List of behavioral economists Even though there is a massive literature on behavioral economics, i couldn't find anywhere a list of economist that focus on that discipline.

Since i am interested in environmental economics, i would be grateful if you could mention those in academia that apply "behavioral theories" into environmental economics.

Thanks in advance!

Top 10% authors in the field of Cognitive & Behavioural Economics

https://ideas.repec.org/top/top.cbe.html

Ernst Fehr, Institut für Volkswirtschaftslehre, Wirtschaftswissenschaftliche Fakutät, Universität Zürich, Zürich, Switzerland
Urs Fischbacher, Fachbereich Wirtschaftswissenschaften, Universität Konstanz, Konstanz, Germany
Robert Sugden, School of Economics, University of East Anglia, Norwich, United Kingdom
Andrew Schotter, Department of Economics, New York University (NYU), New York City, New York (USA)
Peter P. Wakker, Econometrisch Instituut, Faculteit der Economische Wetenschappen, Erasmus Universiteit Rotterdam, Rotterdam, Netherlands
...

And the other 140 economists.


Mythbusters - Determine optimal boarding strategy based on time and satisfaction score Most airlines board passengers starting from the back of the plane and then working their way towards the front (after boarding priority classes and passengers).

In an episode of Mythbusters, Adam and Jamie tested the myth that the boarding strategy favoured by most airlines, back to front, is the least efficient.

The myth was confirmed, and these were the results:

The random no seats strategy is the fastest, followed by the WILMA straight strategy. However, random no seats strategy gives the lowest satisfaction scores.

The highest satisfaction score is given by the reverse pyramid strategy even though it is the fourth fastest.

How might one determine the optimal boarding strategy based solely on times and satisfaction scores given (not including advanced stuff like computing expected aisle or seat interferences)?

I can't seem to think of any kind of unit conversion except to convert the time to seconds and then multiply it with satisfaction score so it's like we're trying to maximise the product of time and satisfaction score:

$$f(t,s) = ts$$

What are some of the advantages or disadvantages of doing this?

One disadvantage seems to be that ranking by product of time and satisfaction score gives the same ranking by satisfaction score.

What else could be done? All that seem to come to mind are products so perhaps I might maximise anything like these:

$$f(t,s) = t^2s$$

$$f(t,s) = ts^{1/2} \text{(eliminating random no seats)}$$

$$f(t,s) = t(s-s_{ave})$$

I'm thinking we're going to have to relate time and satisfaction score to some unit such as money. So, one would have to find some relationship (for example, a linear relationship through linear regression) between boarding time and cost and then another between satisfaction score for boarding today and revenue from flight next month?

Does it have to be something like that?

I was suggested z-scores or something so I tried standardising, I think:

Why did the sum of squares of z's turn out to be 6? Did I do something wrong? Is that the fourth moment or something?

I would start with your generic
$$f(t,s) = t \times s$$
and, instead of add weights or factors to this, I would add other variables related to time and satisfaction like:

boarding time as function of luggage management ($TL$) times baggage number ($B$), and time to go to seat row ($T_g$) vs time to actually sit ($T_s$) for the different seat types (Window, Mid, Aisle) per number of passengers ($N$)
satisfaction as function of easiness of boarding ($E$) (think about the WilMA and families with little kids), functionality of luggage positioning ($F_b$), amount of distraction required ($D$) (e.g. boarding while listen to loud music or boarding while talking at the phone distract from the boarding and leads to errors).

A proposal can be
$$f(TL, Tg, Ts, W, M, A, N) = (TL\times B)+(TG\times N + \frac{N}{3}W + \frac{N}{3}M + \frac{N}{3}A)$$
$$f(E, Fb, D) = E \times Fb \times \frac{1}{D}$$

$$\mbox{boarding strategy score}=f(TL, Tg, Ts, W, M, A, N) \times f(E, Fb, D)$$

and I would start assigning weights while conducting some more simulations (I understood that Mythbusters example refers to single trials only for each strategy).

In my opinion, advantages / disadvantages are not coming from equations itself but from the methodology. Without more robust experimental data, all equations above, and even more factors, are arguable and contestable.

I also would not add "money" in the model, but rather added value for the airline vs added value for the passenger, and things will easily escalate: you may find that keep people stuffed in tunnels and in line waiting to enter the aircraft, or waiting in airports for delays or flight cancellations, can increase exposition time for advertisement boards, therefore potential revenue for airport services, therefore ...utility functions of delays.


Detecting the presence of measurement error When can I know that I'm in the presence of measurement errors? i.e., when can I know that I should use a measurement error model(error-in-variables model)?

Are there any statistical tests for checking that?

Suppose we observe only $\tilde{x}$ and $\tilde{y}$ which are the true values measured with error:
$$ \tilde{x} = x + u$$
$$ \tilde{y} = y + v.$$
We would like to estimate:
$$ y = \beta x + \epsilon$$ but all we can really estimate is
$$\tilde{y} = \hat{\beta} \tilde{x} + \zeta$$

However, if  y is measured without error ($\tilde{y}=y$) we can use reverse regression to bound the measurement error in regressions. The regression is:

$$ y = \hat{\beta} \tilde{x} + \epsilon$$

You also run the reverse regression
$$ \tilde{x} = \hat{\frac{1}{\beta}}y  + \hat{\frac{1}{\beta}}\epsilon+ u$$
Define the reverse regression beta as $\hat{\beta}_r = \frac{1}{\hat{\frac{1}{\beta}}}$. It turns out that the results from
the standard regression and from the reverse regression will bracket the true
coeffcient, i.e. $plim\  \hat{\beta} \leq \beta \leq plim\  \hat{\beta}_r$.

This interval is small when the variance of $u$ and $\epsilon$ are small. If one of the two are zero then one of the inequalities hold exactly. Unfortunately, in practice this interval may be large and therefore this approach not very helpful.


Consumption Set in Arrow-Debreu I'm very inexperienced in mathematical economics, so when I came across the idea of the consumption set in Arrow-Debreu, I was a little confused. So for each element x in consumption set X, what does x represent? This also applies to element y in technology set Y. Are x and y values?

And as a side note, how can one analyze something like a Value Equilibrium on a macroeconomic scale? What kind of variables and analysis goes into looking at equilibrium/efficiency of allocations?

Assuming you are referring to the definition from the original Arrow & Debreu paper $x_i$ represents the units of good $x_i$ consumed.

I would not call $Y$ technology, according to Arrow & Debreu $Y$ is set of production plans - so by that they mean production (which may make some assumptions on technological nature of the process).

Also, from your second paragraph I am a bit confused what do you mean by value. In mathematics value is some definite mathematical object like integer 5. In that sense yes, in the general model they are variables but once you would try to apply it to concrete problem $x$ and $y$ would have some values, if you are referring to some human values then no. In economics human values or preferences are taken as given and expressed through utility function, in Arrow-Debreu paper $u(x)$.

A bit confused what you mean by value equilibrium on macroeconomic scale, assuming you mean general equilibrium, that depends on exact model you are following. Arrow-Debreu model is an example of basic general equilibrium model so you can use the variables you see there. If you would want to estimate it then currently one of the most popular ways is to estimate Dynamic Stochastic General Equilibrium (DSGE) models- these include a lot of variables. In such model you try to explicitly model demand, supply and monetary policy. The number of variables there can be so large that it’s not feasible for me to list it rather you may want to see an example of Smets-Wouters DSGE model for Euro-area economy.


Does the hoarding of perishable goods such as pulses and cereals make economic sense? I was thinking about the idea of hoarding and the kind of news it is making in India. What strikes me is that hoarding of pulses and cereals does not happen the way people thinks because they are a perishable commodity and by hoarding any such product a trader will loose turnover and will have lower return of capital although he can get higher margin. I don't think that except for rice any other produce can be hold for long, as they are perishable. Even rice will turn blackish with time. And if anybody thinks that storing them in cold storage will preserve then then also then it does not make sense as  the traders sunk cost will remain the same whether he hoard or look for more turnover.

So could please anybody help me have a more clear understanding of this hoarding news or is it just a hoax.

It does not make economic sense. Disclaimer that I am responding directly to your Thai example, and not talking about economics of storage. Rice can be safely stored for a period of time, and it is in the United States. Many developing countries struggle with proper storage of grain. For example, the World Bank estimates that 10-20% of South Saharan Africa's grain crop is lost to spoilage each year, roughly equivalent the amount of grain that they import yearly which can feed 48 million people.

The Thai government, not traders, where hoarding rice under their rice pledging scheme. The government has lost billions of dollars under the scheme and the Thai prime minister Yingluck Shinawatra faces possible jail time. During the 2011 crop year, the Thai government announced the rice pledging scheme which would establish the price floor and pay growers 50% over market price (subsidy).

The Thai government also greatly reduced exports, so it had a large stockpile of old crop rice (for several years). Part of the plan was to help their producers, the other possibly to have a hold of the global market (large control over the global supply). What ended up occurring was that their new crops had to compete against the old crops. The "alternative" product brought Thai and world prices down.

In addition, this was a very strange way of the Thai government to give aid to producers since it would have been more efficient to directly pay an equal amount of the additional producer surplus gained from the price floor straight to the producers.

A large percentage of the rice did in fact rot, and the government is now auctioning off rice that is not fit for human consumption. Some of it is also not fit for animal consumption and will have to be auctioned off for ethanol producers.


Credit market debt is $200 trillion? In a recent article by David Stockman he wrote:

  Credit market debt outstanding has soared from \$85 trillion to \$200
  trillion.

What is he talking about? What is the itemization of this debt. The whole US debt is less than \$20 trillion, so I don't see where there could be \$200 trillion in debt.

It's the total credit market, not only US government bonds, but also debt of other countries, also corporate debt etc. See here: http://www.investopedia.com/terms/c/credit_market.asp and here: http://www.investorwords.com/8562/credit_market.html

Maybe this chart (not recent though) clarifies it a bit also: http://warisacrime.org/sites/afterdowningstreet.org/files/images/Credit%20Market%20Outstanding%20by%20Sector%2010012008_0.jpg


Theoretical market capitalisation of a crypto currency? Here’s the context. I want to know what the theoretical market capitalisation (say eg $US / Bitcoin times Q the number of coins - it doesn’t have to be Bitcoin) of a crypto currency? Assume that the crypto currency is uniformly better than fiat. I think it tends to infinity. To see this, let’s say everyone decides that a digital currency is better (eg because you could avoid tax / minimise transaction costs), then in an extreme scenario ppl decide that it’s just better to hold digital currency rather than fiat - so demand for fiat currency in that case tends to 0 and exchange rate between fiat / digital currency tends towards infinity because the fiat becomes worthless. So it’s kind of like holding a stock that’s price goes to 0. And that probably explains the considerable growth of digital market capitalisation. Is my reasoning correct?

