How to justify the treatment and control groups for Difference-In-Difference with staggered implementation of laws? A. Background:
Dong, 2019 and Dasgupta, 2019 used the same way to generate the treatment and control groups because they all learn about the impact of the same laws on different dependent variables. I retrieved the information on how Dasgupta, 2019, p. 2596-2597 create their sample as below:
The baseline identification:
$Y_{it}$ = $\alpha$ + $\beta$ $(Leniency Law)_{kt}$ + $\delta$$X_{ikt}$ + $\theta$$_t$ + $\gamma$$_i$ +$\epsilon$$_{it}$ (1)
where $i$, $k$, and $t$ index firms, countries, and years respectively. $X_{ikt}$ is a vector of the different firm, country, and industry control, while $\gamma$ and $\theta$ are firm and year fixed effects.
The variable of interest here is $(Leniency Law)_{kt}$. Dasgupta, 2019, p. 2597 documented that this variable equals 0 before the passage of the leniency law in country $k$, and 1 afterward. They follow a standard difference-in-difference approach with staggered implementation of laws (Bertrand, 2003) to construct the treatment and control group.
B. Author's description:
Dasgupta, 2019, p. 2597 said that

the treated group comprises all firms that are headquartered in
countries that have passed a leniency law by year t. The control
group comprises firms in countries that never adopted a
leniency law in our sample period and firms headquartered in countries
that adopted a leniency law at some later point of time.

C. Discussion:

Whether the control group includes firms in countries that implemented this law at some earlier point in time? My understanding is: for example, from Table 2, Dasgupta, 2019, p. 2599, Korea passed the laws in 1997, therefore, the control group including all firms in all countries not implementing this law from 1995 to 2002, including the US.
From my point of view, it should be the case because based on this identification, firms in the US will not be affected by the laws passed in Korea based on this identification strategy so these US firms from 1995-1996 and 1998-2002 can be control observations for Korea. But if this is the case, why do they need to write down "adopted a leniency law at some later point of time"? Because in Korea case, the word "our sample period" means "1995-2002" already.

Does it mean that only $(Leniency Law)_{kt}$ of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before the laws being passed receive a value of 0?

Imagining two cases: (1) Both Germany and Brazil implemented this law in the year 2000 and (2) Only Germany implemented this law in 2000. So whether the numbers of observations of the control groups for these two cases are identical?

Australia and Belgium passed the law in 2003 and 2004, accordingly. So, firms in Malaysia only exist once in the control sample? I mean, whether one observation can be the control for many treatments in this DiD setting? (meaning no duplicate control observation).

Above is how he grouped the control and treatment based on his identification, but on page 2600, Dasgupta, 2019, section 3.1 said that "control firms are all firms in the same industry in countries that had not passed a leniency law in the 7 years surrounding the event date". This identification confused me because now we have one more dimension about the industry. I am wondering why they did not put it right in the baseline identification but instead in results section? What is the reason behind that?

[W]hy do they need to write down "adopted a leniency law at some later point of time"? Because in Korea case, the word "our sample period" means "1995-2002" already.

Assuming Korea is the early-adopter country, then all countries theretofore untreated before 1997 may serve as a counterfactual. This includes the countries never adopting a leniency law and those with impending treatment adoption periods.

Does it mean that only $()_{}$ of the treatment group after the laws being passed received a value 1? Other than that, the control group and the treatment group before laws being passed receive a value of 0?

Correct.
The binary treatment variable in this more general setting is not the same variable as in the 'classical' difference-in-differences case. Suppose a leniency law is espoused by all firms within treated countries in the year 2000. In this setting you could write this equation more simply as the interaction between a treatment-control dummy and a post-treatment indicator equal to 1 after the law goes into effect in both groups, 0 otherwise. However, once we move away from this setting and the roll out of treatment is staggered or even switching 'on' and 'off' over time, then the "post-treatment" variable is no longer well-defined. To proceed, we must use the 'generalized' difference-in-differences estimator which defines the product term in a different way.
In the equation you reference, the interaction term is implicit in the coding of $LL_{kt}$ (i.e., leniency law). It is equal to 1 in all firms that are headquartered in countries that have passed a leniency law by year $t$, 0 otherwise. Imagine you have a column of zeros. Input a value of 1 once the firms in country $k$ adopt by year $t$, 0 otherwise! Thus, any country never adopting a leniency law is left as 0 for the entire observation period.

Imagining two cases: (1) Both Germany and Brazil implemented this law in the year 2000 and (2) Only Germany implemented this law in 2000. So whether the numbers of observations of the control groups for these two cases are identical?

From the perspective of this estimator, the treatment variable $LL_{kt}$ makes no distinction between treatment/control groups. We only have countries $k$ switching on their policies by year $t$. Any country $k$ never adopting a leniency law is equal to 0 in all $t$.
Suppose a panel runs from 1995–2002. In example (A) both Germany and Brazil will 'turn on' (i.e., switch from 0 to 1) in the year 2000 and stay on. Thus, Brazil and Germany have 5 pre- and 3 post-treatment time periods. They have "identical" treatment histories. The countries never espousing the new law would serve as a counterfactual.
In scenario (B) only the within-group observations in Germany would switch on in the year 2000, while the within-group observations in Brazil would all equal 0. The countries never espousing the new law, which now includes Brazil, would serve as a counterfactual.

Australia and Belgium passed the law in 2003 and 2004, accordingly. So, firms in Malaysia only exist once in the control sample? I mean, whether one observation can be the control for many treatments in this DiD setting?

Yes.
Suppose Malaysia never adopts a leniency law—ever. Malaysian firms may serve as a counterfactual for Australian and Belgian firms up until the time periods in which they are treated. In other words, the Malaysian sector is a counterfactual for the early- and late-adopters before each of their respective exposure periods.
The data frame below shows how we code $LL_{kt}$ in practice. Note this is a little different since we only see country-year observations. I assume based upon my cursory review of the paper that leniency laws affect all firms within each country. If so, and the authors observe all firms before and after the laws goes into effect, then we can estimate the equation at the firm level or at the country level.
For simplicity, the fictitious data frame below observes three countries from 2000–2006. Germany espouses the law early; $LL_{kt}$ switches from 0 to 1 in 2003 and stays on. Australia adopts late; $LL_{kt}$ switches from 0 to 1 in 2005 and stays on. The Malaysian market never adopts a leniency law. Note how $LL_{kt}$ is 0 for the entire observation period. If we also observed firms $i$ over time within Malaysian industries, then all firms should be coded 0 in every firm-year period.
Again, the variable $LL_{kt}$ is not delineating a specific subset of treated or untreated firms/countries; rather, it simply 'turns on' (i.e., switches from 0 to 1) if a jurisdiction was treated and only during the periods $t$ when the law was actually in effect, 0 otherwise.
The following data frame should help with your intuition:
$$
\begin{array}{ccc}
country & year & LL_{kt} \\
\hline
 \text{Malaysia} & 2000 & 0 \\
 \text{Malaysia} & 2001 & 0 \\
 \text{Malaysia} & 2002 & 0 \\
 \text{Malaysia} & 2003 & 0 \\
 \text{Malaysia} & 2004 & 0 \\
 \text{Malaysia} & 2005 & 0 \\
 \text{Malaysia} & 2006 & 0 \\
\hline
 \text{Germany} & 2000 & 0 \\
 \text{Germany} & 2001 & 0 \\
 \text{Germany} & 2002 & 0 \\
 \text{Germany} & 2003 & 1 \\
 \text{Germany} & 2004 & 1 \\
 \text{Germany} & 2005 & 1 \\
 \text{Germany} & 2006 & 1 \\
\hline
 \text{Australia} & 2000 & 0 \\
 \text{Australia} & 2001 & 0 \\
 \text{Australia} & 2002 & 0 \\
 \text{Australia} & 2003 & 0 \\
 \text{Australia} & 2004 & 0 \\
 \text{Australia} & 2005 & 1 \\
 \text{Australia} & 2006 & 1 \\
\end{array}
$$
Note, before Germany adopts a leniency statute in 2003, their counterfactual history is the Malaysian and Australian sectors. It is also worth highlighting that previously treated countries may also serve as counterfactuals for the late-adopter countries. In other words, when Australia adopts a leniency law later in 2005, the Malaysian firms (i.e., non-adopters) and the German firms (i.e., early-adopters) may approximate their counterfactual history. This estimator is actually averaging all possible 2x2 difference-in-differences estimates.
The downsides of this estimator may serve as a whole separate discussion. But let's address one concern briefly. For instance, the model assumes the absence of time-varying effects. For example, the effect of a leniency law on Germany's outcome trajectory is assumed to be instantaneous and constant. This assumption is tenuous and difficult to defend statistically. Note how we still assume a "common trend" across all groups before their exposure periods. Imagine a scenario where Germany's outcome trend was changing over time after 2003. Once Australia moves into the treatment condition in 2005, then Germany's outcome trajectory may be offset by their earlier exposure. When 'already-treated' countries are allowed to act as controls for 'soon-to-be-treated' countries, then changes in their treatment effects over time get subtracted from the difference-in-differences estimate. So while the two-way fixed effects estimator is yielding a weighted average of treatment effects across all groups and times—some of the weights may be negative. The negative weighting only arises in the presence of heterogeneous treatment effects. Peruse this working paper for more on this topic.

Above is how he grouped the control and treatment based on his identification, but on page 2600, Dasgupta, 2019, section 3.1 said that "control firms are all firms in the same industry in countries that had not passed a leniency law in the 7 years surrounding the event date". This identification confused me because now we have one more dimension about the industry. I am wondering why they did not put it right in the baseline identification but instead in results section? What is the reason behind that?

It is my understanding that the timing of leniency laws did not vary across industries, but I can't be sure until I give the paper a thorough read.
Usually authors may specify a base model and then expand upon it in their results section. For example, they may report lead/lag coefficients in tabular form but do not explicitly show their equation. The authors may also try a series of alternative specifications such as including industry-by-year effects even though this term was excluded from their main empirical equation.


What types of identification are there in Econometrics? Identification of a model is the simple property that when two likelihoods (for the same data), or Joint CDF (for the same random variables), if equal, then they must imply that the parameters must also be equal, i.e. we cannot have two different set of parameter values which fit data. (This is probably imprecise, sorry).

I'm wondering just how many types of identification there are. I've heard of weak, semi-strong, and strong... And what do they mean?

If possible, I would like the answer to also link to a bibliographical reference, where the authors would explain in some detail most of these concepts in a general setting.

Thanks in advance.

I think to answer this it is best to first go over definition of identification.

