Why did John Keynes believe that monetary policy to be less effective than fiscal policy? So why did John Keynes believe that monetary policy to be less effective than fiscal policy for stabilising the level of economic activity, particularly in a recession?
Lowering markets' interest rate encourages borrowing and reduce the incentive to save, which to me it seems that it produces similar economic benefits to fiscal? But I guess that it takes time for the effects of low interest rate to flow through the transmission channels?
I suppose the reason why he favoured fiscal is tied to the time period he lived in?

This was due to the idea of liquidity trap, and due to Keynes thinking that in economic recession it is easy for an economy to slip into the liquidity trap. Liquidity trap is a situation where preference for holding cash becomes virtually infinite. Keynes (1936) in the 'General Theory' states:

There is the possibility...that, after the rate of interest has fallen to a certain level, liquidity-preference may become virtually absolute in the sense that almost everyone prefers cash to holding a debt which yields so low a rate of interest. In this event the monetary authority would have lost effective control over the rate of interest.

Infinite preference for holding money means that more savings will not necessary lead to more investment but rather reduced income (see Blanchard et al. Macroeconomics a European Perspective - box on the 'paradox of savings' on pp 55). Keynes called this the 'paradox of thrift' and in his own words (again quoting from the General Theory):

For although the amount of his own saving is unlikely to have any significant influence on his own income, the reactions of the amount of his consumption on the incomes of others makes it impossible for all individuals simultaneously to save any given sums. Every such attempt to save more by reducing consumption will so affect incomes that the attempt necessarily defeats itself.

Consequently, if an economy is in a liquidity trap monetary policy becomes impotent because monetary policy stimulates economy by lowering interest rate which encourages investment and borrowing and thus spending, as well as putting upward pressure on prices, which in turn helps to 'grease' the wheels of economy by reducing the frictions caused by sticky prices and wages (e.g. wages are very slow to adjust nominally so inflation can force real adjustment even when nominal ones would require long negotiations). These links break in liquidity trap as in such situation changes in money supply tend to just affect holding of money rather than encouraging investment or other spending.
However, government spending can still stimulate economy even in such situation. In fact, paradoxically in such situation government spending becomes extra potent because monetary policy that follows Taylor rule (or similar rule) usually works against government spending, because in normal situation such central bank would raise interest rates when government would pursue expansionary fiscal policy. Nonetheless, in situations as liquidity trap they reinforce each other because in such situations such central bank will keep its interest rate on the minimum possible level even if government pursues fiscal expansion  (see discussion in Blanchard et al reference mentioned above, the Burda and Wyplosz in Macroeconomics ch. 10 have nice discussion of this or if you look for some more nuance then models in texts such as Romer Advanced Macroeconomics - although this is all contemporary treatment).


Exports: How can a small amount of export make really high income I'm currently researching for a school project about Thailand.

I found the wonderful Observatory of Economic Complexity Website.

There are two visualizations I don't quite understand:
https://atlas.media.mit.edu/en/visualize/tree_map/hs92/export/tha/all/show/2015/
https://atlas.media.mit.edu/en/visualize/stacked/hs92/export/tha/all/show/1995.2016/

How can it be that Animal products (light yellow) are only 1.3% of total exports but make the most money.

Maybe I didn't understand the Stacked Visualization right. So maybe that's my mistake.

EDIT: OK; I just found out that I misread the chart. I was a little bit confused obviously. These stacked line charts are completely unlogical to me

What you're asking1 stems from a more general question

  "How can such a small amount of some export $Q$ make the most money out of all the exports?"

To that the answer is to simply recall the standard formula for Total Revenue.

i.e.

$$TR=P \times Q$$

Even though the quantity of export $Q$ is small it could be that the price for $Q$ (be that for whatever reason) be the cause for why it makes more revenue relative to the countries other exports.

Hope this is helpful!

1 I could be wrong


What Is the Economic Meaning of "Structural Reforms" I see tweets like this a lot:

  Poor 2q growth shows its not just a "1q curse". Cannot ever cause eco recovery thru just #QE, low interest rates.  #StructuralReforms

They seem to be saying essentially that structure should be the focus, not interest rates.  What is an economic structural reform?

Economic variables can be "real" or "nominal". So when policymaker discus influencing the economy, they look at implications of influencing these economic variables.

An example of a nominal variable is the price level influenced by money supply. Money is directly created by the government and can take on any value per se and is not "real. E.g. if we multiply all prices and bank notes by 10 nothing in the economy changes.

Real variables are for example output, productivity etc.

However, it turns out that these nominal variables can influence the real ones. Increasing the money supply can increase output (at least temporarily). This is also easy to implement, so monetary policy is often used to fight a recession. Note that lowering the interest rate and increasing the money supply are two sides of the same coin and both part of monetary policy.

An alternative to monetary policy (based on nominal variables) would be "real" economic reforms. One form of "real" policy is a "structural reform". In this case you would try to increase output by changing something in the real side of the economy. For example lowering (wage) costs through changes to labor protection laws. Fiscal policy (government spending or tax cuts) is another form of "real" policy, however it is not a structural break.

To fully understand structural reforms we must introduce the concept of output potential. This is the output that would normally occur in absence of disturbances. It only depends on the underlying structure of the economy. These are factors such as costs (e.g. ease of doing business), labor market flexibility and other things that influence output in general in "real" terms. The economy generally comes back to that level and fluctuates around it in the medium run. (Note this level may be growing over time, e.g. by 1-2% each year).

However, sometimes it happens that our economy is operating below its "natural" potential. In this case we can employ monetary and fiscal policy to get back to that natural level. We can also push the economy beyond the natural level, but we will eventually come back, so this is not a good idea since the side effect that brings is typically more inflation.

If the crisis is because we are simply below potential, we do not need structural reforms. However if we are in a crisis because the potential or "natural" output has risen, then we need "structural reforms". An often cited recent example is the Greek crisis where Greece's wage rate increased a lot over time making its exports too expensive, meaning the economy did not have its past potential to export anymore. In this case many economists suggested "structural" reforms (law changes) that would lead to lower wages.

Monetary and Fiscal policy are often used to correct short-term problems or deviations from output potential. Structural reforms are supposed to change the (medium to long term) output potential itself.

The type of policy best suited often depends on the type of crisis at hand and its underlying causes.


Auctions with affiliation This problem is an example given in Vijay Krishna's Auction Theory (2nd Edition, Chapter-6, Example 6.2). The problem is as follows:

Suppose $S_1,S_2$, and $T $ are uniformly and independently distributed on $[0,1]$. There are two bidders.Bidder 1 receives the signal $X1=S1+T$,and bidder 2 receives the signal $X_2=S_2+T$. The object has a common value for both the bidders, $V=(X_1+X_2)/2$.

Now, we are required to find out the bidding strategy for a first price auction. The equilibrium bidding function is given as $\beta(x)=\int_0^xv(y,y) \,dL(y|x)$. $L(y|x)$ is further equal to $\exp(-\int_y^x\,\frac{g(t|t)}{G(t|t)}\,dt)$.

I have four questions:

  How are $X_1$ and $X_2$ affiliated?
  How is $\frac{g(t|t)}{G(t|t)}$ calculated in this example?
  How do I find the joint density of $X_1$ and $X_2$?
  How do I find the conditional density of $X_2$ given that $X_1=x$?

Roughly speaking, $X_1$ and $X_2$ are affiliated because they have the common component $T$. That is, if $X_1$ is large, $X_2$ tends to be large as well, because a large $X_1$ makes a large $T$ likelier than without this information.
The variables are affiliated if for the joint density $f_{X_1,X_2}$
$$f_{X_1,X_2} (x',y)  f_{X_1,X_2} (x,y') \geq f_{X_1,X_2} (x,y) f_{X_1,X_2} (x',y') \quad \forall x'\geq y, y'\geq y,$$
which here holds with equality. That is, they are weakly affiliated. See below.
In Krishna's book $G$ (resp. $g$) is usually the CDF (resp. density) of the highest value among the other bidders, $Y_1$. Here, there is only one other bidder such that $Y_1=X_2$ and $g(y|x) = f_{X_2|X_1=x}(y |x)$.

Now, my answers only help you if you know how to find the joint density.
Here is what we want to find:

 So, how to get there?

$S_1,S_2, T$ are i.i.d. draws from the uniform distribution.
Let us first consider the joint CDF of $X_1, X_2$ conditional on $T=t$. That is easy, because they are independent now! Once we have this conditional CDF, we use that
$$F_{X_1,X_2} (x,y) = \int_0^1 F_{X_1,X_2|T=t}(x,y|t) f_T (t) dt,$$
where $f_T(t)= 1$ if $t\in [0,1]$ and $f_T(t)= 0$ otherwise.
Then, we take derivatives with respect to $x$ and $y$ to find $f_{X_1,X_2}(x,y)$. This should look like what Krishna has in his picture or alternatively, like the function in the appendix of Avery (ReStud 1998), where the example is from (I guess).
These calculations are cumbersome work with a bunch of case distinctions and thus it is easy to mess up, but you start with
$$F_{X_1,X_2|T=t}(x,y|t) = \mbox{Pr}(X_1<x,X_2<y|T=t) = \mbox{Pr}(S_1<x-t,S_2<y-t)$$
and then pay attention to the support of the uniform distribution. The CDF is $F_{S_1}(s)=s$ for $s\in [0,1]$ and zero for smaller $s$ and one for larger $s$.

Let us take an exemplary region, $y>x>1$:
$$F_{X_1,X_2}(x,y) = \int_0^{x-1} 1 dt + \int_{x-1}^{y-1}(x-t)dt+ \int_{y-1}^1 (x-t)(y-t)dt,$$
which gives us something of which Mathematica says it has a derivateive $\partial x \partial y$ equal to $2-y$. Now, we can verify one of the regions of Krishna's graphic.
One more for fun, if $x<y<1$,
$$f_{X_1,X_2}(x,y) = \frac{\partial^2}{\partial x \partial y} \left( \int_0^x (x-t)(y-t) dt + \int^1_x 0 dt \right) = x.$$
And so on. Now, you can go back to my reply 1.

The conditional density of is given by the equation
$$f_{X_1,X_2} (x,y) = f_{X_1|X_2=y} (x|y) f_{X_2} (y),$$
where
$$f_{X_2}(y) = \begin{cases} y &\mbox{if } y \in [0,1] \\
2-y &\mbox{if } y \in [1,2],
\end{cases} $$
which follows from the fact that $X_2 =S_2+T$, and $S_2$ and $T$ are independently uniformly distributed. You get there with convolution. What you get should look like this plot from the Avery paper

Then, you integrate to find $G(y|x)$ and go back to my reply 2.


(tax with) costs to first order 1) What does it mean that something (in this context the taxation) has (or doesn't) costs to first order?

So, below is an excerpt from Romer's Advanced Macro (p. 598) with such a statement that I've encountered and felt a bit lost. Altough the model developed later is pretty clear, I still don't get what the sentence with this 'costs to first order' really mean.