As there is no practical limit on the number of cryptocurrencies that can be created, and as no one crpytocurrency has a unique advantage over any other, and as no cryptocurrencies are legal tender anywhere, the theoretical market cap of any one cryptocurrency has zero underlying value.

What its actual market cap is, will depend on how many suckers can be lured into purchasing it, and what their combined purchasing power is.

Furthermore, even if all your incorrect assumptions were correct, your conclusion would not be: the total capitalisation of all existing sovereign currencies is finite. If they were all to be replaced by a cryptocurrency, then its capitalisation would be much less than the existing total capitalisation of all sovereign currencies, because economic activity would have taken a huge downturn. That's because the world is not an Optimal Currency Area.

This is not to say that cryptocurrencies will never have any value. But at time of writing, there is no visible legitimate application that gives them value.


Balance of Trade ( Surplus and Deficit) I'm new to economics. I want to ask that what remedies that the government can actually employ to address the deficit/surplus in the fish trade balances over time?

My attempt,

Trade deficit

(i) The government can strengthen the diplomatic relationship with foreign countries by signing five to ten years contract with the purpose of buying more Malaysia’s fish commodities but at a lower price. This action can increase the export of our country’s fish commodities and earn a long term profit.
(ii) Provide the export incentive to local fish industries in order to help them remain competitive in the global market by exporting more fish commodities. Export incentive will reduce the cost of export and therefore increase the competitive force of local industry in global market.

Trade surplus

(i) the government can eliminate the trade tariff in order to induce foreign countries to import more fish commodities in our country. This will make the import and export of Malaysia equivalence.
(ii) The government can buy the fish commodities surplus of local fish industries and store them to reduce the export.

Is my answers correct? Is there better solution? Thanks in advance.

Your first response to the trade deficit would seem to increase imports and not exports, unless I'm misunderstanding and you are writing from the perspective of Malaysia. In that case, the policy might be otherwise flawed due to the lower amount a fisher would supply given a lower price for exportation. While you may reach an agreement with other countries to import fish, your fishers must still, as private individuals, be incentivized to supply fish for this trade. I believe a better solution would just be the opposite of that in your first trade surplus answer, raising tariffs.


Budget Set- closed and boundedness I am fairly new to economics, and we were introduced to budget sets,
The professor mentioned that the budget set $B(p,w) = \{x \in R^{l}_{+}: px \leq w\}$ is non empty and closed  - I could prove the latter (the closed part) by taking sequences, but how do I claim that it is  non empty?
Also, if the price of one commodity goes to zero, what happens to boundedness of $B(p,w)$? - I know zero prices means "free goods", and person can purchase infinite amounts of it. but how to prove it rigorously?

The budget set is always defined given a price vector $p=(p_i)_{i\leq l}$ (it seems like $l$ is the number of goods in your problem) and an income $w$. We usually implicitly assume that prices are strictly positive and the income is finite.
Otherwise, as you correctly point out, we end up with infinite consumption being optimal and we don't even have an economic problem (loosely defining economics as the science of incentives regarding the allocation SCARCE goods).
A set is bounded if there exists an upper and a lower bound. We have the lower bound of $x_i\geq 0$ for all goods $i$ by construction of the problem. We can only have an upper bound on any quantity $x_i$ if $p_i>0$ and a finite $w$. Suppose some finite $\overline x= (\overline x_i)_{i \leq l}$ were an upper bound and $p_i=0$ for some $i$. Then you can see that this $\overline x_i$ cannot be the upper bound, because $\overline x_i+\epsilon$ is also in the budget set for any $\epsilon>0$.
EDIT: Sorry, I ignored the question on non-emptyness. Take any finite income and positive prices and note that $x$ is a continuous choice. Take the maximum price $\overline p= \max_i (p_i)_{i\leq l}$ and then consider a quantity vector $\widehat x= (\widehat x_i)_{i\leq l}$ with $\widehat x_i= \frac{w}{l \overline p}$ then $\sum \widehat x_i p_i \leq \sum \widehat x_i \overline p = l \frac{w}{l \overline p} \underline p=w$. So quantity vector $\widehat x$ is in the budget set such that it cannot be empty.


How do you use a Log-linear model when you have negative Xs? I am trying to us a Log-linear model to derive an elasticity. However, some of my Xs are negative numbers. Being as the model relies on the natural log of the Ys and the Xs, how can this model work when the natural log can only be taken of X>0?

The elasticity of Y with respect to X is often estimated by running a regression like this:
$$ \ln Y = \alpha + \beta \ln X + error$$

However, this isn't always applicable because it is perfectly fine to ask what is the percent change in Y to a percent change in X even if X or Y are negative. There are other issues too, like the fact that the elasticity may not be constant over X.

Let's go back to the textbook (Varian (1992), p. 235) for the definition of the elasticity of $Y$ with respect to $X$:
$$ \epsilon(Y,X) =\frac{\partial Y}{\partial X} \cdot \frac{X}{Y} $$
Which equals another common definition of the elasticity:
$$=\frac{\partial \ln Y}{\partial \ln X}$$
which we often use when estimating the elasticity with linear regressions.

However, the advantage of the first form is that you don't have to use the log function to evaluate it. All you need is a local estimate of the derivative, X, and Y.
One strategy would be just to do that.

However, what you are doing could well be nonsensical because percentage changes are weird when working with negative values. For example, consider X going from -11 to -10. This is a 9.1 percent decrease in X ((-10-11)/-11). However, this is is an increase in X! Going from 11 to 10 is also a 9.1 percent decrease in X. What would be the economic conditions where we think going from -11 to -10 and +11 to +10 would have the same effect on demand?


"Elasticity of marginal utility of consumption ( μ ) with respect to utility." (Green Book quote)? In the Green Book (page 97) it is stated "the elasticity of marginal utility of consumption (μ) with respect to utility".

What is it? Is it an error ? I always heard of the elasticity of the marginal utility with respect to consumption...

Mere googling leads me to find a bunch of references about elasticity of marginal utility.

Here Professor Acemoglu defines the elasticity of marginal utility as the inverse of the intertemporal elasticity of substitution (pagg. 17-18).

Here you can find a question analogous to yours together with an answer explained in an easier way than Acemoglu's lecture.

If you are interested in something more empirical, have a look here: it is a paper by David Evans about the empirical estimation of elasticity of marginal utility in OECD countries.

Hope this is enough. If you have other questions, please ask


Equivalence of inada conditions and non-negativity constraints? In a standard constrained utility maximization problem with an agent's preferences defined over good(s), does the imposition of Inada conditions on the utility function preclude us from adding non-negativity constraints while setting up the Lagrangean? The latter seem redundant because the Inada conditions will guarantee an interior solution.
Thanks!

In a standard growth model, people don't put non-negativity constraints as you have mentionned. Theoretically, concavity of the Hamiltonian (or Lagrangian in discrete time) is the sufficient condition for the optimality of the program.
Just for some additional details, the non-negativity constraints are so usefulin some subfields of economics, like in environmental economics.
For example, if you want to put a ceiling for stock of pollution in atmosphere as $\overline{P}$. You can use it if you want that economy do not pass beyond this critical threshold. So, basically you can make as

$$max \int_{0}^{\infty} u(c,P)  e^{-\rho t} dt $$

s.t

$$\dot{K}=F(K)-c\\
\dot{P}=\epsilon K - \delta P$$

where $\epsilon$ is the emission rate from the use of capital and $\delta$ is the decay rate of pollution.

$$\mathcal{H} = u(c,P) + \lambda (F(K)-c) - \mu (\epsilon K - \delta P) + \alpha (\overline{P} - P) $$

The constraint $\alpha$ is a non-negativity constraint which will ensure that pollution level will not pass beyond the threshold $\overline{P}$. As long as the pollution level is below the threshold, $\alpha$ will be non-binding (and will be equal to zero).

Hint : You can look at the Kuhn-Tucker conditions in order to understand better the role of non-negativity constraint.


Is "management science" a subsection of economics? Reading about the academic journal Management Science, one can read the following:

  Management Science is a scholarly journal that publishes scientific research on the practice of management. (...) Our articles are primarily based on the foundational disciplines of economics, mathematics, psychology, sociology, and statistics, and we encourage cross-functional, multidisciplinary research that reflects the diversity of the management science professions.

So is management science (the scientific research that is published in journals like this) a separate science, similar to the way economics uses Mathematics? What is the view of economists on this?

The journal Management Science is a decent one. Many economists do publish on that journal. In fact, a quick look at its editorial board reveals many notable names, with the recent Nobel laureate in Economics Richard Thaler among the associate editors.

The field of management science is more interdisciplinary in nature. See Wikipedia's entry for it. To me, it's like operations research for business or applied microeconomics in the context of business decision-making. To call it a "separate science" is a bit too much, though.


What are some sectors in which nations benefit from collaborating with other nations? For instance, if a nation is an ally (diplomatic/trade) with another nation, the average payoff each nation gets is higher than the payoff they would gain individually?

For instance, comparing to individuals and chat applications, if one user uses chat app A, and another uses B, they can't communicate. But they can communicate with all friends who use the same chat app. The payoff is directly proportional to the number of friends which use the same application.

Nations can benefit from each other in all three of the broad sectors.

Collaboration in the primary sector (Involving raw materials, fishing, agriculture and mining) can lead to more sustainable extraction of resources. For example Nation A can choose to import Resource Y from Nation B rather than exhausting their own source. Nation A gains Resource Y and Nation B gets money.

Outsourcing is the most obvious example of collaboration in the secondary sector, Australian companies may outsource manufacturing work to Bangladesh, this is beneficial to both nations as money is going to Bangladesh and goods are going to Australia.

The tertiary sector involves providing services to consumers. One example of this could be tourism related services, if a tourist from overseas uses one of these tourist related services, the country they are visiting would receive money in exchange for the service, and the tourist would have their experience improved in some way. This example is small scale but when you consider how many people travel around the world it builds up.


