "Bayesian Econometrics" (2003).


printing money after making the value by setting prices by law Why cant America set prices by law, which would establish the value of the dollar within its borders and then print money and give it to those who qualify.  Making deals with some of those who are over seas could be made with the people who import goods into the USA.

Why cant America set prices by law; and then print money and give it to those who qualify

    This is the main question

So the Fed along with the US government declares that it will start controlling the prices within the US territory. There will be some simple consequences to this decision,

$1)$ Investors who invest in enterprises and corporations, big and small will run away, literally run away. This will drive the investments at an all time low. Ask yourself, why would anyone invest their hard earned cash in the US if they would not be allowed to enjoy the profits. After all, profits revenues and other benefits are a result of price flexibility.

$2)$ No price means no profit, means no investment means soon many private entities will either stop functioning or just find some other country to move to. If there are no other entities to tax on, whom will the government tax on.

$3)$ If prices remain fixed, then who will the government give its money? Who will even want the money. What good use is the money if an individual cannot buy or sell, in short have any purchasing power.

Here you go, http://www.socialstudieshelp.com/economics_basic_concepts.htm


How do transactions in different currencies work? Very basic question but I have not much knowledge in this field so sorry:

Say I have a bank account and over the past year there were transactions in different currencies. I am guessing that a transaction in a different currency than 'mine' (whatever that is, I assume an account has like a 'default' currency) gets booked (immediately / later that day / ...) with the exchange rate (at that time / 4th the end of the day / ... ?) + some fees.

How exactly does booking of a transaction in any currency work?

Let me explain you in the most simple way how it work behind the scenes...

Credit card issuers and payment networks charge cardholders an international transaction fee if the purchase has been made abroad or an overseas bank has to interfere into the transaction process. (it is roughly for about 1 to 3% of the transaction amount, some issuing bank can even avoid charging it). Each credit card issuer uses its own currency conversion rate (per each transaction). I.e. in e-commerce when the transaction occurs (merchant gives you the option to process the transaction in foreign or local currency) and the merchant gets to choose the exchange rate. Paying in local currency means your credit card network will handle the conversion (less costly).
The exchange rate for the customer’s home currency is calculated by adding the wholesale interbank rate and the merchant’s foreign currency fee.

E.g. in the US  international  e-commerce (when accepting payments from other countries with other currencies) merchants have to have an international merchant account/ foreign currency payment gateway but they supposse higher risk of chargebacks (you can file it as a customer) and that is why the rate per transaction tends to be higher /up to 6% for merchant).

What you as a customer do not see is a process behind the scenes --Till the transaction settle, is cleared and authorized it can take even 24 hours or even for about 2 days => one thing is you see on your bank statement the money has been deducted from your account but for e-commerce where the manual confirmation of any order or reservation is needed it has not been settled on merchant’s account  yet and that is why you have to wait until it is authorized and settled (vendor's account set up plays  a vital role as well) and issuing bank , acquiring bank and bank network communicate first and accept or decline the transaction (you can come across a pending transsaction mainly if an an AVS mismatch occur, it is not your failt but it takes time to bank to communicate and validate it, as you have to submit an signed authorization form to prove your billing address).

That explains as well why if you decide to return a product or service is harmful to any business  as the handling and bank transaction fees are always associated with the purchase as the customer will not bear it (has the right to return goods) and you are protected as customer anyway, from merchant’s point of view they have it stipulated in contracts with banks and partners.

You as a final consumer just see a money charged form your card (from the statement from your bank), you can check with retailer to see where is based and which payments options offer and mainly with your bank how much charge for int payment.


If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element Consider a TU game $\langle N,v \rangle$ with $N = \{1,2,\ldots,n\}$ being the set of players and $v : 2^N \to \mathbb R$, $v(\emptyset) = 0$ the characteristic function. The Core of $v$ is defined by undominated allocations $x \in \mathbb R^n$
\begin{align}
\mathcal C(v) = \left\{x \in \mathbb R^n ~ \bigg| ~ \sum_{i \in S}{x_i} \geq v(S) ~ \forall S \subseteq N\right\}.
\end{align}
Let $m_i(S) = v(S \cup \{i\}) - v(S)$ denote the marginal contribution of $i$ to coalition $S$.
We say that the players are symmetric if $m_i(S) = m_j(S)$ for all $i,j \in N$ and $S \subset N$.
The marginal contribution of any agent to any coalition is thus identical.