2) Also: how the last sentence is implied by the sentence before the last one? That is: what exactly (this first-orderness? how?) implies that $C \approx T^2$?

1) "To first order" appears to be a shorthand for "to a first-order approximation", meaning in turn, "to a linear approximation".

2) It is not that "the one implies the other", it is that the one is consistent with the other.

In attempting to arrive at a functional specification for the "tax distortion costs" function, say $C_D=h(T)$, first we reason that if tax revenue is zero (not due to inefficiency of tax collection and/or tax evasion, but because the tax rate is set to zero), then distortions due to tax will also be zero. This implies that $h(T)$ should not contain a constant term, so that $h(0) = 0$.

Moreover if we reason (as D. Romer does) that "distortions by taxes are likely to increase more than proportionally with the amount of revenue raised", this implies that $h(T)$ should not be linear in $T$, but it should be convex (this is what it means to say "more than proportionally").

The natural step then is to consider a quadratic distortion cost function, without a constant term:

$$C_D =  aT^2,\;\; a>0$$

which reflects both our theoretical arguments (and in particular, $h(0) =0$).

But this also implies that "distortion costs are zero to a first-order approximation", because we have

$$h'(T) = 2aT \implies h'(0)=0$$

so a first-order approximation (Taylor expansion) around zero would give

$$h(T) \approx h(0) + h'(0)\cdot T = 0+0\cdot T =0$$


Whats the meaning of this parameter? I was looking the currency exchange board, and I saw the value of "exotic" for the parameter Group, what is the meaning of this value?

I dont know if this is a local figure or is used in currency.

Currencies and currency pairs are sometimes classified as Major, Minor, or Exotic, depending on how heavily traded they are (Major most heavily traded, Exotic least).


Qualitative methods in Economics As a quantitative analyst and armchair economist I have seen little reason to do qualitative analysis as done in the other social sciences.

However I keep running into these rather wordy analyses of economic issues (primarily from social research organisations) which are constantly pushed and presented as holding the same weight as those presented by the quant camp.

I don't want this to turn into a rant and if this question is opinionated please close it.

My question is: Where does qualitative analysis fit in formal economic research?

Good question!

Qualitative methods (QM henceforth) have become more widespread in the recent decades, according to a 2014's survey in the Journal of Economic Surveys. QM include, among others, in-depth interviews, focus groups, case studies, ethnographic studies, and mixed-methods (quanti+quali). They are still not that popular though, and have been very often looked down by the field. This is evident from the JEL coding, which does not include a specific section for QM, but does include one for quantitative methods.

In contrast to BB King comment, these methods have been used in very hard-core economic issues like Macro (e.g. Blinder et. al. 1998), Labour Economics (e.g. Cregan 2005), International Organisation (e.g. Autor, Levy and Murnane 2002), Development Economics (e.g. Valente 2011), etc (for more examples, see Table 1 in survey above). Here is an article from an MIT economist commenting on his experience with using qualitative methods in labour economics research (excuse the fallacy of authority).

If you want to explore more, there is a 2016's book dedicated entirely to the subject. Also, Chapter 7 of this handbook focuses on QM used in Heterodox Economics.

Other references in which QM are used:

Health Economics
Introductory economics teaching
Measurement of poverty
Finance
Input-Output analysis
Economic modelling


Clearly explaining the concept of "normalizing" price and relative prices to undergraduate I am trying to clearly explain to the concept of "normalization" or relative prices. Because students are so used to just think of prices in absolute terms, it is initially tough to make the leap to the "relative" or "normalization" concept.

What is your angle if you were to explain this to an econ 101 student?

For example, the toughest part is to explain the "need" for looking at the ratio of prices. Who cares? Why is it useful?

Thanks bunch.

I usually ask if they would rather get an IPhone or 5000 South African Rands (ZAR). You can also ask if 5000 ZAR is a good enough monthly salary. As we are far from that country usually no one has a clue about the currency exchange rate.  This way you can explain that the number in itself is meaningless, what matters is what you can exchange it for, i.e. relative prices matter.


What has been the reception of Posner & Weyl’s claim that property rights lead to market power? Posner & Weyl claim that property rights lead to monopoly rents:
Property Is Only Another Name for Monopoly
Radical Markets
Have their claims been elaborated, tested, or peer-reviewed?
In situations with assets with unique attributes that cannot be substituted (e.g. a plot on a high street) this seems to be intuitive. An owner cannot be compelled to subject her property to market competition but she can hold out for a higher price than she values the property at. This generates an economic rent resulting from her market power over the plot of land in question. The claim seems to be less obvious where assets can be easily replicated (e.g. a pint of milk). That would imply that property rights may only lead to monopoly rents where they grant property over assets that generate monopoly rents (e.g. land).

In narrow sense of the word the article was definitely peer reviewed as it was published  in Journal of Legal Analysis. But in this narrow sense it was most likely peer reviewed by jurists not economists.
In a broader sense of the word there are economists who had a look at the claims as well. For example, review of their book by Levine (2020) in Journal of Economic literature (one of the top ranking journals), examines their claims about monopolies and also other claims and policy proposals made in the book. The review is not very warm to the ideas of Posner and Weyl. Regarding the idea of property rights causing widespread monopoly power and misallocation Levine writes [emphasis mine]:

The chapter begins with a clear and careful description of the hold-up problem.
Here, indeed, monopoly power is a problem.
It then goes on to suggest that the hold-up
problem is ubiquitous in ownership of property—that all property is monopoly and that
the resulting misallocation might be on the
order of 25 percent of income. Here as elsewhere the book is not strong on data. There
is extensive research indicating that in some
places at some times, misallocation is a problem possibly of this order of magnitude.
Typically this research does not attribute
misallocation to a problem with private property—generally the reverse.

... Nevertheless the economics
profession does not generally view hold-up and private information problems as ubiquitous nor externalities as insoluble. In many
markets, ample substitutes are available—
for example in the housing market—so that
monopoly power is not important. In some
cases the market solves the hold-up problem relatively well—through tender offers
and other ways of packaging properties.

Pushing on: the authors argue that the
problem with the ubiquitous monopolies
created by private property can be solved by
a self-assessment wealth tax with mandatory
sale at the self-assessed price. Self-assessment
with a commitment to sell at that price is
a useful and credible solution to problems
involving assets for which there is substantial
private information. This has been a particular problem for commercial real estate and
for privately held businesses, where often
artificially low transfer prices are used to
avoid taxes. As the authors correctly say, top
researchers such as Harberger (1965) and
Cramton, Gibbons, and Klemperer (1987)
have argued for self-assessment in these
types of circumstances.
It is not immediately clear why a proposal
for solving a particular private information
problem should be viewed as a broad solution to a problem of ubiquitous monopoly...

There is more critique on this in the paper. To give you a wrap up (as mentioned in the comments under your other question), the problem with Posner and Weyl is not that their analysis would be inaccurate in some specific cases. Rather they try to generalize well understood result that holds in some very specific circumstances onto property rights in general.
There is quite large consensus in literature that property rights do not generally lead to monopoly or even market power (see Belleflamme & Peitz Industrial organization: markets and strategies for overview of many models where even monopoly does not guarantee market power and models of competition where property rights are implicit but firms have no market power). In fact as Mankiw put it in his Principles of Economics, economists view property rights as "an important prerequisite for the price system to work".
Furthermore, Acemoglu and Robinson (top developmental economists) in their magnum opus Why Nations Fail, go even so far as to state:

To be inclusive, economic institutions must feature secure private property...

Acemoglu and Robinson go further on proving that sustainable growth and economic development without inclusive institutions (including property rights) is virtually impossible.
Additionally, this work is apparently even criticized in law profession itself (see Wyman, K. M.; 2019), but as I am not a law professional I will have no further comments on that (it just seemed relevant to point that out).
Hence given the above I think it is fair to say that the claims of Posner and Weyl are not received positively by economics profession, to put it very generously.


How do takeover deals create arbitrage opportunities that involve necessarily selling short the acquirer’s stock? Yes, you can get a juicy yield from a pot ETF. But here’s the catch - The Globe and Mail

  Takeover deals create arbitrage opportunities that involve selling short the acquirer’s stock [I emboldened.] and going long on the target company , and the ensuing demand for borrowed shares drives up the rates Horizons can charge, Mr. Noble says.

Why sell short the acquirer's stock? Why assume that the acquirer's stock will fall? Can't it increase like the target company's?

In simple terms, an acquisition usually happens when a successful (read: low-risk) company purchases an unsuccessful (read: high-risk) company. It can be viewed as a transference of risk from the acquiree to the acquirer, and all things being equal, the market demands higher returns from riskier assets. In the short-term this is effected through the price, and one channel for this is that risk-averse holders of the acquiring company will move to safer harbors. The acquirer's stock thus falls, implying a higher return in line with the new level of risk.  The market also demands lower returns from lower-risk assets, and so more investment will flow into the acquiree, driving its price up.

Obviously, there's more going on and this is not investment advice.  It's just a rule of thumb.


Why is it a problem if a considerable amount of currency is hoarded by the enemy of the issuing government? During WWII and after Pearl Harbor, the United States issued the Hawaii overprint notes because they feared the possibility that if the Japanese captured Hawaii they would also control a considerable amount of USD bills [1].

I found this piece of trivia not only interesting but also made me wonder why this could be an issue? Does this mean it is also an issue if say today China starts hoarding stacks of American dollars?

https://en.wikipedia.org/wiki/Hawaii_overprint_note

If the Japanese captured Hawaii and hence also all the dollar bills in Hawaii, then the problem would not be that they "hoarded" these dollars, but that they would have been able to use these dollars (to buy other goods and services).

And so the purpose of the Hawaiian overprint notes was this: they "would be worthless in the hands of the enemy, but could be used as money in Hawaii as long as the U.S. maintained control."

I have noticed that laypersons tend to be unduly worried about the possible effects of "hoarding" (as evidenced for example by the questions on this site about hoarding). There seems to be a myth that hoarding cash can somehow cause damage to other parties. But in fact, under most circumstances, it is those who do the hoarding that suffer, while those who aren't hoarding benefit.*

If China "hoards" US dollars that were acquired through legitimate, peaceful means, then this is no bad thing for the US.

Say China sells T-shirts and other goods to the US in exchange for \$1 trillion in dollar bills. That is, the US receives these goods and simply gives China a bunch of paper money. China then "hoards" these \$1T in paper money. Then this is no bad thing for the US. The US will simply have acquired \$1T worth of T-shirts and other goods, while China will have acquired a lot of paper that is useless so long as it is "hoarded" and unused.

Indeed, the value of this paper will tend to depreciate. In fact, the US can cause the value of this paper to depreciate by simply printing more dollars. And in a state of war, the US could even undertake drastic measures such as declaring those \$1T useless or achieving this effect by requiring all US citizens (and possibly other non-Chinese parties) to exchange their dollar bills for new legal ones.

