Nash Equilibrium in 2 bidder auction I am trying to find Nash Equilibrium of an auction with two bidders in which the highest bidder wins the object but both bidders pay the losing bid. Here every bidder follows the same bidding strategy β which is a strictly increasing and differentiable map from values to bids.

I am unable to find the Nash equilibrium of the game. Can someone help me with this? Is this some kind of standard Auction?

Assuming the auction has a pure strategy equilibrium, we can set up the problem as follows:

Each bidder has a value $v_i\sim F$ on support $[\underline{v},\overline{v}]$. The symmetric equilibrium, we posit, is an increasing continuous function $b(v)$.

Now, imagine bidder $i$ bids as if his value were $\widetilde{v}$. Then he wins iff $v_j<\widetilde{v}$ and $i$'s expected payoff is therefore

$$U_i=-[1-F(\widetilde{v})]b(\widetilde{v})+\int_0^{\widetilde{v}}\! \left(v_i-b(v_j)\right)F'(v_j)\,dv_j$$.

The first term is the payoff if $i$ loses (he then pays his own bid, which is $b(\widetilde{v})$ sinc $i$ is using $\widetilde{v}$'s bidding strategy). The second term is the payoff if he wins (he pays his rival's bid, provided his rival's value is less than $\widetilde{v}$).

Now we compute the best $\widetilde{v}$ for $i$ to choose by computing a first-order condition:
$$\frac{\partial U_i}{\partial \widetilde{v}}=-[1-F(\widetilde{v})] b'(\widetilde{v})+v F'(\widetilde{v})=0.$$

We know that, in equilibrium, it must be optimal for $i$ to choose $\widetilde{v}=v_i$ (i.e., it must be a best response for $i$ to use his intended strategy rather than deviate to someone else's strategy):

$$-[1-F(v)] b'(v)+v F'(v)=0.$$

Thus,
$$b(v)=b(\underline{v})+\int_{\underline{v}}^v\!\frac{xF'(x)}{1-F(x)}\,dx.$$

We can check that $$b'(v)=\frac{vF'(v)}{1-F(v)}>0$$ as required. All we need is a boundary condition to pin-down $b(\underline{v})$. For example, we might think it natural that $b(\underline{v})=0$ because a $\underline{v}$ type knows he can never win the auction.


Is a smartphone app(Non premium or subscribed service like Netflix) a public good? A smartphone app that provides users with information on surf conditions at their location. (“Surf conditions” refers to information on wind, waves, and ocean tides that is useful for surfers).
For this I assume the consumption here is strictly the info on the app also assuming the app is free anybody can use it (like google maps). This is a pure public good then. First. they are non-rival in consumption that is my consumption of the app does not in any way affect other people to use the app as well. Even if a million people is using the app, no one will be affected since the consumption here is only the info. It can be opened simultaneously by many user, and adding more users does not affect any of the users on using the app. Second, they are non-excludable, even if I want to deny someone else to use the app there is no way I can do so. Anybody can download the app and it will be really difficult to ban people from downloading the app
That is my answer.
My professor said the app is excludable. Am I correct to say that the app is non-exlucludeable? My logic behind it was a free app is always non-excludeable since there is always a workaround to accessing/downloading the app. For example a certain app that is not available for a specific region, you can always use a VPN to access/download it. If it is a premium service on the other hand, I completely agree that the it is excludable, however the professor has confirmed that the reasoning behind it being excludable doesn't have anything to do with it being a premium service.
Here is the professor's answer
It is, however, excludable because the app provider can limit you from downloading it to your phone.
1)This example is similar to the Netflix option, where Netflix can deny users access to the streaming service.
2)The definition for excludability it “it is possible to deny someone the opportunity to consume it.” It is possible to deny someone the opportunity to consume the app (no matter what they do, like owning the right smartphone) because it is possible for the app designer to restrict your ability to download the app. As for paying for it, that is a resultof the app being excludable. If it were a public good, the app design could not get people to pay for it.

Am I correct to say that the app is non-excludable?
I am sorry to say but this is wrong. Just assume google removes the application from the store. All android users would be excluded. There are many ways to exclude people from consuming an application. Access can be bound to a minimum age requirement, to a certain country (geo-blocking) or simply to a payment.
The possibility is important not the concrete situation.
Just because it is free now does not imply it is generally non-excludable.
Let’s ask the same question for a classic public good: Could you easily exclude someone from consuming air? Even if you wanted to this would be (nearly) impossible.


Amplification Effects: Negative Denominator Let $0 < \rho < 1$ be the discount rate, $V$ some option value, and $F$ some fundamental.

$$ \rho V = \beta V + F$$

You have access to some option value $V$ that will always deliver you some fundamental flow value $F$, and you get access to another $\beta$ of the initial option value (some sort of amplification of the initial value/claim). Solving for the value of the option, we get

$$ V = \frac{F}{\rho - \beta}$$

Now that's maybe just me being confused talking, but typically here we only check whether $\rho - \beta$ is not exactly zero.

For $\rho > \beta$, we have that $V$ is valued much more than the original $F$. How can I understand cases where $\rho < \beta$? These are very possible, even for both $\beta$ and $\rho$ between zero and one. In that case, the denominator turns negative, and the value of the option $V$ is negative. What is happening?

Extended Example

Per popular request, here is a more general version of the model (still an abstraction, but hopefully this delivers enough context.

Think about $V$ as the value of a vacancy, in a search-and-matching context. Given market tightness $\theta$, you will find a match at rate $q(\theta)$. Vacancies are associated with flow costs $c$.

Now, once you've matched with an unemployed worker, you can decide to either accept that match ($\beta = 1)$, or to reject him ($\beta = 0$). In fact, the whole line $\beta \in [0, 1]$ is allowed, understanding it as a mixed strategy. In fact, the mixed strategy can be understood as population shares, given that the vacancy is representative.

So, given strategy $\beta$, the value of a vacancy is given by

$$ \rho V(\beta) = -c + q(\theta)[ \beta(V(\beta)-J) + (1-\beta)J]\\
\Leftrightarrow V(\beta) = \frac{-c + q(\theta)(1-2\beta)J}{\rho - q(\theta)\beta}
$$

Note that setting $F\equiv - c+ q(\theta)(1-2\beta)J$ almost recovers the initial equation.

Now, I tried to solve this (or rather, a more complicated setup) using a grid for $\beta$. But sometimes, the solution was given by a $\beta$ such that the denominator was negative (and small), and the numerator, too, was small. I tried wrapping my head around that, but I couldn't intuitively understand what that even means.

Note that while the current answer goes a long way on solving the model for me, it does not actually answer my question - about the meaning of a negative denominator. Hence, at the current state, I will not award that bounty and invite more answers.

$\beta$ appears to be a decision variable, at firm level. Then its value should be determined optimally, under some criterion. So what is the criterion, what is the objective function which should be optimized over the value of $\beta$?

It appears that we should maximize the value of the option $V(\beta)$

$$V(\beta) = \frac{-c + q(\theta)(1-2\beta)J}{\rho - q(\theta)\beta}$$

The first derivative is (I will dispense with $\theta$)

$$\frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\cdot[\rho - q\beta]+q\cdot[-c + q(1-2\beta)J]}{[\rho - q\beta]^2}$$

$$\implies \frac {\partial V(\beta)}{\partial \beta} = \frac{-2qJ\rho -cq + q^2J}{[\rho - q\beta]^2} = \frac{q}{[\rho - q\beta]^2}\cdot (-2J\rho -c + qJ)$$

$$\implies {\rm sign}\frac {\partial V(\beta)}{\partial \beta} = {\rm sign}\big\{(q-2\rho)J-c\big\} \tag {1}$$

So $V(\beta)$ is monotonic in $\beta$. Therefore, depending on the values taken by $J,\rho, c, q$, it will be optimal to set $\beta$ equal to either $0$ or $1$.

Assume now that $\partial V(\beta)/\partial \beta >0$. This requires that

$$(q-2\rho)J-c > 0 \implies q > \frac {c}{J} + 2\rho \implies q > \rho \tag {2}$$

Then $\beta^* =1$ and

$$V^*|_{\beta =1} = \frac{-c - qJ}{\rho - q} = \frac{c + qJ}{q-\rho } > 0$$

given $(2)$.

In turn, assume that $\partial V(\beta)/\partial \beta <0$. This requires that

$$(q-2\rho)J-c < 0 \implies q < \frac {c}{J} + 2\rho  \tag {3}$$

Then $\beta^* =0$ and

$$V^*|_{\beta =0} = \frac{-c + qJ}{\rho} $$

To have $V^*|_{\beta =0} >0 $ we need

$$\frac {c}{J} < q < \frac {c}{J} + 2\rho \tag {4}$$

to respect also $(3)$. So here a necessary restriction on the parameters is imposed, in order to obtain a positive value for $V$.

Combining, if we impose the restrictions

$$q > \rho, \;\;\; q > c/J$$

we allow for either $\beta =1$ or $\beta =0$ (it will depend on the exact parameter values), while at the same time guaranteeing that $V>0$ in all cases.
This is not unusual in economic models -i.e. the need to restrict the parameters in such a way so as for the model to provide solutions that are economically meaningful. And we need $V$ to be positive, because otherwise the vacancy would not exist in the first place, as I understand the set up.


Leveraging private capital If an ultra high networth individual, like Bill Gates, Mark Zuckerberg, Carlos Slim, Paul Allen, Elizabeth of Windsor, Amancio Ortega,  Liliane Bettencourt, Stefan Persson, Bernard Arnault etc. would leverage their fortune in the way a bank does with savings, would that multiply their wealth?:

If an UHN individual owns their own 'bank' and puts their fortune in it as savings, they could create e.g. 10 x their fortune in loans.

Its legal to borrow and lend money, but if you become anything close to a bank, (take deposits, make loans, take risks, make a business out of it) then you will be regulated. You will be insured by the FDIC, and since you are insured, you will then be required to follow certain guidelines regarding how you asses risk and which risks you can take or not. Moreover, they will not be allowed to lose money in a large scale, so they would have to recapitalize the 'bank' after any of the charitable projects swallows up capital.

So to answer your question, they would not have more wealth, but they would manage a bigger pot of capital with which to invest. However, at some point they will lose the freedom of doing whatever they want. Since its not their money, then they can't do whatever they want.