Flat Term Structure and Immunized Portfolio Strategy The current term structure is flat at 2%. You have a liability of $500,000 per year for the next five years. You decide to form an immunized portfolio.

  a) Describe your exact strategy if you invest in one through five-year zero-coupon bonds.

What I did is

$$
D_{\rm funding}=w_1(1)+w_2(2)+w_3(3)+w_3(3)+w_4(5)+w_5(5)
$$

but does $D_{\rm funding}$ equal to 5? what does it equal to and how do I solve this equation

Note: This answer is preliminary, as I am unsure about some components of the question. I will note that it is a very long time since I studied portfolio immunization, and so I am describing how this would be looked at in the industry.

The simplest strategy for immunization in this case is to purchase five zero coupon bonds, each with a principal value of \$500,000. This means that your portfolio generates cash flows that exactly match the liability.

(That part of the question is unclear to me. One could attempt to duration match the liability, which would make the answer more complicated.)

The price for \$1 for the i-th zero coupon is: $\frac{1}{(1.02)^i}$, (the zero coupon price formula, using a simple interest rate convention).

The cost for the immunization is the sum of the five costs.

The market value weight of each bond is its market value divided by the sum.


How should one determine the proper number of lags in a time series regression? I am using time series data in economic model estimation. I want determine proper lag for Error Correcting Model (ECM) model for example. I can check AIC, SC and HQ criterion for determine proper lag. But I am not sure about max lag that I must start with it. What is max lag for annual, quarterly and monthly time series.

Is max lag would be changed if I used another econometric approach for example ARDL, VAR or VECM?

I don't have advice specific to error correcting model (ECM) setting, but in undergraduate applied econometric class they gave us the generic advice to continue to extend lags in the model until the residuals of the fitted model were serially uncorrelated. For example, in the US life expectancy data, residuals of male life expectancy is serially uncorrelated in the AR(5) model but not the AR(4) model. You can see this for yourself with the following Stata code:

use http://www.stata-press.com/data/r8/uslifeexp.dta
tsset year, yearly
reg le_male L(1/4).le_male
estat durbinalt, small
reg le_male L(1/5).le_male
estat durbinalt, small

The Stata documentation for the vector error-correction models also seems to roughly follow this approach but it looks like it is automated under the varsoc function and additionally the AIC, HQIC, and SBIC are all generated programmatically.

  To test for cointegration or fit cointegrating VECMs, we must specify
  how many lags to include. Building on the work of Tsay (1984) and
  Paulsen (1984), Nielsen (2001) has shown that the methods implemented
  in varsoc can be used to determine the lag order for a VAR model with
  I(1) variables. As can be seen from (9), the order of the
  corresponding VECM is always one less than the VAR. vec makes this
  adjustment automatically, so we will always refer to the order of the
  underlying VAR. The output below uses varsoc to determine the lag
  order of the VAR of the average housing prices in Dallas and
  Houston....

  We will use two lags for this bivariate model because the Hannan–Quinn
  information criterion (HQIC) method, Schwarz Bayesian information
  criterion (SBIC) method, and sequential likelihood-ratio (LR) test all
  chose two lags, as indicated by the “*” in the output.

. clear all

. use http://www.stata-press.com/data/r13/txhprice

. varsoc dallas houston

   Selection-order criteria
   Sample:  1990m5 - 2003m12                    Number of obs      =       164
  +---------------------------------------------------------------------------+
  |lag |    LL      LR      df    p      FPE       AIC      HQIC      SBIC    |
  |----+----------------------------------------------------------------------|
  |  0 |  299.525                      .000091  -3.62835  -3.61301  -3.59055  |
  |  1 |  577.483  555.92    4  0.000  3.2e-06   -6.9693  -6.92326  -6.85589  |
  |  2 |  590.978  26.991*   4  0.000  2.9e-06*  -7.0851* -7.00837* -6.89608* |
  |  3 |  593.437   4.918    4  0.296  2.9e-06  -7.06631  -6.95888  -6.80168  |
  |  4 |  596.364  5.8532    4  0.210  3.0e-06  -7.05322   -6.9151  -6.71299  |
  +---------------------------------------------------------------------------+
   Endogenous:  dallas houston
    Exogenous:  _cons

If, as @GraemeWalsh suggests, you would like to use an auto-regressive distributed lags methodology (ARDL) you can do so without having to code it up yourself.

use http://www.stata-press.com/data/r13/txhprice
sort t
net install ardl.pkg
ardl dallas houston,  maxlag(4)

ARDL regression
Model: level

Sample:  1990m5 - 2003m12
Number of obs  = 164
Log likelihood = 313.86816
R-squared      = .96315461
Adj R-squared  = .96246376
Root MSE       = .03613756

------------------------------------------------------------------------------
      dallas |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      dallas |
         L1. |    .427744   .0789194     5.42   0.000      .271886     .583602
         L2. |   .1747019   .0720507     2.42   0.016     .0324089    .3169948
             |
     houston |   .3404766   .0567884     6.00   0.000     .2283252     .452628
       _cons |   .7276476   .2061803     3.53   0.001     .3204618    1.134833
------------------------------------------------------------------------------


On the (rather strange) notion of strategies in repeated games As undergraduates often note, the definition of a strategy appears to be rather strange in the context of repeated games. To illustrate, consider a very simple game in which one player needs to make a decision twice. Every time they need to make the decision, they have 2 possible choices -- call them A or B. Ordinarily, we would think of this person as having $4$ possible (pure) strategies: choose A twice, choose B twice, choose A then B, or choose B then A. According to the standard definition, however, we would say that there are 2 possible histories (A or B); and that a strategy must specify what happens following every possible history. As a result, the player really has $2^3 = 8$ pure strategies! Moreover, each of these strategies specifies what would happen in an impossible situation; namely, if the person were to have made a different choice in the first stage than the one they planned to make.
In case this doesn't seem strange to you, let me provide an example of a pure strategy (I'll put it verbally to convey the full force of the strangeness):

In the first round, I will choose action A. After choosing A, I will then choose A again. However, if I had chosen B in the first round, then I would then have chosen B again in the second round.

My question is why we need to include this last sentence.
Before finally posing my question, I also want to note that this question is not specific to the rather trivial game above. Obviously, one can devise similar examples for more complex multi-player games.
Question: Why does the standard notion of 'strategy' force players to specify what they would do in situations that they know will never arise given their choices earlier in the game? And can we prove standard results (e.g. existence of a Nash equilibrium in extensive games) without making use of this rather un-natural concept of a strategy?

Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper

Rubinstein, Ariel. "Comments on the interpretation of game theory."
Econometrica: Journal of the Econometric Society (1991): 909-924.

Some concepts become harder to define, such as backward induction in games of perfect information. However, one can still do pretty much all of game theory. The book " Rationality in extensive form games" by Andres Perea uses the definition as a plan of action throughout.


Formal definition of perfect information I am looking for the formal definition of 'perfect information' in game theory.

Please direct me to a book or preferably an online paper where I can find it.

On a related note:
The Wikipedia page for the term is not very useful. It only offers an informal definition:

  In game theory, an extensive-form game has perfect information if each player, when making any decision, is perfectly informed of all the events that have previously occurred

Given this definition the simultaneous move examples seem strange. The games mentioned (e.g. iterated prisoner's dilemma) could easily be altered to have sequential moves where the second mover is simply not aware of the first move. This game would have the same extensive form but would no longer fit the informal definition.

A simultaneous-move game is not a game of perfect information. It is a game of imperfect information.

Let me quote Gibbons (Chapter 2, p.58).

  The key features of a dynamic game of complete and perfect information are
  that
   (i) the moves occur in sequence,
   (ii) all previous moves are observed before the next move is chosen, and
   (iii) the players' payoffs from each feasible combination of moves are common knowledge.

Picking up the discussion in the comments, let me again quote Gibbons, p.122, Footnote 19:

  This characterization of perfect and imperfect information in terms of singleton
  and nonsingleton information sets is restricted to games of complete information
  because, as we will see in Chapter 4, the extensive-form representation of
  a game with perfect but incomplete information has a nonsingleton information
  set. In this chapter, however, we restrict attention to complete information.


Translated Selten Subgame Perfect Eqm Paper Does anyone know where I can get a translated copy of Selten's original work on subgame perfection?

Do you mean this article?

Selten, R. Reexamination of the perfectness concept for equilibrium points in extensive games, Int J Game Theory (1975) 4: 25.


How do central banks regulate the money supply to avoid inflation? It is a tricky problem. Historically, poor control money supply led to inflation. Out of control money supply led to hyperinflation. Central banks seem to do a wonderful job in recent years with modern economies seeing low inflation despite quantitative easing.

How do they do it?

The answer provided by Mick is pretty good, but I'll put my own spin on this.

1st: What is inflation?

Roughly speaking it is when the ratio of money to the things money buys increases.  A bumper food crop can reduce inflation.  Crop failure can increase it.  But certainly the most volatile component of inflation is the supply of money itself.  Economists love to exaggerate the importance of the former (productivity) to cover-up the mess they are making with the latter (money supply).

2nd: What is the supply of money?

A difficult answer, as money is whatever we accept.  But the most common forms of money however are classified into large aggregates by economists.

MB: Aka the monetary base in the USA (all money created by the
government...coins, dollar bills, us notes fed deposits)
M1: Bank demand depots (promises of MB) plus cash outside of the banking system
M2: M1 + interest bearing deposits and other checking substitutes

Because private banks can get away with promising more money then they have...and we accept this...then banks actually create money.  And because the higher aggregates like M2 are much larger than MB, it is plain to see that banks are more important variables to the money supply equation and the supply of government money (aka MB).

3rd: How does the Fed regulate the money supply?

It is confusing because the Fed directly dictates the amount of base money in the economy, but also indirectly dictates how much bank money is in an economy.

The primary means by which MB changes include discount loans to banks, buying and selling repos from big banks, and special bailouts.

The primary means by which the Fed controls the supply of bank money is through the open market.  The idea is if the banks have more base money, they will be able to pyramid more deposits on top of that, and vice versa.  To facilitate this the Fed will buy repos from a primary dealer ("special banks") with fabricated money.  The primary dealer than resells the new reserves to lesser banks which then allows them to create more bank money.