I am wondering whether the following Proposition is true and how one proves it formally.

  Proposition: If the players are symmetric and the core is nonempty  $\mathcal C(v) \neq \emptyset$, then the equal division of the worth of the grand coalition is an element of the core $(v(N)/n,\ldots,v(N)/n) \in \mathcal C(v)$.

The center allocation $z_{i}:=(v(N)/|N|)$ for all $i \in N$ is in the core of the symmetric game $v$, whenever $z(S) = \sum_{i \in S} z_{i} \ge v(S) $ holds for all $S \subseteq N$. Now, if $v(N)/|N| \ge v(S)/|S|$ for all $S \subseteq N$, then it holds $z(S) = \sum_{i \in S} z_{i} = |S|\cdot v(N)/|N| \ge v(S)$. Thus, $\mathbf{z} \in C(v)$.

We observe that for a symmetric game $v$, the core is non-empty if $v(N)/|N| \ge v(S)/S$ holds for all $S \subseteq N$. Otherwise, the core is empty.


Increase in the price of skateboard wheels cause a decrease/no change in the demand of skateboards? (2 separate markets w/ 2 separate buyers) Would an increase in the price of skateboard wheels cause a decrease/no change in the demand of skateboards? (2 separate markets with 2 separate buyers)
Note: I'm going to be mainly focused on the demand curve and its shifting rather than supply curve and markets, etc. Trying to simplify the thing I'm trying to figure out here. I'm asking this question more to double-check, but I could be having a major inconsistency in my basic understanding.
So, let's say there's a skateboard-making company that buys skateboard wheels from another company in order to make its product (skateboards). Let's say that other company selling skateboard wheels to the skateboard-making company increases the price of its skateboard wheels. This would cause a movement along the demand curve where the skateboard-making company would decrease the quantity it demands of skateboard wheels. ( Here skateboard wheels are an intermediate good and this is a B2B market (Business to Business) by the way. So no regular consumers involved here! )
This would basically mean the skateboard-making company would have fewer skateboard wheels in its inventory, etc. Now onto the market for skateboards: this would not shift the demand curve to the left/inwards, right? Rather, the demand curve would be unchanged (not shifted left or right). Obviously, there would be a shortage of skateboard, but I'm not so focused on that. I'm just double-checking that for the first market I mentioned, the skateboard-making company was the buyer and due to a change in price there was a movement along the curve. In the second market I mentioned, we have regular people who buy skateboards as the buyers and other than there being a shortage in the number of skateboards being produced the demand curve wouldn't have shifted since they weren't directly affected by the change in the price of skateboard wheels. They still love buying skateboards and continue to do so (only they will be a bit disappointed to know there is a shortage of skateboards because of the skateboard wheel probably explicitly affected the skateboard-making company's quantity demanded)
Yes, I am aware the quantity demanded and demand (market/indiv. demand curve) is different. I've tried to be explicit in which one I am referring to above in my question.

Since there is no change in consumer preferences, the demand function will not change.
The firms supplying skateboards face increased costs due to the wheels being more expensive. This will alter their supply function and thus the equilibrium price in the skateboard market may change, and hence the equilibrium quantity demanded may also change.


Useful software for notes and graphs in economics? I'm looking for a software which may help me to both take notes and draw graphs during university lectures. It must have the ability to switch between texts and graphs, add other formats such as Office related format and pdf. Also, the the graphs should be easily tunable (preferably on the fly) and of course, shareable. Being able to draw matrices and operate with them would be great too.

I can't seem to find an application/software for laptop which makes all of those easy in an economics related setting. Any idea?

Thanks a lot

I've been trying to find a solution for this same question.  As mentioned by others, most of the solutions are a bit too time consuming to do on the spot if you're taking notes, so I still think it's easier to draw out the graphs by hand on the spot.

There's a good post referencing some of the different software options, along with how to do this with R and ggplot if you're interested on Andrew Heist's blog

Also, see Create simple graphs without functional forms

Among these options, I'm trying to learn to do everything with TiKZ, but it's a work in progess...