Following Stachurski (2016) identification or identifiability (I omit formal description since its also in that wikipedia article you provided in your comment):

  means that the parameter vector associated with unknown distribution can eventually be distinguised from the data.

Hence identification is more or less what people usually call estimation. For example, in OLS ($y= X\beta+e$) where the $\beta$ coefficient is:

$$\hat{\beta} = (X' X)^{-1}X'y$$

it can be proven that $\hat{\beta}$ can only be identified when the $X'X$ matrix is invertible otherwise $(X'X)^{-1}$ is not defined and you simply wont be able to calculate the $\beta$ or R or Python or Stata would give you error message, like for example where you have perfect multicolinearity.

Every model you can think of has some identification conditions - hence its not really appropriate to talk about types of identification, identification means the model can estimate the parameters and every model has its own conditions for identification of the parameters.

Consequently the process of estimating any coefficients is usually called identification strategy. If by asking about different types of identification you mean different models/ identificaiton strategies, a good-all-purpose textbook is Pesaran (2015). Verbeek (2008) is good intermediate introduction textbook, if you look only for time series analysis Hamilton (1994) is classic although bit dated now. For treatment evaluation I recommend Angrist & Pischke (2008). I wont go over all possible types of models as nowadays you have such a variety its impossible to make exhaustive list in answer on SE.

Next when we talk about weak or strong identification we mean by weak identification that the estimators and test statistics are not well approximated by their standard asymptotic limits because of limited information in the data. Basically, the point is that identification in itself does not guarantee the coefficients are consistent and unbiased only that they can be estimated and usually the term weak identification is applied when the aforementioned are not guaranteed and strong identification when they are. Also I mostly seen these terms being applied to the IV, GMM or other instrument using estimator where weak identification is often used as a synonym for the fact that the first stage or instruments are weak. But I dont see people denoting weak or strong as a type so I dont know if this is what you mean by that. To see when the coefficient estimates are consistent and unbiased you can again for each separate model see in the literature I recommended above.

References:

Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press.

Hamilton, J. D. (1994). Time series analysis (Vol. 2, pp. 690-696). New Jersey: Princeton.

Pesaran, M. H. (2015). Time series and panel data econometrics. Oxford University Press.

Stachurski, J. (2016). A primer in econometric theory. Mit Press.

Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.


Generalizing demand for perfect substitutes utility function I have the utility function:
$U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero)
$\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ where $K<\infty$
Where the problem is:
$\max_{x_1,...,x_n}U\;\;\;\;\;\;\;\;c.t.\;\;\sum_{i=1}^{n}p_ix_i=m\;\;\;\;\;$ (With positive prices and income)
Which is the general case for perfect substitutes preferences for finding Walrasian-Marshallian demands. In the particular case $n=2$ the solution could be found geometrically, and the optimum turns out to be one of three cases (where two of them are corner solutions) depending on the relation between prices and parameters $a$. When $p_1/p_2>a_1/a_2$ solution is $(x_1^*=0, x_2^*=m/p_2)$, and the opposite case she spends all income in $x_1$, and the third case is a continuum of feasible points that satisfy the budget constraint.
So, here's the question: Is there a way to obtain a generalized Walrasian-Marshallian demand function (I imagine it would be by parts) for $n$-goods in this case, just like in Cobb-Douglas preferences that it turns out to be $x_i^m=\frac{a_im}{p_i\sum_{i=1}^{n}a_i}$ (please prove).

Let $b_i := \frac{a_i}{p_i}$, now choose the good that maximize $b_i$, let it be $b_j$ i.e. the $j$th good. Optimal choice is $x_j=m/p_j$, and $x_i = 0$ for $i\neq j$.


How to tell if F-Stat is significant? And at which significance level? This question has been posted on

http://forums.eviews.com/viewtopic.php?f=18&t=12403

But no reply.

Question:
How do I tell if the F-stat is significant at 1%, 5%, or 10% level?
Or not significant?

You get:

F-statistic          24.60231
Prob(F-statistic)    0.000000

You observe that the p-value is 0.00, so this means that the F-statistic is significant at at 99+%, and also obviously at 95% and 90%.


Consumer choice: How could a consumer weakly prefer A to B *and* B to A? From Perloff, Microecomics, 6th edition (Pg. 75):

A consumer chooses between bundles of goods by ranking them as to the pleasure
the consumer gets from consuming each. We summarize a consumer’s ranking using
a preference relation, such as the consumer weakly prefers Bundle a to Bundle b, if the consumer likes Bundle a at least as much as Bundle b.

Given this weak preference relation, we can derive two other relations. If the consumer weakly prefers Bundle a to b, but the consumer does not weakly prefer
b to a, then we say that the consumer strictly prefers a to b—would definitely
choose a rather than b if given a choice—which we write as a b.

If the consumer weakly prefers a to b and weakly prefers b to a —— then we say that the consumer is indifferent between the bundles, or likes the
two bundles equally.

My question is: When you have declared that a consumer weakly prefers A to B, how could you then argue that a consumer could also prefer (weakly) B to A? Aren't they mutually exclusive?

No, weak orderings are not mutually exclusive, but strict orderings are, exactly as for the usual comparison of real numbers: you can have $x \geq y$ and $y \geq x$ (if $x=y$), but $x>y$ and $y>x$ are incompatible with each other.

The weak orderings $A \succeq B$ and $B \succeq A$ mean that the decision-maker is indifferent between $A$ and $B$: he might therefore choose any of these options indifferently.


Why do banks and stock exchanges close/have fixed hours? Banks and stock exchanges all around the world seem to adhere to quite a strict regime of 9-5, Monday - Friday, in the countries local time.

This can have the effect of where an event that happens overnight, causes the market to open at an immediate dip as happened recently on Wall Street in response to economic data from China.

The question is - in today's always-on, all connected world, why does this system still remain? I would have thought that for industry that is all about making money, they'd be the first to be always on.

The answer is simple and it is due to the conduct of business and conditions relating to the early days of exchanges and banks. Then, there was no interest in trading  or providing liquidity to companies 24 hours a day and there was no coordination with the other parts of the world. Today, it is more like preserving the tradition, and just look at how NYSE opens at every morning, the good old bell is there to kick start the business. Today, it may make sense to have 24/7 financial services provision as we are living in the age of internet.

OTC FOREX trading is thriving 24/7. As long as there are willing traders/market makers to remain on the floor on different shifts during working hours.


When do firms pursue the cost minimization objective instead of profit maximization? Are there any intuitive/practical examples/reasons? I understand that profit maximization implies cost minimization, that is, if a firm is maximizing profits, it will definitely minimize costs.

However, under what conditions does cost minimization become an objective in itself?

Firms can be said to pursue cost minimization when they do not have control over demand of their product. In the short run, this implies also that unit prices are fixed.

A standard example here is a Utility, say for electricity. Except in extreme situations of very high demand that reaches the network capacity, consumers consume electricity on demand for a given unit price or price schedule.  So "quantity produced" is not determined by the firm, and so the Cost-minimization problem is the most appropriate (minimize costs given output).


Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization? I read that if we have quasilinear utility for all consumers, then any pareto optimal allocation maximizes the sum of utility levels of all consumers. That is:

$\textbf{What we know:}$
$$1)\quad u^i(m^i,x^i)=m^i+\phi^i(x^i)\; \quad \forall i=1,...,I$$
$$2)\quad\phi^i(\;)\;\text{is continous and strictly increasing (but not necessarily differentiable)}$$
$$3)\quad \text{An allocation,}\,x\, \text{satisfies}\;\neg\,\exists\,\hat{x}\; s.t. \;\hat{m}^i+\phi^i(\hat{x}^i)\geq m^i+\phi(x^i)\;\forall i$$
$$\text{and} \quad \hat{m}^i+\phi^i(\hat{x}^i)> m^i+\phi(x^i)\,\text{for some}\,i$$

$\textbf{What to show:}$
$$x\;\text{solves}\;max\sum_{i=1}^Im^i+\phi^i(x^i)$$

Can anyone provide a proof of this? Any help would be greatly appreciated!

$\textbf{Edit:}\,$I don't know if this is the right path, but by the strict increasing property of $\phi(\,)$, preferences satisfy local non-satiation, which implies they satisfy the first welfare theorem. Now, If I could figure out whether all pareto optimal allocations are competitive equilibria with quasilinear utility, I may be on to something!

Edit: Edge cases suck; see comments. See also MWG Chapter 10 section C, D.

Suppose $(\vec x^*, \vec m^*)$ solves

$$\max \sum^I_{i=1} m_i + \phi_i(x_i)$$

but is not Pareto optimal.

$$\begin{align}
\implies \exists \ (x_i', m_i') \quad \text{s.t.} \quad & u_i(x_i', m_i') \geq u_i(x_i^*, m_i^*) \quad \forall \ i = 1,\cdots,I \\
& u_i(x_i', m_i') > u_i(x_i^*, m_i^*) \quad \text{for some} \ i
\end{align}$$

$$\implies \sum^I_{i=1} m'_i + \phi_i(x'_i) > \sum^I_{i=1} m^*_i + \phi_i(x^*_i)$$

which is a contradiction. If we have a solution to the utility maximization problem, it must be Pareto optimal.

(Note that this comes form continuous and increasing properties of $\phi(\cdot)$)

Suppose $(\vec x^*, \vec m^*)$ is a feasible Pareto optimal allocation, but does not solve

$$\max \sum^I_{i=1} m_i + \phi_i(x_i)$$

Because we treat $m_i$ as numeraire and $\phi_i(\cdot)$ is strictly increasing, we know $u_i(\cdot)$ is locally non-satiated. The Pareto allocation should be just feasible.

$$\exists \ (x_i', m_i') \quad \text{s.t.} \quad \sum^I_{i=1} m'_i + \phi_i(x'_i) > \sum^I_{i=1} m^*_i + \phi_i(x^*_i)\\
\implies \boxed{ \sum^I_{i=1} \phi_i(x'_i) > \sum^I_{i=1} \phi_i(x^*_i)}$$

If this is true because this alternative allocation simply gives an individual more of $x$, for all else equal, then the alternative allocation is infeasible. So we'd have a contradiction.

If this is true because in the alternative allocation, someone else is allocated more $x$ and just one other person is allocated less, then the original allocation would not be Pareto optimal. Suppose it was. If you took the original allocation and shifted $x$ in the way of the new allocation, then you would need a corresponding trade in the numeraire good, $m$, to keep whoever is losing $x$ at least at the same utility level. But trades in just the numeraire good can never change summed aggregate utility. From the original allocation, if you can trade $m$ for $x$ and make someone better off without hurting anyone, you weren't at a Pareto optimum, and if you can't trade $m$ for $x$ to make someone better off, you can't increase summed aggregate utility, which means the original allocation was a solution to the maximization problem.