*One important exception to this is the underground economy. When cash is hoarded, this can negatively affect those in the underground economy who rely heavily on cash to facilitate transactions.


Adam Smith: If humans are self-interested individuals, why should we reinvest our wealth? According to Adam Smith, humans are self-interested individuals. If this is true, why does he argue that people should reinvest their wealth? How does this benefit a self-interested individual? Does morality play a role here? What role does productive and unproductive labor play here precisely?

Economics have evolved considerably since Adam Smith and his ideas have been refined. I can think of two approaches to your question:

Why would a self-interested individual re-invest wealth?

1) Financial perspective

Whenever you invest money instead of spending it now, you get paid interest. If that interest is higher than the satisfaction from spending the money now instead of later, then re-investing is in your self-interest.

2) Game-theoretical and behavioral perspective

A self-interested person wants to maximize his or her payoffs. Payoffs can differ from monetary outcomes. If I re-invest my 1000 dollars into bonds for a year I can get 1050 dollars at the end of the year. But if I give 1000 dollars to charity, I may get a lot more satisfaction from that, provided that I am an altruistic person. So if try to measure my payoff from the charitable donation, it may be worth the equivalent of 2000 dollars to me. It is in my self-interest to donate instead of re-investing then. The same reasoning can be applied to saving and spending. If it gives me pleasure to hoard money then I will choose to re-invest rather than spend. Using this approach you can justify any behavior as self-interested or rational.

You can take financial perspective to explore how productive and unproductive labor relates to your question, but if you take more general game-theoretical perspective, then it will be difficult for you to even define "unproductive labor" as even piling money and setting it on fire may be considered productive in some situation. But I would take the latter. After all we should study economics to better understand the world in all its complexity, not to take the easier path.


echange rate from LOC/EUR to EUR I have import values in LOCEUR (LOC is local currency).
How can I obtain these values only in EUR? My data are from Eurostat, where external trade indicator is "Exports in million of ECU/EURO"

If you mean you want to convert ECU to EURO, they are essentially the same currency; one was a basket of European currencies used before 1 Jan 1999, the other is used after. The exchange rate is exactly 1 ECU = 1 EURO (reference) So you should be able to use that as EUROs.

Note: This came from my comment above, which I take that it helped the OP. Just to boost up answered questions ratio for this site.


What levers are typically available to a Government to exit an economic downturn? What levers are typically available to a Government to exit an economic downturn?

I can think of currency devaluation, efficiency improvements and infrastructure spending (possibly using funds raised by borrowing).

The usual policy options are.

Monetary policy - lower interest rates and/or expand the monetary base.
Fiscal policy - expand the fiscal deficit (government spends more and/or taxes less).
If the value of the currency is controlled, devalue.
If there are controls on lending (now rare in the developed countries), loosen them.

Something like efficiency improvements (how can the government endure this?) may help long-term growth rates, but it is unclear whether they will have an immediate impact. For example, if increasing efficiency involves firing workers, there would be a further reduction in household demand, which will not be corrected until they can find new jobs.


Corporate Finance: From External Financing Needed(EFN) to Sustainable Growth Rate Well in this link I get the EFN definition that I know of, with the spontaneous liabilities term included. I'm trying to derive the Sustainable Growth Rate for a corporation, from this definition of EFN.

How does one calculate the liabilities-that-change-directly-with-sales term, using just previous year's sales (S), total assets(A), total debt(D), total equity(E), projected growth in sales(g), profit margin(PM) and retention ratio(b) as variables?

I found this link(page 11-12) that does the required derivation, but I don't understand the reasoning...

Any help would be appreciated.

We can replace growth rate in the formulation as below
$$\frac{A_0}{S_0}gS_0-\frac{L_0}{S_0}gS_0-PM(1+g)S_0b=0$$
$$(A_0-L_0)g-PMS_0b-PMS_0bg=0$$
$$g=\frac{bPMS_0}{A_0-L_0-bPMS_0}$$

In the paper you linked the author assumed $A_0$ and $L_0$ to be total assets and total debt (pg.11); therefore $A_0-L_0=E_0$
$$g=\frac{bPMS_0}{E_0-bPMS_0}$$
$$g=\frac{b\frac{PMS_0}{E_0}}{1-b\frac{PMS_0}{E_0}}$$
$$g=\frac{b\times ROE}{1-b\times ROE}$$


Econ Intuition for Jacobian inverse in demand system Consider the following simple linear demand system (in vector notation) with n different products

Demand: $\quad\mathbf{q=B\left(a-p\right)}$

Inverse demand: $\quad\mathbf{p=a-B^{-1}q}$

where $\mathbf{p}$ is the vector of prices and $\mathbf{q}$ is the vector of quantities supplied. The Jacobians of these two equations

$\dfrac{\partial\mathbf{q}}{\partial\mathbf{p}}=-\mathbf{B}\quad$
  and  $\quad\dfrac{\partial\mathbf{p}}{\partial\mathbf{q}}=-\mathbf{B}^{-1}$

tell you how prices react to a change in quantity supplied and vice-versa. Now, consider the example:

$\mathbf{B}=\left[\begin{array}{cc}
\frac{4}{3} & -\frac{2}{3}\\
-\frac{2}{3} & \frac{4}{3}
\end{array}\right]\quad$ so that $\quad\mathbf{B}^{-1}=\left[\begin{array}{cc}
1 & .5\\
.5 & 1
\end{array}\right]$

then it is easy to see that, for some product $i$:

$\dfrac{\partial q_{i}}{\partial p_{i}}\neq\left(\dfrac{\partial p_{i}}{\partial q_{i}}\right)^{-1}$

I understand that that is true mathematically and will generally be the case, but what does it mean in terms of economic intuition? Why does the demand curve for product $i$  slope differently depending on whether I put $q_{i}$ on the vertical axis rather than the horizontal axis?

For the 2x2 case being considered, write
$$\mathbf{B}=\left[\begin{array}{cc}
b_{1,1} & b_{1,2}\\
b_{2,1} & b_{2,2}
\end{array}\right].\quad$$

It follows that the element (1,1) in $B^{-1}$ is given by $\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}}$. Notice that
$$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 })^{-1}$$
implies
$$b_{1,1}=(\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}})^{-1},$$
which can only occur if either $b_{1,2}=0$, $b_{2,1}=0,$ or both.

To better understand the intuition for this result, consider the demand function $q_1(p_1,p_2).$ Suppose we are interested in the demand curve for good 1 when $p_2=\tilde{p_2}$, which is a plot of the function $q_1(p_1,\tilde{p_2})$ with $p_1$ on the vertical axis and $q_1$ on the horizontal axis. The slope of this demand curve is given by $(\frac{\partial q_1(p_1,p_2)}{\partial p_1})^{-1}=-\frac{1}{b_{1,1}}$.

Now consider the inverse demand for good 1, which gives us maximum marginal willingness to pay for good 1 conditional on quantities of good 1 and good 2. If we want to plot this demand curve we must fix the quantity of good 2, say at $x_2=\tilde{x_2}.$ Suppose we want this demand curve to correspond to the demand curve we plotted for $q_1(p_1,\tilde{p_2})$, meaning $\tilde{x_2}$ must be the quantity demanded at $\tilde{p_2}$. However, to find the quantity demanded of good 2 at $\tilde{p_2}$,we must condition on the price of good 1. Likewise, to find the marginal willingness to pay of good 2 at quantity $\tilde{x_2}$, we must condition on the quantity of good 1. Both of the previous statements make no sense, as we are trying to plot the relationship between $x_1$ and $p_1$, so we cannot fix either of them.

The basic intuition is that when we invert $B$ to solve for inverse demands, we are accounting for the effects of cross-price effects in writing our marginal willingness to pay (inverse demand) functions. In economics terms, the following two expressions are not equal unless cross-price effects are zero:

Marginal effect of the quantity of good 1 on willingness-to-pay for good 1, conditional on the  quantity of good 2.
Inverse of the marginal effect of the price of good 1 on quantity demanded of good 1, conditional on the price of good 2.


Submodularity property in congestion games? Let $G$ be a $n$-players and $m$-elements congestion game.

For an equilibrium $e$, denote by $$SUP(e)\triangleq<sup_1(e),sup_2(e),\ldots, sup_n(e)>$$

Where $sup_i(e)$ contains the support of the $i$'th player playing $e$ (the set of strategies $i$ play with positive probability).

Also, we say that $SUP(e)\subseteq SUP(e')$ iff $\forall i\in[n]: sup_i(e)\subseteq sup_i(e')$, that is every player in $e$ randomizes his action on a subset of the actions he could have chosen playing $e'$.

One last definition is the social cost, $SC(e)$ which is defined to be the sum of costs for the players.

Let $e,e'$ be two (possibly mixed) equilibriums for $G$.

  Does $$SUP(e)\subseteq SUP(e')$$ imply $$SC(e) \leq SC(e')$$?

This proposition is in general not true. One can show that it is true in the case $n=2$ and $m=2$. Here, I exhibit a counter example when $n=3$ and $m=2$.

A brief comment. We can rephrase the question in words: does a Nash equilibrium that is "more random" ($e'$ versus $e$) is less efficient? Intuitively, as more mixed strategies are played, the realized outcome is more random and it can be very inefficient due to a lack of coordination between agents. When agents play pure strategies, we can think that we reduce the coordination problem given that we consider Nash equilibria. This intuition does not hold if the proposition is false, as I will show when $n=3$ and $m=2$.

Denote $A$ and $B$ the two possible actions. The delay functions are defined as follows:
$d_A(1)=5$, $d_A(2)=7$, $d_A(3)=10$ and $d_B(1)=1$, $d_B(2)=6$, $d_B(3)=7$.
It means that when $x$ agents play $A$ (resp. $B$), they receive the payoff $-d_A(x)$ (resp. $-d_B(x)$). This is a (symmetric) congestion game as long as the delay functions are increasing.

Define $e$ as the equilibrium when 1 agent plays $A$ and 2 agents play $B$. Define $e'$ as the equilibrium when 1 agent always plays $B$, and the 2 others plays $A$ with probability $\mu=2/3$ and $B$ with probability $1-\mu=1/3$. It satisfies the property  $sup(e) \subseteq sup(e')$.

First, we show that $e$ is a Nash equilibrium. The agent who plays $A$ is maximizing her payoff given the two other players' strategy when choosing $A$ is better than choosing $B$, $d_A(1)<d_B(3)$ (i.e. $5<7$). Both agents who play $B$ are playing optimally if $d_B(2)<d_A(2)$ (i.e. $6<7$). $e$ is thus a Nash equilibrium and its social cost is $d_A(1)+2d_B(2)=17=\frac{153}{9}$.