Historically, controlling the supply of bank deposits through a reserve ratio was a technique used (and also capital requirements), but these techniques are not considered fashionable among entrenched interests.

Has the Fed been doing a good job?

This is a complicated question.  QE did not create massive inflation because the banking sector was in collapse and a large amount of bank deposits had been destroyed (meaning we had less money).  Whether private for-profit interests should have had privileged access to government resources is a very good question.  Certainly the problem of 2008 was created by the banks themselves. Simply put, they made promises to depositors and creditors they couldn't keep.  Banks operate by mismatching short term/low yield debt to long term/high yield debt.  This is a very profitable but risky practice in which banks are not held proportionately responsible when things go wrong.

Classifying inflation is also very tricky because of international trade.  The dollar is a popular currency overseas, so foreigners help sponge up the damage from inflation.  Trade also obfuscates inflation. If you double say the number of dollars, not all prices will increase proportionately.  International commodities can be brought in multiple currencies, whereas local products (like healthcare and college education) can't and can only be purchased with dollars so they increase much more rapidly.

For a fascinating look at the politics of CPI/inflation, checkout: http://www.shadowstats.com/article/no-438-public-comment-on-inflation-measurement

The most paramount question unasked is if the Fed is seeding the conditions by which another future crisis will happen, to which I fear the answer is yes.


Why would China and Russia selling the US' debt ruin the economy? I was listening to an interview with David Harvey and he mentioned that if Russia and China were to sell the debt owed to them by the US, the US economy would take a serious dive. How would they sell the debt and why would that impact the US economy so negatively?

It wouldn't "ruin the US economy". The US (the country as a whole, not the government) bought goods and services from rest of the world by selling them IOUs. So far, the world has been content to let the US roll over these obligations. If the world decided that they wanted to trade these IOUs for goods and services, the US would have to start running current account surpluses and shipping goods abroad to pay for its past current account deficits. These surpluses have to come from somewhere, and they would result in less consumption and less investment in the US economy.

Of course, an economy can't borrow its way in global capital markets forever - the US does eventually have to ship some stuff over to China in exchange for all of the stuff that China shipped over to the US in exchange for (electronic) green pieces of paper. If China decides to unilaterally impose capital controls and stop lending goods to the US, then this hurts both China and the US - the reason capital in China was flowing to the US is that it could get higher returns there. (The functioning of domestic capital markets in China has become a lot better over the past decade, which is why China can now allocate its savings to productive uses domestically instead of lending large amounts of it to foreign countries.)

What some people are specifically afraid about ties into the USD's status as the international reserve currency - the Chinese government and central bank own plenty of US government debt as part of their foreign exchange reserves, and if they want to redeem this debt in exchange for goods and not roll it over, the scenario I mentioned above comes to pass. This seems to me like a grossly exaggerated concern at a time when global real interest rates are quite low. If China wanted to sell its US government bonds and the US had no intention of stopping the rollover process, returns would rise until the market for US external debt cleared, and there's no reason to think that the return response would be large given the low level of global real interest rates. The US (as a country) would have to pay more interest on its debt outstanding, which isn't great, but it's not a disaster of a scale that would "ruin the US economy".


Why is the U.S. birth rate considered below replacement if more people are being born than are dying annually? In 2018 there were 3.85 million births and 2.81 million deaths.
My understanding is that the U.S. birth rate is near or below replacement rate, but I don't understand why. If more people are being born than are dying, won't the population continue to grow?

When you hear that "the U.S. birth rate is near or below replacement rate", what is probably meant is that the total fertility rate (TFR) is below 2.1 (replacement rate).
It is possible that the TFR is below 2.1, while births exceed deaths. One way this can happen is if there's a rising share of women entering reproductive age.
This phenomenon is sometimes called population momentum (more details: 1, 2, 3).
This is actually a fairly common phenomenon: Many countries have had TFR below 2.1 for many years but their populations have continued to grow. For example, China has had TFR below 2.1 since 1992 but births have continued to exceed deaths every year.


Using margins after probit estimation to equal probabilities between almost identical individuals I'm considering a Probit model for the probability that a student will finish the course based on their hours of study, age, sex, origin, how they passed the previous course and labor market situation for students.

probit finish hours age women inmigrant previous_courses

Now with these estimates, I have to compute the number of hours of study necessary so that the probability of completing the course of a student who works is the same as that of another student with identical characteristics but who does not work.

I know I have to use the command margins but I can't figure out how. Any clue?

Thanks in advance!

Let me give an example inspired by Richard William's presentation, and the Stata command margins.

webuse nhanes2f, clear
keep if !missing(diabetes, black, female, age, age2, agegrp)
gen femage = female*age
label variable femage "female * age interaction"
probit diabetes black female age, nolog

Probit regression                               Number of obs     =     10,335
                                                LR chi2(3)        =     380.15
                                                Prob > chi2       =     0.0000
Log likelihood =  -1808.992                     Pseudo R2         =     0.0951

------------------------------------------------------------------------------
    diabetes |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       black |   .3567259   .0638644     5.59   0.000      .231554    .4818978
      female |   .0859841   .0450391     1.91   0.056    -.0022909    .1742592
         age |   .0271895   .0016318    16.66   0.000     .0239912    .0303879
       _cons |  -3.215875   .1015044   -31.68   0.000     -3.41482    -3.01693
------------------------------------------------------------------------------

Among other things, the results show that getting older is bad for your health (surprise!) – but just how bad is it?

Adjusted predictions (also known as predictive margins) can make these results more tangible.  With adjusted predictions, you specify values for each of the independent variables in the model, and then compute the probability of the event occurring for an individual who has those values. Using the mean values for the other independent variables (female, black) that are in the model, we get:

margins, at(age=(20(10)70)) atmeans

Adjusted predictions                            Number of obs     =     10,335
Model VCE    : OIM
Expression   : Pr(diabetes), predict()

1._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          20
2._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          30
3._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          40
4._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          50
5._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          60
6._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          70
 ------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .0063084   .0009888     6.38   0.000     .0043703    .0082465
          2  |   .0113751   .0013794     8.25   0.000     .0086715    .0140786
          3  |   .0204274   .0017892    11.42   0.000     .0169206    .0239342
          4  |   .0364184   .0021437    16.99   0.000     .0322167      .04062
          5  |   .0641081   .0028498    22.50   0.000     .0585226    .0696935
          6  |   .1104379    .005868    18.82   0.000     .0989369     .121939
------------------------------------------------------------------------------

The results show that a 20 year old has less than a 1 percent chance of having diabetes (.0063084), while an otherwise comparable 70 year old has an 11 percent chance (.1104379).


Constraints on a symmetric pareto allocation under uncertainty I've been trying to figure out how the author came up with the constraints for this liquidity model in a textbook I'm reading.

details: http://imgur.com/TpVjg4w

$U = \pi_1u(C_1) + \pi_2u(C_2)$ where $\pi_i$ denotes the probability of an agent being of type $i$ and $C_t$ denotes consumption at time $t$.  Agents of type $1$ consume at time $1$ and agents of type $2$ consume at time $2$.

In autarky, $C_1=1-I+lI$ and $C_2 = 1-I+RI$ where $0\leq I \leq 1$, $l < 1$, $R > 1$.

The setup for the unique ex ante symmetric Pareto-optimal allocation (not autarky) is

$\max \pi_1u(C_1) + \pi_2u(C_2)$

subject to the following constraints, which is what I don't understand:

$\pi_1C_1=1 - I$ and $\pi_2C_2 = RI$

There's a footnote on the word symmetric which says:

  Since agents are ex ante identical, we only consider symmetric allocations $(C_1, C_2)$, where an agent's consumption profile does not depend on the agent's identity.

I think the symmetry definition might be part of what's tripping me up.  It seems to me that since an agent of type $1$ can only consume at time $1$ his bundle should be $(C_1, 0)$, and the bundle for a type $2$ agent should be $(0, C_2)$, which would mean that the only symmetric bundle is $(0,0)$, but clearly this is not the author's intention.

The idea is that since agents are ex-ante identical, they should have the same (ex-ante) expected utility.

Ex-ante identical means that even the agent does not know what's his type until resolution of type at date-1. Because they have the same preferences each agent's consumption profile must be identical. And the analysis can then focus on symmetric equilibrium.

What symmetric means, as explained in the footnote, is that the consumption profile must be identical. The consumption profile $(C_1, C_2)$ represents the consumption for the agent at date 1 (if he is type-1) $C_1$ and the consumption for the agent at date 2 (if he is type-2) $C_2$. Otherwise stated $(C_1, C_2)$ does not depend on the type and each agent has the same $C_1$ and $C_2$. Since the types coincide with the date it's a little bit trivial but it really says nothing more than that.

For the constraint, there is a mass-1 continuum of agents, each endowed with one unit of wealth, so the total wealth is 1. Then from that total wealth an amount $I$ will be invested in the productive technology whose return is $R$ per unit invested. Then it follows that :

at date 1, an amount $1-I$ is available, and the demand will be $\pi_1C_1$, so this yields : $\pi_1C_1 = 1-I$
at date 2, an amount $RI$ is available, and the demand will be $\pi_2C_2$, so this yields : $\pi_2C_2 = RI$


Can m0 be created as a loan? Are the following true?

m0 is only created by destroying something else (federal reserve deposits)?
m0 can be created as a loan?

Related questions: can something other than federal reserve deposits be exchanged for m0? what happens to the FRD, they get destroyed?

M0 are notes (e.g. dollar bills in the US) and coins Held by the public. It seems likely that almost all the notes and coins come from the public withdrawing them from banks.

Banks get the notes and coins via exchanging settlement balances at the Federal Reserve (“reserves”) for them, which are put in vaults for distribution.

The only alternative is for the Federal government to directly pay out notes and coins, which is possible, but this does not appear to be common practice.

If we ignore that alternative creation mechanism, we can answer your questions.

Firstly, an increase of M0 is reliant on the public withdrawing notes and coins from a bank. This is only possible if the bank has notes and coins in its vault, so this process is limited. In order to be sustained, the vaults need to be replenished.

Yes, banks need to exchange reserves for notes and coins.
A bank could borrow reserves from another bank or at the discount window to get vault cash, but those are two transactions.
Settlement balances are destroyed when exchanged for notes and couns.