This logic applies no matter how you rearrange $x$ between multiple people.

$\square$


Gali and van Rens: The Vanishing Procyclicality of Labor Productivity Let's see how questions like the following are taken here.

Gali and van Rens, 2014 show that empirically, the correlation between $Y, Y/L$ has been declining over time. In fact, it only was a "real thing" when the RBC theory was formed, and is now not statistically  different from zero.

Is that an end to RBC theory as we know it? And by RBC I mean anything that sets driven by a TFP shock / uses TFP shocks to explain empirical phenomena: Standard neoclassical models for sure, even Neokeynesian models rely (to a lesser extend) on these. Also the whole Diamonds-Mortensen-Pissarides literature, where the surplus of a match comes from labor productivity, seems to be affected.

Alternatively, one could argue that that the correlation has become smaller, but amplification mechanism are larger now. Or attack the measurement of Gali and van Rens (but their results seem to be pretty robust).

What are potential views on this, what arguments does the literature emphasize?

A napkin theoretical exploration of the issue, to find possible directions to look for explanations, could go like this:

According to the paper, the volatility of output and productivity has fallen. So for the correlation to have gone down, the covariance must have fell steeply.
Modeling output as

$$Y = AK^a(h\cdot L)^{1-a}$$

where $h$ is the "effort per unit" variable, we have

$${\rm Cov}(Y, Y/L) = E(Y^2/L) - E(Y)E(Y/L)$$

Now

$$Y^2/L =   A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}$$

$$Y/L =   A(K/L)^{a}\cdot h^{1-a}$$

So

$${\rm Cov}(Y, Y/L) = E\left[A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}\right] -E\left[A(K/L)^{a}\cdot L\cdot h^{1-a}\right]E\left[A(K/L)^{a}\cdot h^{1-a}\right]$$

We observe ${\rm Cov}(Y, Y/L) \rightarrow 0$. Now, I am in the mood of assuming that the capital-labor ratio is essentially a constant, as is $A$. More over, that the effort-per-unit variable is independent of the amount of labor hours. Under this scenario, we have

$${\rm Cov}(Y, Y/L) = A^2(K/L)^{2a} \cdot E(L)\cdot \big[ E\left[(h^{1-a})^2\right] -\left[E(h^{1-a})\right]^2\big]$$

$$\implies {\rm Cov}(Y, Y/L) \rightarrow 0 \implies {\rm Var}(h^{1-a}) \rightarrow 0$$

So here, the effort variable should be approximately a constant too.

--

In a more agnostic approach, using lower case letters to denote logarithms, we would have

$${\rm Cov}(y, y-\ell) = Var(y) - {\rm Cov}(y, \ell)$$

If what we observe is

$${\rm Cov}(y, y-\ell) \approx 0 \implies  Var(y) \approx {\rm Cov}(y, \ell)$$

But this means that in a least-squares regression of $\ell$ on a constant $\gamma_0$ and $y$
$$\ell = \gamma_0 + \gamma_1y + u$$
we would obtain $\gamma_1 =1$ and the relation

$$\hat \ell = \hat \gamma_0 + y \implies \approx Y = e^{-\gamma_0}L$$

(with $\gamma_0 <0$ most probably). This is consistent with the previous approach since the constant $e^{-\gamma_0}$, may be seen as containing the $A(K/L)^ah^{1-a}$ terms, which were assumed or emerged as constants for the result of zero-covariance to occur. Here, there is more flexibility since we only require that their product is constant, not each one separately.


Pure Nash Equilibria 3 players game I'm trying to solve this pure-strategy Nash equilibria of this game below:

I highlighted the best pay off for player 1 and 2. But I don't get it when it comes to player 3. The correct answer is (A)

I tried to solve it as a gaming tree. But still I don't get it.

How shall I think?

Three-player games are notoriously tricky to analyze. The ideal way to display them would be a three-dimensional array of cells, each containing three payoffs. But this is difficult to write down on two-dimensional paper. So typically an $n\times m\times l$-game is displayed as $l$ different $n\times m$-matrices. In your example you might think about it in this way:

Player 1 chooses the row (upper row, $A$, or lower row, $B$)
Player 2 chooses the column (left column, $A$, or right column, $B$)
Player 3 chooses the matrix (upper matrix, $A$, or lower matrix, $B$)

Therefore, to highlight the best payoff of player 3, for each of the 4 choices of players 1 and 2 (for each of the 4 cells in the matrices) you have to compare the player-3-payoffs between the upper and the lower matrix. So in the $(A,A)$-cell you highlight the $70$ in the upper matrix, since it is greater than the $60$ in the lower matrix. For the $(A,B)$-cell you highlight $23$, which is greater than $0$, and so on. In the end, the $(A,A)$-cell of the upper matrix (corresponding to the choice of $A$ by player 3) will have all three payoffs highlighted, and the same will be true for the $(B,B)$-cell of the upper matrix (again corresponding to the choice of $A$ by player 3). So the pure NE are $(A,A,A)$ and $(B,B,A)$.


How does economics cope with production processes that have large lumpy investment requirements? If seems that all equations (e.g. Price elasticity of supply ) assume that there is a marginal cost of production that is nicely related to the level of output.  But clearly you can’t buy half a new machine.

Also if you buy the new machine then the price decreased you can’t afford to pay the interest on the loan, so manager loses his job with the company closing down, compared to the manager getting a small incentive for the increase in production.

Therefore how is "real life" coped with!

Indeed, a lot of the marginality conditions cannot be used directly in the presence of discrete choice levels, or fixed costs of adjustment. There are many: number of children, working or quitting, number of cars, paying the fixed cost of adjusting capital, consuming or selling durable goods, number of hours worked, workers hired, units produced, delivery trips, etc.

Marginal conditions are an approximation to reality. Economists would argue that for a plant that hires 1000 workers, we can think of the decision of hiring the 1001 worker as best though of as adding a marginal until of labor.

There are a wealth of studies into these topics: Scarf's K convexity and the S-s inventory replenishing policies, Caballero and Engel's Fixed Costs of Adjustment of Capital, etc.

A book that tries to unify some of these ideas is "The economics of inaction".

From a dynamic programming point of view, there are two useful intuitions that might help: a) In the presence of fixed costs, the firm's policy function is discontinuous and the value function is continuous and convex, but not smooth; b) the problems with fixed costs are best split into two: do I act? If I act, where do I go?

Empirical studies use this last intuition to argue that if you want to study marginally optimal capital investment decisions, you should look at situations in which firms have decided to adjust their capital level. Presumably, after paying the fixed cost of acting, firms will choose a marginally optimal policy.


What does Piketty mean by the C90/C10 ratio? I am currently readings Piketty's "The Economics of Inequality", where he states in chapter 3:

  The P90/P10 wage ratio in the United States in the 1990s is 4.5, whereas it is “only” 3.2 in France (Table 1.7). From this one might deduce that in order to create as many jobs in France as in the United States, and in particular in order to stop the labor share of value added from decreasing further in France, one ought to increase the C90/C10 ratio between the 90th and 10th percentiles of wages plus social charges in France by about 40 percent, which would mean eliminating all employer-paid social charges on low wages and reassigning them to high wages.

I am aware of what the P90/P10 ratio is, but I have never heard of a C90/C10 ratio. The term is mentioned again several times in the following paragraphs, but receives no further explanation in the remainder of the book. An online search for an explanation did not return any results for me, so I am hoping someone here will be able to help me out.

My understanding is that Piketty uses the term "P90/P10" only to measure wage inequality  and the term "C90/C10" for wages plus social charges. The underlying distributions are different, but they both measure the ratio of the 90th percentile to the 10th percentile.


Indifferent choice assumption in economics Take, for instance, that you have a game similar to the one proposed by Levitt and Duggan (2000) in sumo wrestling (link: http://www.nber.org/papers/w7798.pdf, section II (pages 9-10) in which the paper describes the model).

Summarizing, a fighter maximizes a expected return in a non-rigged match. For a given fighter consider taking the bribe, the other minimizes a side payment (bribe) to the point of indifference between taking and not taking the bribe -- so, why would the fighter setting the side payment would not set the bribe so it would be marginally better to accept the bribe than not taking it? Why do the authors include the point of indifference among the possible solutions?

Because otherwise you would not have subgame perfect equilibria. Suppose I have an item you need and the item is worthless to me. I would be happy to give you the item for any positive amount of money. If you give me no money for it, I am indifferent, both giving you the item and not giving you the item is an optimal choice for me. Just to be clear: we are not bargaining. You set I price and I will take it or leave it.

If you offer price 0 and I give you the item that is an equilibrium. No one could have made a better choice given the other person's choice.
 If you offer price 0 and I do not give you the item that is not an equilibrium, because I would give you the item for any small positive payment and you would be willing to pay that, so you could have made a better move.
 If you offer price $2a>0$ I will give you the item. But this is not an equilibrium. You could also have offered me $a$, which is also positive, and I would also have given you the item then. So you could have saved some money, your original move was not optimal.

Because there is no smallest positive number, the only equilibrium in such cases is when one side is indifferent.


Liquidity trap and consumers' reaction to an increase in money supply I've understood that when the nominal interest rate reaches the zero lower bound there is nothing that the monetary policy can do to increase the output level of a certain economy. Because $i=0$, no one is interested in bonds anymore, and people decide to hold more money in correspondence to the same interest rate. But if they hold more money, why they decide to keep them, instead of increasing their level of consumption and thus increase also the level of production?

It may be a stupid question, but I really cannot figure out the answer. Hope you can help me. Thank you in advance.

Good question. The answer depends on what exactly you mean by an increase in the money supply and how it is implemented.

Because standard monetary policy (ie. open market operations) is implemented through banks, it is functionally quite different from a pure infusion of cash to consumers, which is more the realm of fiscal policy, depending on how it is financed. Consider a central bank that buys bonds from commercial banks by crediting their accounts with reserves. Banks now have excess reserves which they are able to lend should they so choose. If interest rates are above zero, interest rates will fall enabling banks to offload the excess reserves. However, if interest rates are already at a nominal lower bound of zero, the price of loanable funds cannot fall to stimulate increased demand, and hence banks cannot offload the excess reserves.

It also depends on whether the increase in the money supply is temporary or permanent. A permanent increase in the money supply will permanently increase the price level, thereby inducing a one-period spike in inflation and a contraction in real interest rates ($i^R=i^N-\pi$). The quantity of loanable funds demanded will increase. If the increase in the money supply is temporary, though, this will not work. This is why economists view the implications of permanent versus temporary changes to the money supply very differently during zero bound episodes.