Second, we show that $e'$ is a Nash equilibrium. On one hand, the agent who plays $B$ is maximizing her payoff when the two others play mixed strategy if she is better off playing $B$ than $A$,
$$(1-\mu)^2d_B(3)+2\mu(1-\mu)d_B(2)+\mu^2d_B(1)<(1-\mu)^2d_A(1)+2\mu(1-\mu)d_A(2)+\mu^2d_A(3)$$
i.e. $\frac{1}{9}5+\frac{4}{9}7+\frac{4}{9}10<\frac{1}{9}7+\frac{4}{9}6+\frac{4}{9}1$, which is true.
On the other hand, each one of the agents playing the mixed strategy is indifferent between choosing $A$ or $B$ if
$$\mu d_A(2)+(1-\mu)d_A(1)=\mu d_B(2)+(1-\mu)d_B(3)$$
i.e. $\frac{19}{3}=\frac{19}{3}$.
$e'$ is then a Nash equilibrium and its social cost is
$$(1-\mu)^2 [3d_B(3)]+2\mu(1-\mu)[d_A(1)+2d_B(2)]+\mu^2[2d_A(2)+d_B(1)]$$
 which is equal to $\frac{1}{9}21+\frac{4}{9}17+\frac{4}{9}15=\frac{149}{9}$.

Finally, we have shown that $sup(e) \subseteq sup(e')$ but $SC(e) > SC(e')$. The mixed-strategy Nash equilibrium results in a lower social cost than the pure-strategy one.


Do current Central Banks need to target a monetary aggregate as they set interest rates for a target inflation? On reading a textbook, the author state that after the 80's, and the serious consequences of thinking that inflation could be controlled by just targeting a monetary aggregate, modern central banks(CB) have shifted their paradigm to a targeting inflation by setting the interest rate.
My question is if the CB sets the interest rate, does he need to target M0? Or does the CB just print money according to money demand?

Any help would be appreciated.

The demand for liquidity is usually defined as
$$
L(Y,i)
$$
where $Y$ is income and $i$ is the nominal interest rate. In equilibrium this equals the aggregate money supply $M$ which is function of several things, like $M_0$ the reserve rate, etc. Let us assume that everything except $i$, $M_0$ and $M = f(M_0)$ is constant. Then
$$
L(Y,i) = M = f(M_0).
$$
Demand for liquidity decreases in $i$ and $M$ increases in $M_0$. Thus for any given value of $i$ there is at most one equilibrium value of $M_0$, hence the central bank cannot simoultaneously set both $i$ and $M_0$ and always expect an equilibrium outcome. Lacking equilibrium either $i$ or $M$ will shift.


What graph theory textbook should I use? I'm doing my Msc dissertation on production networks and business cycles and I want to learn graph theory. I need to be able to model the evolution of the network and identify equilibrium graphs.

Could someone recommend a textbook for me?

Social and Economic Networks, by Matthew O. Jackson, and its associated MOOC at https://www.coursera.org/course/networksonline (videos from the 2014 session available, new live session starting September 2015).


Commodity bundles as vectors By definition,vectors are quantities having magnitude as well as direction ,especially as determining the position of one point in space relative to another

We study consumption bundles by taking them as vectors.My question is why can't we think of them as scalar quantities to study them.Why do we think of consumption bundles as locations in commodity space.What is its use?

In real world, whenever we think of commodities , only their quantities come to our mind.And when we compare two commodity bundles, we only compare their quantities.Then,why do we have to bring in the concept of vectors here...are there any cases in the real world where we are concerned about the displacement of the commodity bundles.I am unable to understand this.

Thanks.

First of all, you need to expand the definition of a vector. It is not only quantities having maginitude and direction. While I agree that is the representation in $\mathbb R^2$, it is not always the case. Think of a vector as a representation of any ordered group of numbers $(x_1,x_2,...,x_n)$ and forget completely that you ever thought these implied magnitud and direction.

We think of commodity bundles as vectors and not as quantities by themselves because we want to be able to explain consumption choices as bundles. Let me give you an example with a bundle of 2 goods, $(x_1, x_2)$. Think that $x_1$ is the amount of coffees that you can buy whereas $x_2$ is the amount of packets of sugar that you can buy. We want to be able to compare, say, $(2,1)$ with $(1,2)$. I agree, we could analyze both the choice of both goods separately, but ultimately we want to know what is the optimal consumption bundle and not the optimal amount of sugar and coffee separated. Now, did I say something that entails magnitude? Or direction? No.

Bottomline, yes, we are thinking about quantities of goods, but when we say vectors, it means bundles of quantities of goods. That is ultimately what happens in the real world.


Meaning of regression coefficients I am having some trouble understanding exactly what is meant by the "true" Regression coefficients. Let's say it is stated that "the true regression coefficients are given as $y=a+bx+e$ where the error term fulfills all the OLS assumptions. Does this mean

a) Given knowledge of the value x takes one would describe their belief about the value y takes with the probability distribution on the right hand side.

b)a one unit change of x in a certain instance would cause the expected value of y to change by b.

a) not sure what exactly you mean here by probability distribution but if you mean the distribution of potential Y given the distribution of estimated coefficient b and a it would be correct, although usually we look at expectation to get single number than on distribution in which case it would be $E(Y|X)$

b) is correct


Derivation of a Demand Equation A consumer's utility function is $U(x,y)=\sqrt x + y$. Assuming we have an interior solution, I need to show that the demand for $x$ does not depend on income.

I know that the consumer consumes where the marginal utilities per dollar of $x$ and $y$ are equal, so I have found the following two equations:

$\frac 1 {2P_x\sqrt x}$ = $\frac 1 {P_y}$
${P_x}x + {P_y}y = I$

Then, I am stuck. I do not see how I can get rid of both $y$ and $I$ in this case? And how does knowing that we have an interior solution help?

hint:

Solve for optimal $x^*,y^*$ then substitute the solutions into budget constraint and then solve the budget constraint for $x^*,y^*$ respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.

Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.


Equilibrium value of $b$ (market tightness) in a search model Consider the following one-shot version of a labour market matching model. Let the labour force be normalized at 1, who, because there is only one period, all start out as unemployed. There is a very large number of firms who can enter the market and search for a worker. Firms who engage in search first have to pay a fixed cost, $k$. If a measure $v$ of firms enters the labour market, a constant returns to scale matching function $m(1,v)$ gives us the total measure of matches in the economy.

Within each match, the firm and the worker bargain for the wage, $w$, so that the workers get a constant proportion of $y$. Denote this proportion by $\beta$, which is interpreted as the bargaining power of the worker. Assume $\frac{k}{y} < 1 - \beta$ for the firm.

Define market tightness as $ b \equiv \frac{1}{v}$ and assume that the arrival rate for a firm is given by: $a_{F} = 1 - e^{-b}$.

Consider the firms can enter the labour market freely if they pay the entry cost, then what is the equilibrium value of $b$? Describe it graphically? Does it always exist? Is it unique?

My solution: The value of a vacancy: $$V = -k + a_{F}(b)(J-V),$$
and the value of a filled job: $$J = y-w.$$
If the firms enter the labour market freely then $V=0$. Then from these two equations I am left with this equation $$ 1 - e^{-b} = a_{F}(b) = \frac{k}{y (1-\beta)}.$$

Graphically, the function looks something like this:

From the graph, it looks like that $b^*$ is unique, but how do I know if it always exists or not?

From

$$ 1 - e^{-b} = a_{F}(b) = \frac{k}{y (1-\beta)}$$

we get

$$b = -\ln\left(1-\frac{k}{y (1-\beta)}\right) \tag{1}$$

For this to exist (be a real number) it must be the case that

$$1-\frac{k}{y (1-\beta)} > 0 \implies \frac k y < (1-\beta) \tag{2}$$

which is already assumed.

Then, uniqueness of the solution of $(1)$ is guaranteed because all arguments/parameters in the right-hand side enter only once. So for any vector $(y,k,\beta)$ satisfying what it needs to satisfy, we get a unique $b$.


Why does an increase in GDP lead to an increase average incomes? I am a high-school student...sorry if this is a very simple question.

Is it due to the fact that, as there is more money in the economy, firms can afford to pay higher wages/employ more people? But surely this effect is more or less cancelled out by the increases in average price level within an economy?

Thank you in advance.

In a nutshell, a higher GDP is the very same as an increase in average incomes, at least if the population remains constant. If there is more stuff produced and consumed, overall income is higher and average income as well.

As you rightly point out, since income is measured in monetary terms, one has to correct for price increases. And this is usually done. Whenever you hear 'GDP has increased', this better be in real terms, i.e. in constant prices. In nominal terms, GDP (as well as tax revenues) are increasing almost every year, which renders nominal comparisons usually meaningless.


Does a General Equilibrium here require Pareto Optimality? There are two consumers $A$ and $B$, and two producers $X$ and $Y$. he consumers are endowed with labour(L) and capital(K) $L_A,K_A$, and $L_B,K_B$ respectively. The preferences of the two consumers have the utility functions $U_A=X_A^4Y_A$ and $U_B=X_BY_B$. The technologies of the two producers are given by $X=\sqrt{K_XL_X}/2$ and $Y=2\sqrt{K_YL_Y}$. Setting the wage rate $w=1$, what are the competitive general equilibrium relative prices?

This is inspired from a homework problem, but I wish to understand how to solve this general class of questions. Does this require pareto optimality? How does calculus feature in this question? Any help would be great! Thanks.

Competitive equilibrium is the price vector $(p_x, p_y, w =1, r)$ such that it solves the following system of equations:

Demand for $X$  = Supply of $X$
Demand for $Y$ = Supply of $Y$
Demand for $L$ = Supply of $L$
Demand for $K$ = Supply of $K$

where these demands and supplies are either exogenously given or are derived by solving utility maximization problem of the consumers, and profit maximization problem of the firm in a standard way.

Method 1: One way to solve the posted problem is to find these demands and supplies and solve the resulting system of equations.

Method 2: Another way is to use first welfare theorem. First welfare theorem says that if the utilities are increasing then competitive equilibrium is Pareto efficient. In the posted problem, both the consumers have increasing utility function. So, we can use it to determine the competitive equilibrium prices. Equilibrium prices and allocation satisfy the following:

Efficiency in production: MRTS$^X_{L,K}$ = MRTS$^Y_{L,K}$ = $\frac{w}{r}$. Use this to derive Production Possibility Frontier (PPF).
Efficiency in consumption: MRS$^A_{X,Y}$ = MRS$^B_{X,Y}$ = MRT$_{X,Y}$ = $\frac{p_X}{p_Y}$.

Solution (using method 2): Given the data in question, we see that both the production functions exhibit constant returns to scale and the PPF will be of the form $4x + y =$  constant. Therefore, the price ratio $\frac{p_x}{p_y}$ in equilibrium will be 4 regardless of the initial endowments. This will be true as long as both the inputs $L$ and $K$ are available in positive quantities. Equilibrium can be fully determined once the details about endowments are specified. To find $\frac{w}{r}$ ratio we need to know the total endowment of labor and capital in the economy, and to find the absolute prices of $X$ and $Y$ given $w = 1$ we need the data on how much of the total endowment of labor and capital belongs to each consumer.