Are all market demands residual demands to some extent? I’ve been thinking about how the introduction of a new product, say a new substitute, impacts market demand. Intuitively I know that it would decrease demand and increase elasticity. I am curious about the equations.
When thinking about demand shifters and related goods we usually only consider their price in the demand equation because they already exist.
For example - before new entry - we g as the good in question, rg1 related good 1:
(1) Q = $100 - 2p_{g} + 2p_{rg1}$
After new entry, would the equation be something like:
(2) Q = $90 - 3p_{g} + 2p_{rg1} + p_{rg2}$
So, the constant falls and the own slope increases in magnitude. My questions are as follows: (1) For equation (1), can we think of the 100 as the amount consumed given what you consume of the related food? (2) Can we think of a given market demand as a sort of residual demand given the existing products in the market? This is similar to how we think of things with monopolistic competition. I was just wondering if it was implicitly the same for the market as a whole. Thanks.

I agree, in the real world all markets are residual. unlike the theory laboratory where a product is clearly distinct from all other products, in reality there are always substitutes, and there are fine lines between one product and another.


New-Keynesian Model: Log-linearizing the firm's FOC In Gali's book (chapter 3), the FOC of a firm is given by:

$$(\sum_{k=0}^\infty \theta^k E_k(Q_{t,t+k} Y_{t+k|t} (P_t^*/P_{t-1} - \alpha MC_{t,t+k} \beta_{t-1,t+k}))) = 0 $$

Basically, the first order condition for maximizing a firm's profit. $0< \theta <=1$, $\alpha$ is markup over competitive price. MC is marginal cost. And $beta = P_{t+k}/P_t$.

At the zero inflation steady state (the point we're supposed to linearize this around),

$P_t^*/P_{t-1} = 1$
$beta = P_{t+k}/P_t = 1$
$P^*=P_{t+k}$
Thus Y is constant. $Y_{t+k|t} = Y$ and so is MC.
Q is stocastic discount factor and equals $B^k$ around steady.
Similarly, Marginal cost (MC) is the reciprocal of the markup. i.e. $MC = 1/\alpha$

I need to expand this lovely expression. The problem I'm having is that if I do an expansion around the steady state, everything seems to be cancelling out, leaving me without anything that looks like the intended solution to the problem.

$$(\sum_{k=0}^\infty \theta^k E_k(Q_{t,t+k} Y_{t+k|t} (P_t^*/P_{t-1} - \alpha MC_{t,t+k} \beta_{t-1,t+k}))) = 0 $$

Log linearize around the zero inflation steady state.

$$p_t^* - p_{t-1} = (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k} - p_{t-1}) $$

Where $$\widehat{mc}_{t+k|t} \equiv {mc}_{t+k|t} - mc$$
In other words,
$$p_t^* = \gamma + (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k}) $$
where $\gamma \equiv log \frac{\epsilon}{\epsilon + 1} $

If I remember correctly, this model uses Calvo pricing, so inflation is just from wage stickiness. So if you set $\theta = 0$ (no price stickiness)
$$p_t^* = \gamma + mc_t + p_t$$

See what you can work with from there.

Edit: If you're wondering how this is done, then "An Old Man in the Sea"'s resource in the comments is helpful, in sections 3.2 and 3.3. It will show how to do the First-Order Taylor Expansion to derive the result.


Why is fractional reserve banking allowed? I learned about banks' fractional reserves and that revolted me. How is that not considered to be fraud?

The idea doesn't seem too sophisticated, on the contrary, it sounds very simple: the bank just loans money they don't even have and charges an interest rate (which is not low at all) on that virtual money. When I go to a bank to take out a loan, they will just type the amount of money in my account (but the money does not exist) and I will introduce the money into the economy (when I use that money to buy something).

Why are fractional reserves not considered fraud?

I recommend a careful read of the Wikipedia page for Fractional-reserve banking because it seems you are confused about how fractional reserve banking works.
Banks only lend out money they have under fractional reserve banking and, in fact, strictly less than they have. Those funds are a mix of deposits, funds from other debt issuance, and bank equity. They don't lend out funds they don't have and they don't simply credit accounts at will in the way you suggest. When they credit a borrower's account with funds, they have to debit those funds from elsewhere on their balance sheet. So, lend-able funds go down as borrower's funds go up, and then the resulting loan is a new asset of the firm. While they do create money this way, they do so at the expense of these lone-able funds so the loaning out doesn't create new assets for the bank.


Can an overall improvement in technology make the owners of one factor worse off? In a closed two-factor economy, suppose there is a biased technological improvement that increases the productivity of both factors, but one more than the other. Can this reduce the aggregate returns (and not just the share of returns) to one factor? If so, how is that consistent with the usual rules of factor pricing? Doesn't (or can't) the marginal productivity of both factors rise with their respective total productivity?

Finally, if biased technology improvement can result either in increased payments to both factors (in real terms) or in increase to one and decrease to the other, how might one characterize quantitatively the conditions for (partially) immiserizing technological growth?

1x2 model

Consider a mode where production of a single good is given by a constant returns to scale CES production function:

$$Y=A(\alpha L^\rho +(1-\alpha)K^\rho)^{\frac{1}{\rho}}$$

where the elasticity of substitution between the two factors is

$$ \sigma = \frac{1}{1-\rho} $$

It can be shown that the marginal product of labour (equal to the real wage in a competitive economy) is

$$ MP_{L} = A\alpha \left( \frac{Y}{L}\right)^{1-\rho} $$

To understand the consequences of biased technical change on real wages, we need to look at the three components separately.

For simplicity, let's assume a fixed supply of labour, $\bar{L}$. Now, a technical change that improves productivity of both factors, but more that of capital is such that $A$ increases (the neutral component), and $\alpha$ falls (the bias). On top of this, any increase in productivity (regardless of its bias) will increase $Y$. Yet, labour is fixed. Then, you have that the three determinants of the real wage change. In particular, two are increasing and one is falling. The change in the real wage - and the nature of growth (enriching, immiserizing, neutral) depend on how these play-out. In particular, the greater the bias, the smaller the neutral technical change, and the higher the elasticity of substitution (higher $\rho$), the more likely you will have immiserizing growth. If $\rho = 1$, the third term disappears altogether.

2x2 model

You can get similar results by having a 2x2 model (2 goods and 2 inputs). For instance, say that capital productivity increases more than labour productivity. Furthermore, say that the elasticity of consumption among the two goods ($Y_{a}$ and $Y_{b}$) is very high. For example,  individuals are willing to consume a lot of the former and little of the latter. If the production of $Y_{a}$ is very intensive on capital, and the technology has a high degree of substitution between capital and labour (so firms could produce by expanding capital and not much expanding labour), then you will end up with a reduction in aggregate demand for labour. This will lead to a fall in the real wage, and an absolute loss of income for workers (i.e. immiserizing growth).

Thus, in the end you can get any type of behaviour depending on:

technology used in either good, and in particular, the degree of substitution between factors
preferences of consumers, and in particular, the degree of substitution between goods

Think of robots increasing the productivity of capital by much more than that of workers. If consumers are happy to rebalance their consumption bundles toward more robot-produced goods and away from labour-intensive goods, then the wage rate will fall, affecting workers.


The use of preference relations in choice theory and the $\succsim$ symbol At least as from Edgeworth and Pareto we think about utility in mathematical terms. My question twofold

(i)  about the start of the usage of binary relations to model preferences in economics, and hence the start of the axiomatic utility representation exercise.

(ii) about the start of usage of the symbol $\succsim$ to represent weak preference.

About (i) the earlier reference I could find is Wold's (1943): "A synthesis of pure demand analysis I-III" pioneering work on utility representation; and about (ii) is Herstein and Milnor (1953): "An axiomatic approach to measurable utility" work on expected utility.

Does anyone know about earlier uses??

Ant further comments, references or ideas are most welcome.

About your second question, Gérard Debreu, in his book Theory of Value, says in the notes of chapter 1 that the $\succsim$ notation is due to Herstein and Milnor. So, probably, this is the origin of symbol for preferences.


Does SARP imply WARP? and GARP imply SARP? It’s obvious that WARP does not imply SARP, since WARP does not rule out cyclic choices, whereas SARP does.

The term “Strong” axiom suggests that it encompasses the “weak” axiom. But this is not obvious to me from the definition.

Does any choice structure that satisfies SARP also satisfy WARP?

Similarly, the "generalized" axiom of revealed preference (GARP) seems to encompass SARP, but again, I can't see this from the definition.

  WARP: If $x,y\in B$, and $x\in C(B)$, then we cannot find a $B_2$ such that $x,y\in B_2$ and $y\in C(B_2)$ but not $x\in C(B_2)$.

  SARP: Assume for all $B$ the choice $c(B)$ is only one element. If $x_i,x_{i+1}\in B_i$, and $x_i = c(B_i)$, for all $i\in \{1,N-1\}$, then $x_1=c(B_1)\notin B_N$.

  GARP: if $p_i\cdot x_{i+1}\leq p_i\cdot x_i$ for $i\in \{1, N-1\}$ and $p_N\cdot x_{1}\leq p_N\cdot x_N$, then those inequalities must be equalities.

(GARP specifically assumes $p$ are price vectors and $x$ are consumption bundles, whereas the other two axioms apply to any type of choice structure.)

Yes, SARP implies WARP. Since all choices are singletons under SARP, we abuse notation and write $C$ as a function that takes alternatives as values.
We prove the contrapositive, every violation of WARP in a setting where choices are unique give rise to a violation of SARP. So assume there are sets $B$, $B'$ containing $x$ and $y$ such that $c(B)=x$ and $C(B')=y$. Letting $B_1=B$ and $B_2=B'$, you get a direct violation of SARP (I take it you require $x_i\neq x_{i+1}$ in the definition of SARP).

In the context of budget sets, SARP implies GARP. Let $B_i=\{x\mid p_ix\leq p_ix_i\}$. Then SARP says that if $p_i x_{i+1}\leq p_i x_i$ but $x_i$ is the unique choice from $B_i$ for $i\in\{1,\ldots,N_1\}$, then $p_N x_i>p_N x_N$. So the case $p_N x_i\leq p_N x_N$ can never occur; GARP cannot be violated.