A few links that may be helpful:

http://www.voxeu.org/article/combatting-eurozone-deflation-qe-people
http://www.bruegel.org/nc/blog/detail/article/1527-permanent-qe-and-helicopter-money/


Is the implicit function theorem needed to relate the EMP to UMP? My professor says the expenditure function and the indirect utility function are inverses of each other. But how is this possible? Consider each.

$$v:(p_x,p_y,m) \rightarrow \Bbb{R}$$

$$e:(p_x,p_y,\bar{U}) \rightarrow \Bbb{R}$$

These are both mappings from $\Bbb{R^3}\rightarrow \Bbb{R}$. How can one be the inverse of the other?

Only if one was a scalar field and the other a vector field would this work.

My Question:

What is the relationship between the expenditure minimization function and the indirect utility function?

Fix prices at some arbitrary positive values $p_x^0,p_y^0$ and view the functions $v$ and $e$ as $v:\mathbb{R}\rightarrow \mathbb{R}$ and $e:\mathbb{R}\rightarrow \mathbb{R}$. Then they are inverses.


Exercise where lagrangian is needed? I teach a general equilibrium class in my university and I want to have an exercise that is not too difficult where the Lagrangian multiplier is needed. I was under the impression that with Cobb Douglas and 3 goods I could force them to use it but in the end it turned out that they could easily solve it without it. Is there a rule of thumb or an exercise for what kind of problems force students to use the Lagrangian? To be clear, I want an example with exchange.

I think a situation in which corner solutions may occur would be more fruitful. Cobb-Douglass ensures an interior solution, so they can use the expenditure share trick pretty easily, for any number of goods. Perhaps a really simple CES function, like $x^.5+y^.5+z^{.5}$? Or, if you still want an interior solution, pick something that's not a well-known function, but where marginal utility becomes infinite as either good approaches 0, like $\ln(x) - \frac{1}{\sqrt{y}} - \frac{1}{z}$ You'll still get tidy derivatives, but substitution won't be possible.


What would happen to US economy before US government lose control over its national debt? From this wiki page we can clearly see that US national debt is growing steadily after 2010 and now it's larger than US GDP already (at the end of 2019).

Still it's commonly believed the debt is maintainable. At least, neither the Obama administration nor the Trump administration care to spend (enough) effort to eliminate the debt.

To overly simplify, the absolute death sentence to this debt problem is that the whole government budget is not enough to pay the interest, but I suppose the actual critical point arrives much earlier than that.

Regarding US debt problem today, the only referenceable example of to me is Hyperinflation in the Weimar Republic
. Yet they are only remotely parallel: Weimar Republic back then was hardly a major world power (politically and economically) while US being today's only global hegemony, let alone its Petrodollar policy and other global influence.

So I wonder, being such super power, what would happen to US domestic economy along the debt growing towards such critical point?

To make this question simpler, let's assume that during such progress

There is no war between US and the other major powers against each other.
The external force disturbing US policy remains minimal.

It depends.  Mathematically, you just need GDP to grow at the same rate as the debt grows, and then you can sustain deficit spending indefinitely. But this simply shifts the line of questioning to "can GDP grow indefinitely", and for most people the obvious answer is "no".

Ironically, the band-aid solution that most developing nations have taken is to grow the population through immigration (ironically because this approach is seeing significant geopolitical pushback).  It's well-known that more affluent countries have lower birth rates, and so immigration is the "obvious" solution to this.  But it clearly carries its own challenges.


Pure exchange economy with two consumers and non differentiable utility functions We have a pure exchange economy, two consumers $A,B$ and two goods $x,y$. The utility functions are as follows $$u_A=\min\{x_A,y_A\}\qquad u_B=\min\{x_B,\sqrt{y_B}\}$$
The endowments are $$\omega_A=(30,0)\qquad \omega_B=(0,20)$$

I want to derive the equilibrium price and the equilibrium allocation. Now, I can understand that if the vector of prices $\mathbf{p}>>\mathbf{0}$ the equilibrium does not exist because the offer curves do not intersect (and this is evident once you have drawn the Edgeworth Box). My problem is that I don't know how to approach the case in which one of the two prices is zero. How should I study this situation? Should I derive formally the two offer curves as functions of prices?

I would suggest that ask yourself the following questions (hopefully, this should help you figure out how to solve the problem) :

If good $z \in (x,y)$ was free, what would be the demand for both agents?
Is the conjunction of these demands feasible given the endowments? (this should allow you to rule out one of the cases)
If the demands are feasible, which price(s) for the other good would support such demands? What could be the equilibrium(a) allocation(s)?

Notice that if good $z$ is free, one of the agents is not able to consume the other good, and she is therefore indifferent between consuming any quantity of good $z$.

Hope this helps.


In what sense is economic growth cumulative? In my economics syllabus, one of the student competences is to "Show the cumulative nature of growth and measure average and annual rates of growth". What I don't understand is how a country's GDP growth can be subject to cumulative or compounding effects?

The basic idea is simple. If you grow 1% each year for five years you end up with a GDP that is (1.01)^5 of what you started: the 1% growth of each year accumulates over time.


What is the difference between contract theory and mechanism design? Can someone explain the difference between contract theory and mechanism design?

Contracts are a subset of all mechanisms where agreements are enforcable.

An example of a mechanism that is not a contract:
A second price auction (or Vickrey auction) is a truth-telling mechanism where the enforcability of contracts is not required. In the truth-telling equilibrium no one has any incentive to change their bid, no matter the outcome. This is not because changing the bid would be a violation of contract and result in penalties.


Why do we add Net Foreign Factor Income instead of substracting it while calculating GDP? (It's assumed that we use the income method.)

GDP is supposed to include all incomes received on territory of given country, no matter if receivers of income are citizens or not. So incomes of foreigners in given country (I will write them as FH, stands for "foreigners here") are included into GDP, but incomes of citizens abroad(I will write them as CA, stands for "citizens abroad") are NOT included. GDP=Wages+Rent+Interest+Profits+Sales Taxes+Deprecation+Net Foreign Factor Income. For sake of simplicity I will replace "Wages+Rent+Interest+Profits+Sales Taxes+Deprecation" with "A". The formula can be rewritten as GDP=A+NFFI

As for Net Foreign Factor Income, it's the difference between income that citizens of given country earned abroad and income that foreigners earned in given country. Or in my notation, NFFI=CA-FH.
Considering this, the formula of GDP can be rewritten as GDP=A+CA-FH. But it's wrong! This way we will include incomes of citizens aborad and exclude incomes of foreigners in given country, but it should be reverse of that, we should include incomes of foreigners in given country and exclude incomes of citizens abroad. In short, it should be GDP=A-CA+FH. We can get this effect either if we change how NFFI is calculated (NFFI=-CA+FH would do the trick) OR if we substracted NFFI from A, instead of adding it.

Seems like I got incorrect information about how to calculate NFFI.

I was told by a worker of the BEA that NFFI would be calculated as NFFI=FH-CA. In this case everything is okay.


Why is every action considered to be in the support while calculating mixed strategy equilibrium? While evaluating the mixed strategy equilibrium, in most of the cases, we consider that all actions of a player are in the support of the equilibrium.Is it always so?
I was going through a problem (36.1, Chapter 3 in Ariel Rubinstein’s A Course in Game Theory) where the mixed strategy equilibrium was calculated by considering all the possible actions in the support of the probability distribution. I was able to reach to this solution using the basic theory, but I cannot prove that every action will be in the support.
Here I am borrowing the definition of support from Ariel Rubinstein’s A Course in Game Theory.

It is not, only in intro problems.

You have already seen counterexamples: The prisoner's dilemma has an equilibrium, yet it does not have full support. (But it is a pure strategy equilibrium! So? All pure strategies are just very particular mixed strategies.)

Mixed strategies with full support are called completely mixed strategies.

It is a good exercise to show that strictly dominated pure strategies are never in the support of mixed equilibrium strategies. Using this you yourself can construct a simple example where the only equilibrium is a mixed but not completely mixed strategy profile.


Why do businesses care about rate(s) of change? I'd like to introduce math-students to the concept of average and instantaneous rate of change, in order to get them interested in the topic.
I found the following example:

  Based on data from 1990 to 2000, the average price for a half gallon of prepackaged ice cream may be modeled by
  $$
C(t) = 0.0142*t^2 - 0.0272*t + 2.53
$$
  where t is the number of years since 1990. (Source: Modeled from Statistical Abstract of the United States, 2001, Table 706, p. 468.)
  To help project future revenue, a creamery wants to know what the average annual increase in the price of ice cream was from 1990 to 2000 and at what rate the price will be increasing at the end of 2000.

So, in other words, we are looking for the average and instantaneous rate of change.

My question is the following:
Why is the rate of change important for economics? I know this sounds silly, but if my goal was to project the price of ice cream for let's say 2015, all I had to do was insert the value 25 for t and I'd be done, right?
So why do we care about the rate of change in economics? What is to be gained by this information and what would be decided differently after finding out the rate of change?

In economics, a firm wants to maximize profit. Your ice cream shop has some costs that vary depending on how many half-gallon ice cream packages are made, and some benefits, the revenue from selling ice cream. The additional cost of making one more package of ice cream is called the marginal cost, and it can change based on how much ice cream you are already making. The marginal benefit of course the price that you sell each package for.

In order to maximize profit, you have to choose some quantity of ice cream packages to produce. You do this by setting marginal cost equal to marginal benefit, and then solving for production. So if you have a formula for what the price of ice cream will be over time, then you can predict your marginal benefit  of production over time, and can anticipate what your production will look like in the future.


How to calculate the quantity of an incumbent that sets the profit of a challenger equal to 0? I have the inverted market demand of $P=4Z-(q_1+q_2)$ and cost function of both firms is equal to $C=cq_i+f$ for $i=1,2$

So now, the existing firm wants to discourage a possible challenger to entry and sets an output that would make the profits of the challenger equal to 0.

My approach would be, to set up the profit function for the challenger:

Step 1: $(4Z-q_1-q_2)*q_2 - cq_2 = 0$ and then solve for $q_1$ in terms of $q_2$.

Step2: Then I can substitute $q_1$ in the profit function of firm 1 and take the derivative of the profit function with regard to $q_2$

Step3: The result of step 2 I would insert in step 1 in order to receive the quantity that firm one needs to produce in order to set the profit function of the challenger equal to zero.

Is this approach correct? I am a bit confused because of the strange algebra, but I could not come up with a different solution.

Some steps are a bit unclear to me, but I would say that they are wrong. For example step 1 is wrong because the meaning of this equality is: I will set the q1 that makes the challenger do zero profits once (after) it enters, but the point of this problem is that we have to block it before it enters.

so here my version.