Why is the inverted yield curve a good predictor of impending economic recession? The inverted yield curve has often been cited as a good predictor of impending economic recession. What are the reasons to explain why is it a good predictor? Is the inverted yield curve a cause or symptom of recession?

There is a fair amount of ambiguity to this question. The first question is: what is the yield curve? A fixed income investor may refer to the yields across all maturities as the yield curve, while economists pick the difference in yields between two arbitrary maturities as the yield curve. Which two? Typical choices are the spread between the 2-year and 10-year bond, or the spread between the 3-month Treasury bill rate and the 10-year. The quality of a recession indicator depends which pair you choose.

The second question is: what country are we talking about? The yield curve is often described as a good recession indcator for the United States (which can be seen by looking at the time series of the 2-year/10-year Treasury spread, and compare them to recession dates. It is less clear how well it works for other countries.

There is a simple theoretical reason why it works. Under any modern financial model of the risk-free interest rate curve, we can see that term interest rates are roughly equal to the average of the expected overnight rate to its maturity, plus a term premium. (Technically, the average is a geometric average, since we are compounding an investment at the overnight rate.) As long as the term premium is not significantly negative, an inverted yield curve implies that the expected path of short-term rates is heading lower. The usual reason for a central bank to cut rates significantly is a recession (either domestically, or in trading partners).

In other words, if an inverted yield curve is a good recession indicator, it is just telling us that bond investors may be good at forecasting recession odds. Do we have reason to expect that they are perfect forecasters? Probably not. There’s any number of things that could distort bond market pricing, or else bond investors could herd into a poor forecast. Attempting to do statistical analysis of the yield curve without taking into account the changing institutional structure of the bond market is problematic.

I am not aware of any reason that suggests the inverted yield curve causes recessions under current institutional arrangements. (I am admittedly biased.) This is because developed country banking systes have much tighter control of interest rate risk now. This was not the case earlier (e.g., 1970s), and it may be that for some countries now, interest rate risk is not being hedged.

If you are interested in that question, I suggest asking it in a separate question, and you should specify what countries and eras you are interested in. The title of the question refers to prediction; a lot of readers would not relate that to causing recessions.


How does the notion of utility differ from that of value? Is utility merely the notion of value in the subjectivist/marginalist (aka neoclassical) school?

This turns out to be a surprisingly complicated issue. On one hand

  ‘value depends entirely upon utility’ (Jevons, 1970, pp. 139 and 77, respectively). [...]

  For the marginalists of the first generation, a marginal utility curve incorporated the relation between a commodity and human wants; total utility was a derived concept and corresponded geometrically to the area beneath the curve (the integral, in mathematical terms).

  Gradually, it became more natural to consider utility, right from the beginning, as a a property of consumption bundle, rather than individual consumption goods. An important step in this direction was made by Edgeworth (1881). [...] Utility of a consumption basket, rather than marginal utility of individual goods, thus became the ‘primary’ concept.

On the other hand the more commonly used notion (nowadays) of von Neumann-Morgenstern (cardinal) utility isn't the same thing as the neoclassical one, but is close enough:

  What is controversial is whether the utility function derived by the N-M method can
  be taken to reflect the true subjective utility of the individual in the sense of the
  neoclassical economists such as Edgeworth. A firm negative answer has been given:

    "what relationship, if any, does the N-M cardinal utility theory have to that of the
    neoclassical utility theorists? It is generally (though not universally) agreed that
    there is none - the two utility measures have nothing in common insofar as their
    cardinality is concerned" (Baumol 1977, p. 431; see also Banmol 1951, 1958).

  [...]

  Baumol is right, at least formally. The N-M hypothesis does not show that the
  utility function derived, though predictive of the objective choice of the individual,
  actually reflects his subjective utility in the sense of the neoclassical. On the other hand,
  most practising economists are also right, at least effectively, in accepting the N-M
  utility indices as subjective. This is so since, as is shown below, using the same set of
  axioms as the N-M hypothesis, with only the recognition of finite sensibility (that
  human beings are not infinitely sensitive or perfectly discriminative), it can be shown
  that the utility function derived from the N-M method is in fact a subjective utility
  function in the sense of Edgeworth.

Also, the step (back) from cardinal to ordinal utility is due to Pareto

  Pareto inverted, so to speak, Edgeworth’s problem: rather than deriving the indifference curves from a prior knowledge of a utility function, he did it the other way round – his ‘primary’ concept was a series of indifference curves (later to be called ‘preference ordering’) and from them an analytical description of utility (ofelimità, in his terminology) was derived. What mattered now was the index of the indifference curves, a higher index corresponding to a higher utility.

  He (correctly) remarked that a surface having I as ordinate was not univocally determined: in so doing he discovered what came to be known as ‘ordinal utility’: a result that first-generation marginalists would perhaps have much appreciated!


What are the assumptions behind calculating the market equilibrium from supply and demand curves? Imagine that there are 5 buyers (B1,...,B5) with maximum willingness to pay as follows:

B1 = $5
B2 = $4
B3 = $3
B4 = $2
B5 = $1

Imagine that there are 5 sellers (S1,...,S5) with minimum selling prices as follows:

S1 = $5
S2 = $4
S3 = $3
S4 = $2
S5 = $1

I now present two hypothetical situations:

SITUATION 1: Each buyer is matched to the seller with the corresponding price.

In this case, the quantity traded = 5 units

SITUATION 2: Match B1 with S5, B2 with S4, B3 with S3, B4 with S2 and B5 with S1.

In this case, the quantity traded is equal to 3 units since the last two trades cannot happen.

DISCUSSION AND QUESTION

Obviously, these are two extreme situations. However, it seems to highlight an interesting point, namely, what assumptions do we have to make about "reserve prices" of buyers and sellers, and the way they interact with each other, in order for us to be able to calculate the equilibrium point? In other words, there seems to be more than meets the eye with the idea of calculating the equilibrium point from the supply and demand curves alone.

Public information about prices--not necessarily about costs or willingness to pay--is indeed the key here.

If S5 is willing to sell to B5 at $1, she must also be willing to sell to any other buyer at this price. Given the low price, there would be an over-demand for the unit sold by S5, and a resulting upward pressure on the price asked by S5.

Similarly, if B1 is willing to pay S1 $5 for a unit, he must also be willing pay the same amount to the other sellers for one unit. Then Sellers with lower costs will rush to compete for B1's business, thereby creating a downward pressure on the amount bid by B1.

The process settles when the market reaches equilibrium described by your second scenario, where three units are traded at $3/unit.

In Vernon Smith's canonical market experiment (and countless replications thereafter) wherein buyers and sellers with varying values/costs trade in a double auction,  the equilibrium outcome predicted by theory, rather than the assortative matching outcome described in your situation 1, is observed.

This is also related to the Law of One Price.


Can anyone solve this indifference curve question? Draw the indifference curves for a preference relation which is neither monotonous
or strictly convex, yet convex.
My solution was to draw a circle but I'm pretty sure that is wrong.

Consider the utility function $u(x, y)= -|x-5|-|y-5|$.
Indifference map for $u$ is as follows :


Definition of Pareto efficiency and prisoner's dilemma Pareto efficiency is defined in Wikipedia as:

  Pareto efficiency or Pareto optimality is a state of allocation of
  resources from which it is impossible to reallocate so as to make any
  one individual or preference criterion better off without making at
  least one individual or preference criterion worse off.

And it has similar definitions in other sources.

Let's say in a Prisoner's dilemma game, A is confession, B is tie.

The strategies with results are:

(A, A) = -6, -6
(A, B) = 0, -10
(B, A) = -10, 0
(B, B) = -1, -1

Nash equilibrium is (A, A), also both confessing, and it's not Pareto efficient because by moving from (A, A) to (B, B), both could improve the result (form -6 to -1, also by 5).

But why is (A, B) and (B, A) not Pareto efficient? In both cases, no player can be made better off without making another player worse off.

Moving from (A, B) to (B, B) makes "player 2" better off (from -10 to -1) but makes "player 1" worse off (from 0 to -1).

Moving from (A, B) to (A, A) makes "player 2" better off (from -10 to -6) but makes "player 1" worse off (from 0 to -6).

Moving from (A, B) to (B, A) makes "player 2" better off (from -10 to 0) but makes "player 1" worse off (from 0 to -10).

So moving from (A, B) to any other combination makes one player worse off. That's the case with (B, A) too.

So why is (A, B) and (B, A) not Pareto efficient?

(A, B) and (B, A) are in fact Pareto efficient.
I believe that your confusion may be because when discussing the Pareto inefficiency of the Prisoner's dilemma equilibrium, we always discuss (B,B) as the Pareto efficient alternative to (A,A) and (almost) never discuss (A,B) or (B,A).
Keep in mind that Pareto efficiency always requires a starting point to be evaluated. I.e. one way to determine if a case is Pareto efficient is to ask "Starting from this point, is there any other allocation that is a Pareto improvement?". Here a Pareto improvement is a case making at least one person better off and no one worse off.
A natural starting point of discussion is the Nash equilibrium (A,A). Starting from there, only (B,B) is a Pareto improvement, which suffices to show that (A,A) is not Pareto efficient. That is what economists like to emphasize about the Prisoner's Dilemma and why textbook discussions focus on (B,B).
However, if your starting point is (A,B) or (B,A) then there are no Pareto improvements possible, as you argue. Hence, both those cases are Pareto efficient as well. However, note that these are not very interesting starting points anyway, because they are not Nash equilibria and therefore a bit irrelevant to discuss.


What is an example application of a quasilinear utility function? I am told a quasilinear utility function is a function like $$U(x,y) = \sqrt{x}+y$$

My Question:

Can someone provide a real world example of a quasilinear utility function?

Quasilinear utility functions are useful in much of the demand estimation literature, particularly in discrete choice. For instance, check out Berry 1994,Berry Levinsohn Pakes 1995 and the many applications in Nevo's papers on demand estimation (here's a "practicioner's guide"). Ken Train's book on it is available for free here!

To summarize, they can lead to indirect utility of the form $$u_{ijt}=\alpha_i\underbrace{(y_i-p_i)}_\text{real income}+\underbrace{X{jt}\beta_i}_\text{observed product characteristics$*\beta_i$}+\underbrace{\xi_{jt}}_\text{unobserved product characteristics}+\underbrace{\epsilon_{ijt}}_\text{mean zero stochastic term}$$
where $i$ represents individuals $i=1,\dots,I_t$ in each of the $t=1,\dots,T$ markets selling $j=1,\dots,J$ products. Here, $\alpha_i$ represents the marginal utility of income and $\beta_i$ represents the marginal utility from the product characteristics observed in $X_{jt}$.