Now GARP as usually formulated is a condition on data sets consisting of price-choice combinations. It is not required that the same price system leads to the same choice, so GARP does not imply SARP.


What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy? In an experimental setting, how could you effectively incentivize the subjects to not to adopt mixed strategy?
I would like to re-emphasize that the question in concern is "how to prevent people from using mixed strategy" such that only pure strategies are adopted. The mixed strategies must be theoretically adoptable, and we have a mechanism to force people to think about pure strategies only. In a non-repeated game, if a person plays strategy "H", in general you don't know if it is pure strategy "H" or a mixed strategy with positive probability on "H". The current answers are very useful and well-prepared; however, what I am always looking for is a proven (either theoretically or experimentally) method that constrains the choice set from a mixture space to a doublet.
Technically, in a game theory experiment, the set of alternatives is a mixture set. I want to restrict the set of alternative to two objects, $\{H, T\}$, only.
Of course, you could just post a title of a paper which includes an incentive mechanism or an experimental design. This can be a perfect answer despite of its length. Opinions are welcome but opinions are not answers.

Let's consider a one-period game where the first player choose $H$ or $T$. The game satisfies the following two conditions:

The equilibrium is unique at $\frac{2}{3} H+\frac 1 3 T$.

If the player is restricted from choosing mixed strategy, $T$ becomes the optimal choice for player 1.

I've chatted with a few people and they all believe that it makes sense to consider those two cases separately; by "separately" they mean that, a game problem restricting the player from choosing mixed strategy also make sense, we can compare $H$ and $T$ in isolation of those mixed strategies. What is the philosophy behind this?
I, on the other hand, believe that the latter case doesn't make sense in real life: one cannot just abandon the mixed strategy completely. In an experimental setting, how could you prevent the players from adopting a mixed strategy?
One awkward way to impose this restriction, I think, is to instruct the player that it is a super game repeated for 10 times (or even life-time); you can only choose the same $H$ or $T$ for the lifetime and never change your choice. But this way, people can still think about mixed strategy.

Clarification: I am not advocating that people must use mixed strategy only. I am just saying that, I cannot find a good experiment that restricts people from including mixed strategies in their menu. Similarly, I cannot find a good experiment that restricts people from including pure strategies in their choice sets. So I think, in analysis, we must think about all strategies together, and it is pointless to consider pure or mixed strategies only.
PS: Assume the player is rational, as this is economics SE.

If in equilibrium, a player "chooses a mixed strategy" that plays $H$ and $T$ with positive probability, $H$, and $T$ must be both optimal choices. It is a standard result that for a (subjective or objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected utilities being linear in probabilities. So the problem you mentioned can never occur.
So expected utility maximizers do not ever have a strict incentive to randomize, which raises the question of how one can interpret Nash equilibria in mixed strategies. One popular interpretation is that a player's mixed strategy is really representing a shared probabilistic belief other players have over the choices of that player. One can then define Nash equilibrium as a consistency condition on beliefs, though it is not clear why this condition should obtain in practice. Oner idea, Harsanyi's purification idea is that with a certain probability the player has private information that makes $H$ or $T$ uniquely optimal, but for the other players it will look like choosing $H$
and $T$ randomly. Especially in zero-sum games, there is also the idea that players deliberately use randomization devices such as coins so that their plans cannot be found out. Lastly, there is the "mass action" interpretation that can already be found in Nash's thesis. According to this interpretation, you are randomly matched with players from a huge population that play a fixed pure strategy, and the mixing probabilities represent the population fractions playing each pure strategy.


Quantifying economic mobilization Total war requires mobilization of the economy.
The degree of a nation's mobilization can be quantified in two major ways:

Negatively: as, e.g., the reduction of "non-essential consumer production" --- describing the privations of the population.
Positively: as, e.g., the increase in "war production" (food and military heavy industrial equipment?) --- describing the increased fighting capability.

What was the degree of mobilization of major World War (1&2) combatants by year?

Motivation:

I have seen claims that the degree of "negative mobilization" of Germany was higher in WW1 than in WW2 because Hitler wanted to avoid the collapse of the home front that lead to the defeat in the WW1 (this claim seems to suggest that Hitler did not quite believe himself in the Stab-in-the-back myth).
The common theme in the Soviet propaganda is that USSR won the war because the Socialist Economy is best suitable to mobilization. Indeed, the "negative mobilization" was probably the highest in the USSR (of course, "positive mobilization" is more important to war fighting).

Complications:

It is probably impossible to find any reliable numbers about USSR.
Soviet economy has been always geared towards war production, so the changes due to war were not as dramatic as in other countries.
Similarly, German "positive mobilization" started way before 1939.

PS. I asked this on History, got nothing.

It is my impression that the subject has seen its fair share of scholarly research already. Check the references in the wikipedia article for WWW-II (the article itself is not very good), and maybe the book The Economics of World War II: Six Great Powers in International Comparison.


Non-unique utilities in a competitive equilibrium In continuation to this question: Uniqueness of utilities in competitive equilibrium I think I found a simple example in which the utilities in equilibrium are not unique, and wanted to check if it is true.

This is an exchange economy with two goods and two agents with utilities:

$$u_1(x,y)=2x+y$$
$$u_2(x,y)=3x+y$$

Initially, each agent has 1 unit of $x$ and 5 units of $y$.

Suppose $p_y=1$. Then, whenever $p_x\in[2,3]$, the economy is in equilibrium, since agent 1 wants to sell all his $x$ endowment and agent 2 wants to buy all $x$. So the equilibrium utilities will be:

$$u_1 = 5+p_x$$
$$u_2 = 11-p_x$$

This shows that utilities are not unique.

A. Are my calculations correct?

B. If they are correct, what is going to happen in a real competitive economy? Since the equilibria are essentially different, which of them will prevail?

EDIT: After the price theory book again, I am not sure about my calculations. The book defines a competitive equilibirum as a situation in which the sum of demands exactly equals the sum of endowments. The demand of an agent is defined as the optimal bundle he can afford using his initial endowment. Now:

Whenever $p_x<3$, the demand of agent 2 is $(1+5/p_x,0)$ - he wants to buy $x$ with all his endowment of $y$. Since the total amount of $x$ is 2, this implies $p_x\geq 5$ - a contradiction.
Whenever $p_x>3$, the demand of both agents is $(0,5+p_x)$ - both want to buy only $y$, which is obviously not an equilibrium.
Hence we must have $p_x=3$. Now agent 2 is indifferent between buying or selling $x$, while agent 1 wants to buy only $y$. Hence, the demand of agent 1 is $(0,5+p_x)=(0,8)$.
So, in this case there is a unique eqilibrium in which $p_x=3$, agent 1 gets $(0,8)$ and agent 2 gets $(2,2)$; the utilities are 8 and 8.

Now I am confused: is the utility vector unique or not?

This answer is in response to the edited (2015.10.16 ca. 12:30 CET) question.

I think the concept of equilibrium means that given the circumstances you have no incentive to alter your behavior. (This is pretty much based on the Nash-equilibrium.)
 In this situation equilibrium consists of a price and an allocation of goods. Given the price everyone makes a trade decision. The decision has to be feasible both with respect to the budget and the physical availability of the goods, i.e. you cannot buy goods that do not exist. We are in equilibrium if everyone's decision was optimal (for themselves).
In your situation for any $p_x \in [0,2]$ there exists an equilibrium allocation, where consumer 2 bought the good $x$ from consumer 1 for $p_x$ units of good $y$. There is no better feasible decision for either of them.

If you were to adhere to the strict/dogmatic interpretation there would be no equilibrium at all, because for any $p_x <3$ consumer 2 wants to buy an infinite amount of good $x$ and for any $p_x > 2$ consumer 1 wants to sell an infinite amount of good $x$. Since there is no $p_x$ such that $p_x \leq 2, \ p_x \geq 3$ you could not have an equilibrium with finite amounts.


Proof that two utility functions $u(x_1,x_2) \text{ and } v(x_1,x_2)$ with the same MRS represent the same preference I am having some difficulty proving that two utility functions with the same marginal rate of substitution represent the same preference. I think I need two use some argument related to strictly increasing monotonic transformation, but I am really lost. If someone could help me, it would be great.
Thank you very much!

I think that you need to also assume that the utilities are non-decreasing in the goods. That is, if $x'_1>x_1$, then $u(x'_1,x_2)>u(x_1,x_2)$.
The utility functions having the "same preferences" means
$\forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) > u (x'_1,x'_2) \text{ iff } v(x_1,x_2) > v(x'_1,x'_2) $
So let's assume $u(x_1,x_2) > u (x'_1,x'_2) $. Take a curve going from $(x_1,x_2)$ to a point $(x'_1, x''_2)$ by changing $x_1$ to $x'_1$ and having the second coordinate change according the MRS so that this curve has constant value of $u$ [1]. Then since $u$ is constant along this curve, $u(x'_1,x''_2) = u(x_1,x_2)$. Since $u(x_1,x_2) > u(x'_1,x'_2)$, $u(x'_1,x''_2) > u(x'_1,x'_2)$, and from the non-decreasing utility property, we have $x''_2 >x'_2$.
Since this curve was defined by the MRS of $u$, and the MRS of $u$ is equal to that of $v$, we also have that $v(x'_1,x''_2) = v(x_1,x_2)$, and so  $v(x_1,x_2) > v(x'_1,x'_2)$
[1] This can be done, for example, by taking $C = (x_1+ t, x_2+\int_0^tMRS_{x_1x_2}dt)$. Then taking $t=x'_1-x_1$, we get the point $\left(x'_1, \int_0^{x'_1-x_1}MRS_{x_1x_2}dt \right)$


testing the relationship between stock price and macroeconomic variables I'm in the midst of preparing data to test the relationship between stock index and macroeconomic variables. The variables include m1, cpi, ipi, interest rate, exchange rate. I have some question below:

Is seasonality adjustment on the data required? If one variable is seasonality adjusted,  is that all variables in the models have to be seasonality adjusted as well? If all are not seasonality adjusted,  will there be a problem?
Would it be alright if all data are not inflation adjusted,  that is nominal data?  And if one variable  are inflation adjusted,  is that all must be so to be consistent?
The base period for CPI and IPI is not the same. Is it a requirement for the base period to be the same in the model ?