The first thing to do is to calculate the challenger best response function, which you obtain by taking the derivative to respect to q2 (challenger)with given q1(incumbent). You will then obtain the optimal q2 which you will substitute in the challenger profit function, and only now you choose the q1 by setting the challenger profit equal to zero.

intuition: once the challenger enters, he will face a price and a quantity given by the incumbent, by doing the derivatives, he will also choose a quantity that best help him maximize his profit. So we substitute the optimal q2 in his profit function

The incumbent knowing this, and knowing the challenger best response function, will choose a q1 that set the opponent profit function to zero.

Now you may wonder what is the difference with step 1 of your solution.
Notice that in your solution the incumbent doesn't know what quantity the challenger will set ex-ante, but only ex-post, while in my analysis the incumbent knows that the challenger is rational and will set an optimal q2 which depends on the quantity q1, so in this case the incumbent knows ex-ante.


Type - I Error & Type - II Error: Pregnancy test analogy - is it legit? I found this picture in my stats book but I'm now confused to what 'positive' and 'negative' is referring to.

As seen in the table below, Type 1 error is the error that its H0 is actually true but FALSEly claims that it's false. Type 2 error, on the other hand, is the error that its H0 is actually false but FALSEly claims that it's true.

So my question is, how do the pregnancy analogy and whole 'false positive' & 'false negative' thing make sense?

For the first picture to be a type 1 error,  H0 (null hypothesis) should be "The person is NOT pregnant" so that "You're pregnant" statement becomes false.

However, the second picture has the complete opposite H0, where H0 should be "The person is pregnant" so that "You're not pregnant" statement becomes false.

I thought it was really confusing because I thought false POSITIVE and false NEGATIVE corresponded to "You're pregnant"(positive) / "You're NOT pregnant"(negative)

But based on the Chart given below, that doesn't seem to make any sense.

So the question is, is there anything that I'm missing here or is it just that textbook's analogy sucks?

Presumably here

the null hypothesis is $H_0:$ You are not pregnant
the alternative hypothesis is $H_1:$ You are pregnant

so being pregnant would be the positive result.

You take a pregnancy test

if the pregnancy test gives a positive result when you are not pregnant then this is a false positive, a Type I error when the null hypothesis $H_0$ is in fact true but has been rejected by the test
if the pregnancy test gives a negative result when you are pregnant then this is a false negative, a Type II error when the null hypothesis $H_0$ is in fact untrue but has not been rejected by the test

So in a statement of being a true/false positive/negative test, the true/false part is about the accuracy of the test while the positive/negative part is about the result of the test rather than being the real situation


Labour Market: simple macroeconomic modeling I just started reviewing macroeconomic modeling for an upcoming entry exam, and I'm not sure about my answers in this application:

  Suppose that the economy of a country has a representative firm with
  the following production function $y = f(N) = 2N - (1/2)N^2$ knowing
  that $N$ is the size of employment.

  1) Write the profit of this firm knowing that the price of the good is $P$ and the wage is $W$.
  2) Determine the demand for labour which is the optimal size of this firm. Comment on your results.
  3) From the last question, deduce the optimal size of production which is the optimal supply in volume of the good produced by this
  firm. Comment on your results.
  4) Suppose the total supply of labour is exogenous, and is given by: $N^o = (W/P)$. Define the notion of labour market equilibrium, and
  calculate the equilibrium real wage of this economy.
  5) At this general equilibrium, calculate total production (in volume), the total real payroll, total real profits (profits deflated
  by the price), and comment on the distribution in this economy.

Admittingly, I'm still quite weak when it comes to macroeconomic modeling, I haven't yet found a good enough source so all my knowledge comes from scattered sources all over the web.

But here's my attempt:

1) I know profits are
$$\pi = Revenue -Cost \equiv P.y - W.N$$
Where P is the price of the good, y the quantity produced, W the wage and N the size of employees.

Since the quantity produced is given as a function of N I re-wrote that expression into:  $$ P.y - W.N \equiv P. 2N-(1/2)N^2 - W.N$$

Which can simplify to: $$\pi = P.N(\dfrac{-N+4}{2}-w)$$

Not sure if this is what they mean but I stopped here.

2) I've read somewhere that optimal demand for labour is reached when $Marginal Benefit = Marginal Cost$

Marginal Benefit is defined as $Marginal Product \times Price$ so in our case that should be the derivative of the Production Function (marginal product) times price, or $MB = (2-N)\times P$
I've read that Marginal Cost is the real wage $\dfrac{W}{P}$
So at equilibrium $(2-N)P = \dfrac{W}{P}$ which after some computation we get $$N=-\dfrac{W-2P^2}{P^2}$$ which should be the demand for labour at equilibrium.

3) Since N we found is (supposedly) the optimal demand for labour, and production of the good is a function of N alone, I thought we'd get the optimal size of production by inputting the expression of $N$ we just found in the production function $y$ that was given to us. After some long computations, I got: $$y=-\dfrac{W^2}{2P^4}+2$$

4) The real wage at equilibrium is obtained when Demand for Labour equals Supply for Labour. Using the exogenous expression for Labour Supply we were given this means:
$$\dfrac{W}{P}=-\dfrac{W-2P^2}{P^2}$$

Which when solved get $$p=1, w=1$$ and thus real wage would also be equal to $1$

5) Replacing values in various expressions we derived in prior questions I got: $$N=1$$  $$y=\dfrac{3}{2}$$   $$payroll=\dfrac{W}{P}.N = 1$$ $$\pi =\dfrac{1}{2}$$

Now I'd be very surprised if I had any answer right since I'm very new to this, so I'd appreciate some feedback on my answers. Also, I'd appreciate some help with the comments on the answers especially the one about the distribution.

Thanks.

To find demand for labor, solve the firm's profit maximization problem :
\begin{eqnarray*} \max_N & \ P(2N - 0.5N^2) - WN\end{eqnarray*}
Differentiating the objective with respect to $N$ and setting it equal to 0 yields the demand function for labor :
\begin{eqnarray*} N^d = 2 - \frac{W}{P}\end{eqnarray*}
Corresponding supply of output is

\begin{eqnarray*} Y = f\left(2 - \frac{W}{P}\right) = \left(2 - \frac{W}{P}\right)\left(1 + \frac{W}{2P}\right) =  2 - \frac{1}{2}\left(\frac{W}{P}\right)^2\end{eqnarray*}

Given the supply of labor is
\begin{eqnarray*} N^s = \frac{W}{P}\end{eqnarray*}

Equilibrium in the labor market occurs where $N^d = N^s$. Solving it, we get equilibrium employment, real wage, output, profits as follows :
\begin{eqnarray*} N^* & = & 1 \\ \left(\frac{W}{P}\right)^* & = & 1 \\ Y^* & = & 1.5 \\ \pi^* & = & 0.5 \end{eqnarray*}


Can Dynare solve general equilibrium (GE) models with non-convex adjustment costs? I know that Dynare (which sits on top of Matlab) can solve many kinds of dynamic stochastic general equilibrium (DSGE) and overlapping generations (OLG) models. I also know that Dynare can handle some sorts of adjustment costs. For example, I have seen convex adjustment cost examples in Dynare. In particular, The Macroeconomic Model Data Base provides on the order of 50 models compatible with Dynare and the user manual indicates several models (e.g. NK_IR04 and US_NFED0) with quadratic (a type of convex) adjustment costs.

Can Dynare solve models with non-convex adjustment costs like An equilibrium model of lumpy housing investment (Iacoviello and Pavan (2008)) or Housing and debt over the life cycle and over the business cycle (Iacoviello and Pavan (2013))? Non-convex has a specific mathematical meaning, but in the context of these papers it indicates that adjustment costs that are not proportional to the amount of adjustment. Instead, the adjustment costs have a fixed-cost proportional to the current asset value.  However, there are other forms of non-convex adjustment cost. If Dynare can solve any model with any sort of non-convex adjustment costs that is of interest.

If models with these adjustment costs can be solved with Dynare, please provide an example or a link to an example (if possible).  If Dynare currently cannot solve these models is there any published code that can do so? Even sample code for a specific model solution rather than a general product like Dynare would be helpful.

More details on non-convex adjustment costs:

I draw my language here from a
A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence (Flavin and Nakagawa (2008))

  At the instant the house is sold, the household pays a transactions
  cost proportional to the value of the house sold, so that wealth also
  changes discontinuously....The housing model developed in Section I
  invokes a fourth set of assumptions: utility depends nonseparably on
  nondurable consumption and on housing, nondurable consumption is
  costlessly adjustable, but housing is subject to a nonconvex
  adjustment cost ($\lambda > 0 $).

Perhaps this language is non-standard but that's a quote from a paper in the AER, and when I've discussed it with others people seem to know what I'm talking about. The two mentioned papers don't use that language but do have the same rough form, that transaction costs are not increasing in the degree of the adjustment but rather that any use of adjustment (other than a small bit, perhaps for depreciation or unit improvement perhaps) triggers a cost related to the state variables instead of the control variables. The paper On the Nature of Capital Adjustment Costs (Cooper and Haltiwanger (2005)) seems to use nonconvex adjustment costs in the same way in a firm capital setting.

  Building upon the analysis of Abel and Eberly [1999], Cooper,
  Haltiwanger and Power [1999] and Caballero and Engel [1999], during
  periods of investment plants incur a fixed adjustment cost. Generally,
  these non-convex costs of adjustment are intended to capture
  indivisibilities in capital, increasing returns to the installation of
  new capital and increasing returns to retraining and restructuring of
  production activity. These fixed adjustment costs represent the need
  for plant restructuring, worker retraining and organizational
  restructuring during periods of intensive investment

Short answer: no.

Dynare, and linearization/perturbation methods in general, are designed for solving

smooth models
approximated around a single point in state space (the steady state).

A model with fixed cost is typically non-smooth, and its behavior away from the steady state may be very different, if e.g. the firm switches from investing to not investing. On the most practical level, a model with fixed cost will typically include equation such as

$$
V = \max \left\{ V^{\text{invest}}, V^{\text{not invest}} \right\},
$$

which cannot be entered into Dynare, because max operator is not supported. On the other hand, first order conditions for convex (e.g. quadratic) adjustment cost are still smooth (one simply adds additional terms to Euler equation for investment) and thus can be easily solved with Dynare.

To actually compute optimal policy with fixed costs, one has typically to use global method, e.g. value function iteration. I'm not aware of any standardized toolbox for solving such problems, so you may need to code your own.