Suppose that we restrict the heterogeneity across consumers to only enter through the stochastic term $\epsilon_{ijt}$. Then both the individual specific parameters $(\alpha_i,\beta_i)$ must be equal to $(\alpha,\beta)$ and the market share of each good can be represented by
$$s_{jt}=\frac{exp(X_{jt}\beta-\alpha p_{jt}+\epsilon_{jt})}{1+\Sigma_{k=1}^{J}exp(X_{kt}\beta-\alpha p_{kt}+\xi_{kt})}$$

The equation for the market share is a function of variables that only vary at the product-market level, so you only need information on prices and quantities (and product characteristics) for a crude estimation. However it gives some quirky results. Particularly regarding the elasticity of market share wrt own price and cross price (market share) elasticities, and consumer subsitution patterns. Things can get more robust the more you read on demand estimation, you can introduce consumer characteristics and get results that have more desirable substitution properties.

There are also many criticisms of the conclusions that this type of modeling can draw, but I'll leave that up to you to imagine should you want to read on the topic.


Struggling with uncompensated/compensated demand I'm working on a problem set for my intermediate microeconomics course, but I'm having trouble deriving the compensated and uncompensated demand functions. This is the utility function:

$U(x, y, z) = aln(x) + bln(y) + z$, with goods $x$, $y$, $z$ and income $I$. I found the following optimal values:

$x = (P_z/P_x)a$, $y = (P_z/P_y)b$, $z = (I/P_z) - a - b$

Even though I followed the steps, I'm not completely sure this is right. Moreover, I have to find cross price effects and both the compensated and uncompensated demand functions, but I'm having some serious trouble solving this problem.

First, we want to find the optimal good baskets, ie how much to buy of goods $x$,$y$ and $z$ to get the maximum value of U out of it.

Let's write down the $MRS$ for $x$ and $y$ against $z$. For this we need to differentiate the utility function for each of the variables:
$d_xU(x,y,z)=a/x$ , $d_yU(x,y,z)=b/y$,$d_zU(z,y,z)=1$. We hence deduce the rates of substitution to be $-1a/x$ and $-b/y$ respectively.

So we have:
$p_x/p_z=a/x$ and $p_y/p_z=b/y$

The budget constraint is $p_xx+p_yy+p_zz=I$ which we rewrite
$p_x(ap_z/p_x)+p_y(bp_z/p_y)+p_zz=I$, which simplfies in
$ap_z+bp_z+zp_z=I$

$z^*=(I/p_z)-a-b$

$(x^*,y^*,z^*)=(ap_z/p_x, bp_z/p_y,(I/p_z)-a-b)$

Good news, you were right :).

The cross price effect is the rise of demand in good $x$ following a rise in the price of good $z$. Just calculate the difference between $x^*$ for $p_z$ and $x^*$ for $p_z'=p_z+\epsilon$ (using $x^*=ap_z/p_x$).

Finally, the six curves asked are how demand change when price change holding income ($I$) constant, or utility ($U$) constant. Draw same by taking two prices fixed for each situation.


Income Elasticity of Demand vs. shift in demand curve I've just read an econ textbook and want to verify a statement/definition about income elasticity of demand.

Changes in income shift the demand curve, and we can measure the responsiveness of demand to income changes by calculating income elasticity of demand as follows:

Question: Where does this change in quantity demanded come from? Is it from a movement along demand curve, or from a shift of demand curve?

Short answer:

Denoting income by $I$, price by $p$, the demand function by $D(I,p)$ and income elasticity of demand by $\eta$, the definition of point elasticity of income is

$$
\eta = \frac{\text{d}D(I,p)}{\text{d}I}\frac{I}{D(I,p)}.
$$

So the change in demand comes from a change in income.

Long answer:

Without specifying your coordinate system the terms "movement along the curve" and "shift of the curve" are meaningless.

Consider the function $f(a,x) = a/x$. Suppose this function determines the value of a variable $y$, that is $y = f(a,x)$. Given $a$ we can plot $f(a,x)$ in the $(x,y)$ coordinate system. If $x$ changes there is movement along the curve in this coordinate system. If $a$ changes the curve shifts in this coordinate system.
However treating $a$ as a variable and given $x$ we could also plot $f(a,x)$ in the $(a,y)$ coordinate system. Then change in $a$ would mean movement along the curve and change in $x$ would shift the curve.

As quantity demanded usually depends on both income and price, you face a similar situation, where $q = D(I,p)$. Income elasticity examines a change in income, but without specifying if your curve is ploted in $(p,q)$ or $(I,q)$ you cannot classify the change in $I$ as either "movement along the curve" or "shift of the curve".


How does a Pareto efficient equilibrium scale for n individuals? Say I have a starting situation with Pareto efficient allocations between 2 individuals, now a third one comes in and changes the allocations. Now a fourth one, etc. Up untill a nth individual comes in. Do we now have to resolve (n-1) equations or n^(n-1) equilibria? How does the "difficulty" scale for n individuals? I don't know if that makes sense? Thanks.

If you are doing pairwise comparisons, then with $n$ individuals there will be $$\dfrac{n(n-1)}{2} $$ such comparisons.  You can write this as $\binom{n}{2}$ or as $\frac12n^2-\frac12n$ if you prefer.

$1$ for two people, $3$ for three, $6$ for four and so on, known as triangular numbers


Search and matching model: effect of hiring cost on wage curve What effect does the hiring cost have on the wage bargaining outcome (wage curve)?

In Cahuc, Carcillo, and Zylberberg: Labor Economics, they derive the following wage curve:
$$w = z + (y-z) \frac{\gamma[r+q+\theta m(\theta)]}{r+q+\gamma \theta m(\theta)}$$

where $z$ = unemployment benefit, $y$ = marginal product of labor, $\gamma$ = bargaining power of the worker, $r$ = interest rate, $q$ = rate of exogenous job destruction, $\theta$ = labor market tightness, $m(\theta)$ rate of job filling.

On the other hand, following Pissarides (2000) we get the following wage curve:
$$w = (1-\gamma)z + \theta \gamma c + \gamma y$$
where $c$ = hiring cost.

The first wage curve (from Cahuc et al.) would suggest that $c$ does not affect the wage curve while the second one (from Pissarides) would suggest that it does.

My thoughts so far are:
1. The wage curve tells us the outcome of bargaining but $c$ does not affect bargaining because $c$ is only directly relevant for $\Pi_V$ (the employer's outside option) but $\Pi_V$ is always zero in equilibrium. Therefore, the employer's outside option is not any worse if $c$ is higher. 2. In deriving the wage curve, Pissarides substitutes labor demand into the wage curve. So, $c$ enters the Pissarides' wage curve only through labor demand and not in any other way, suggesting, again, that bargaining itself is unaffected by $c$.

So, what is the effect of a change in $c$ on the wage curve? Does an increase in $c$ shift the wage curve?

It depends on what you define as wage curve. I did not check in the textbooks, but you suggest that Pissarides and Cahuc et al. do not use the same mathemaical expression for the "wage curve".

1) In Cahuc et al., the expression combines the surplus sharing equation with the Bellman equations without using the free entry condition of firms. The wage $w$ is increasing in $\theta$ because a tighter market tightness improves the worker's bargaining position. There is no labor demand effect in this wage equation. Therefore, the vacancy cost does not appear.

2) The Pissarides' wage curve is less straightforward to interpret. Given the positive relationship between $w$ and $c$, one could erroneously think that the vacancy cost increases the wage. The interpretation is the following.
At $\theta$ fixed, a higher vacancy cost means that firms spend more to recruit. More spendings capture the fact that firms make larger job profits (free-entry condition). When firms make larger profits, workers also obtain a larger surplus due to the Nash bargaining. Thus, the wage is higher.

This definition of the wage curve does not depend on the matching technology ($m$). To understand this result, we can write the model with an abstract matching market.
With your notations, we define the value of unemployment $U$, the value of employment $W$ and the value of a job for a firm $J$.
We can write the three Bellman equations, and the Nash bargaining condition with the wage:
\begin{align}
    & rU = z + R\\
    & rW = w+q(U-W)\\
    & rJ = y-w-qJ\\
    & w=\gamma y + (1-\gamma)rU
\end{align}
If a worker is employed, she produces $y$ and receive $w$, with a risk to lose the job $q$.
If the worker is not employed, she receives $z$ plus the returns from participating to a matching market $R$.

In the full model $R=\theta m(\theta) (W-U)$. Here, we do not use this structure.

We assume that each unemployed worker participating to the "matching market" produces $G$ and obtain $R$. To participate to the market and obtain part of the gains, firms have to pay a cost $c$. If we assume that $G$ is Nash-bargained, each worker obtains $R=\gamma G$, and recruiting firms share $(1-\gamma)Gu$. The average returns for a firm is $\frac{(1-\gamma)G u}{v}=(1-\gamma)\frac{G}{\theta}$.
The free entry condition implies that firms make no profits from participating to the market. Therefore, $(1-\gamma)\frac{G}{\theta}=c$.
Using the wage bargaining solution, we obtain Pissarides' wage curve using the wage bargaining solution and eliminating $rU$, $R$ and $G$.

The wage curve holds without assuming how the "matching market" works, we only assume that the outcome is Nash-bargained. To come back to a change in $c$. An increase in the vacancy cost $c$ necessarily implies that the gain $G$ is higher, ans so the wage.

3) This is partial equilibrium analysis. If you solve for the equilibrium level of wages, you will find that $c$ actually reduces the equilibrium level of wages.


VECM interpretation in Matlab? After testing for cointegration with Johansen test using the following code

[h,pValue,stat,cValue,mles] = jcitest(ret,'model','H1*','lags',4,'display','params')

I received this output:

Data: prices
Effective sample size: 56
Model: H1*
Lags: 4
Statistic: trace
Significance level: 0.05

r  h  stat      cValue   pValue   eigVal
----------------------------------------
0  1  40.5214   35.1929  0.0161   0.3350
1  0  14.6781   20.2619  0.1453   0.1667
2  0  5.4670    9.1644   0.1580   0.1091

So it rejects no cointegration. So I am running the VECM with rank 1 restriction:

B = mles.r1.paramVals.B % Cointegrating relations with rank = 1 restriction

I get the following output:

r = 1
------
A =
-0.6259
-0.2261
-0.0222

B =
0.7081
1.6282
-2.4581

B1 =
0.0579 1.0824 -0.8718
0.1182 0.4993 -0.5415
0.1050 0.1667 -0.1600

B2 =
-0.5462 2.2436 -1.7723
-0.1518 0.6605 -0.6169
-0.1610 0.5966 -0.5366

c0 =
2.2351

c1 =-0.0366
0.0872
0.1444

How would I interpret this output and how do I move forward after estimating the parameters? (I have only displayed the r=1, and these are not the actual results, by I couldn't find any explanation behind them)

Based on the official documentation, you have found that there is one cointegration relationship with 95% probability. Given your selection of Model: H1* and Lags: 4, the model you are estimating is:

$$ \Delta y_t = A(B´y_{t−1}+c_0) + B_1\Delta y_{t-1} + B_2\Delta y_{t-2} + B_3\Delta y_{t-3} + B_4\Delta y_{t-4} + \epsilon_t $$

The matrix B is of dimension $n \times r$, and it contains as columns the cointegration vector(s). In your case, $r=1$, and the cointegration vector is

B =
0.7081
1.6282
-2.4581

The Matrix A represent the "error-correction" matrix. For a system to converge towards the equilibrium cointegration relation (i.e. to be stable), you want these numbers to be in between -2 and 0 (as in your case).

c0 is just a constant of the cointegration relationship (like in a standard function $y=c0+bx$).