What kind of tax diagram can show the full consumer tax burden? If a government increases a purchase tax on an apartment, the tax will be fully burdened by the consumer, as the "producer" will simply be the owner of the house.
But in a diagram, should this be shown by a perfectly inelastic demand diagram where the consumer has full tax burden? However, how can we justify the demand for apartments being "perfectly inelastic?" Are there other types of diagram that could be used to show full consumer tax burden without an extreme demand/supply?

I would recommend that you read what Harvey S. Rosen and Ted Gayer have to say about taxation and tax burdens on their book, Public Finance.
Your hypotheses for perfectly inelastic demand might be flawed, to the extent that (1) the house market is competing with Airbnb-type companies and (2) -for inelastic supply- there is an affluence in housing available (that is, there are vacant houses -which is most always true in most first-world countries).


Fractional Reserve Banking Highly contrived example with 3% reserve ratio:
Lets say I deposit $100 in a bank.
Bank loans $97 PersonA, keeps $3 as reserve
PersonA buys something from PersonB for $97
PersonB deposits it in Bank, bank loans 97%of $97 to PersonC, keeps 3% in reserves.
PersonC buys something from PersonD for $94
This cycle goes on until until 100/0.03=$3333 is in deposits,
$3233 in loans and $100 in reserves.

Is it accurate to say, if a bank starts with $100 as reserves, they can loan out $3333 assuming reserve ratio is 0.03%?
In the above example, if everyone repays the loan ($3233) can bank lend it again? That doesn't sound correct. Where am I going wrong here?

Yes with reserve requirement of $0.03\%$ the maximum amount of money that can be created will be given by $\frac{100}{0.03} = {\\\$}3333.33$, with ${\\\$} 100$ reserves and ${\\\$}3233.33$ new money.
Yes if everyone repays the ${\\\$}3233.33$  and there is still demand for loans bank can lend it again. To be clear the ${\\\$}3233.33$ will not become new reserve, so the reserve ratio would still have to be maintained. If people start paying the loans back with no demand for new loans the money supply will start contracting.
However, with presently reserve requirements being set to zero by Fed the only constraint is the demand for loans. Furthermore, while in past banks tended to hold almost no excess reserves (see this graph from Fred), after 2008 they started holding excess reserves and in such situation the multiplier idea becomes less useful abstraction as banks can always borrow extra reserves from other banks that have extra reserves if they need to. For example, if system holds excess reserves when the ${\\\$}3233.33$ would be repaid (assuming all this is done through lending and borrowing with single bank) bank could just get extra reserves from other banks holding excess reserves, even when  banking system as whole would not be allowed to create new reserve (without central bank agreeing to that).  McLeay, Radia, & Thomas(2014), money creation in the modern economy is good source for explaining how fractional reserve system works in such situation.


Intepretation of $dC_1=-\theta(dT_1+\frac{dT_2}{1+r})=0\Longleftrightarrow dT_1=-\frac{dT_2}{1+r}$? We are studying the Ricardian equivalence in a macro course and we are asked to show that a tax cut today will be matched by a tax hike tomorrow. I believe the above is supposed to be the solution, however I do not quite understand the total differential and even less so the notation. I have three questions:

How exactly do I take the total derivative of $C_1=\theta(V_1+Y^L_1-T_1+\frac{Y^L_2-T_2}{1+r})$ with respect to $T_1$ and $T_2$?
How would I then get to $dC_1=-\theta(dT_1+\frac{dT_2}{1+r})$?
What is the interpretation of $dT_1$ and $dT_2$? Specifically, what is the interpretation of $dT_1=-\frac{dT_2}{1+r}$?

Basic intro to total differentiation

Suppose we have a function $$f(x_1,x_2,\ldots,x_n).$$ The total derivative is given by $$\frac{\partial f}{\partial x_1}dx_1+\frac{\partial f}{\partial x_2}dx_2+\ldots+\frac{\partial f}{\partial x_n}dx_n,$$
where the $\partial$ terms are partial derivatives and the $dx$ terms are so-called "differentials".

Conceptually, what is going on is the following: we ask "if I change every variable $x_i$ by a small amount $dx_i$, what is the effect on the overall value of $f$?"

To answer this question, we can think about the variables one-by-one. Start with $x_1$. A unit increase in $x_1$ causes $f$ to change by $\partial f/\partial x_1$. We are making a small $dx_1$ unit change in $x_1$. Thus, the total change in $f$ due to the change in $x_1$ is $\frac{\partial f}{\partial x_1}dx_1$. At the same time, we change $x_2$ by $dx_2$, resulting in a $\frac{\partial f}{\partial x_2}dx_2$ unit change in $f$ and so on.

If you want to read more on the topic, I'd recommend the relevant chapter of "Fundamental Methods of Mathematical Economics" by Alpha Chiang.

Aside on why this might be useful (can be skipped without loss)

Let's look at an example of how this could be useful. Suppose we have a game where players $i=1,2$ each choose an action $x_i$. Player $i$'s payoff is

$$u_i=f(x_i,a)+g(x_i,x_j,a)$$

where $a$ is a parameter and $x_j$ is the other player's action. Thus, a player's best action is given by the first-order condition

$$f_1(x_i,a)+g_1(x_i,x_j,a)=0$$
where $f_i$ means the partial derivative of $f$ wrt its $i^{th}$ argument.

It is hard to know how a change in the parameter $a$ will affect $i$'s choice of $x_i$ because the change in $a$ will affect $x_i$ not only directly, but also by changing $x_j$. When all of these changes have settled back to equilibrium, what happens? It seems like we can't go any further because we need to know the functional form of $f$ and $g$ to be able to explicitly solve for $x_i$ and see how they depend on each other and on $a$.

But total differentiation can help us understand the properties of the solution here. Let's totally differentiate the first order condition, allowing all of $x_i$, $x_j$, and $a$ to change at once (I omit the arguments of the functions for brevity):

$$[f_{11}+g_{11}]dx_i+g_{12}dx_j+[f_{12}+g_{13}]da=0.$$

Now I can solve for $\frac{dx_i}{da}$:

$$\frac{dx_i}{da}=-\frac{f_{12}+g_{13}}{f_{11}+g_{11}}.$$
So long as I know something about the individual properties of $f$ and $g$, I can use this equation to tell me what the sign of $fx_i/da$ will be!

Coming back to your example

For notational convenience, I give your function a name ($f$):

$$f(C_1,T_1,T_2)\equiv\theta(V_1+Y^L_1-T_1+\frac{Y^L_2-T_2}{1+r})-C_1=0,$$

we apply the basic rules of total differentiation:

$$f_1 dC_1+f_2dT_1+f_3dT_2=0.$$

We compute $f_1=\frac{\partial f}{\partial C_1}=-1$,$f_2=\frac{\partial f}{\partial T_1}=-\theta$, $f_3=\frac{\partial f}{\partial T_2}=-\theta\frac{1}{1+r}$. This therefore gives $$dC_1=-\theta\left(dT_1+\frac{dT_2}{1+r}\right).$$

In words, this equation tells us that the change in $C_1$ must be equal to $-\theta\left(dT_1+\frac{dT_2}{1+r}\right)$, where $dT_1$ and $dT_2$ are the changes in $T_1$ and $T_2$. Thus, if we want to say that $C_1$ doesn't change then it must be that $dT_1=-\frac{dT_2}{1+r}$ so that the changes in $T_1$ and $T_2$ 'cancel-out'.


Updating MRW (1992) with PWT 9.0 I'm trying to rework Mankiw, Romer, Weil (1992) with fresh data.

From their paper:

n: workforce growth rate

  We measure n as the average rate of growth of the working-age population, where working age is defined as 15 to 64.

s: rate of savings

  We measure s as the average share of real investment (including government investment) in real GDP...

Y/L: output per worker

  ...and Y/L as real GDP in 1985 divided by the working-age population in that year.

They also assume...

  ... that g+δ=0.05

As for human capital:

  We use a proxy for the rate of human-capital accumulation (Sh) that measures approximately the percentage of the working-age population that is in secondary school. We begin with data on the fraction of the eligible population (aged 12 to 17) enrolled in secondary school, which we obtained from the UNESCO yearbook. We then multiply this enrollment rate by the fraction of the working-age population that is of school age (aged 15 to 19).

Considering the latest version of the Penn World Tables, should I use emp and the corresponding growth rate for n, rgdpo/emp for Y/L, and hc for human capital (Sh)? What about s? I'm a little perplexed as MRW include government investment.

Excerpt from the PWT 9.0 legend for convenience.

Real GDP, employment and population levels

  rgdpe Expenditure-side real GDP at chained PPPs (in mil. 2011US$)

  rgdpo Output-side real GDP at chained PPPs (in mil. 2011US$)

  pop   Population (in millions)

  emp   Number of persons engaged (in millions)

  avh   Average annual hours worked by persons engaged

  hc    Human capital index, based on years of schooling and returns to education; see Human capital in PWT9.

Current price GDP, capital and TFP

  ccon  Real consumption of households and government, at current PPPs (in mil. 2011US$)

  cda   Real domestic absorption, (real consumption plus investment), at current PPPs (in mil. 2011US$)

  cgdpe Expenditure-side real GDP at current PPPs (in mil. 2011US$)

  cgdpo Output-side real GDP at current PPPs (in mil. 2011US$)

  ck    Capital stock at current PPPs (in mil. 2011US$)

  ctfp  TFP level at current PPPs (USA=1)

  cwtfp Welfare-relevant TFP levels at current PPPs (USA=1)

National accounts-based variables

  rgdpna    Real GDP at constant 2011 national prices (in mil. 2011US$)

  rconna    Real consumption at constant 2011 national prices (in mil. 2011US$)

  rdana Real domestic absorption at constant 2011 national prices (in mil. 2011US$)

  rkna  Capital stock at constant 2011 national prices (in mil. 2011US$)

  rtfpna    TFP at constant national prices (2011=1)

  rwtfpna   Welfare-relevant TFP at constant national prices (2011=1)

  labsh Share of labour compensation in GDP at current national prices

  delta Average depreciation rate of the capital stock

Shares in CGDPo

  csh_c Share of household consumption at current PPPs

  csh_i Share of gross capital formation at current PPPs

  csh_g Share of government consumption at current PPPs

  csh_x Share of merchandise exports at current PPPs

  csh_m Share of merchandise imports at current PPPs

  csh_r Share of residual trade and GDP statistical discrepancy at current PPPs

Your assumptions seem correct. For savings rate, the consistent definition with MRW would be

$$ \frac{ccon - cda}{cgdpo} $$

ccon is domestic absortion, equal to $ C + G+ I$, whereas cda is $C+G$. Therefore, $I$ is investment, which by definition includes both private and public sector.