PS: there are some modelling tricks that make the problem smoother, typically in a setting with many, possibly heterogeneous agents/firms. For example, Thomas (2002) keeps track of number of firms depending on how long they didn't invest, and solves the model with standard linearization on this extended state space. Khan & Thomas (2007) assume that the fixed cost is random and iid over time and across firms, so one can average over the realization of fixed cost to obtain smooth value functions. Miao & Wang (2014) use a similar approach in a model with constant returns to scale and show how it aggregates to a version of representative-firm model with only convex adjustment costs.


Relationship between trade deficit and borrowing I have seen lots of essays saying that trade deficit means the nation needs foreign funds to finance their imports. In other words, they need to borrow from the foreigners. However, it sounds very unintuitive to me. My concern is that why would do they have to borrow? I always thought financing trade deficit was all about transferring the ownership of financial assets. Of course, if you count the sale of financial assets as borrowing, we are good.

We count the sale of financial assets as borrowing. You simply sell a document indicating that you will pay the buyer in the future, which is simply a debt.

Actually, sales of financial assets cause capital account to increase, so that, current account balance is compensated by the sales of financial assets which is basically a capital account surplus. Hence

$$
Current Account + Capital Account = 0
$$


Is there always at most one full-support extreme symmetric equilibrium? Given a ($n$-player) symmetric game and two equilibriums $s_1,s_2$, is
  it true that if the support (the set of strategies with positive
  probabilities) of $s_1$ is identical to the support of $s_2$ then
  $\frac{s_1+s_2}{2}$ is also a symmetric equilibrium?

By $\frac{s_1+s_2}{2}$ I mean that a strategy $i$ will be selected by a player with a probability which equals the average of the probability of $i$ under $s_1$ and the probability of $i$ under $s_2$.

This seems obvious.  If $\sigma_1,\sigma_2$ are in the joint support of $s_1,s_2$ and $\tau$ is not, then (writing $P$ for the first player's payoff) you need
$$P(\tau,s)\le P(\sigma_1,s)=P(\sigma_2,s)$$

where $s$ is a convex combination of the $s_i$.  But if you replace $s$ with $s_1$ or $s_2$, this holds, so it still holds after you average over $s_1$ and $s_2$.

What am I missing?


How much inflation will changes to official wage benchmarks cause? For example, right now Livable Wage hovers around \$20 an hour. That's around \$40,000 a year.

Let's say the US somehow will give \$50,000 a year as a "Guaranteed basic income".

How high will the "Livable Wage" go?

.

From Investopedia

  Inflation is the rate at which the general level of prices for goods and services is rising and, consequently, the purchasing power of currency is falling. Central banks attempt to limit inflation, and avoid deflation, in order to keep the economy running smoothly.

The Consumer Price Index (CPI) is your best measure of inflation which is (put simply) the change in cost of a representative basket of goods. Each country measures this in a slightly different way.

If you are looking for a numerical increase in inflation, you'll be disappointed. A Basic Income has quite a few effects making this difficult to model. There are anticipated significant impacts on labour/leisure as well as consumption bundles. Also, inflation itself is difficult to model. To make things worse, many reserve banks target inflation with their monetary policy. So you're unlikely to ever get any empirical evidence as well.

Anyway:

Why would increasing income of lower income earners have an inflationary impact at all?

An increase in welfare should result in higher incomes. Higher income leads to higher consumption (or quantity demanded) which should lead to an increase in prices. But, this is only a one-off increase, not an ongoing source of inflation. I would be hesitant to call this inflation and would prefer to call it a permanent demand shock.

Unfortunately, I am not aware of any estimates of this impact but I would expect it the long term impact quite small. The higher the basic income, the greater the impact.

Also worth noting:

I can't think why this would be any more inflationary than any other form of welfare.


Good Site for Practicing Advanced Problems In the past I have used Khan Academy to great success to practice advanced math problems. It is useful because it gives simple questions in a multiple choice fashion then it quickly tells whether your were right or wrong, and it will walk you through the correct approach as needed.

Now I am in my first semester of a PhD in economics. Khan Academy has a series on economics, but it is no longer sufficient. The series is not accompanied with practice problems and it is done largely in algebra rather than calculus. Finally, there are a number of graduate topics it entirely misses such as:

Phase diagrams
Lagrangian multipliers to solve for optimal allocation
Compare/contrast of using different models for the same question (Austrian, Solow, Diamond, RCK, DSGE...)

If it helps, a primary course text is Romer's Advanced Macroeconomics, 4ed.

What are some online resources to practice such problems?

There are lots of good courses (economics, math etc.) on MIT OpenCourseware or interestingly, there are also very very good lectures in YouTube.
But I think the web site for MIT opencourseware is a real treasure. (at least for me).


Can CPI and real GDP have high correlation? I have historical data where the CPI and GDP are correlated. Does this make sense? If not, how do I test for it?

There was a degree of economic stability between the UK's exit from the Exchange Rate Mechanism in 1992 and the World Financial Crisis in 2008: positive real GDP growth typically between 2% and 4% and low steady inflation typically between 1% and 3%.

The Governor of the Bank of England called this "The Great Moderation".

The effect is what you have observed: the level of real GDP and the level of the CPI both grew, apparently highly correlated with each other, as shown in the red line below based on quarterly data.

But this correlation was an artefact of each indicator growing itself over time and of auto-correlation, as well as not having a recession in that period or sudden spikes in prices; looking from 1991-2009 would have produced a less smooth chart.

If instead you look at the percentage changes over the previous 12 months, the apparent linearity collapses, as shown in the blue line below.


Is there a systematic approach to find equilibria in sequential games? I know that one can use backward induction to find one particular subgame perfect NE. And I know that wherever possible one can represent the game in normalform and then find all NE.

But is there a systematic way of finding all extensive form specific subtypes of NE, like sequential or weak perfect Bayesian equilibria? What about the rest of the subgame perfect equilibria that can't be found by backward induction?

Do I always have to guess and proof that it really is the kind of equilibrium I think it is?

You can first find all NE. Then you check which ones are subgame-perfect. Then you proceed and check for which of the NE you can find beliefs that are consistent with the definition of PBE. You can go on and refine the set of equilibria further by kicking out all equilibria that do not satisfy the additional requirements of your stricter equilibrium concept.

If by "backward induction" you mean solving the game backwards subgame-by-subgame, then by definition you find all the SPNE (= all NE in which the equilibrium strategy profile also constitutes NE in all subgames).


What's the use of '% to GDP' type of variables? In my study I will look for the relationship between the Gini coefficient and trade, FDI and other variables. However, as I was regressing it... the result turn out to be insignificant. My data that I used then were 'trade as a percentage to GDP' and 'FDI as a percentage to GDP'.

On the other hand, when I regress it with just the total of trade and FDI... it brings a somewhat significant result.

Can I drop the 'as a perentage of GDP'? And just use the total trade and FDI without that?

Plus I only have 16 years worth of data, though.

Imagine that I am trying to determine whether eating corn has any effect on your height. I see that in the US, total corn consumption is 20 million tons per year (made up number, all others will be made up as well) and adding up the heights of the 300 million citizens we get 500 million meters. Similar statistics for France are 4 million tons per year and 100 million meters.

Total American corn consumption was higher, as well as total American height.
Is this because corn has an effect on height? Or is it that both statistics are likely to be proportional to the size of the country (measured here by population)?

Depending on what your exact hypothesis is, you may want to filter out size effects by measuring these statistics of the national accounts as % of GDP.


Gamification and incentives Is there any (theoretical?) work on "gamification" in game theory?
I do not have a proper definition of gamification, but StackExchange would be an example. Contributers like us provide a costly service without monetary compensation, but in exchange for fictional reputation points or badges that do not mean much -- other than it being fun to obtain them. I certainly would not post anything without the "game aspect" implying a certain non-monetary payoff that can be designed.
Moreover, it is different than other non-monetary services like organizing workshops or refereeing, which at least have real-life consequences for my professional reputation.

It seems that there is already some prior work on theory of gamification from game theory perspective. For example, Easley & Ghosh (2016), provide theory on optimal badge design even explicitly mentioning sites like StackOverflow and their paper seems to be highly cited.  Hamari, Huotari & Tolvanen (2015) have article titled "Gamification and economics" which offers a survey of uses of gamification in various economic concepts. The paper itself does not seem to offer any novel models but it offers a literature review with a chapter specifically focusing on game theory.


Is there always a pure Nash equilibrium in a resource selection game? Denote $[r]\triangleq\{1,2,\ldots,r\}$.
Consider a game with $n$ players, $[n]$, each has $m$ strategies, $[m]$.
Each player $i$ has an associated payoff function, which considers only his selected strategy, and the number of players selected the same strategy:
$$U_i:[m]\times[n]\to[0,1]$$
Furthermore, the utility function is monotonically decreasing in the number of players which picked the same strategy, i.e.
$$\forall i\in[n],j\in[m],k\in[n-1]:U_i(j,k)\geq U_i(j,k+1)$$

Does this game always have a pure Nash equilibrium?
Can we (computationally) find it efficiently?

Notice that the special case, where all players are symmetric ($\forall i,j\in[n]: U_i\equiv U_j\equiv U$), the game reduces to an exact potential game and therefore is guaranteed to have a pure Nash equilibrium.
The potential function for the symmetric case would be, given a strategy profile $s$:
$$\phi(s) = \sum_{j\in[m]}\sum_{k=1}^{\#_j(s)} U(j,k)$$
Where $\#_j(s)$ is the number of players in $s$ playing strategy $j$.

Yes, there is always a pure Nash equilibrium. See:

I Milchtaich (1996). Congestion games with player-specific payoff functions.
Games and economic behavior 13 (1), 111-124.

You are interested in the special case of singleton congestion games with player-specific payoff functions.

And yes, they can be computed in polynomial time. See Corollary 7 in:

Heiner Ackermann, Heiko Röglin, Berthold Vöcking (2009). Pure Nash equilibria in player-specific and weighted congestion games. Theoretical Computer Science 410 (17), 1552-1563.


How do I calculate consumer surplus and producer surplus from this? Calculate the consumer surplus, producer surplus and total surplus in the market equilibrium.

Market for beef and home demand and home supply:

$$ Q_d(p) = 50 - p $$

and

$$ Q_s(p) = p $$

With $p$ representing the price of 1 kg of beef

I understand that consumer surplus is consumer's willingness amount to pay minus consumer actual paying price.

But how do I derive from only information from above?

In equilibrium, we have supply equal to demand. So $Q_{s} = Q_{d} \implies 50 = 2p$, so $p^{*} = 25$ is the equilibrium price. Consumer surplus is defined as:

$$\int_{p^{*}}^{p_{max}} Q_{d}(p) dp$$

Here, $p_{max} = 50$, as $Q_{s} = 0$ when $p = 50$. So evaluate the integral.