Will the services rendered by a worker from an unorganized sector be counted in GDP? My guess is it won't be because the unorganized sector isn't captured in GDP? but I read somewhere it will be. not sure if I am wrong or it was an error in the answer key.

please help

If by ‘un-organized’ sector you mean gray and black market then these activities are not directly recorded in GDP. For example, if you pay 10e to a neighborhood kid to mow your lawn it won’t show on GDP as a spending. Although, you could say those activities still indirectly can increase GDP, as if then that kid goes and buys a lego at a store that would otherwise not be bought that purchase would raise GDP. But I don’t know what exactly you mean by unorganized if. If the spending/incomes in a sector are tracked by statistical office it will show up on GDP.


Purpose of Semidefinite Integral I want to know the meaning of Semidefinite Integral.

I am used to read definite and indefinite integral but I want to know the meaning of such equation :

$\pi(e)\left(1-F\left[-\frac{a}{\pi(e)}\right]\right) \left(a+\frac{\int_{-\frac{a}{\pi(e)}}\mu f(\mu)d\mu }{\left(1-F\left[-\frac{a}{\pi(e)}\right]\right)}\right)$

Where :

$\pi(e):$ is the probability of finding the information $\mu$ depending of the effort $e$ of the agent.
$\left(1-F\left[-\frac{a}{\pi(e)}\right]\right)$ is the condition for the agent to search this information depending on the parameter $a$ which can be viewed as a reward.

$\mu$ is a stochastic term distributed by $f(\mu)$, the expected value of $\mu$ is 0.

I do not understand the second term of the equation where there is only the lower bound:
$\int_{-\frac{a}{\pi(e)}}\mu f(\mu)d\mu$

It is just an opaque way to denote that the non-definite integration limit will take the value of the variable itself. Namely,

$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$

... where $s$ is just the dummy variable of integration. Since sometimes we use the notation $\int _D f(x) dx$ to denote fully definite integration "over the domain $D$ of $x$", I would avoid the left-hand side notation and I would stick with the more clear right-hand-side one.

I do hope that the authors where you find this notation mean what I wrote above, because this is what mathematicians mean by it.


Pooling equilbrium refinement I am analyzing a model in the signaling game, There are three stages of a game. there are two participants in the game, a supplier and a buyer. The supplier has private information about its capacity level. In the first stage, the supplier sets a wholesale price. In the second stage, the buyer gets the wholesale price as a signal and decides on its order quantity and finally, in stage 3 both agents decide on selling quantity to the market.
If the supplier has high capacity, it may have incentive to pretend to be of the low capacity.
The problem that I have is, I get the range of pooling equilibrium, Is there any equilibrium refinement that allows me to choose the best pooling equilibrium?

There are many refinements of pooling equilibria. For starters, you want to make sure all your equilibria are Perfect Bayesian Equilibria. After that, you can also require any equilibrium to survive the intuitive criterion or other forms of forward induction. Lastly, you could also require things like efficiency or symmetry.

The choice of refinements is not always straight-forward, it depends on what you mean by "best" and how well the assumptions implied by each refinement match the story of in your model.


Does finding real income adjust for inflation? I have been reading from this textbook which talks about CPI and inflation

http://faculty.washington.edu/cnelson/Chap04.pdf

On pages 5 and 6, it talks about how they are deflating the nominal income using the CPI. But in page 6, they work out the percentage difference in real income between two years, and they state that is the net difference of her increase in real income and inflation

But I thought using the CPI was getting rid of the effect of inflation? What am I not undersranding?

Indeed deflating nominal income using CPI does get rid of the effect of inflation. I think you're just getting confused by the phrasing.

I assume you are referring to the following sentence on pg. 6:

  We  have  shown  that  the  net  result  of  her  raise  and  inflation  was  an  increase of 7.9% in real income.

Let me rephrase this sentence: we have shown that after raising her salary by 10% and then adjusting for inflation, the net effect is that her real income increased by 7.9%.