Lagrangian: when to discount budget constraint? I am getting a bit confused about setting up the Lagrangian in intertemporal constrained optimization problems. The confusion is as to when is the one-period budget constraint multiplied by the discount factor Beta and/or when is it included in the summation? Here is what I mean. In this first example:

the constraint is not multiplied by Beta, nor part of the summation. Now, in example 2:

the constraint is multiplied by the discount factor Beta and part of the intertemporal summation. Why this difference? Aren't the two cases pretty much equal, both maximizing subject to one-period constraints? The only difference I see is when the interest rate on the assets is realized, but that should not matter I believe(?)
I also found a third case (will upload an image later if I find it) in which the one-period constraint is summed up intertemporally, but not discounted by Beta.
So to conclude: what is the rule here? What am I missing?

EDIT:

By looking at the FOCs that follow example 1, it becomes clear that the author means the Lagrange Multiplier to be part of the summation, but not discounted by Beta. This is clear from the following:

So this really looks to me like a different formulation to the second example. The "funny" thing, however, is that in both cases you get the same result (i.e. the same Euler Equation).

The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".

So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).

The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.


Difference-in-differences in 2SLS regression Usually when we do a difference-in-differences estimation, we do it in a OLS reduced form as follows:
$$
Y_{it}=\alpha After_t+\gamma Treatment_i+\delta After*Treatment_{i,t}+X_{it}\beta+\epsilon_{i,t}
$$
However, I was wondering, if the $Treatment$ group is endogenous (e.g. self-selected), but we can define an "eligible" group for the treatment, whether it would be more precise to estimate a diff-in-diff in a OLS/2SLS form as:
$$
Treatment_{i,t}=constant+\alpha After_t+\gamma Eligible_i+\delta After*Eligible_{i,t}+\epsilon_{i,t}
$$and get $\hat{Treatment_{i,t}}$, then

$$
Y_{i,t}=X_{it}\beta+\delta\hat{Treatment_{i,t}}+\epsilon_{i,t}
$$

How should we understand the diff-in-diff in a OLS/2SLS form? Are there any paper using this particular identification strategy that I could take a look?

Thank you very much in advance!

Well, if you believe that treatment is endogenous (which depends on the problem at hand here and is not an inherent feature of the model), then using eligibility as an instrumental variable will help you to get rid of the biases due to the safe selection in treatment. (Incidentally, DID is intended to do the same, but won't do as good a job as a well chosen instrument, so there is some doubts whether applying both of them is better then resorting to only one). However it is up to you to decide whether eligibility is exogenous, as it well may be, that those who are expecting higher return to treatment made sure to be eligible.

Taking that we believe that there are some biases that arenot eliminated by DID and that eligibility can help us, there is still considerations of efficiency. In many cases eligibility may happen to be a weak instrument and then the reduction is bias will come at a cost of significant efficiency loss.

And taking a look at the particular specification that you have sugested, it seems not very reasonable in general setting. You may choose when you believe that eligibility is changing quickly, or the interaction term in second equation will be generally unhelpful. Inclusion of time After in that equation can have even more drastic consequences, as it is likely to be endogenous and will weaken the bias reduction effect. If not endogenous, it is likeliy to be negligible as well as interaction, unless Treatment is rapidly changing on it's own.

So in this case I would recommend leaving only the eligibility as an instrument in the first equation and specifying the third one in a DID form.

With respect to interpretation, my specification does not allow for a nice interpretation of difference in changes in two subgroups and should be interpreted as a difference in changes in two hypothetical subgroup where each person is divided between them with some weights.

Your specification, however, loses all interpretation as DID, because you do not use the resulting interaction coefficient, but just employ more variables as instruments for treatment.

Unfortunately, probably due to the aformentioned reasons, I was unable to recall or find any appropriate paper, sorry about that.


Are coupons all semi-annual? How do coupons get priced in together?

Edit this answer was written for the original question, before user completely edited whole question to something completely different and unrelated to original question for some reason:

How is it that the coupon rate is higher than the interest rate?
How can the coupon rate be higher than the interest rate?
Doesn't that mean that after one year, you're beating the interest rate. For example, if the coupon rate was 8% and the interest rate was 4%. On a $100 priced bond, I would get 8 dollars in the first year, which is 8% returns, as opposed to 5% returns on the interest rate. Am I missing something? If I'm not missing something, then is this scenario not possible?

I believe that at the heart of this question is confusion between the price of the bond and face value of the bond.
A general bond pricing formula (see here an example) can be expressed as as a discounted stream of future cash flows from the coupon and the value of the bond itself. Hence we have:
$$ P = \sum^T_{t=1} \frac{C_t}{(1+i)^t}+\frac{M}{(1+i)^T}$$
where $P$ is the price of the bond, $C$ is the coupon, $i$ is the interest rate, and $M$ is the value at maturity (i.e. the face value — the value printed out on the bond).
Based on the above formula if the coupon was $8\%$ of the $\\\$100$ face value (i.e. $\\\$8$), the interest rate $4\%$, and we assume $T=2$ the actual price of the bond simply cannot be $\\\$100$.
The correct price would be:
$$ P = \sum^2_{t=1} \frac{\\\$8}{(1+0.04)^t}+\frac{\\\$100}{(1+0.04)^2} \approx \\\$107.54$$
If the price would  be lower than $\\\$107.54$ people would increase their demand for the bond until it would reach its equilibrium at that price and vice versa if the original price would be above.
It is simply not viable for the price to stay at $\\\$100$ in free bond market. In real life there might be some mispricing occurring at some times (although to discuss why is outside the scope of this answer) that can create arbitrage opportunity but as people start taking advantage of it, it will eventually disappear unless the mispricing is deliberately encouraged by some policy.


Identification of switching costs from price shocks What, if anything, can we learn about customer switching costs by looking at price, revenue, profit, and quantity responses of producers to cost shocks?

For example, we can define the profit equation as:

$\Pi = \sum^\infty_{t=0} [ -\alpha S + \beta^t(P - C(q))] \cdot  q(P,C,S)$

Where $q(P,C,S)$ is demand as as function of prices, costs, and switching costs respectively, $C(q)$ is total cost as a function of quantity produced, and $\beta$ is the discount rate of the firm, and $\alpha$ is the fraction of the switching cost refunded to the purchaser. Are there published or just worked out examples of choices of $q(P,C,S)$ and $C(q)$  that allow identification of S from $\partial\pi/\partial C$, $\partial P^*/\partial C$, and $\partial q^*/\partial C$?

Consider the cost shock as an exogenous shock, but I have in mind a setting that allows a structural identification.
By switching costs, I mean that if a customer wants to first buy from firm A and later switch to firm B he pays $P_B+S$, in the period of the switch then $P_B,P_B\ldots$ in subsequent periods instead of $P_A, P_A, \ldots$.
The $-\alpha S$ term is a refund to only the purchasers in the period of initial purchase while $S$ is paid in the final period of doing business with a firm. An example of $-\alpha S$ could be getting a special deal on your Verizon cellphone because you can't use that phone with any other network or a free toaster when you open a bank account. I fixed it in the profit equation so that $S$ was multiplied by $q$ to indicate that $S$ is is the switching costs per unit of $q$. I had in mind that customers purchase either $0$ or $1$ unit of the service and $q$ aggregates their individual decisions. But I'm not wedded that refund, if it gets me somewhere I'm happy to assume that $\alpha =0$

This is not a complete argument but I'm going to give a brief sketch. Assume that producers set prices instead of quantity (which seems to be the model above). Assume that cost shocks do not affect demand, in other words $q(P,C,S) = q(P,S)$, and that they are not correlated with some kind of variation in switching costs. For simplicity, imagine also that this is a constant cost model. Then:

$$\frac{d q^\star}{d C}= \frac{\delta q}{\delta P}\frac{d P^\star}{d C}$$

Assuming switching costs are such that $\frac{\delta q}{\delta P}$ is low for high switching costs, this could give you an idea. If you enough form for $q$ you should be able to pin down $S$ using just-identified GMM (1 moment, 1 parameter).

Let's take a more concrete example, using a more concrete problem. Assume a two period model. In the first period, prices are announced, in the second period there is a surprise shock to the costs of $B$ (WLOG they get cheaper) which $A$ does not experience. This causes a switch from $P_B$ to $P_B^\prime$. Suppose a continuum of consumers, $i$, each with a time invariant valuation of each good, $j$, $V_i^j$, jointly distributed according to $F$. Then people who switch to $B$ from $A$ this period are those individuals for whom:

$$U(V^B_i - P_B^\prime -S)> U(V^A_i - P_A)$$
but:
$$U(V^B_i - P_A - S)< U(V_i^A-P_A)$$
This means:
$$q^\prime - q = \int_V1_{U(V^B - P_B^\prime - S)> U(V^A-P_A) >U(V^B - P_A - S)}dF(V^A,V^B)$$
Note that this is basically analogous to the argument above using derivatives. With form on $V$ and $U$ estimation is straightforward.

More complicated setups, (for instance, consumers with infinite horizons) will require more complicated arguments, but the basic identification will be the same. In this setting what you really need are forms for demand and not costs. As far as demand goes, with only this data you are restricted in what you can estimate. With linear utility, and one of the goods valuations normalized to 0, you could probably pin down the mean of $V_i^j$, and maybe a bit of its cross-sectional distribution. More detail would be possible with consumer level data.


How does your dominant strategy change in a second price auction if there are two items? Suppose there are 4 individuals and two items in this particular(?) auction. I understand that the dominant strategy in second price auction is to bid your true value if there is a single item but I'm having a hard time understanding why that would change at all if there were two items.

This question is hard to answer, because it is not clear what the appropriate multi-unit version of the single-unit second-price auction (SPA) actually is and which economic environment you have in mind. There are several possibilities, and I mention some.
There are 4 bidders and two units.