Is seasonality adjustment on the data required? If one variable is seasonality adjusted, is that all variables in the models have to be seasonality adjusted as well? If all are not seasonality adjusted, will there be a problem?

Seasonal adjustment of data is required in your VAR/VECM, you can do this by employing seasonal dummies (dummies 11 if you have monthly data and 4 if you have quaterly data).

All your data has to initially be seasonally unadjusted, because the inclusion of dummies will affect all the data that you include in your model. You will end up having skewed estimates of your already adjusted variables influence on a given process.

  Would it be alright if all data are not inflation adjusted, that is nominal data? And if one variable are inflation adjusted, is that all must be so to be consistent?

Again if you want your estimates to be consistent you have to make sure all of your monetary variables are inflation adjusted. practically you may want to use real exchange rate instead of nominal exchange rate data for this.

  The base period for CPI and IPI is not the same. Is it a requirement for the base period to be the same in the model ?

No this is not necessary and your estimates will be fine. Just remember to keep this in mind when your run your model.

Hope this helps


One-shot deviation principle for infinite repeated games and dynamic programming In a context that future return is discounted by a constant parameter, one-shot deviation principle holds for both repeated games and dynamic programming.

Because, in repeated games, a one-shot deviation refers to one history, so on equilibrium path, a one-shot deviation could produce a play that differs on more than one stages from the original equilibrium path.

Is it true for the sequence of state variables and control variables in dynamic programming? In other words, can a one-shot deviation generate an aforementioned sequence that differs for more than one stage?

A deviation (one-shot or not) can certainly generate a sequence that differs from the optimal one for an arbitrary number of periods.

You could treat a dynamic programming problem as a repeated game between one player and chance. The one-shot deviation principle should then carry over from repeated games to dynamic programming.


Limits to Growth: Is environmental collapse by 2030 a likely scenerio? According to a simulation from 1972, the results of which were published in 'The Limits to Growth' environmental collapse is due mid 21st century. The Guardian reports that Dr. Graham Turner reran the simulations using recent data and the collapse is actually due sooner. Here is the referred paper by Dr. Graham Turner:
Quoting from his profile by the Guardian:

Dr. Graham Turner is principal research fellow at the University of Melbourne's Melbourne Sustainable Society Institute. He has a background in applied physics and his current work involves whole-of-system analysis on the long-term physical sustainability of the environment and economy.

As economists do you think Dr. Turner is right?

We know one core element of the Guardian article is nonsense - the discussion of Peak Oil.

The Peak Oil lament, fashionable a decade ago, was that we didn't have enough oil, and that was going to become a major problem very soon.

However, we know from climate science that the reverse is true - we have far too much fossil fuel reserves (disclosure - that's a paper by my colleagues). Our challenge isn't that we won't extract enough to continue economic growth - it's that we'll extract too much and cause a major environmental collapse. Turner's report makes a significant show about the age of cheap and easy oil being over (pp 11,12) - and that's a prediction that was made to look very silly, quickly, as the report's publication in August 2014 happened just as the 18-month oil-price crash caused by huge global over-supply was beginning.

Is environmental collapse possible? Yes.

Are we causing huge ongoing damage to the environment? Yes.

Were all of the significant predictions of the Club of Rome correct? No. Not only has geometric population growth stopped, but even arithmetic growth is in decline - and the growth wasn't capped by disaster, but by better healthcare, better access to contraception, and more education and emancipation of women. Food production per capita has way exceeded predictions. And the renewable alternatives to fossil-fuel energy are already economic and scalable for static applications, and becoming so for transport too.

Are we still on a trajectory of yesterday's brown economy, that will require a trajectory change if we are to avoid environmental collapse?
Yes.
Do we have enough evidence to say that the date for that collapse is 2030?
No.
Do we have the technical means to transition to a modern, clean economy and avoid that collapse?
Yes.


Convexity of Walrasian Demand A follow up to my previous question:

  Given $u(x) = (x_1-b_1)^\alpha (x_2-b_2)^\beta(x_3-b_3)^\gamma$

and ending up with from a maximization that

$$x(p,w) = (b_1, b_2, b_3) + (w - (b \cdot p))\left(\frac{\alpha}{p_1},\frac{\beta}{p_2},\frac{\gamma}{p_3}\right)$$

How do I show that $x(p, w)$ is convex? I already showed that it is homogeneous of degree zero and satisfies Walras' Law. The book from Mas-Colell says that the solution is obvious, but I have not gotten anywhere with it. My setup for showing convexity is that I want to show:

$$x(tp + (1-t)p', tw + (1-t)w') < tx(p,w) + (1-t)x(p',w')$$

But I'm not finding anyway to simplify this expression to show it is true. Any help would be appreciated, as I feel like I am missing something very obvious.

I am not sure whether the question asks about the function being convex, or simply the demand correspondence/function being a convex set. The latter is often asked and if that is the case here is an answer.

This is a general answer to your question which may be more useful to you and future readers than a specific one.

The following theorem holds in general:

Theorem: If preferences are convex, then the marshallian demand x(p,w) is a convex set for every p>>0 and $w \ge 0$.
Note: p>>0 here means that every element of the price vector is strictly positive.

Proof:

The Budget set $B(p,w) \equiv \{ x \in \mathbb{R}_+^n:px \le w \} $ is a convex set.

If two points $ x, x' \in x(p,w )$, then x ~ x' (the consumer must be indifferent between the two as they are both part of x(p,w), i.e. his best choice. If one of them would be better than the other, then only the best one would be part of x(p,w) as the marshallian demand is derived from maximization.)

For all $\alpha \in [0,1]$ we have that $\alpha x + (1-\alpha)x' \in B(p,w)$ by convexity of B(p,w). Furthermore, $\alpha x + (1-\alpha)x'\succeq x$ by the convexity of preferences. Therefore $\alpha x + (1-\alpha)x'\in x(p,w)$
Q.E.D.

Your preference relation is convex, so the theorem applies and your set x(p,w) is convex. Note that a preference relation is (strictly) convex if and only if the utility function  is (strictly) quasiconcave. Your utility function as far as I can see is concave, which implies it is quasiconcave, which implies convex preferences.

Further note that if preferences are strictly convex, then x(p,w) is single-valued for every p>>0 and $w \ge 0$. Hence in that case x(p,w) would be a convex set as a point is a convex set.


Has anyone ever done an analysis of the flexibility of the demand curve for summer camps? I'm trying to support an argument being made with respect to camp fee changes, and I would need to know if anyone has ever come up with a "rule of thumb" for demand drop due to price increases for a standard 1-week sleep-away summer camp?  No fancy expert training, or limited-access development, just standard summer camp  activities.

It seems not, judging for the results of this and this search (but you might want to take a look yourself).

There are some more informal analysis of the market for summer camps (e.g. here) and some organisations recollecting some form of data (e.g. here). The former is interesting because it shows, unsurprisingly, that the services proved by a summer camp are very heterogeneous. They are also targeted to different audiences. This means that the good itself is heterogeneous, so thinking of a unified "demand for summer camps" might be problematic.

Still, there are studies estimating demand for such heterogeneous goods. Here (subscription-wall) is an example which perhaps might be related to your interest (not the least because it uses the word camp too, which is how I found it, btw). The study estimates the demand for camping park's amenities. The key is that these amenities differ across camping sites (e.g. electricity access, natural beauty, etc). The authors estimate, among other things, the "price and income elasticity" for 48 camping parks in a region of Canada. The main finding with respect to elasticities are:

  ... it is immediately evident that there exists no definite pattern with regard
  to the price elasticity; there appear to be almost as many cases of
  inelastic as there are instances of elastic demand. [...] Since our results also indicate a negative income elasticity, higher user fees would represent a regressive form of taxation. The principle of user pay may still be justified on efficiency, not equity, grounds.

If you want to do some form of econometric analysis of the market for summer camps, I think this method is a possible approach.


Causes of income inequality in the US I read an article in BusinessWeek that talked about income inequality in the US.

The article attributed the difference to the Intel chip that came in 1971. Since that time big hi-tech companies have grown in US, which caused income growth among wealthy people and also didn't lead to a relatively large growth in jobs because a lot of work was outsourced to places like China.

To what extent is this explanation valid in the US... or is it insignificant in accounting for the income differences between the rich and the poor?

The reason you bring forward belongs to technological directed change, which is regarded one of the main explanations for wage growth differentials. Keep in mind it's not exactly the way you're phrasing it: The growth in tech companies rewards people who are skilled well for their kind of jobs (as opposed to "wealthy people").

Card and DiNardo have a nice summary article on the issue:

  The recent rise in wage inequality is usually attributed to skill-biased technical change (SBTC), associated with new computer technologies. We review the evidence for this hypothesis, focusing on the implica- tions of SBTC for overall wage inequality and for changes in wage differentials between groups. A key problem for the SBTC hypothesis is that wage inequality stabilized in the 1990s despite continuing ad- vances in computer technology; SBTC also fails to explain the evo- lution of other dimensions of wage inequality, including the gender and racial wage gaps and the age gradient in the return to education.


mean variance utility function for risk loving person Let $X$ denote wealth. The mean variance utility for a risk-averse person is given by $E(X)-\frac{r}{2}Var(X)$ where $r$ is degree of risk-version. $r=0$ implies that person is risk-neutral. How does the mean variance utility function look like for a risk-lover?

For a risk lover, r is negative. Making the term $-\frac{r}{2}Var(X)$ positive. Thus, as variance increases, $E(x)-\frac{r}{2}Var(X)$ increases, which means their utility increases.


How long into the past can we trace the effect of institutions? In one of my courses we recently went through several institutional economic papers (The colonial origins of comparative development, The Slave Trade and the Origins of Mistrust in Africa, The Long-Term effects of Afirca's slave trades, ...). In these papers the authors are able to (or at least claim to) find significant effects of historical events, developments and so on, that still have an effect on today's institutions or economic outcomes. Some of the historical causes go back a few hundred years.

This is where my question comes in, was there an attempt made or is there some vague understanding of how long we can see significant effects? Obviously, there won't be an exact number, but is there some research going on, that asks about this? Or what are the longest time spans researchers have convincingly traced back effects of institutions, culture and so on?