The matrices Bi show how previous period shocks translate through the system over time.


Strike time in a game Suppose A forwards a puzzle to B where if B solves the puzzle within 24 hours then it gets a payoff of M (M>0), if not then it gets nothing i.e. 0. Consider A locks the money with the bank for the 24hours and if B solves the puzzle it withdraws the money from the bank. If not bank refunds M to A after 24 hours. If B solves the puzzle within 10seconds (strike time 10), it gets payoff of M. If it solves at 23rd hour (strike time 23rd hr) it gets a payoff of M. Can there be someway to disincentivize delay in solving the puzzle so that A need not keep his money locked for long period of time? One strategy which I think is that ask B to lock a certain amount as well, so that it is forced to solve the puzzle quickly. How do I calculate the strike time at which B solves so that any delay beyond this will not be profitable?

Calculating a precise strike time without additional details  is - in my opinion quite clearly - impossible. On the question of how to do this with additional info/assumptions:

A straightforward solution would be to make M time dependent - the larger the time elapsed, the smaller M is. Sounds like you do not want to do this.
Base problem
Let us now assume M is not time dependent, and B maximizes the present value of her payoff (assume a constant interest rate $r$) minus the cost of the effort of completing the task at time $t$ (denoted by $C_t$), so the goal of B is
$$
\max_t \frac{\text{M}}{(1+r)^t} - C_t.
$$
$C_t$ not dependent on time
If we also assume that $C_t$ is constant in time, then this is not only decreasing in $t$, but decreasing at a decreasing rate. (The second derivative w.r.t. $t$ is positive.) This means that B will either not want to lose any present value and solve the puzzle at the earliest possible time (when $r$ is quite large), or that B will not care about the loss very much and solve at the deadline (when $r$ is quite small).
B also puts money down
This is unchanged by B also escrowing (locking up) an amount of money ($\text{M}_B$), as the loss
$$
\text{M}_B - \frac{\text{M}_B}{(1+r)^t}
$$
resulting from this is also decreasing at a decreasing rate.
$C_t$ dependent on time
Let us now assume that $C_t$ is time dependent, e.g. B has to be at the movies for the next three hours and can only solve the puzzle afterwards, or the computational costs make it costly to lease additional computing power and cut down on time. If there are instances where $C_t$ increases at an increasing rate, that is $C_{t+2} - C_{t+1} > C_{t+1} - C_{t}$, then it is possible that the maximization problem will have an interior solution, i.e. the optimal time will be larger than zero but smaller than the deadline. However, one can only calculate this if one knows the interest rate $r$ as well as the cost function of B, so one needs to get quite a bit of information from B.
$r$ dependent on time
Oddly if we allow $r$ to vary, that is the interest rate is $r_t > 0$ at time $t$, not much changes. In the $C_t$ not dependent on time case, the problem will again be solved immediately or just at the deadline. If $C_t$ is dependent on time, an interior solution is possible, but to calculate the optimal time we need a bunch of additional information/assumptions.


How to make deliberate devaluation or depreciation of a currency? It is said in Wikipedia, that devaluation can be done as monetary policy act.

But HOW this can be done if there is no fixed rate?

What are actual tools Central Bank can use?

One thing I can imagine -- currency intervention.

What are others?

A country can increase the supply of its currency, through some combination of increasing the amount of base money, or loosening the capital / reserve requirements of banks.

It can decrease interest rates.

It can decrease the supply of the currency it wants to depreciate against (for example by buying and holding lots of that currency, or equivalent instruments).


References where fewer rules resulted in better governance I am trying to find references that support an idea that I think I read in a book on theory of institutions. I can't find the original reference. Basically, I'd like to find anything that talks about number of rules and the strictness of enforcement as variables in controlling outcomes.

In the reference I can't find as I remember it, there was a case study related to OSHA or Worker's Compensation where increasingly strict safety policies caused an increase in accidents. The analysis ran that the existing policies were too stringent to allow workers to get their jobs done, and that the workers were deciding which rules they could ignore to get tasks completed on schedule. In this case, they had better outcomes by restricting the set of rules to a smaller set that were realistic and strictly enforced. Basically, too many rules resulted in workers prioritizing the rules to follow, while decreasing the rules allowed management to make the decision about which rules were truly the most valuable.

I think in the literature of uncertainty I once read that increasing the number of laws increases the potential source of uncertainty. However, it was probably more related to investment than to working performance. However, if you are interested "Measuring economic policy uncertainty" by Baker, Bloom and David (2013) might be a good start.


Is there a way to adjust trade data between China and EU member countries for the Rotterdam effect? Basically Eurostat trade data for individual EU member countries with China seems to have some substantial artefacts probably because most EU imports from China arrive in Dutch ports, even though their ultimate destination is probably all over the EU. It's hard to explain otherwise the huge figure for Dutch imports (and thus trade balance) from/with China relative to Dutch population, economy, etc.

Are there any data sources or papers that try to adjust for this Rotterdam effect, i.e. figure out where those Chinese imports are really headed in the EU (and adjust the trade figures per member country accordingly)?
(Thanks to Henry for pointing out the right term.) Eurostat actually acknowledges the issue in a FAQ, but does not try to quantify it:

The quasi-transit is known to impact mostly the Member States with big ports at the external border to the European Union and in particular the Netherlands. This is why its impact on figures is known as the ‘Rotterdam effect’. In case of imports, the goods destined to other EU Member States arriving in Dutch ports are recorded, according to Community rules, as extra-EU imports by the Netherlands (the country where goods are released for free circulation) and as dispatches from the Netherlands to the Member States of actual destination even though there is no link with economy of the Netherlands.
The quasi-transit is known to impact more the imports but exports are also affected. In exceptional cases, the customs clearance occurs not in the actual Member State of export but in the Member State of exit, i.e. in the Member State from which the goods are taken out of the EU customs territory.

This seems relevant: China and the Dutch Economy

I have only a limited experience in this, but here are some approaches you could look at.

Re-export data

UN Comtrade has re-exports data (products that the Netherlands import and export in the same form to other countries). Using this, you could make an assumption that x% of goods imported from China are re-exported to the EU. Assuming that this share of re-export actual source countries are the same for Dutch exports to the EU, you could get a rough estimate of how much goods from China goes to each EU country.

Using aggregate data, this could be a very far-off estimate. Belgium could import a lot from Netherlands, but all it imports are tulip flowers. To account for this, you can look at this on a HS-code level and you should get a reasonable estimate.

TiVA

Another approach is to look at the TiVA (trade in value-added) concept. This should tell you each country's value-added into an import of a country. OECD has this data, but I don't believe it covers all EU-28. Here's a link.


How would you call a market for kidney donors and recipients? I was wondering how you would call a market for kidney donors and recipients, where you have many 'buyers' and many 'sellers'. However, the 'seller' can only 'sell' his/her 'product' once. Since it's illegal to sell kidneys, the market structure is a little bit vague for me to categorize. Suppose selling and buying kidneys would be legal, how would that market be called?

Generally, these types of markets are called matching markets. That's also the same term given to a variety of economic decisions and market structures, from school choice to some types of marriage problems.


Will the monopolist inevitably increase the price if demand grows? Supposing that demand for a product in a monopolised market has grown, is it possible that equilibrium price will go down?

In most cases we simply draw curves D, MR, MC and make conclusions. I usually deal with linear demand and constant marginal costs, so nothing interesting.

It is naturally to say that the monopolist will increase the price to gain profit, even for inferior goods this seems true.

In this post, we find that the optimal monopoly price can be written as

$$P^* =  \frac {|\eta|}{|\eta|-1} MC $$

where $\eta$ is the elasticity of demand with respect to price, and so optimal price will necessarily be at a point where $\eta$ is higher than unity in absolute terms.

From this one can deduce some unambiguous and some ambiguous scenarios when the demand schedule shifts outwards, e.g.

SCENARIO A: constant price elasticity of demand, convex costs: $\implies$ Price and quantity will increase.

SCENARIO B: constant price elasticity of demand, linear costs: $\implies$ Price will stay put, quantity will increase.

The floor is yours.

PS: Note that to have constant price elasticity of demand, the demand function will necessarily be non-linear.


Euler equation through tangency conditions I am rather new to economics in general and to the Neoclassical Growth Model in particular and I was wondering if there was a way to get the Euler equation for consumption without using the Lagrangian multiplier? Assuming no corner solutions, let's say the agent solves  $$\max_{c_t, k_t} \sum_{t=0}^{\infty} \beta^t U(c_t)$$  subject to $$c_t + k_{t+1} \leq f(k_t) \\ c_t \geq 0 \quad \forall t \\ k_{t+1} \geq 0 \quad \forall t$$

The Lagrangian is given by $$L = \beta^t [U(c_t) + \lambda_t ( f(k_t)- c_t - k_{t+1})]$$ which can then be solved for the Euler equation $U'(c_t) = \beta U'(c_{t+1})f'(k_{t+1})$.

Why is that if I try to use the tangency condition of the gradient of $U$ and the resource constraint I don't get the same EE? Taking partials with respect to consumption and investment ($\delta =0$ so $k_{t+1}=i_t$)

$\nabla U = \langle \beta^tU'(c_t), 0\rangle$ and

$\nabla RC = \langle -\beta^t,  -\beta^t+\beta^{t+1}f'(k_{t+1})\rangle$.

Using tangency I should get $\frac{U'(c_t)}{-1} = \frac{0}{\beta f'(k_{t+1})-1} \Rightarrow U'(c_t) = \beta U'(c_{t})f'(k_{t+1})$. Clearly my time subscript is off. Can someone tell me if my approach is wrong? or if I made a mistake in the algebra? I think the optimizing condition should hold with tangency so its not obvious to me why this approach gives a result different from the Lagrangian.

In answer to your first question (can you get the Euler equation without a Lagrangean?), the answer is 'yes'. At least, there are less formal ways of deriving it. In general, assuming that a consumer chooses to consume strictly positive quantities of two perfectly divisible goods $x$ and $y$, then it must be that:

$$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

(If this were not true, then the consumer could increase their utility by shifting spending a little more on one good and a little less on the other.)

The so-called 'Euler equation' is nothing more than an application of this result, viewing the 'goods' and $C_t$ and $C_{t+1}$. If you delay your consumption by one period, you put your cash in the bank for one period, earning real interest at rate $r_{t+1}$. Hence, if we normalise $P_t = 1$, then effectively $P_{t+1}=1/(1+r_{t+1})$. Consumption tomorrow is cheaper than consumption today (assuming $r_{t+1} > 0$) since delaying your consumption allows you to earn some interest.