All bidders draw a two values $v_{1i},v_{2i}$ representing the willingness-to-pay for unit $k\in\{1,2\}$, having both units is valued $v_{1i}+v_{2i}$ (only one draw such that $v_{1i}=v_{2i}$ is also possible). There are two separate SPAs and in each one every bidder truthfully bids their value in equilibrium.

All bidders draw  two values $v_{1i}\geq v_{2i}$ representing the willingness-to-pay, where having one unit is valued $v_{1i}$, having both units is valued $v_{1i}+v_{2i}$. There is one uniform price auction, i.e., all bidders submit a bid vector $(b_1,b_2)$, the highest two bids win and the payment per unit is the highest losing bid. It is a dominant strategy to bid truthfully on the first unit $b_1=v_{1i}$, but not necessarily for the second unit, because the second bid could determine the price which a winner of one unit must pay.

All bidders draw  two values $v_{1i}\geq v_{2i}$ representing the willingness-to-pay, where having one unit is valued $v_{1i}$, having both units is valued $v_{1i}+v_{2i}$. There is one Vickrey auction, i.e., all bidders submit a bid vector $(b_1,b_2)$, the highest two bids win and the payments are the highest losing bids of the opponents. It is a dominant strategy to bid truthfully on both units with the same argument as with a single unit.

All bidders have single-unit demand and draw a single value $v_i$. Two SPAs are executed sequentially. The three remaining bidders bid truthfully in the final auction, but none of the four bidders in the first auction do, because there is a positive option value of losing the first auction: participating in the second auction.

All bidders have single-unit demand and draw a single value $v_i$. A generalized second-price auction is executed, i.e., the highest bidder wins and pays the second-highest bid, the second-highest bidder wins and pays the third-highest bid. If two bidders submit high bids and the other two submit low bids, the highest bidder would prefer to bid lower in order to win the second unit instead to then pay the third-highest and lower bid.


Why did the Swiss National Bank end the currency peg? The title says it all.  Why did the Swiss National Bank end the currency peg?  The issue obviously has many aspects, so I'm looking for a concise summary with references to explanations for each of those crucial aspects.

From the Economist:

  The SNB suddenly dropped the cap last week for several reasons. First, many Swiss are angry that the SNB has built up such large foreign-exchange reserves. Printing all those francs, they say, will eventually lead to hyperinflation. Those fears are probably unfounded: Swiss inflation is too low, not too high. But it is a hot political issue. In November there was a referendum which, had it passed, would have made it difficult for the SNB to increase its reserves. Second, the SNB risked irritating its critics even more, thanks to something that is happening this Thursday: many expect the European Central Bank to introduce “quantitative easing”. This entails the creation of money to buy the government debt of euro-zone countries. That will push down the value of the euro, which might have required the SNB to print lots more francs to maintain the cap. But there is also a third reason behind the SNB’s decision. During 2014 the euro depreciated against other major currencies. As a result, the franc (being pegged to the euro) has depreciated too: in 2014 it lost about 12% of its value against the dollar and 10% against the rupee (though it appreciated against both currencies following the SNB's decision). A cheaper franc boosts exports to America and India, which together make up about 20% of Swiss exports. If the Swiss franc is not so overvalued, the SNB argues, then it has no reason to continue trying to weaken it.

I have only several things to add to this commentary, all contained here.


Business cycle stylized facts are not country specific? Business cycle and macroeconomic stylized facts are country-specific, right? But in my macro classes, they teach it as something universally true whiles using US data...:). For example, after an RBC model, we only check for US data where $\frac{C_t}{Y_t} < 1$, end of story...:). How about other countries where $\frac{C_t}{Y_t} > 1$? The model is not good for that country?

There are several aspects to this question:
General aspects:

What determines a stylized fact?
Are stylized facts country-specific?
Why do economic studies often use the United States as a benchmark?

Specific aspects:

What do we know about the macroeconomic quantity $C_t / Y_t$?

It will benefit our understanding to consider each of these questions separately.
1.
The term "stylized fact" is not very precisely defined in economics, and it is understandable that it confuses you. In my personal experience, most economists have an intuitive understanding of what constitutes a stylized fact, but would struggle to provide a precise definition when prompted.
Intuitively, stylized facts are the closest economics, and social science at large, can come to empirical predictions arising from scientific laws. The study of what precisely determines a scientific law constitutes a whole branch of philosophy of science by itself. Let me, for the sake of discussion, simplistically assume that scientific laws are statements about causation.
Since most social systems, such as the economy, are extremely complex, it is practically impossible to observe phenomena under controlled conditions. Any causal force that can be observed is usually counteracted by myriad other causal forces, and scientific experimentation aimed to isolate a single causal effect is usually ethically and practically out of bounds. As a result, it is exceedingly difficult to falsify economic "laws" (insofar as these exist) by way of precise empirical predictions, as is for example common practice in physics (see, e.g., the long history of attempts to falsify Einstein's general relativity theory). Stylized facts constitute a less precise measuring stick for the falsification of economic laws: they are robust empirical regularities observed in a certain social system, which any proposed causal mechanisms (in the context of a scientific law or model) characterizing that system have to generate as a baseline. In your case, if I write down a model of the macroeconomy, I typically want it to generate $C_t / Y_t < 1$, unless I have a good reason otherwise -- that is, unless I can posit a causal mechanism that explains my deviation from that stylized fact.
The linked philosophy of science article provides a more formal discussion of the use of the term stylized fact in the social sciences. The author distinguishes four features of stylized facts, which they say involve (p.607-608):

Claims about the kinds of things that exist in the world: the categories that divide the world into an analytical ontology.
A way to operationalize those categories for analysis, usually in relation to a particular dataset or statistical classification (e.g. $C_t / Y_t$ only has meaning in the context of national accounting).
A claim of non-robust dependence ("Y usually depends on X", "$C_t / Y_t$ is usually smaller than 1").
A normative claim that the "particular regularities identified are the ones most important to study and are preferable to other potential characterizations of the evidence" (p.608). (A silly example: $C_t / Y_t$, the consumption share of output, tells us something important about an economy; the share of the value of cat food in output less so.)

2
Stylized facts are not supposed to be country-specific. Rather, they are meant to capture some general feature of an arbitrary economic system. However, due to the fact that economics often focuses on select sets of countries, and that stylized facts typically hold generally but not universally, it may appear differently. This brings us to...
3
It is true economics has traditionally focused excessively on advanced economies, and in particular the US. This has shifted dramatically in recent years, though. Still, a lot of economics teaching has not caught up with this shift in research focus, and it may certainly seem strange to students in countries / economies that are very different from the US that it is so often used as a benchmark.
This recent article in The Economist discusses this question in detail, and provides several reasons for economics's focus on particular countries:

The size of a country's economy accounts for 80% of variation in research attention.
The quality and availability of data.
A country's use of the English language.
Professional incentives: papers focusing on the United States have been published more frequently in top journals.

4
Now, to get to your question about $C_t / Y_t$. Though your other questions were interesting and important in their own right, they didn't really need to be answered: simple logical thinking could have already gotten you a long way.
$Y_t$ denotes income, $C_t$ consumption. Typically, it is not possible for a country to consume more than it earns, except through borrowing or transfers. Hence, as a baseline, $C_t / Y_t < 1$. This is actually a very unsurprising empirical regularity, and not that interesting of its own right (as a matter of fact, the deviations from the baseline are much more interesting). See the data here and here (depending on your definition of C_t) to discover which countries constitute the exception to the rule.


Revealed preference when there is no substitution effect I've started reading about revealed preference, and in particular, how it works in the context of price changes.
I understand that revealed preference can show us the general direction of the substitution effect. However, can revealed preference help us if there is no substitution effect present? As in, is there any information we can discern from it about the final choice?
For example, consider a scenario where Px and Py increase by the same amount. (I've drawn this diagram below). If they initially chose Bundle X1, then they would choose it again on the "hypothetical" bundle after income is compensated for.

However, what does this mean for the final choice, if anything? Would they consume at X = 4 also? Or can we not know anything else, as the 0 substitution effect means we can't know the direction of the final choice?
My apologies if I've missed something obvious, but it's a problem that's been bugging me for a while!

The short answer is no. Reveal preference only determines which combination of goods is preferred, it does not affect in any way the allocation of goods, which has more to do with the functional form of the utility function, the types of goods and the types of changes made.

For example, the 2 budget lines represent different budget sets when there is a price change in either good. The weak axiom of revealed preference (WARP) posits that $x^i$ is preferred to $x^j$ insofar as good 1 is revealed preferred to good 2. It does not give any information on the allocation of the consumption bundle, however. If there is a simultaneous change in both the price of good 1 and 2 so that there exists another budget line that is parallel with $p^i$, the allocation of good in this case has more to do with the optimization of utility.
I have attempted to derive the change here but I am not entirely sure if this is the correct way to do it or whether I have done all the steps correctly so take it with a pinch of salt.
Assuming the consumer is maximizing his/her utility function, which is standard in consumer theory, we want to find the highest point tangent to the budget line. The slope of the indifference curve, after the price adjustment, depends on the types of adjustment made, the type of goods, and the functional form of the utility function. As I understand it, you are asking about the case of relative price change (since the slope of the budget line remains the same). In this case, the total change in each type of goods can be derived from the Slutsky equation
\begin{align}
{d x_A\over dp} &= {\partial x_A\over \partial p_A} + {\partial x_A \over \partial p_B}\\
 &= \underbrace{\partial h_A\over \partial p_A}_{\text{substitution effect good A (?)}}-\underbrace{{\partial x_A\over \partial m}x_A}_{\text{income effect good A (?)}} + \underbrace{\partial h_A\over \partial p_B}_{\text{substitution effect good B (?)}}-\underbrace{{\partial x_A\over \partial m}x_B}_{\text{income effect good B (?)}}\\
\end{align}
where $x_i,h_i,p_i$ and $m$ respectively denote the Marshallian demand, Hicksian demand, the price of good $i$, and the total budget. The ?'s in the parentheses are the sign of the term. As in the case here where the types of goods are ambiguous, I can say little about the direction of change.
In the case of normal good (both are normal goods) and standard Cobb-Douglas utility, the substitution effect of own good (A to A) is negative, the income effect of own good (A to A) is negative, the substitution effect of the other good (A to B) is positive while the income effect of the other good (A to B) is negative. The substitution effect of own good (A to A) and of the other good (A to B) cancel out each other, leaving only the negative terms. This generally translates into a lower Marshallian demand for good A, and similarly for good B. The total change is then, ${dx_A\over dp}=-{\partial x_A\over \partial m}(x_A+x_B)$. In levels,
$$\Delta x_A=-{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\\
\Delta x_B=-{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)
$$
Taking the ratio,
$${x_A-\Delta x_A\over x_B-\Delta x_B}={x_A+{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\over x_B+{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)}$$
This will remain unchanged if and only if $\Delta x_A/\Delta x_B=x_A/x_B$, which simplifies to ${\partial x_A\over \partial m}/{\partial x_B\over \partial m}={x_A\over x_B}$. This holds true when with the CES form of Cobb-Douglas utility wherein the coefficient of good A and B are equal to 0.5.