Institutions are likely having a permanent effects as they create path dependency which implies that even temporary shocks would have permanent effect. At least that is what is argued by many developmental economists including Acemoglu and Robinson (see Why Nations Fail). Furthermore, they claim that this hold not just for development itself but that the political institutions are themselves path dependent and reinforcing.

However, permanent effects and path dependency should not be confused for some sort of historic fatalism. Just because some countries had (or were imposed on them) extractive (i.e. in Acemoglu and Robinson's view 'bad') institutions  does not mean that they are forever doomed to be underdeveloped. It just means that the loss of welfare that occurred as a result to the 'shock' (i.e. adoption of extractive institution itself) wont be recovered.

For example, below I simulated an example of an one time adoption of extractive institution. I tried to simulate the sort of extractive institutions imposed by the king Leopold II on Congo which arguably both affected the production capabilities of Congo and its growth.

Below you can see that the effect is permanent even if country will eventually grow and attain the level of output it had before the extractive institution was implement.

Even if the old institution will be reversed and replaced by better one the effect of the old institution will be still permanent as you can see the simulated examples of growth where the old extractive institution was replaced by inclusive (left picture) vs where extractive institution was never in place (right picture) but inclusive was implemented show that the GDP per capita will be permanently affected.

Hence to sum up, if you buy into the theory that development is path dependent which Acemoglu and Robinson do then the answer is that the effect of those institutions is permanent. Thus you can trace the effects as back as recorded history goes. Even some ancient event such as the Bronze Age Collapse or the fall of Roman Empire (or as Brian mentions in his +1 comment even ancient road network patterns) and changes in institutions caused by those have still significant effect today as if we could avoid them today's output would arguably be significantly different.


How to analyze small human activities from the economical standpoint? If a company has project A and project B it can decide what is more profitable. The math in this example is straightforward.

If human has activity A and activity B, how can someone calculate what is more efficient from economical standpoint?
For example, take a taxi or use public transit to get from A and B. If human takes taxi, it will cost him 18 dollars, he or she will spend 20 minutes mostly sitting. If human takes public transit, he or she will spend $2.5, it will take 45 minutes including 10 minutes walking.

How to account "energy", "motivation", "missing time", etc?

A useful framework for analysing personal activities involving both monetary and time costs is a household production model. A simple example of such a model (adapted from Chiappori & Lewbel (2015) pp 411-2) is below. Note that a "household" could be one person (the assumption that members of a multi-person household cooperate to maximise their joint utility can be regarded as an optional extra of the model).

The model distinguishes between commodities, which directly affect utility, and goods, which affect utility only indirectly.  The taxi ride, for example, would be a good, not a commodity, because it is just a means of getting from A to B. A household is assumed to maximise utility which is a function of commodities:

$$max\quad U(Z_1,\dots,Z_m)$$

For each commodity there is a production function:

$$Z_i=f_i(\mathbf{x}_i,T_i)$$

Here $\mathbf{x}$ is a vector of goods and $T$ is time taken by a household member in producing the commodity using the goods. Production here should be understood broadly: it would include, for example, using a combination of public transport and time to "produce" a journey from A to B. Leisure activities such as visiting an entertainment can also be produced in this sense.

The goods used in producing $Z_i$ can be bought at prices $\mathbf{p}_i$, and income can be earned, over any chosen time period, at a wage $w$ (this is a simplification, eg because working hours are often fixed by an employer). Considering a period of, say, a week (and simplifying again by ignoring saving and storage of goods between weeks), the monetary constraint is ($T_w$ is time worked):

$$\Sigma_{i=1}^m \mathbf{p}_i\mathbf{x}_i\leq wT_w$$

The time constraint is ($T_{tot}$ is total time available):

$$T_w + \Sigma_{i=1}^m T_i \leq T_{tot}$$

The two constraints can be combined (by substituting for $T_w$) as:

$$\Sigma_{i=1}^m \mathbf{p}_i\mathbf{x}_i \leq w[T_{tot} - \Sigma_{i=1}^m T_i]$$

This formalises the trade-off between money and time: more time used as an input to commodities means less time available for wage-earning work and so less income with which to purchase goods inputs to commodities.

One way to bring personal energy and motivation within the model would be via the value assigned to $T_{tot}$.  Starting from 168 hours in a week, deductions might be made for sleeping time, quiet relaxation, etc to arrive at what might be termed effective time available for activities, with smaller deductions for a more energetic and motivated person.


Do countries with centrally-planned economy have Entrepreneurship as a factor of production? Let's don't count self-employed people without other employees (like "lone wolf" craftsmen) and people who have an illegal business as entrepreneurs. By centrally-planed economy I mean an economy without legal private sector (or where its contribution to GDP is so small, that it can safely be ignored).
If the answer is "yes", then how? For an example, can we count a state-owned factory director as an entrepreneur?

This is hard to answer because it is disputable if entrepreneurship is a factor of production just to begin with. There are economists who would argue that any entrepreneurial activity already falls under labor/human capital.
Setting the caveat above aside there are two major views on how entrepreneurship as a factor should be treated (see here). In one view entrepreneurship cannot be divorced from ownership of assets since as argument goes people can take entrepreneurial risk only if they actually have 'skin in the game'. In centrally planned economy asset are collectively owned state and all market risks are socialized. Under this view there can be no entrepreneurship in centrally planed economy. Moreover, managers would not count under such view either.
A second view would treat entrepreneurship as purely an intellectual activity of combining different factors of production (to be clear the previous view also agrees on this part but they add assets as additional requirement). For example, in Fed St. Louis series for students the entrepreneurship as a factor of production is defined as:

The fourth factor of production is entrepreneurship. An entrepreneur is a person who combines the other factors of production - land, labor, and capital - to earn a profit.

The above definition does not have asset ownership as a part of requirement and manager's work here could be considered to be an entrepreneurial factor. The additional important requirement is that this is done to earn profit but as far as I understand here it does not matter if the payment is considered profit or wage in accounting manner but rather from economic perspective. Furthermore, generally speaking profits can exist in centrally planned economies, to the extent these are shared with the managers in some explicit or implicit way, you could argue that under this definition entrepreneurship can exist in centrally planed economy.
This being said, centrally planned economies most likely don't have much of it. According to highly cited work of Audretsch (2007):

Measurement of entrepreneurship capital is no less complicated than is measuring
the traditional factors of production. Just as measurement of physical capital, labour,
and knowledge invokes numerous assumptions and simplifications, creating a metric for
entrepreneurship capital has also presented a challenge. Many of the elements that constitute
entrepreneurship capital defy quantification. In any case, entrepreneurship capital, like all
of the other types of capital, is multifaceted and heterogeneous. However, entrepreneurship
capital manifests itself in a singular way—the start-up of new enterprises.

The emphasis was added by me. Hence an entrepreneurship can be measured by the rate at which new enterprises form (and in fact this is the way in which most literature measures it). It goes without saying that historically speaking centrally planned economies are not exactly known for high rates of new enterprise formation.


How should I adjust data for both inflation and foreign exchange rate? I’m not sure if this is typically done, but suppose my data point is in 2013 JPY and I want to express in 2016 USD. Do I apply CPI to express data in 2016 JPY, and then apply exchange rate to 2016 USD? Or do I apply exchange rate to obtain 2013 USD first, then apply CPI to bring forward to 2016 USD? Or is this all redundant?

NO, it is not redundant. Controlling for nominal effects gives you a better picture, and makes more sense on a theoretical level. To transform assets into USD use

$\textit{asset}_{USD}=\frac{s_t P_{t,JPY}}{P_{t,USD}} \textit{asset}_{JPY}$

where the fraction is the real exchange rate, $P_t$ is the respective price level and $s_t$ the nominal exchange rate.

Alternatively, you can work with differences

$\Delta rer =\frac{\Delta s_t \pi_{t,JPY}}{\pi_{t,USD}}$.

Note that the real exchange rate is far from perfect, but it is common practice in macroeconomics. Also, it does not matter which transformation you do first, as its purely multiplicative.


When should a receiver randomize across actions in a signaling game? Suppose there is a signaling game with a finite message space $M$, finite action space $A$, and finite type space $T$. Even simpler, all sender types have identical preferences (the receiver just prefers different actions in response to different types). Can the receiver ever do strictly better by randomizing across responses? When an equilibrium exists where the receiver only takes pure actions?

Ubiquitous summarized my question nicely, "Is it ever the case that the equilibrium with the highest receiver payoffs necessarily involves mixed strategies?"

Let's go with sequential equilibrium.
If you'd like some notation to start with.

$\sigma_{t}(m)$ is the probability that $t\in T$ sends $m\in M$.

$\sigma_R^m(a)$ is the probability that the receiver responds to $m$ with $a\in A.$
$\mu^m \in \Delta T$ gives the receiver's beliefs after observing $m$.

A sequential equilibrium requires $\sigma_t$ give optimal responses given $\sigma_R$, $\sigma_R$ is optimal given $\mu$ and $\mu$ is Bayesian given $\sigma$. This is really the definition of a weak sequential, but there is no distinction in a signaling game.

My intuition says no when there exists a equilibrium where the receiver only plays pure actions, but I've always been horrible with this kind of stuff. Maybe we also have to stipulate that it is not a zero-sum game, but I'm only saying that because I remember players being better off with the ability to randomize in those games. Perhaps this is a footnote in a paper somewhere?

Consider the game below where sender preferences are not identical. I apologize for the low quality. There are three sender types, each equally likely. We can create what I believe is the receiver (player 2) optimal equilibrium only if they randomize upon receiving message 1. Then types 1 and 3 will play $m_2$, creating a separating equilibrium. If the receiver uses a pure strategy in response to $m_1$, then a type 1 or 2 would deviate and make the receiver worse off.

$\sigma_R^{m_1}(a)=.5=\sigma_R^{m_1}(r)=.5$

I think this cannot happen with risk averse senders, risk neutral receiver, and $A$ rich enough.

For example, and to stick to the canonical signaling model, suppose that $A$ is the positive real line and senders' utility $u$ is increasing in $a$ while receiver's have linear utility decreasing in $a$.