Producer surplus is similarly defined as:

$$\int_{0}^{p^{*}} Q_{s}(p) dp$$


Continuous Preference Relation Imply Continuous Utility Fn Existence I am reading MWG's explanation in Chapter 3 when showing continuous preference relation implies the existence of continuous utility function.

First, the authors show $u(.)$ is continuous by using the definition that the image under $u(.)$ of a convergent sequence is convergent. Consider a sequences $x_n\rightarrow x$. They first claim that $u(x_n)$ must have a convergent subsequence.

I understand the big picture: Since $x_n$ converges to x, for some large N, $u(x_n)$ must all lie in some compact set, and any infinite sequence in a compact set must have a convergent subsequence.

The part I am having trouble is when they use monotonicity to show this compact set. The exact excerpt is:

  "By monotonicity, for any $\epsilon>0$, $\alpha(x')$ lies in a compact subset of $\mathbb{R_+}$, [$\alpha_0,\alpha_1$], for all $x'$ such that $\parallel x'-x\parallel$$\leq0$."

Here I used $u(.)$ and $\alpha(.)$ interchangeably to represent the utility function. Can somebody elaborate the above claim just little more in detail please?

I understand monotonicity implies local nonsatiation, hence, in any given small ball, you always have some bundle that you prefer that to x. Part of my confusion comes from the Figure they present, which is they put the bundle x on the indifference curve ($\{y\in X:y\sim x$}). But isn't $\alpha(x')$ on the diagnoal line Z?

Please help. Thank you.

I think you're overthinking this proof. MWG only say that such a compact set exists because, by monotonicity, you can always find $\alpha_0$ and $\alpha_1$ such that $\alpha_0*e<\alpha(x')*e<\alpha_1*e$ (with $<$ I mean strictly preferred). Remember that, by monotonicity, if $y>>x$, then $y$ is strictly preferred to $x$. Therefore, we only need to find some $\alpha_0$ and $\alpha_1$ such that $\alpha_0<\alpha(x')<\alpha_1$ for all $x'$.

This is because the set $\{x':||x'-x||\le\epsilon\}$ is bounded and $x'∼\alpha(x')*e$.

(Note that with local non-satiation the result does not immediately follow).

You're right, $\alpha(x')*e$  is on the diagonal line Z and $x'$ is not (note that $\alpha(x')$ is mutiplied by $e$, a vector filled by 1's). However, working with $\alpha(x')$ reduces the complexity of the problem. Instead of working with two (or more) dimensions, we just work with one (the diagonal line Z or one axis). We can do that because for any element $x\in \Bbb R_+$, there is $x∼\alpha(x)*e$ which is in the diagonal line.


Why are constant price elasticities reported? Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity.
Then why is everywhere (research papers, textbooks etc.)  a constant estimate provided?
Also, doesn't the non-constancy of the price elasticity render the comparative statics $e_{P^*,\alpha} = \dfrac{e_{D,\alpha}}{e_{S,P}- e_{D,P}}$ invalid?

Premise of your question is simply false. You state (emphasis mine):

Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity.

Incorrect. Consider trivial counter example. A perfectly reasonable demand can be given by:
$$ Q  = A p^{-\epsilon} $$
where $Q$ is quantity, $p$ price and $A$ and $\epsilon$ are constant parameters.
Hence the price elasticity of demand $e$  will be given by:
$$e = \frac{dQ}{dp} \cdot \frac{p}{Q} = -\epsilon  A p^{-\epsilon-1} \frac{p}{ A p^{-\epsilon}} = -\epsilon $$
Consequently, clearly elasticity does not always depend on the levels of price and quantity and can be completely constant.

Then why is everywhere (research papers, textbooks etc.) a constant estimate provided?

Again false premise, in several textbooks and papers you will find elasticity being expressed as a function of price and quantity. Since you made statement about everywhere it is enough to show one example to the contrary which would be Varian Microeconomic Analysis, where there are several examples where elasticity in a model won't be constant.
In empirical papers elasticity will be most often estimated as a single number, but this is due to the way how empirical models are set up. It is too broad to discuss all empirical models in a single question, but typically coefficients estimated by model are values that hold on average. For example, in a regression $y_i = \beta_0 + \beta_1 E_i + e_i$ where $y$ would be income and $E$ years of education, the $\beta_1$ would tell us how much on average $y$ increases with education, it would not tell us what happen to every individual kids. That is if $\beta_1=10$ that does not mean that if your kid stays in school one more year they are guaranteed that their income will increase by 10. That is just a value we would expect for average kid.  Similarly, even if researcher assumes elasticity is not constant because they are using linear demand, they might be forced to estimate elasticity at a point rather than as a whole function and convention is to estimate point elasticity at average quantity and price (see discussion in Espey  Espey  & Shaw (1997).


Bayesian Nash Equilibrium - Mixed Strategies What's the correct way to solve BNE in mixed strategies? I've found two conflicting methods used.

Two players. Player 1 knows which game is being played, player 2 knows the game is chosen with probability $\mu$.

$
\begin{array}{c|c|c}
  \ & A & B \\
  \hline
  L & 1, 1 & 0, 0 \\
  \hline
  R & 0, 0 & 0, 0
 \end{array}
$

or

$
\begin{array}{c|c|c}
  \ & A & B \\
  \hline
  L & 0, 0 & 0, 0 \\
  \hline
  R & 0, 0 & 2, 2
 \end{array}
$

So for pure strategies I am finding a consistent method. Form a normal form game:

$
\begin{array}{c|c|c}
  \ & A & B \\
  \hline
  LL & \mu, \mu & 0, 0 \\
  \hline
  LR & \mu, \mu & 2\mu, 2\mu \\
  \hline
  RL & 0, 0 & 0, 0 \\
  \hline
  RR & 0, 0 & 2\mu,2\mu
 \end{array}
$

This allows us to find the pure strategy solution by using the normal form. How do we calculate the mixed strategies?

I've seen two different methods.

1: Look at mixing over (L, R) in game 1 with probability (a, 1-a) and (L, R) in game 2 with probability (b, 1-b).

Look at mixing over (LL, LR, RL, RR) with probability (a, b, c, 1-a-b-c).

Any advice/reading is much appreciated.

I believe that the answer given by @denesp is incorrect. The second method involves simply writing the game in strategic of "normal" form. I believe
the first method is better (easier to use), but I think that they can both be used.
In the answer given by @desesp, the following explanation is given.

  The reason why method two is flawed is that the probabilities $a$, $b$
  and $c$ are not  independent as $$ a = p \cdot q, \hskip 20pt b = p
 \cdot (1 - q), \hskip 20pt c = (1 - p) \cdot q, \hskip 20pt  1 - a - b
 - c = (1 - p) \cdot (1 - q). $$

Here, it appears that mixing is occurring over L in game 1 (with probability $p$) and L in game 2 (with probability $q$). This interpretation does make sense. I believe that @denesp is confusing conditional and unconditional probabilities. To better understand this, I'm going to start with a discussion of actions versus strategies. Then I'll discuss how the set of strategies considered in methods 1 is included in method 2. I'll note that method 2 contains a larger strategy set, which may or may not be useful. I'll conclude with an example of how both methods can produce the same answers.

Difference between actions and strategies

In the explanation given above, it may appear that mixing is occurring over actions. This is not a
correct interpretation. It is technically incorrect because the player is not mixing over actions but mixing over strategies. This is because a player chooses strategies, not actions. A strategy is a plan
that denotes that actions that a player takes in any and every contingency.
We can think of it as mapping information sets to actions. This is important because we would like player 1's actions to depend on the state of nature---we want them to depend on which game he/she is playing.

For reference, we can find definitions of actions and strategies in the first chapter of Rasmusen's book, Games and Information (4th edition). The relevant text is given here:

What are the sets of strategies considered in each method?

In the case of the game that you have given, the pure strategies available can be written succinctly (LL, LR, RL, RR), as you have already done in method 2.
Therefore, the method that you described in method two mixes over the pure strategies, with probabilities: $a$, $b$, $c$, and $1 -a-b-c$.

What strategies, then, are we mixing over in method 1? Suppose that $p$
is the probability of choosing L is game 1 and $q$ is the probability of choosing L in game 2. Then in method 1, we can see that we are choosing
the conditional probability of taking each action in each contingency.
Suppose that game 1 is denoted $G_1$ and that game 2 is denoted $G_2$.
When we specify $p$ and $q$, we are really specifying
$$
p=P(L|G_1)\\ q=P(L|G_2).
$$
Now, in order to show that these two methods are equivalent, we need to show that the sets of strategies represented by each of these sets is the same.

Method 2 contains the same strategies as method 1, plus more.

First, note that the pure strategies LL, LR, RL, and RR can be represented in method 1 by setting $p$ and $q$ to zero or 1.

Suppose that we are using method 2 and that we choose a particular $a$,$b$, and $c$, as defined above. This can be represented in method 1
with
\begin{align*}
p &= a + b \\
q &= a + c.
\end{align*}

However, suppose we choose a particular $p$ and $q$ in method 1. It can be represented in method 2, but not uniquely.
Suppose $p=1/2$ and $q=1/2$. Then two possibilities are $(a,b,c) = (1/2,0,0)$
or another is $(a,b,c)=(0,1/2,1/2)$.

Why does method 2 contain more strategies?

Method 2 contains more strategies because it allows more flexibility
to specify off-equilibrium behavior. This can end up capturing non-credible
threats. Depending on which equilibrium concept you're using, you may or may not want to include these. If you're only interested in Bayesian Nash equilibria, then you want to include these. If you're interested in sub-game perfect Nash equilibria or Bayesian sequential equilibria, then you don't want them.

Here is an example of how method 1 can miss some equilibria

The following game is again take from Rasmusen's book. Suppose that in this game
Smith moves first.

Then, Jones must choose among 4 strategies. In each of these strategies, he specifies his actions in each contingency. The 4 strategies are listed here and the game is represented in strategic or "normal" form.

There are three equilibria, denoted $E_1$, $E_2$, and $E_3$.
If we were simply interested in the Nash equilibria of this game,
we would include all of these equilbria. However, if we are interested
in only the subgame perfect equilibria, we would only want $E_2$.
That is because $E_1$ and $E_3$ involve non-credible threats.

I believe that if we were to try to solve this game using method 1, we would not be able
to identify all three of these equilibria.