We assume that utility is constant, time-separable and exponentially discounted (as in your set-up). Thus, we need to remember to put a $\beta$ before $MU_{t+1}$. We now write $u'(c_t)$ for $MU_x$, $\beta u'(c_{t+1})$ for $MU_y$ and $(1+r_{t+1})$ for the price ratio.

Finally, observe that if firms maximise profits taking the cost of capital $r_{t+1}$ as given with total depreciation (as you assume), they hire capital until $f'(k_{t+1})=1+r_{t+1}$. Plugging in yields the Euler equation.

In answer to your second question (how can you derive it with a Lagrangean?), I would recommend that you try the following steps:

First of all, you should really have a summation before your Lagrangean (summing over an infinite number of periods). This is a dynamic problem.
Differentiate the objective function with respect to $c_t$.
Roll forward by one period and combine the equations. You should get:

$u'(c_t)/\beta u'(c_{t+1}) = \lambda _t/\lambda _{t+1}$

Then differentiate the objective function with respect to $k_{t+1}$. Don't forget the $f(k_t)$ term! This may look irrelevant, but the Lagrangean should be an infinite sum so it shows up when you 'roll forward' by one period. This should give you:

$\lambda _t = \lambda _{t+1}f'(k_{t+1})$

Plug that in and you have it! The equation on which so much pointless macroeconomics built.


Energy Costs by Source over Time I'm looking for a comparison I can make between "traditional" energy sources and solar energy.

Ideally, I'd like to show how much solar energy has yet to come down in price before it is cheaper than, say, gas or coal. I would like to know the historical rate at which this has been currently happening.

The question is then what metric is good for this. Levelized cost of electricity by source in time would be my preference, however, I cannot find data for this over time and by source (e.g solar, coal & gas)..

Any advice would be appreciated.

PV is already cheaper than gas or coal in most of the world, on a like-for-like basis. So that sort of answers your main underlying question.

It's estimated that new PV will be cheaper than existing coal and gas within the next 2-5 years, depending on where in the world we're talking about.

LCoE is a hypothetical construct resting on a bunch of assumptions that often don't hold, so isn't wildly useful.

For real data, look at actual auction results for new generation that's priced per unit electricity supplied to the grid. That's pretty easy for PV, but you may struggle to find recent comparable auction data for dirty old generators such as gas and coal, because those are rarely procured as auctions priced per unit electricty.

You then need to adjust those costs for the factors not represented in the auction prices.

For PV, that's the cost of integration into the grid (balancing intermittency, reserve capacity, reactive power, and so on). Those costs are pretty tiny - well below USD 10/MWh - but will vary, depending on the particular grid. Look out for additional subsidies such as the CELs (tradeable clean energy certificates) in Mexican auctions, tax-breaks, free land, state-subsidised loans, etc.

For dirty generators, you'll need to add in the unpriced social costs, including the insurance risks of accidents, and the waste costs: greenhouse-gas marginal social costs are around USD 100 /tCO2e (see peer-reviewed work by Chris Hope & others).


Positive Monotonic Transformations and Nested Functions Suppose there is an economic agent with the utility function $u(x,y)$. A second agent has the utility function $h(g(f(u(x,y))))$.

Am I correct in thinking that if $f'(x)>0$, $g'(x)>0$, and $h'(x)>0$, then  $h(g(f(u(x,y))))$ must also be positive and therefore  $h(g(f(u(x,y))))$ is a positive monotonic transformation of $u(x,y)$?

If not, how come?

The value of $h(g(f(u(x,y))$ is not necessarily positive but the transformation $u \rightarrow h \circ g \circ f \circ u$ is indeed a positive monotone transformation.

To see this, take any $(x,y,z,w)$. We have the following chain of equivalences:
\begin{align}
u(x,y) \geq u(z,w) & \Leftrightarrow f(u(x,y)) \geq f(u(z,w)) \text{ since } f \text{ is increasing} \\
& \Leftrightarrow g(f(u(x,y)) \geq g(f(u(z,w)) \text{ since } g \text{ is increasing} \\
& \Leftrightarrow h(g(f(u(x,y))) \geq h(g(f(u(z,w))) \text{ since } h \text{ is increasing}
\end{align}
Thus, the transformation $u \rightarrow h \circ g \circ f \circ u$ preserves the ordering, i.e. it is a positive monotone transformation.

Another way to see it is to assume that all functions are differentiable and notice that
\begin{equation*}
\frac{d (h \circ g \circ f )(x)}{dx}= f'(x) g'(f(x)) h'(g(f(x))
\end{equation*}
which is positive since $f'>0,g'>0,h'>0$.


Decentralized Mechanism Design From what I've read in mechanism design theory, it seems to me that the mechanism designer always plays an essential role in a mechanism. Information is sent to the designer who designs incentives that aim to enforce the truthful revelation of information from the strategic agents of the mechanism.
But, within the mechanism design theory, is it possible to remove the requirement of agents communicating their information to a central authority (aka the designer)?
For example, design a framework where the agents communicate with each other based on certain rules and figure out by themselves the designer's desirable outcome without any communication between the agents and the designer.

I am not sure if this is the answer you were aiming at, but... what about the revelation principle?
There is no requirement to communicate with the designer. However, researchers are ofter first interested in what the optimal allocation rule is. In that search, it is wlog to consider direct mechanisms by the revelation principle: Suppose an indirect mechanism without communication to the designer implements the optimal allocation, then we can obtain the same allocation if every agent simply reports their type to the mechanism designer who then computes the equilibrium allocation of this indirect mechanism and implements it. See my reply with more details here.
If you want to know if there is a general approach of how to find equivalent indirect mechanisms for the optimal direct revelation mechanism, I am not aware of an answer.


How is "opportunity cost" an useful concept? Even when it does exist? How is "opportunity cost" an useful concept? Even when it does exist?

What use does it serve?

Probably the most useful aspect of the concept of "opportunity cost" is to distinguish between choice problems where a particular choice is available, but the alternatives differ.  The concept of opportunity cost stresses that when we make a choice of some action, we forego alternatives, and so the gain from that action is the difference between the benefit of that action, and the benefit of the next-most-preferred alternative.

For example, you might have two choice problems (e.g., two nights of the week) where you have the opportunity to go to the movies on your own.  You get a certain amount of enjoyment from this, and so you might consider making this decision by thinking about how much you enjoy this activity.  The concept of opportunity cost would tell you to bear in mind the available alternatives, so that you remember you are foregoing other possible activities if you choose to go to the movies.  In one of these cases (e.g., the first night) you might not have any exciting alternatives, and so going to the movies is pretty attractive (the opportunity cost is low).  However, on another night, you might have been invited to an exciting party, with lots of close friends, and so now going to the movies and missing the party seems like a relatively bad idea (the opportunity cost is high).

All that "opportunity cost" is really doing is reminding you of the fact that decision problems must be considered on the basis of all the available actions, and the relative value of the optimal action is affected by the value of the best foregone alternative.  Most people understand this implicitly (for the most part), but economists are particularly aware of this, and have developed a concept around it.

One consequence of considering opportunity cost is to highlight the sunk cost fallacy, where a person makes a decision to pursue some action even though it is less desirable than the available alternatives, because they have already committed resources to that course of action, and they do not wish to accrue losses from a change of action (i.e., they have loss aversion).  In the example above, suppose you had already pre-paid for your movie-ticket before you find out about the party, and you can't get a refund.  The payment you made for the ticket is a "sunk cost".  You might be inclined to go to the movies, and miss the party, just so that you don't "waste" the money on the ticket.  However, doing this would just compound one mistake (buying the ticket) with another mistake (choosing to see the movie over a party you would have enjoyed more).  The concept of opportunity cost tells you to compare the available actions against the alternatives, noting that the sunk cost occurs under every case (and is therefore not relevant to a particular action).


Block pricing for different products How is a block price decided when there are many products to be sold?
For example the buffet in various restaurants or combos in McDonalds or KFC (like the bucket buddy).
Can somebody point me to research papers or any link or book I can read to get into the logistics and mathematics behind pricing such combos?

I am not exactly sure what you mean by "block price" and I don't know the bucket buddy. However, I feel like you are talking about what microeconomists term bundling, the practice of selling two (or more) different goods together in one package.

Interestingly, bundling can be profitable even without any production-cost related efficiency arguments. It can serve as a tool to sort consumer types and thus can be used to price-discriminate.

The classical paper on bundling by a monopolist is McAfee, McMillan and Whinston (1989). For an introductory treatment of the topic read, e.g., Belleflamme and Peitz, Chapter 11 - or the corresponding chapter of basically any other textbook on industrial organization or microeconomics.


Why not say the buyer will *definitely* buy at price X? In supply-and-demand curve, we draw curves of indifference. Say the demand goes through (1, \$1). Then Bob the Buyer is indifferent between spending \$1 for 1 apple or just keeping the \$1. If Sally the Seller had 1 apple to sell, then the equilibrium quantity sold would be 0 or 1—we cannot say. Seen differently, Bob will buy 1 apple if the price is in the interval [\$0, \$1) (and he may buy it if the price is exactly $1).

Would it not be simpler to use closed intervals? In our example, Bob will buy 1 apple is the price is in the interval [\$0, \$1]. The equilibrium quantity sold will definitely be \$0. Seen differently, the demand curve represents the maximum price buyers will pay, and the supply curve represents the minimum sellers will sell for.

It's (literally) an infitesmal of a difference, but it seems, at least for discrete situations like the above, it would be nice to say exactly what will happen.

Why don't we do that? There must be some elegant reason.

Under Yatharth's approach, the set of prices at which Bob will definitely buy an apple is the closed interval $[\$0,\$1]$, which is nice. However, the set of prices at which Bob will definitely not buy an apple now becomes an open interval $(\$1,\infty]$.

In contrast, under the conventional approach, the set of prices at which Bob may buy an apple is closed: $[\$0,\$1]$. So too is the set of prices at which he may not buy an apple: $[\$1,\infty)$. So, under the conventional approach, we're dealing solely with closed intervals.

Both approaches have their merit. But I believe that all things considered, fewer inconveniences arise with the conventional approach than with your suggested approach. In general, it's "nicer" to not have to deal with any open intervals.

Example 1. Under Yatharth's approach, there is no answer to the question "What is the lowest price at which Bob may not buy an apple?"

Example 2. Under Yatharth's approach, the demand curve will be discontinuous and with jumps. In contrast, under the conventional approach, it is continuous (albeit with kinks).

Elaborating on Example 2.

The demand function $D:\mathbb{R}_{0}^{+}\rightarrow\mathcal{P}\left(\mathbb{R}_{0}^{+}\right)$ maps each price (a non-negative number) to a set of quantities demanded (a subset of the non-negative numbers). The economic interpretation is that at each price, $D$ tells us what numbers of units of the good the consumer most prefers to buy.

We assume that the good is infinitely divisible.

Then here are the graphs of the demand function under the two approaches:

The demand function is continuous under the conventional approach, but not under Yatharth's. (Nonetheless, under Yatharth's, it is lower semi-continuous.)