(Admittedly, this is only a partial answer as the framework is much less general that the one in your question, so it might not be satisfactory to you. I still provide an argument in case you were ok with these assumptions)

To derive a contradiction, suppose that at an equilibrium $\sigma^m_R(a') > 0$ and $\sigma^m_R(a'') > 0$ for some $a' \neq a'' \in A$. Let

$$a''' \equiv \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') }  a' + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } a''.$$

By risk aversion

$$ u[ a''' ] >  \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a') + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a'').$$
$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a''' ) >  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

Under some continuity assumption, there must also exist

$$ a '''' < a''' $$

such that

$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a'''' ) =  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

So consider $\sigma^m_R{'}$ constructed in the following way

$\sigma^m_R{'}(a') = \sigma^m_R{'}(a'') = 0$,
$\sigma^m_R{'}(a'''') = \sigma^m_R(a'''') + [\sigma^m_R(a') + \sigma^m_R(a'')]$
For all other $\tilde{a}$, $\sigma^m_R{'}(\tilde{a}) = \sigma^m_R(\tilde{a})$

Receivers would prefer $\sigma^m_R{'}$ over $\sigma^m_R$ if it did not alter the signals sent by the senders, because it involves lower expected compensations. But by construction, senders are indifferent between $\sigma^m_R{'}$ and $\sigma^m_R$, so they should send the same signals as in $\sigma^m_R$.  Thus $\sigma^m_R$ cannot be an equilibrium which shows that we cannot have two different actions played with positive probability at an equilibrium.


Equity-efficiency tradeoff examples What could be an example of equity-efficiency tradeoff in healthcare, social protection or defense? For environment I did think of pollution permits that can cause geographical differences in pollution. But for other sectors of government spending I do have a trouble with coming up with an example.

Publicly funded healthcare and social welfare schemes typically serve redistributive functions and therefore can be considered as promoting equity (among the well-off and the poor). At the same time, these programs must be financed through taxation such as income tax that's usually distortive, thereby weighing down on efficiency.


Why does 'government consumption'in NK models not add to utility In almost all NK models public government consumption (like coastal defences) or private government consumption (like primary and secondary education) is by default 'wasteful'. Government consumption as we measure it is however defined as production of goods and services by the government which are used/consumed by households). But it seems silly to assume that a privately paid nurse is, by definition, productive while a government paid nurse (working for instance at the NHS in the UK) has zero productivity. So, why is government consumption (again: government production which is consumed by households) not included in 'utility'? Same for government investments. Addendum: not including it means that institutional differences between countries (the NHS!) make it impossible to use NK models to compare countries.

You can include government consumption in the utility function of the representative consumer, but why? When we add exogenous government purchase shocks to a macro model, we enquire about the dynamics of the agent's consumption and labour supply change. This concept is clear and easy to measure.

Let's assume that you are including government spending in an additively separable utility function. For example, the representative consumer is happy because of the government is investing in coastal defences. But he is unhappy about the state's Olympic games project. It's all OK, but how are you going to defend these assumptions empirically speaking? I think you should introduce a few agents plus. How do you describe the preferences of agents about the different government spending projects (defence forces, park building, etc.)? It's a normative assumption, what makes this research Economics no more. This research becomes Political Science. Unless you find or do a reliable research on the preference of agents about government spending projects. But keeping in mind, political preferences are very far-fetched from the so-called rationality. There is always an emotional factor, which makes a spending project likeable or not. How to make these "soft" questions into "hard" science? It's a very hard question.

Anyway, what is the purpose of including include government consumption in the utility function? I can imagine an OLG model with a social choice problem. E.g. children want playground, adults park, and elders hospital.

To answer your sub-question, government purchase is wasteful by default in Macroeconomics, unless this purchase don't effect the production function in a positively. In any other case, the welfare of the economy can be suboptimal. If government buys tablets for every pupils in schools, it may makes the workforce more productive in the long run.

You can measure the effectiveness of government spending considering the fiscal multipliers indicator. It's a stylized fact that the effectiveness of government spending depends on the countries structural features (income, openness,  exchange rate regime, etc.).

See also: Enrique G. Mendoza, Carlos A. Vegh , Ethan Ilzetzki - How big are fiscal multipliers? New evidence from new data (01 October 2009)


Usefulness of the Convexity Axiom I'm asked to write an essay supporting the statement which says the convexity axiom has little economic content and should be eliminated from the economic models of consumer theory.

I'm supposed to find literature which supports my point with theoretical and empirical evidence but I've been searching for weeks and all I have found so far are papers which stress the importance of the convexity axiom. Perhaps you all can assist me in locating the material I'm looking for.

Regards

Consider a set of consumption bundles $X$, which is convex and let $\succeq$ denote 'at least as good as', then convex preferences may be defined mathematically as follows.

Convex preferences. For any $x,y,z\in X$ and $\lambda\in[0,1]$ we have that $$x\succeq z\text{ and }y\succeq z\Rightarrow \lambda x+(1-\lambda)y\succeq z.$$

Informally we may say that convex preferences captures the concept of averages being at least as good as extremes.

Critique 1. Below is an excerpt from the direct link where the author points out the fact that under certain circumstances, the assumption of convexity may not be adequately assumed as a property or preferences.

  That is, [convex preferences says that] if an agent is indifferent between $x$ and $y$ and has convex preferences, then she will like $x/2+y/2$ at least as much as either $x$ or $y$. Of course, this doesn't always make sense. You might like a beer or wine, but not a mixture

However, it depends on the interpretation.

  Whether convexity makes sense often depends on the interpretation of the
  goods space. For example, if the components of $x$ are rates of consumption, then a half-half mixture of beer and wine might mean drinking beer half the time and wine half the time. Convexity of preferences seems more plausible in that interpretation than in the previous one. Similarly, some find convexity easier to rationalize if the "goods" are more highly aggregated – for instance, if the goods are "food" and "clothing," than if goods are highly specific.

Critique 2. The link directs you to McCloskey's critique of economics. She criticizes how economists focus too much on assumptions and mathematical proofs and too little on empirical facts. In this sense, if microeconomics or mathematical economics is to be about general properties of consumer behaviour, then the assumption of convex preferences is not always suitable from a methodological perspective, as non-convexity may hold in a given economy. Also, following McCloskey, one may say that the assumption of convex preferences, while it may guarantee that certain mathematical propositions are true (i.e., one may deduce them from the assumption of convex preferences and simpler axioms), does not by itself say something about economic behavior. It surely implies certain economic behavior $\mathcal B$, but we cannot from the observation of such economic behavior $\mathcal B$ draw the conclusion that economical agents' preferences are such that they are convex over economical entities like goods.

Critique 3. Instead of assuming convex preferences, one may study the wants and sentiments of individuals,  as Adam Smith does in his Theory of Moral Sentiments and later in his Wealth of Nations, or study cognitive biases, like Daniel Kahneman does in his best-selling Thinking, Fast and Slow. These two authors does not, to my knowledge, directly criticize the concept of convex preferences, but they show how economic analysis performed in another way may lead to other conclusions about the economy. If their conclusions do not agree with conclusions derived from axioms like that of convex preferences, then there is implicit criticism in the sense that there is a mismatch in theoretical conclusions.


What are "the fundamental categories of economics"? I had an introductory class in marketing last week, and the professor wanted to test our previous knowledge of economics (we are juniors).
He asked us this simple question: "What are the fundamental/basic categories in economics?"

Confused by the vagueness of the question, most of us said labor, capital, profit, markets... All sorts of things, but none were apparently correct. In the end, he said that the correct answer is:

cost
result (as in profit or revenue)
time

He continued to say that we define everything in economics on the basis of these three categories.

My questions now are: is this true? Do I really have such a significant knowledge gap that I somehow missed this crucial piece? Where can I find the confirmation of his statement?

If this is not true, can you please refer me to where I can find the answer to this question?

Thanks

Cost covers all the inputs you mentioned and markets don't need to exist. Rent seeking is a perfectly valid alternatives to markets. So having to choose (cost/spend) between different alternatives (profit) is all there is. You can even include time as a cost and reduce it to two factors.


To profit from rolling, mustn't you be much more bullish or bearish than before? Aren't you likelier to lose money from rolling, e.g. scenarios 3-5 below? How exactly can rolling profit you?

What if you aren't more bullish or bearish than before? What if you surmise that the underlying stock will attain only a certain range of prices, and de/in-crease no further?

I'll brainstorm three cases. Imagine that:

you bought a call. Your underlying asset's price hasn't risen above the spot price on purchase date.

you bought a put. Your underlying's price hasn't dropped below the spot price on purchase date.

In both cases, your call's and put's prices will have dropped, and you've lost money.

your underlying's price de/in-creases beyond the purchase spot price. Then all options' IV may rise. Then even puts at lower strike prices and calls at higher strike prices will cost more.

If markets are efficient, the option premium will equal the expected return on the option. This is independent of the past history of the underlying security.
In order for the past history to matter, you will need to demonstrate market inefficiency. It is possible to demonstrate some market inefficiencies (e.g., the fixed income pricing literature is generally premised on the existence of a term risk premium, the issue is how to estimate it). However, future returns getting a risk premium based on recent historical price movements would be equivalent to proving that technical analysis can reliably outperform (on a probabilistic basis). Finding validation of such a possibility has been a major challenge for researchers.


Question regarding preferences in Gale and Shapley (1962) Is it correct to say that preferences in the classic Gale and Shapley College Admissions problem are quasi-linear?
Or is this something thats introduced later in the literature, vis a vis Shapley and Shubik (1971), Kelso and Crawford (1982), etc?
Thanks

Gale and Shapley barely make any assumptions about preferences. They don't need a functional form, simply an ordinal ranking of the options. Moreover, there are no transfers in this setting, only the discrete matching options.
Even Kelso and Crawford do not use quasi-linear utility. They assume some utility function that is increasing and continuous in salary, but not necessarily linearly increasing.
In Shapley and Shubik (1971) monetary transfers enter linearly.
However, I wouldn't call this quasi-linear utility, because in quasi-linear utility most people have the following functional form over a good $x$ and money $m$ in mind: $u(x,m) = v(x) - m$, where $v$ is a continuous function.
In their model, the $v(x)$ is simply the money value of house $x$, where the houses are district options.