Difference between long run coefficient and non stochastic steady state coefficient ARDL model I am a little bit confused on the definition of long run equilibrium coefficient. Suppose I have an ARDL model as:
$y_t = \rho_1 y_{t-1} + \rho_2 y_{t-2} + \beta_1x_{t-1} + \beta_2x_{t-2} $
The steady state non stochastic solution is:
$y=\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}x$
What is the definition of long run equilibrium coefficient for $x$ here? Is it defined as the coefficient of the steady state non stochastic solution $\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$? What is the difference between the two concepts?

yes, the term that you showed for the ALDR non-stochastic steady state:
$$\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$$
is long-run multiplier or sometimes also called long run equilibrium coefficient (see Verbeek 2008 Guide to Modern Econometrics 4th ed.  pp 340). As  far as I can understand there is not much difference between the two concepts in the context of ARDL model and they are used interchangeably.
The term non-stochastic steady state has wider application especially in macroeconomics (e.g. in growth theory), but often steady state results are just called long-run equilibrium because in most models steady state is attained only in long-run equilibrium.


Why is the Euro a bad idea compared to other currencies? Many economists, including Nobel laureates Paul Krugman and Joseph Stiglitz, have said that the Euro was a bad idea from the beginning and is a major reason for all the current economic problems. My general understanding of this issue is that different countries with different monetary policies would have more control over how to control their debt and financial obligations instead of being tied to the same rules that all the other countries have to follow. This makes sense to me.

However why does the USA and China not have similar issues even though they are very large countries with a single currency as well? Should the US and China actually be using different currencies for different regions?

One of the key requirements for a currency area to be successful is that there must be a sustainable political mandate to support fiscal transfers between regions, to provide the balancing mechanism that would otherwise have been performed by fluctuating currency rates between those regions.

Another is that labour can move freely between regions.

The Eurozone, it turns out, met neither of those requirements:

Language barriers have impaired freedom of movement.

And although Northern Europe was happy to reap the benefits of currency union during the good times, it has proven to be unwilling to make the fiscal transfers to the South during the bad times.

Exacerbating this, Greek admission to the Euro zone was created on the back of some fraudulent accounting courtesy of Goldman Sachs. Based on the real numbers, Greece would never have been admitted.


General Equilibrium Modelling I have a question regarding GE modelling.
The firm's problem is to:
$$
\max\pi=pf(k,l)-wl-rk
$$

If this is true, we choose $l$ and $k$ to maximize the difference
between costs and revenues. This yields the familiar FONCs for $l$
and $k.$ However, in equilibrim, we set the profits equal to 0.

My intuition tells me that after we solve the above problem, the equilibrium
condition implies our selecting a price $p$ such that $\pi$ equal
$0$ in equilibrium. Is this correct?

In a typical GE model, price (including wage and rent) is determined by the market clearing condition, which requires that supply equals demand. Specifically,

labor supplied (determined by household solving utility maximization problem) equals labor demanded (determined by firm's profit maximization problem)
goods supplied (determined by firm's optimal total production) equals goods demanded (determined by household's consumption and investment decisions)
capital supplied (household's investment decision) equals capital demanded (firm's optimal choice of $k$)

The zero-profit condition is derived from the assumption of perfectly competitive behavior (or both firms and households are price takers). This condition may contribute to the determination of equilibrium prices. But to say that $p$ is selected to ensure zero-profit of the firm just sounds incorrect.


Can destruction be profitable? We often see news that some company destroys items they couldn't sell in time.

Also we can see news of supermarkets destroying food just because it's not fresh although it's perfectly edible.

If someone destroy goods, he would get $0 for it or even need pay for the disposal. How can this act be profitable for a company?

It can be profitable for the monopolist to do so. For the conventional producer who is a price taker the profit objective function looks like this:
$$\max_{q} \Pi^c $$
where $\Pi^c = P \cdot q - C(q)$.

That is, they seek to maximize profits, facing an exogenous price to sell goods and where costs are a function of amount produced. If everything is nice and differentiable and concave this gives a first order maximizing condition of:
$$ 0 = \frac{\partial\Pi^c}{\partial q} = P - \frac{\partial C(q)}{\partial q} \rightarrow q = g(P) $$  where $g(P)$ is the inverse of function $\frac{\partial C(q)}{\partial q}$ (increase q from zero until marginal cost equals the price).

But for a monopolist $\Pi^m = P(q) \cdot q - C(q)$. That is, the monopolist is not a price taker, they know that when they produce more prices will fall. Resulting FOC:

$$ 0 = \frac{\partial\Pi^m}{\partial q} = P(q)+q(\frac{\partial P(q)}{\partial q}) - \frac{\partial C(q)}{\partial q}$$

As a result, a monopolist in this setting will not want to sell where price equals marginal cost but rather a higher price than marginal cost:
 $$ \frac{\partial C(q)}{\partial q} - q(\frac{\partial P(q)}{\partial q})= P(q) $$
Recall: $- q(\frac{\partial P(q)}{\partial q}) > 0$

Indeed, it is this selling of goods above their marginal cost that is the source of the monopolist's monopoly rents.