Strategic, "Normal" form

In the question you've given, method 2 is essentially transforming this
into a static game in which we consider all the strategies. This means that we are considering the "normal" form of the game. This method is easy and appropriate if you're interested in finding the pure strategy equilibria. It can probably also used to find the mixed strategy BNE, but is perhaps more complicated then what is described in methods 2.
For reference,
here are some notes on the topic.
These notes give instructions on how to solve for the pure strategy Nash equilibria using the transformation that you've given. It also demonstrates how to solve the mixed strategy equilibria using method 1. (See http://www.sas.upenn.edu/~ordonez/pdfs/ECON%20201/NoteBAYES.pdf .)

Further reference

You can also use this online tool to test how the methods can give you the same answers. This is a tool to solve for the Nash equilibria of n by n games. I would recommend using this tool on the examples given in the previous section.
I found this tool referenced in this other question.


What is the relation between probability of default and country's interest rates? Is probability of default expected to increase with increase in the interest rates set by the central bank?

The answer depends on the situation for the country in question. Since the answer is correspondingly very large if we covered all cases, I am only attempting to offer an overview of the cases. Feel free to ask follow up questions on more specific questions.

I will cover three important sinplified cases.

“Gold Standard”/“hard currency borrowing”.
Free-floating fiat currency.
Euro area.

(1) The “gold standard” example is not currently of interest, but if you want to look at historical governmental defaults, they were associated with the gold standard, or the countries borrowing in a foreign currency (“hard currency”). (Countries in currency pegs are also in a similar situation.)

In a gold standard, the government’s money in some sense can be converted into gold. Ultimately, it needs to have enough tax revenue in its local currency to force the private sector to turn in enough currency so that the consolidated government (fiscal arm/central bank) can honour the conversion parity. If that level of taxes is not politically sustainable, the government will be forced to default.

The usual operating procedure in a gold standard is for the central bank to raise interest rates to attract gold inflows. However, this raises the interest expense of the government, and so it makes it harder to sustain the gold parity price. So in this case, higher interest rates can raise default rates. However, the causality is ambiguous - the central bank is often forced to raise interest rates to defend the currency value, snd the currency is falling in value because of an expected default. Therefore, one could argue that high interest rates in a currency peg country are the result of an elevated default risk.

For countries with a currency peg or borrow in a foreign currency, they are in a similar boat, as they need to impose taxes in order to be able to obtain the external instrument.

(2) A floating currency sovereign (e.g. the United States, the United Kingdom, Japan, Canada...) does not attempt to fix the value of its currency versus external instruments. It can create arbitrary amounts of its currency unit without using any real resources. The only real concern is the purchasing power of the currency unit (inflation).

Since there are no external constraints (lack of gold), default is essentially a political decision.

There are operating procedures that might entail default (e.g. the debt ceiling in the United States). It is a political decision to not override those procedures.
Interest spending may be such a high part of the budget that politicians would rather default than pay the interest.
Government debt may be viewed as illegitimate. For example, debts incurred by a radical government that was later overthrown.

High interest rates might help push a country towards default, but that was not the case in the 1970s. The number of defaults of countries in this situation is so low and usually associated with other problems that there are almost no data to draw conclusions from.

3) The euro area is a special case: the currency itself is a fiat currency, but it is under the control of a central bank that is a supranational entity. The euro area nations are not sovereign in the same sense that the federal Government of Canada is, rather they are in a situation similar to a Canadian province.


which is an example of demand curve that is always inelastic/elastic? Which class of functions have the property of being always elastic?and instead always inelastic?

I don’t think that there are any classes of functions where you can ex ante say that it is elastic or inelastic, since this will depend on the function parameters.

However, there is a class of functions for which you can say the elasticity is always constant, these are isoelastic functions.

For example, consider function:

$$Q(p)=ap^b$$

By definition elasticity for this function is:

$$\frac{Q’(p)p}{Q(p)}=b$$

Now this kind of function will be always elastic ($b$>1), inelastic ($b$<1) or unit elastic ($b$=1) depending on the actual value of the parameter $b$


How is the differential definition of money supply, velocity, and GDP defined? I was reading: This article on money velocity (see the summary section)
Where two formulas are presented:
first an equality:
$$\text{Quantity of Money} (M) \times \text{Velocity of Money} (V)  =  \text{Real GDP} (Y) \times \text{Prices} (P)  $$
or just stated as
$$ MV = YP$$
then its derivative is stated in the article as:
$$ \Delta M + \Delta V =  \Delta Y + \Delta P $$
This second statement makes no sense to me, since if I apply the product rule I should get
$$ M \Delta V + V\Delta M = Y \Delta P + P \Delta Y$$
And I see no way to easily drop the non differential terms. Where did I go wrong here?

That is not a differential form of the equation. $\Delta$ there stands just for changes not differential. You can derive the equation in such form by log-linearization.
Take the original equation:
$$MV=YP$$
Apply natural log to both sides:
$$ \ln M + \ln V = \ln Y + \ln P$$
And now you are done, since in an equation that is in log-log form you can interpret everything in terms of changes.
Hence:
$$ \ln M + \ln V = \ln Y + \ln P \implies \Delta M + \Delta V = \Delta Y + \Delta P$$


Fixed costs in the long run Can anyone please help me understand how fixed costs in the long run are zero? In Microeconomics by J. M. Perloff, the author gives the example of a factory's rent and says that if we are talking about the short run (till the time the rent agreement is valid), the company has to pay rent even if it stops its operation. However, in the long run (when the agreement expires), the company can choose to close down the factory and stop paying the fixed cost and hence we can assume that in the long run all fixed costs are zero.

This example doesn't bode well with me. We are doing analysis of costs on the first hand to plan for our long term use. If an example talks about making our fixed costs zero by shutting the whole production, then it itself defeats the purpose of planning. I, therefore, haven't been able to understand as to how we can have zero fixed costs in the long run (enough time to change all the factors of production) if we continue to work.

Any help will be appreciated

You should quote directly from the book you're citing, because I think you might be mis-reading it.

At the Econ101 level, there are two important frames for thinking about fixed costs: one is that in the long run, the contribution of fixed costs to average cost falls to zero.  You can see this in the standard textbook graph, which will typically look something like this:

Source: Wikipedia

Note that Average Fixed Cost tends to zero as more units are produced.  True to the circular nature of much economic reasoning, often "the long run" is defined as the period for which average fixed costs fall to zero.

The other frame is around what constitutes fixed costs in the first place.  In the short run - which, in your example, is the contract period - rents are fixed costs because they are contractually unavoidable. When faced with the decision about whether to operate, in the short run a factory treats fixed costs as "sunk" and cares only about recovering average variable cost (if I can produce widgets and sell them at cost, I am indifferent to producing or not producing).

Where it gets confusing is that in the "long run", defined as the period exceeding the term of the contract, the firm is free to negotiate rents and so these are now treated as variable costs.  This isn't really the same as "fixed costs falling to zero" because it's not as if the firm can somehow escape paying rent for buildings.  Rather, it reflects the fact that in the long run, the firm has some ability to control what it pays for rents.

It's pretty normal to be confused by this at the entry level.  In my opinion, it's not great nomenclature. The takeaway is that you need to be very clear about the context you're working in, before you can be clear about the meaning of the terms you're dealing with.  In one context, "fixed costs" are "one-time sunk costs" whose impact on average cost of production falls to zero in the long run because that one-time cost is distributed over a growing number of widgets.  In the other context, "fixed costs" are unavoidable costs in the short run that, in the long run, don't fall to zero but rather are redefined as "variable costs" because the firm gains the ability to control them in some limited sense.

In neither case does the firm avoid fixed costs by not paying them, unless it really does shut down.


Reverse Auction bidding strategy A buyer wants to buy contiguous land plots from three landowners (sellers) who own fixed size plots. Sellers have their private valuation of their plots. The buyer will hold a closed bid procurement auction. The profit for the buyer is greater if he purchases all plots of land as compared to (any) 2 plots(without any specific order), which itself is greater than profit for a single plot. Does someone have an idea of how strategies will be developed for the sellers (in equilibrium)? I will later extend it to multiple rounds reverse auctions.

Given the very general description of the problem, I can think of the following (also very general) way of formulating it mathematically.

Let $v_n$ be the buyer's value from owning $n$ plots of land, with $v_3>v_2>v_1$.
Let $c_i$ be seller $i$'s private valuation of the plot she owns. Let $b_i$ be seller $i$'s bid, and we assume the buyer will purchase $n$ plots of land whenever $v_n$ is greater than or equal to the sum of the $n$ smallest $b_i$'s, and each seller gets the amount she asks for in the bid.

Seller 1's objective is to choose $b_1$ to maximize the following:
\begin{align}
&\Pr(b_1+b_2+b_3\le v_3)b_1
\\&\qquad+\Pr(b_1\le\max\{b_1,b_2,b_3\}\text{ and }b_1+\min\{b_2,b_3\}\le v_2\text{ and }b_1+b_2+b_3> v_3)b_1
\\&\qquad+\Pr(b_1=\min\{b_1,b_2,b_3\}\text{ and }b_1\le v_1\text{ and }b_1+\min\{b_2,b_3\}> v_2)b_1
\end{align}
The problems for the other two sellers would be symmetric.

If you assume that $c_i$'s are i.i.d. draws from a distribution $F$, then it seems plausible to solve for a symmetric Bayesian Nash equilibrium, where every seller bids according to the same bidding function, $b_i=\beta(c_i)$ for each $i$.

Using $b_i=\beta(c_i)$, and perhaps with the assumption that it is strictly increasing, you should be able simplify the $\Pr(\cdot)$'s in the objective function. The rest is just to solve for the bidding function. I will leave you to finish the remaining steps.


If an employee works overtime (and gets paid for it), how does this affect a nation's GDP? GDP = Consumption + Government Expenditure + Investment + Exports - Imports

My cases are:

(1) If the employee is getting paid for his extra-work, his consumption would increase.

(2) This employee might take away the job of someone else --and thus his increase in spending could get nullified

By  definition, GDP = the total value of all goods and services produced.

(1) If even after working overtime, the company produces the same amount of goods / service --by definition, GDP hasn't risen at all.

You are correct: an increase in the wage does not by itself imply an increase in GDP. So, if you pay a worker \$20 for an hour of overtime which you could have bought from another worker for \$10, but they both produce the same thing, GDP does not change by switching from one arrangement to the other.

You can also observe that from the 'income perspective' in the circular flow of the economy: if nothing else changes, a higher wage for a  worker results in lower profits for the firm, but the net effect on income is 0.

However, a change in which hours are labeled as overtime and which are not could have important effects in the economy. You'd expect that on average firms will prefer to have relatively more workers working relatively few hours and so avoid paying overtime. This should increase employment, but not average worker income, it should reduce inequality among workers, etc.

Potentially there can be large effects are on the health insurance costs of firms of changing overtime laws.