Example in which revealed preference doesn't imply preference I'm studying Varian's Intermediate Microeconomics, in which the principle of revealed principle is stated as follows (for now we're assuming strictly convex indifference curves):

Let $(x_1, x_2)$ be the chosen bundle when prices are $(p_1, p_2)$, and let $(y_1, y_2)$ be some other bundle such that $p_1x_1+p_2x_2 \geq p_1y_1+p_2y_2$. Then if the consumer is choosing the most preferred bundle she can afford, we must have $(x_1,x_2) \succ (y_1,y_2)$.

It's further stated that "if the consumer chooses the best bundles she can afford, then the revealed preference implies preference". I can't get an intuitive grasp on this. Even if a bundle $X$ is chosen to $Y$ $(Y \neq X)$, and even if $X$ isn't the best affordable bundle, shouldn't that at least imply that $X \succ Y$? Because if not, then there must be some example where $Y \succeq X$ and still the consumer chose $X$.

Is there some counterexample in which revealed preference of $X$ does not imply preference, provided that the chosen bundle $X$ isn't the best affordable one?

The text basically explains that consumer is assumed to be rational. This means the consumer can identify the best bundle and picks that as her choice.

If the consumer was unable to identify the bundle that is best for her or was simply choosing randomly rather than based on her preference then her choice would not reveal anything about her preferences.


Autocorrelation function of a random walk process What is the intuition behind the result that the autocorrelation function of a random walk process $y_{t}=y_{t-1}+e_{t}$ tends to 1 as $t\rightarrow 0$? Thank you.

By recursive substitution you obtain $y_t = \sum_{j=0}^t e_{t-j}.$ Thus $\forall p \in \mathbb{N} \hspace{.2cm} y_{t-p} = \sum_{j=0}^{t-p} e_{t-p-j}.$

Under the usual white noise assumption for the errors $i \neq j \Rightarrow E(e_je_i) =0\Rightarrow Cov(y_t, y_{t-p}) = E(\sum_{j=0}^{t-p} e_{t-p-j}^2) = (t-p+1)\sigma_e^2.$

$Var(y_t) = E(\sum_{j=0}^{t} e_{t}^2) = (t+1)\sigma_e^2 \hspace{.2cm} \wedge Var(y_{t-p}) = E(\sum_{j=0}^{t-p} e_{t-p-j}^2)(t-p+1)\sigma_e^2.$

$\Rightarrow \rho_t(p) = \frac{(t-p+1)\sigma_e^2}{\sqrt{(t+1)\sigma_e^2}\sqrt{(t-p+1)\sigma_e^2}} = \frac{\sqrt{(t-p+1)}}{\sqrt{(t+1)}}.$

$\Rightarrow lim_{p\rightarrow 0} \rho_t(p) = 1 \hspace{.2cm} \wedge \Rightarrow lim_{p\rightarrow t} \rho_t(p) =  \frac{1}{\sqrt{(t+1)}}$.

$\Rightarrow$ for fixed $p$ $lim_{t\rightarrow \infty} \rho_t(p) = lim_{t\rightarrow \infty} \frac{\sqrt{(t-p+1)}}{\sqrt{(t+1)}}= 1$


Should Prices (or Price Indices) be modelled with deterministic trend? I always face a dilemma on whether to assume prices to have a time trend or not while modelling. It is also partly a statistics problem. Let me explain. Assume I have time series, $y_t$ of price of a commodity which on plotting appears to be having an upward slope (which is after all true for majority of items).
When I apply unit root test without drift, the result is non-rejection of unit root. However, if I include a drift term, the result is rejection of unit root.
Now, if the true process is random walk without drift, inclusion of drift is very likely to incorrectly reject the null (below is a simulation study that shows that if the true process is pure random walk, inclusion of drift rejects the null around 34% of times). The red-line is the distribution of t-statistic when ADF test is applied with drift, and blue-line when applied without drift (which is the true process).

(the vertical line is the 0.05 quantile value for DF-distribution - blue curve)
Now if the series did have a time trend (without unit root), ADF test (without drift) almost never rejects the null of unit root.
Hence the dilemma. My take is that the answer (whether trend is stochastic or deterministic) has to come from theory. So the question boils down to Why should prices have a deterministic trend? What economic dynamics would justify modelling the series as, say AR(1) process with time trend?
Code:
'%>%'=magrittr::'%>%'
n=1e4
tt_coef = 0.1

ar1 = function(phi, n=1000, tt=0, tt_coef=0,sd) {
  y = numeric(1000+100)
  for (i in 2:length(y)) {
    y[i] = phi*y[i-1] + tt*tt_coef*i + rnorm(1,mean = 0, sd = 1)
    }
  return(tail(y,n)) # first 100 observations burned to remove impact of initial condtns
}

samples_tt = replicate(n,ar1(phi = 0.7,tt = 1, tt_coef = tt_coef, sd = 8),simplify = F)
df_stats_tt = lapply(samples_tt, function(x) urca::ur.df(x,lags = 0,
                                                         type = 'none')@teststat) %>%
  unlist()

samples_rw = replicate(n,ar1(phi = 1, sd = 1),simplify = F)
df_stats_rw = lapply(samples_rw, function(x) urca::ur.df(x,lags = 0,
                                                         type = 'none')@teststat) %>%
  unlist()

df_stats_drift = lapply(samples_rw, function(x) urca::ur.df(x,lags = 0,
                                                            type = 'drift')@teststat[,1]) %>%
  unlist()

sum(df_stats_drift<quantile(df_stats_rw, probs = 0.05))/n
sum(df_stats_tt < quantile(df_stats_rw, probs = 0.05))/n

plot(density(df_stats_drift), type = 'l', lwd = 2,
     col = 'red', main = 'DF-Distribution', xlab = '')
lines(density(df_stats_rw), type = 'l', lwd = 2, col = 'blue')
abline(v = quantile(df_stats_rw, prob = 0.05))
legend('topleft', legend = c('Without Drift', 'With Drift'),
       col = c('blue', 'red'), lwd = 2, bty = 'n')

Should Prices (or Price Indices) be modelled with deterministic trend?

Often yes. Prices could have time trend for various reasons. For example, the most obvious one is inflation. While inflation is the increase in aggregate prices not necessary in every individual one, aggregate price level would not increase if significant number of prices would not. Thus you would expect prices to have upward time trend in many prices in any country experiencing inflation. For this reason when people analyze CPI they typically test unit root hypothesis against trend stationary (often even with non-linear trends like in Bierens 1997)
The CPI logic might not be applicable to every single individual  price, some of them might happen to be stable, but it can be extended to many individual prices.
The rationale for trend in CPI is simply that central banks target increase in CPI, so you would expect it to have positive trend because based on economic theory central banks are able to control (with more or less precision) inflation (see various rigorous models of this in Romer Advanced Macroeconomics). If on the other hand central banks would target $0\%$ inflation you would not want to include the trend.

Edit: In response to question about inclusion of drift term where there is no trend it does not lead to bias in general. To explain this consider the Dickey-Fuller test and its mechanics:
The DF test with drift is given as:
$$\Delta Y_t = \delta + (\theta-1) Y_{t-1} +\epsilon$$
Now this is at its hart estimated just by OLS. Consequently, if the model is supposed to have drift but you exclude it and estimate $$\Delta Y_t =  (\theta-1) Y_{t-1} +\epsilon$$ you will bias your estimation of $(\theta-1)$, but the converse will not hold. If you estimate  $\Delta Y_t = \delta + (\theta-1) Y_{t-1} +\epsilon$ but series has no drift then in expectations $\delta$ will just be zero and estimation of $(\theta-1)$ will not be biased. This is crucial because even though we do not care about $(\theta-1)$ per se but about value of the DF test, if $(\theta-1)$ is estimated in a biased manner any subsequent DF statistics is useless (also note running auxiliary DF test and examining if $\delta$ is significant and close to zero or not is cheap way of quickly assessing if it is important).
Now when it comes to $t$ values from DF test you have to realize that you are not using critical values based of DF distribution. Under the null the DF test distribution is non-standard so appropriate critical values have to be tabulated using Monte-Carlo simulations for the non-standard distribution that holds under the null (see Verbeek, A Guide to Modern Econometrics. pp 292).
The correct critical values for DF test with trend and with drift are given in the table below (from ibid).

The table does not include appropriate value for specification with no drift but I found that here.
You will note if you apply these appropriate critical values to the distribution of $t$-statistics of DF test from your simulations you will find that areas in the tail are virtually identical between the DF test with and without drift. The area left of green line  $t^* = -2.87$ below red dist. is very close to the area left of orange line $t^*=−1.942$ below blue dist*.
The areas wont be exactly same of course, and using DF test with drift will in marginal cases lead to incorrect inferences, so you are right into being interested in theoretical reasons for whether there should be drift or no if you want to be pedantic about it (which you should - don't get me wrong). However, this being said, whenever in doubt you should use DF with drift, because excluding it might lead to bias in $\theta-1$ itself and at that point any inferences are completely useless and it is not possible to even get right $t$-stat since $DF_t= \frac{\hat{\theta}-1}{se(\hat{\theta})}$ - if you get numerator wrong then that is the end.

*: note I modified the code to have only 500 observations so I can apply appropriate critical values. In addition, I change the series to be just random walk because there is no qualm that non-drift version is wrong if there is a drift.


If prices stay the same, how's lending money the same as lending real goods? I was watching American TV. They interviewed an economist from Ivy League university. He said

It's called a real interest rate because if prices stay the same, lending money is the same as lending real goods.

Please see the title of this post. That's my question 1.

How does "if prices stay the same, lending money is the same as lending real goods" explain why the real interest rate is called the real interest rate?