Now back to the question itself which asks about when "company destroys items they couldn't sell in time." We can think of inventory as a good with low or zero marginal cost, so yes, by the argument above, it can be profit maximizing to not sell goods with a price above costs as long as it lowers the profits on the other goods you sell enough to offset the additional revenue.

Other forces are likely at play as well. Stores often worry about their brand value and low quality, low price, goods may harm their relationship with their customers. Many people shop at stores specifically for the high quality goods and others at different stores for low prices and it may be too costly to move spoiling goods from one venue to the other and cheaper just to toss it. We can also think of the muffin stumps of the TV show Seinfeld as an example of this high transaction costs / missing markets problem.

That said, there are sometimes innovative alternative methods of disposal:

  Rotisserie chickens have been around for a while. I used to bypass
  them and roast my own, until I noticed something: The rotisserie
  chickens were actually cheaper than buying and roasting my own.

  Cat Vasko noticed the same thing and decided to figure out why. The
  answer makes a surprising amount of sense: Grocery stores make them
  out of unsold chicken that is about to pass its expiration date. It's
  an elegant way to make a profit out of food that would otherwise be a
  net loss. And it's not just chicken -- according to Vasko, the
  ever-expanding prepared-foods section of the supermarket uses up all
  sorts of unsold produce and meat. It is, as she says, a bit like
  hunter-gatherers using every inch of the animal.

Everyone Wins When You Buy a Rotisserie Chicken


OLS and 2SLS normal equations For a system of equations with $M=2$ endogenous variables, $ Y=\begin{bmatrix}  y_1 & y_2 \end{bmatrix}$ and $K=3$ exogenous variables, $X=\begin{bmatrix}  x_1 & x_2 & x_3 \end{bmatrix}$.
The first equation of the system is given by:
$y_{1i}=\gamma_{12}y_{2i}+\beta_{11}x_{1i}+\epsilon_{1i}$.
The data matrices yield, $X'X$, $X'Y$ and $Y'Y$, which are of dimension, $3*3$, $3*2$ and $2*2$ respectively

I need to

Write the OLS and 2SLS normal equations in terms of cross products of the data matrices.

I tried using the standard procedure for deriving normal equations, that is, by minimizing sum of squared residuals, but I fail to understand how the equations can be modelled in form of the cross products of the given data matrices. I think I'm missing the point of the exercise. Please help me in understanding what the question demands. Thanks!

OLS

The OLS part is clear: $\sum_{i=1}^n y_{2i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$ and $\sum_{i=1}^n x_{1i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$, where $\hat\gamma_{12}$ and $\hat\beta_{11}$ are the OLS estimators.

2SLS

For 2SLS, I will assume that $(x_{1i}, x_{2i}, x_{3i})$ is used as IV. The corresponding 2SLS is identical to the IV estimator using $\hat{y}_{2i}$ and $x_{1i}$ as IV. Thus, (I think) the "normal equations" are
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $\tilde\gamma_{12}$ and $\tilde\beta_{11}$ are the 2SLS estimators using $x_{1i}, x_{2i}, x_{3i}$ as IV, and $\hat{y}_{2i}$ is the fitted value obtained by regressing $y_{2i}$ on the instruments $x_{1i}$, $x_{2i}$ and $x_{3i}$.

In the above, I understood 2SLS as the IV estimator using $(\hat{y}_{2i}, x_{1i})$ as IV. We can also understand 2SLS as the OLS estimator of $y_{1i}$ on $(\hat{y}_{2i}, x_{1i})$. In that case, the normal equations can be written as
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $y_{2i}$ is replaced with $\hat{y}_{2i}$. The two sets of normal equations for 2SLS (one in terms of $y_{2i}$ and the other in terms of $\hat{y}_{2i}$) are identical.

Intercept

I have noticed that the intercept is excluded in the question. If the model is $y_{1i} = \beta_{10} + \gamma_{12} y_{2i} + \beta_{11} x_{1i} + \epsilon_i$ instead, you need equations for the intercept as well.