Income gap in Ramsey model Ramsey model with income tax

Question:

For this question you are required to setup a Ransey-Cass-Koopmans model economy with a Cobb-Douglas constant returns to scale production technology, with inputs of capital (K) and labour (L). Setup the model with a constant depreciation rate, constant rate of labour- augmenting technological progress and constant rate of population growth. Also incorporate a government that finances its spending each period via a proportional income tax system. Assume that both labour and capital income are taxed at the same rate – say τ. You can assume that government spending is completely wasteful; or you can make some other assumption about how the tax revenue is used, however do not allow government spending to generate endogenous growth.

I am asking only this past

    Use the model to explain the income gap that exists between wealthy and poor countries.

———

What I did:

I solved this model with hamiltonian. And I got the following results (I posted picture of my results, because this is only extra information that helps to solve question.)

And the phase diagram

And at steady state, saving rate is

$$s^*={(delta +n + x)(1-τ)a\over p+delta+θx}$$

x is growth rath of technology. (Productivity growth rate)

Delta depration rate. and production function $Y=K^a(AL)^{1-a}$

My interpretation:

Economic growth is measured only by x. So only x and A affect the income.

Assume that we have $k_i<k^*$ and every parameters are constant for two countries except for x . Poor country has lower x.

When x increases, saving decreases, MPK decreases, so capital rate of return decreases, capital stock increases. And eventually they have the same growth rate. (This is substitution effect)

But I could not know how to explain this question better. Any help is appreciated. Really I need a help too much!

I think your solution is on the right direction. But allow me to make a few observations.

The question talked about income gap and wanted you to use a Ramsey model, with exogenous technology growth, and no endogenous growth mechanism through public spending. Given these, I would think that the focus should not entirely be on the growth rate of technology. Using your example, if the two countries have the exact same set of parameters, then the country with higher x will have lower long run $k^*$.
See the starting point of my (1) and your analysis: we assumed that the countries have identical parameters. For Rich(R) and Poor(P) countries, this is not a reasonable assumption. Furthermore, I am not sure if assuming that technological progress in R is faster than P is correct.

Might I suggest the following. Pick a parameter (say the tax rate). Make the tax rate higher in one of the countries. Which automatically divides the countries into R and P. Then even if you do not have different x, the income gap will remain.

I assumed the following parameter values (L=Low, H=High):

$\delta = 0.1; n = 0.01; \rho = 0.05; \alpha = 0.3; \theta = 1/0.3$

$t_H = 0.2; t_L = 0.1; x_H = 0.5; x_L = 0.2;$

My Utility function is $\frac{C^{1-\theta}}{1-\theta}$

The corresponding long run $k$ values are:

(i) $ k^*_{t_H x_H} = 0.583336 $

(ii) $k^*_{t_H x_L} = 0.943351  $

(iii) $k^*_{t_L x_H} = 0.69023 $

(iv) $k^*_{t_L x_L} = 1.11622 $

Now you can pick whichever pair among (i)-(iv) as your R and P. Suppose you say (ii) is P and (iv) is R. Then say both countries have the exact tech improvement and x rises from $x_L $ to $x_H$. (ii) moves to 0.58 and (iv) moves to 0.69. The gap (R-P) before the shock was 0.172869. After the shock income gap (R-P) is 0.106894. The gap went down.

My code (Mathematica):

kdot = (1 - t)*k^a - c - (n + d + x)*k;

Ham = c^(1 - \[Theta])/(
   1 - \[Theta]) + \[Lambda]*((1 - t)*k^a - c - (n + d + x)*k);

(*Costate1=\[Lambda]dot/\[Lambda]-(\[Rho]-n*(1-\[Theta])-x*(1-\[Theta]\
))==-D[Ham,k];*)

FOCc1 = D[Ham, c] == 0;

FOCc2 = FOCc1[[1]] + \[Lambda] == \[Lambda];

FOCc3 = FOCc2 /. {c -> c[t], \[Lambda] -> \[Lambda][t]};

FOCc4 = D[FOCc3, t][[1]]/c[t]^-\[Theta] ==
   D[FOCc3, t][[2]]/c[t]^-\[Theta];

FOCc5 = FOCc4 //. {c[t]^\[Theta] -> \[Lambda][t]^-1} /. {\[Lambda]'[
      t]/\[Lambda][t] -> \[Lambda]dot/\[Lambda]};

Euler = -((\[Theta] Derivative[1][c][t])/
     c[t]) - (\[Rho] - n*(1 - \[Theta]) - x*(1 - \[Theta])) == -D[Ham,
       k]/\[Lambda];

Euler2 = chat /. Flatten[Solve[Euler /. {c'[t]/c[t] -> chat}, chat]];

kstar = k /. Flatten[Solve[Euler2 == 0, k]];

csol = c /. Flatten[Solve[kdot == 0, c]];

cstar = csol /. {k -> kstar};

(*Parameters*)

d = 0.1; n = 0.01; \[Rho] = 0.05; a = 0.3; \[Theta] = 1/0.3;

Ht = 0.2; Lt = 0.1; Hx = 0.05; Lx = 0.02;

kstarHtauHx = kstar /. {t -> Ht, x -> Hx}
kstarHtauLx = kstar /. {t -> Ht, x -> Lx}
kstarLtauHx = kstar /. {t -> Lt, x -> Hx}
kstarLtauLx = kstar /. {t -> Lt, x -> Lx}


Does the WTO monitor the trade of players in association football? Could they? So I understand the WTO is able to settle trade disputes. If the German Bundesliga said English players had a 1% luxury fee that went towards operation costs (A \$100,000 transfer would cost \$101,000, with \$1,000 going to the Bundesliga), would the WTO be able to step in if England went to them? What if the German government were to impose a 1% tax instead? If these aren't under the purview of the WTO, could they be in a decade?

No, for several reasons.

WTO deals with trade disputes between nations when one nation creates legislature that breaks WTO rules or some pre-existing trade agreement. For example, one of the most important WTO rules is the most favored nation principle which says that (WTO, 2020):

countries cannot normally discriminate between their trading partners. Grant someone a special favour (such as a lower customs duty rate for one of their products) and you have to do the same for all other WTO members.

But again this all applies to countries not companies. Companies can charge different prices to people in different countries if they want to.

Trading players would likely not even qualify as a trade. If one company releases player from a contract and that player then signs contract with another company that is no more trade in economic terms than when your company fires you and other company in another country hires you. For this to qualify as a trade the players would still have to be employees of the UK company and just provide service to the German company  (when this would become trade in services).

Even if there would be trade in services and $1\%$ government imposed tax this would not be violation of WTO rules as long as all countries Germany trades with are subject to this $1\%$ tax or if it would be part of some of the WTO exceptions (for example, WTO allows trading blocks to set among themselves lower barriers than just under pure most favored nation principle).

If these aren't under the purview of the WTO, could they be in a decade?

I doubt this would ever be under the purview of the WTO given that WTO's objective is to promote free trade by removing tariffs and non-tariff barriers to trade and trade cooperation amongst nations. WTO's objective is not to regulate/police trade among private firms. In addition depending on how the 'trading of players' is set up it does not even qualifies as a trade (e.g. if players change employer).


Implicit Differentiation & Profit Function Here is the question:

A firm produces one output commodity. The cost of producing and selling $x$ units and spending $y$ dollars on advertising is $C = cx + y + d$. The resulting quantity demanded is given by $x = \gamma ap + b + R(y)$ where $p$ is the price per unit. We assume that $R(0) = 0, R′(y) > 0$ and $R′′(y) < 0$. The constants $a, b, c, d$ are all positive.

Determine the profit function from selling $x$ units and spending $y$ dollars on advertising; call it $\pi(x, y)$.
Write and solve the following system of equations:
$$\frac{\partial \pi(x^*,y^*)}{\partial x} = 0$$
$$\frac{\partial \pi(x^*,y^*)}{\partial y} = 0$$

where $x^* > 0$ and $y^* > 0$ represent the number of units sold and amount spent on advertisement that maximize profits. In this case, solving the system means finding one equation that only depends on $y$, the function $R(\cdot)$ (and its derivative), and the parameters of the problem $(a, b,...)$ - in other words an equation where $x^*$ has disappeared.

The equation found in the previous question defines $y^*$ implicitly as function of $a, b,$ and $c$. Find $\frac{\partial y}{\partial b}$ by implicit differentiation.

Embarrassingly, my problem seems to be with the profit function. If we get a revenue function by $p \cdot x$ we are left only with an $x$ in the cost function, I cannot figure out how to get a profit function which would result in $y^*$ as a function of $b$. Anyone know what I'm doing wrong?

I will only set up the first part of your question. See if you can move on from there.

If you try and set revenue as $p \cdot x$ immediately, you will get

$$p \cdot x = p \cdot (\gamma ap + b + R(y))$$

which as you note means $x$ will not be in this term. Profit could technically still be expressed in terms of $x, y$ in this case, but...

$$\pi(x, y) = p \cdot (\gamma ap + b + R(y)) - (cx + y + d)$$

is awkward since $x$ can be expressed in terms of $p$ and vice versa, so why would we have both in our equation?

So instead, solve for the inverse demand function:

$$x = \gamma ap + b + R(y) \implies \boxed{p = \frac{x - b - R(y)}{\gamma a}}$$

So you can solve for $p \cdot x$, but instead of substituting for $x$, you substitute for $p$. Once you solve for your first order conditions, you should be able to solve for $y$ as a function of variables including $b$.

Hope this helps.