Lending money is on fundamental level equivalent to lending goods because you can use the money to buy some goods and services. If a truck cost \$100,000 and you finance that truck purchase via getting bank loan, then on fundamental level that is equivalent of the bank just lending you the truck.
Next, in economics 'real' essentially means inflation/deflation adjusted. Opposite of 'real' variable (e.g. real consumption, real investment, real spending, real output etc) is not imaginary or unreal variable but nominal variable (that is variable that does not take into account change in price level e.g. nominal consumption, nominal investment, nominal spending, nominal output etc.).
If prices do not change from period to period (i.e. there is no inflation or deflation), then any interest rate on a loan will represent real interest rate because by Fisher equation (see Mankiw Macroeconomics pp 110) the real interest rate is given by:
$$r \approx i - \pi$$
where $r$ is real interest rate, $i$ is nominal interest rate and $\pi$ is inflation. If $\pi=0$ then any nominal interest rate you pay at the bank is virtually equivalent to real interest rate.
Moreover, since prices do not change borrowing money from bank is on fundamental level equivalent to borrowing real goods from the bank since if you borrow \$100  you can finance \$100 worth of real consumer spending.


which type of goods maximum utility function represent? I am not sure, which type of goods does the maximum utility function represent i.e., $U(X_1, X_2) =\max(X_1, X_2)$.
As the $U(X_1, X_2) =\min(X_1, X_2)$ represent the complementary goods, and $U(X_1, X_2) =X_1+ X_2$ represent the substitute goods, I think it represents the substitute goods as the maximum of both matter. So am I correct?
Please clarify this doubt, thank you.

$u = \max(x, y)$ represents the preferences over two substitute goods that cannot be consumed together. For example - tea and coffee. In the event that the consumer gets x quantity of tea and y quantity of coffee, consumer choose to consume only one of the them depending on the quantity. He always choose the one that is offered in larger quantity and throws the one that is offered in smaller quantity.


Computational economics What exactly is computational economics? For example, does certain software for calculating/predicting/visualizing cost of manufacture falls under computational economics?

Taken right from the society for computational economics:

  Computational economics explores the intersection of economics and
  computation. These areas include agent-based computational modeling,
  computational econometrics and statistics, computational finance,
  computational modeling of dynamic macroeconomic systems, computational
  tools for the design of automated Internet markets, programming tools
  specifically designed for computational economics, and pedagogical
  tools for the teaching of computational economics. Some of these areas
  are unique to computational economics, while others extend traditional
  areas of economics to new areas through computational techniques.

I would say that calculating cost of manufacture is too trivial to fall under this category - unless it's a monopolist who has to take into account equilibrium responses of his suppliers and demanders to his choices.


Optimal taxing in case of negative externalities Suppose an individual $i$ has the utility function

$U= f(x(i)) - k$(sum of all $x$ with index not equal to $i$)

Where $x(i)$ denotes the miles driven by $i$, and $k$ is a positive constant.

The function $f$ is such that $f(0)=0$, and has a negative double derivative. It crosses the x axis once after increasing at $x=0$.

We're asked, that if a social planner taxes individuals on basis of $t$ dollars per mile driven, how will such an optimal tax change as we increase the number of individuals? It is given that the planner maximizes sum of all utilities.

I started out by finding the best response of individual $i$, which is
$f'(x)=t$. Hence every person has the same best response, so miles driven by everyone are the same. Hence maximising utility of any person will suffice.

Max. $f(x) - f'(x)*x - k(n-1)x$

Which gives me $x=(n-1)k/f''(x)$

I don't know how to proceed further.

I would have thought the social planner might want the value of $x_i$ to be such that $f'(x_i)=k(n-1)$ at which point the marginal gain to each individual offsets the marginal total cost to the others

while individual driver $i$ might want the value of $x_i$ to be such that $f'(x_i)=t$ at which point the marginal gain offsets the marginal tax

To me that suggests the tax rate should be $t=k(n-1)$


When do trades between two countries not result in gains for either of them? So are there any circumstances where two countries trade with each other and they both do not gain anything? is it possible? and if you could also explain why that is the case? I have looked around and haven't really been able to find any answers.

Thanks in advance

The reason you probably find it hard to think of a case where this happens is because rarely do countries go around trying to make poor trades (read: never). If you trade real goods, there will almost always be a comparative advantage of trade to be had as long as countries have different production possibility frontiers in different ratios. Different opportunity costs of making more of one good means you might want another countries' labor to offer that service.

I can't give any specific examples of trades that were bad for each party where each party also was trying to act in there own self interest. Maybe if two countries were trading public debt with each other before a recession, they might end up with riskier assets than they bargained for, so even though the expected value of the debt was worth enough before a negative growth shock, you might get something less favorable in practice. Even then I don't think a situation like that is very likely to happen.


Devaluation of Venezuelan Bolívar - Will it increase or fight hyperinflation? According to recent news, Venezuela is lopping 5 zeros out of its regular Bolívar, and devaluing it, and pegging it to a cryptocurrence issued by Venezuela, to fight the economic situation (hyperinflation, low GDP, etc.)

However, if I remember well, won't the devaluation of a currency increase inflation by increasing prices of imports, and in some cases also increasing prices from domestic products by increasing exports?

What's the rationale here for this measure?

lopping 5 zeros out of its regular Bolívar

This will make the numbers used in daily shopping more manageable.

  However, if I remember well, won't the devaluation of a currency increase inflation by increasing prices of imports, and in some cases also increasing prices from domestic products by increasing exports?

Devaluing the currency might help make imports possible again.
The exchange rate vs the USD is held superficially high by the government. However, it is not possible to exchange Bolivars for USD because of the shortage of foreign currency caused by a drop in oil income. Getting paid in overvalued Bolivars is not very appealing, so imports have essentially ground to a halt causing massive shortages in products that aren't produced locally.

  What's the rationale here for this measure?

The government hopes that this will stabilize the currency and reduce inflation to manageable levels. The market usually requires real change for that to happen, so in my opinion not much will change. There will still be hyperinflation and shortages.

References: https://en.wikipedia.org/wiki/Shortages_in_Venezuela


Does working abroad cause less unemployment? Or is it actually harmful (directly or opportunity cost)? This question is from subjective origins but is (supposed to be) objective.
The context of this question arises from people in third world countries going abroad to work. Often they are seen as money-hungry people who don't 'love' their country (unless they are, say, physicists who do research abroad due to poor facilities or engineers or financiers simply looking for higher salary or something like that).
I was thinking that those people help their country through foreign remittances. I later additionally thought another help done is that whatever job they would have done in the country is likely ('likely' is probably the key word here) going to be replaced by someone else who would otherwise have been unemployed or underemployed would have that job which in turn likely (hooray key words) means that someone else who would otherwise have been unemployed or underemployed would have that job, etc.

If the answer is no (working abroad necessarily causes less unemployment), under what kinds of situations might this be the case?
If the answer is no (again, working abroad necessarily causes less unemployment) due to geographical variations, you may consider third world countries.

Edited to add: Also my concern is understanding whether or not like: are the ones going abroad necessarily like money-hungry people who do not 'love' their country or something? If you choose to work in your country, then do you necessarily 'love' your country more or 'serve' your countrypeople more than if you didn't work in your country? Of course 'love' and 'serve' are subjective, but it's like I wanna understand if somehow working abroad is like more beneficial to the individual but harmful to the country or something. To me, it seems like going abroad benefits everyone at least from 3rd to 1st world.

On remittances:
It is not unusual to hear about income being sent home by, say, US immigrants. However, the product of their labor and the side effects of their intellectual capital, sometimes called positive spillover, do not occur in the home country.  As such, they are not as beneficial to the home country as some counterfactual world where an equally profitable job existed at home. The immigrants are then beneficial to the host country.  The remittances still benefit the home country, however, and are particularly important when no employment is available at home.
On unemployment:
Broadly, immigration does not appear to have much of an impact on the unemployment in the host country. Or at least, something happens which masks that effect and makes it tremendously difficult to measure.  There has been a lot of controversy on this study but it helps highlight a lot of the issues if you take the chain of papers and responses seriously: https://www.jstor.org/stable/2523702?seq=1#metadata_info_tab_contents
On the type of people immigrating:
"Are the ones going abroad necessarily like money-hungry people who do not love their country or something."  This isn't really the type of question economists answer. Everyone is money-hungry and equally self-interested, by default, and that is fine.  This helps (in some ways) avoid stereotyping immigrants. You may want to open another question about the types of people who like to immigrate.


How do I choose an appropriate Instrument to solve the endogeneity issue? I have a quadratic regression model and one of the variables is endogenous. How do I choose an instrument to deal with this issue?

Generally, following Verbeek's guide to modern econometrics there are three main attributes a good instrument should have:

It should be relevant. The instrument should be able to explain your the variable being instrumented and through it also the explanatory variable. For example, in two stages least squares (TSLS) which is a model from the instrumental variable (IV) family the estimator converges to  $\beta_2 = \frac{cov(z_i, y_i)}{
cov(z_i, x_i)} $ where $y$ is the dependent variable, $x$ the independent variable that is endogenous and $z$ the instrument. If there is no covariance between $z_i$ and $x_i$ then the estimator is simply not defined.

Instrument should not be 'weak'. Generally it is not  enough that simply $cov(z,x)\neq 0$, instrument also has to be significant and generally the first stage of regression has to be significant itself. This can be tested by looking at $F$-statistics. A rule of thumb  for TSLS is that instrument is weak if the first stage $F$-statistics is less than $10$ (see mostly harmless econometrics by Angrist and Pischke).

Instrument should satisfy exclusion assumption. It should have effect on $y$ only through $x$ not in other ways. The instrument $z$ also has to be independent.

Moreover, more on a practical note if you are using a quadratic model the program will ask you also to provide an instrument for both the variable itself and  the quadratic term. As mentioned by Michael in his +1 comment you should check if the above assumptions still hold for the quadratic term as well.


How does using HP filter allows us to estimate long-run values for variables such as output and unemployement? I'm reading the paper "Okun's Law: Fit at 50?" written by Laurence Ball, Daniel Leigh and Prakash Loungani.
In it, in order to estimate Okun's Law in its level form, they use the HP filter to find the long-run values of the variable. I quote:

In this case, the tricky problem is to measure the natural rate $U_t^*$
and potential output $Y_t^*$. In most of our analysis, we use the most
obvious method: we smooth the output and unemployment series with the
Hodrick Prescott (HP) filter.

Could anyone explain to me how that works? From what I could gather, the HP filter is used to find the cyclical and trend components of a time series. How does one go from that to estimating long-run values?
Thanks!

I've found the answer to my question. By using the HP filter, we are able to extract the long-term component of a time-series, namely it's trend component. Using that, we can then take the difference between the series and its trend component at every observation. This leaves us with a cyclical component; if we are interested in estimating gaps between observed values and potential values or natural rates, this is what we are looking for.
Of course, this is the english explanation of what happens; for a mathematical/statistical approach, I recommend reading the original paper.