Matrix notations

Using matrix notations, let $y$ be the $n\times 1$ matrix of the LHS variable, $X$ the $n\times 2$ matrix of $(y_{2i}, x_{1i})$ and $Z$ the $n\times 3$ matrix of $(x_{i1}, x_{i2}, x_{i3})$. Let $\beta = (\gamma_{12}, \beta_{11})'$. (I am considering the model without the intercept. For the model with the intercept, $X$ and $Z$ contain a column of ones.) Then the OLS estimator is $\hat\beta = (X'X)^{-1} X'y$, which solves the normal equations $X'(y-X\hat\beta)=\boldsymbol 0$. To see this, note that $$X'(y-X\hat\beta)=\boldsymbol 0 \\
X'y - X'X\hat\beta = \boldsymbol 0\\
X'X\hat\beta = X'y\\
\hat\beta = (X'X)^{-1}X'y$$

The 2SLS (using $Z$ as instruments) estimator is $$\tilde\beta = [X'Z(Z'Z)^{-1} Z'X]^{-1} X'Z(Z'Z)^{-1} Z'y = (\hat{X}'X)^{-1} \hat{X}'y$$ where $\hat{X} = Z(Z'Z)^{-1} Z'X$. This 2SLS solves $\hat{X}' (y-X\tilde\beta)=\boldsymbol 0$, the normal equations for this 2SLS. Again, to see this, note that
$$\hat{X}' (y-X\tilde\beta)=\boldsymbol 0\\
\hat X'y - \hat X'X\tilde\beta = \boldsymbol 0\\
\hat X'X\tilde\beta = \hat X' y\\
\tilde\beta = (\hat X'X)^{-1} \hat X'y $$
You will see that $\hat{X}$ is the $n\times 2$ matrix of $(\hat{y}_{2i}, x_{1i})$. It is also true that $\tilde\beta = (\hat{X}'\hat{X})^{-1} \hat{X}'y$, and the normal equations are also written as $\hat{X}' (y-\hat{X}\tilde\beta)=0$. The two different expressions are identical because $\hat{X}'X = \hat{X}'\hat{X}$.


Labor market model with subsidies homework I would appreciate some help with my homework in comparative statics.

L is the demand of labor
F(L) is the production function
$ \frac{\partial F}{\partial L} > 0 > \frac{\partial^2 F}{\partial L^2} $
P is the price of goods
W is the wage
S is subsidy per hired worker
AA(W) is the supply of labor
$ \frac{\partial AA}{\partial W} > 0 $
$ max_{L} P*F(L) - (W - S)*L $ is what companies want to maximize

a) Analyze the effect of an increase in subsidies on the equilibriums of W and L.
b) Illustrate your findings in a graph. Chose Quantity of labor and W as axis.
c) Analyze the effect of an increase in subsidies on the slope of L in W.
My ideas for a):
The equilibrium conditions are $ L = AA(W) \Leftrightarrow L - AA(W) = 0 $. So lets call it $ H(W; L) = 0 $ and let´s call the profit function G so we get as the second condition $ \frac{ \partial G}{ \partial L} = P * \frac{\partial F(L)}{\partial L} - (W - S) = 0$, so we have $ g(L; P, W, S) = 0$.
I used the IFT:
$ \frac{\partial G}{\partial S} = - \frac{\frac{\partial g}{\partial S}}{\frac{\partial g}{\partial L}} = - \frac{ S}{P* \frac{\partial^2 F}{\partial L^2}} > 0$. This means L increases if S increases.
Now let´s take a look at the labor market:
$ - \frac{\frac{\partial H}{\partial L}}{\frac{\partial H}{\partial W}} = - \frac{1}{\frac{ - \partial AA(W)}{\partial W}} > 0$. Since L increases the wage also increases.
Regarding c):
I have to find $\frac{\partial L(W)}{\partial W \partial S} $, but I don´t really have an idea how to get it.

Does this make sense?:
$$\frac{\partial L_d}{\partial W} = \frac{-\partial g/\partial W}{\partial g /\partial L}=\frac{1}{PF_{LL}(L_d)}$$
Now from solution of (a) we know that as $S \uparrow$, $L_d \uparrow \implies F_{LL} \downarrow \implies \partial L_d/\partial W \uparrow$
Therefore, $\partial^2 L_d/\partial W\partial S >0$
EDIT: Note that I have assumed above that $F_{LLL} <0$. Clearly this is not given in your question statement. So as per the answer above, what happens to the slope should depend on sign of $F_{LLL}$. If, for example, it is zero, then the curve simply shifts with no change in slope.


Causes of interlocking directorship connections between firms I need to interpret some empirical results telling me that firms (intended as profit-maximising entities) prefer to be controlled (e.g. through interlocking directorships) by smaller and less profitable companies. I couldn't find any piece of literature helping me in this sense. Do you have some advice on that? Any hint would be really appreciated.

I've heard of this phenomenon but not for the reasons you suggest:

  The largest corporations tend to have the most interlocks (Table 4).
  This may occur because the directors of the largest corporations are
  the most knowledgeable, the most capable, and the most accomplished
  men available. Other corporations would naturally seek their advice
  and would rather have them on their board than men of less ability.
  This may also occur, however, because of factors unrelated to
  managerial ability. The director of a giant corporation undoubtedly
  has more personal influence with other companies,  with potential
  investors, and with the government than the common man. Having the
  director from a large corporation on your board may also lead to
  profitable business with that corporation.

Dooley, P. C. (1969). The interlocking directorate. American Economic Review 59(3), 314–323


Why plug deficits with bonds rather than printing money? The UK is running a budget deficit £107 billion a year. Inflation is currently running at 1%.

Given these facts, why do they plug the gap by selling debt rather than just creating the money to cover the gap?

Just creating the money seems like it has a lot of upside and not a lot of downside when inflation is so low. Why do we not see countries doing this?

There are several issues with this approach. One is that any changes in the quantity of money - including those considered 'acceptable amounts of inflation' - act to distort the price signal being communicated to all economic participants which is an extremely undesirable side effect.

The other is a little more insidious. Governments can only print cash or asset money. The vast majority, over 98% in most modern banking systems of money being used is liability money - money that is represented as deposits in the banking system. Were the government for example to print physical cash and deposit it in a commercial bank the book keeping operation would be [debit cash, credit deposit]. In almost all cases, it is the deposit money that is actually spent as the government pays salaries etc.

So if a government prints money and its banking regime relies on a framework where assets act as a regulatory control on lending (and consequent deposit creation), and there is no other regulatory control, the result is hyper-inflation. The problem is not just the money created by the government, it is the consequent multiplication that results from banks increasing their credit/money creation. (This form of regulation is generally referred to as the reserve requirement in economic literature.) Typically the resulting inflation then leads the government to print more money, and the result is a rapid spiral that quickly destroys the usefulness of the currency concerned for any economic transaction.

Most modern banking systems use a combination of reserve requirements and capital requirements, and this is why the quantitative easing interventions have so far had no inflationary effect on the economies using them. Even though the US government printed a huge amount of money for the TARP intervention, the capital controls intervened to prevent the runaway hyperinflation that would have occurred under earlier regimes. However it's still a dangerous thing to do, and the long term behaviour of banking systems that rely on basel capital controls is poorly understood.

Finally, it's worth noting that the alternative available to any government besides borrowing, is raising taxes or controlling its' expenditure. Ultimately the government's use of social resources has to be regulated by something, and keeping a more or less balanced budget isn't a bad place to start.


How do oil prices rise when crude oil inventories fall? I am new to economics, I am reading a post on CNBC which states

Weekly crude oil inventories unexpectedly fell, supporting further gains in oil prices.

1) Why does crude oil inventories unexpetedly falling make oil prices rise?

2) By Crude oil inventories falling do they mean stock running out due to mass purchase?

3) What could cause oil inventories to unexpectedly fall?

I am assuming a purchase from somebody lowering stock overall could cause panic buying from others therefore temp demand making prices rise? - Probably wrong.

1) Supply and Demand. When supply falls, what supply remains becomes more valuable

2) There are massive storage tanks across the US (Cushing, Oklahoma comes to mind) where oil companies can purchase/rent storage space. Refineries cannot process crude as fast as it is extracted, so the backlog must be stored somewhere.

3) Any number of reasons. Increased consumption from industry or consumers. "unexpectedly fell" could actually mean that inventories grew slower than expected because drillers have reduced output. Perhaps a refinery came online after being shut down and helped draw down the crude stockpile.