2 versions of the koyck distributed lag In econometrics, ( this material is in a lot of the literature but, IMHO, can most clearly be found in Harvey's text, "econometric analysis of time series"), there are generally two versions of a specific lagged dependent variable which is usually  termed the koyck distributed lag. ( It's a very special case of a distributed lag ).

I write down version 1 and version 2 below.

Version 1 is shown below.

1):   $y_t = \rho \times y_{t-1} + \beta \times x_t + \epsilon_t$

where $\epsilon_t = (v_t - \rho v_{t-1}) \sim N(0, \sigma^2) $.

There is also version 2 which is

2):     $y_t = \rho \times y_{t-1} + \beta \times x_t + \epsilon_t$

where $\epsilon_t \sim N(0, \sigma^2) $.

So, in version 1, the error term is such that $v_t$ is AR(1) and in version 2), the error term is pure noise.

My question is the following: Is there a way to write to a version where the error term is MA(1) ? I initially figured there must be some kind of symmetry because  we have the pure noise case and the AR(1) case so I figured there must be a version with an MA(1) error term. But now I'm not so sure and actually don't think do.  Thanks for any insights.

EDIT BASED ON GOOD QUESTION IN COMMENT:

Hi: $\frac{\epsilon_t}{(1 - \rho L)} = v_t$ where $L$ is the lag operator.

Therefore, $v_{t}$ can be thought of as an exponentially smoothed average of the past white noise error terms $\epsilon_{i}$.

Now, version 1) of the model can be re-written in the following way:

$y_t = \rho \times y_{t-1} + \beta \times x_t + (1-\rho L) v_{t} $

which can be re-written as

$y_{t}(1 - \rho L) = \beta \times x_t + (1 - \rho L) v_{t} $

Then, dividing the whole the equation by $(1 - \rho L)$ results in what I call the long form of version 1 of the model:

$y_{t} =  \beta \times \sum_{i=0}^{\infty} \rho^{i} x_{t-i} +  v_{t}$

So, the model implies that response is $\beta$ times an exponentially smoothed average of the past $x_{i}$  plus v_{t}. But, we can re-write the last term which results in

$y_{t} =  \beta \times \sum_{i=0}^{\infty} \rho^{i} \times x_{t-i} + \sum_{i=0}^\infty \rho^{i} \epsilon_{t-i}$

So, in version 1), the response can be thought of as an exponentially smoothed version of the past $x_{t}$ plus the exponentially smoothed version of the past error terms, namely the $\epsilon_{i}$.

I won't write it out, but, in  the long form of version 2, the error term is not an exponentially smoothed average of the past $\epsilon_{i}$. The past error terms  are not involved and the error term is just $\epsilon_{t}$.

Thanks for good question.

SECOND EDIT BASED ON GOOD QUESTION IN COMMENT:

When your question made me write it out in the long form, it hit me that
an MA(1) structure could just be:

$y_{t} =  \beta \times \sum_{i=0}^{\infty} \rho^{i} \times x_{t-i} + \epsilon_t + \rho \epsilon_{t-1} $

This would be an MA(1) error term for the koyck distributed lag. I don't think it simplifies in any way by writing it in the short form but it's still a nice way of thinking of a third possibility. In one case, the error term lasts on period. in another case, the error term is  is an exp smoothed average of past errors and in the last case, the error is linear combination of the last two errors. Clearly your question totally led to this and I think what you're saying in your comment is the same as here, so thanks much for insight.

As I explained in the edited question, the comment-question made by @caverac made me write the models out in their respective long form versions. Doing this made it obvious that an MA(1) structure can be imposed and the model is shown below.  As far as I can tell, the long form of this model cannot be transformed to a lagged dependent variable model (what I refer to as the short form ) nicely like can be in the  AR(1) and pure noise cases.

Long Form of Koyck Distributed Lag Model With MA(1) error term:

$y_{t} = \beta \times \sum_{i=0}^{\infty} \rho^{i} x_{t-i} + \epsilon_t + \rho \times \epsilon_{t-1}$


How is marginal benefit measured? In my economics class, we're learning about the production possibilities frontier, marginal benefits, and marginal costs. The book uses the example of a company that can choose between producing colas, pizzas, or some combination of both:

The book then goes on to note that while marginal cost can be directly derived from this graph in terms of the slope (which makes sense), the marginal benefit is separate data that depends on preference. Here is the data they present for the marginal benefit of producing one more unit of pizza:

Question: I would greatly appreciate if someone could please clarify why the marginal benefit of producing one more unit of pizza is measured as "Willingness to pay cans of soda per pizza).

To clarify: I understand why marginal cost would be measured in terms of cans of cola. This is because producing one more unit of pizza incurs a loss (cost) of a certain number of cans of cola that we could have otherwise produced.

But for marginal benefit, it's unclear how we're measuring the "benefit" we derive in terms of... cans of cola. That makes no sense.

The reason why marginal benefit is measured in cans of soda is that this economy only has two goods: pizza and soda. So instead of using money we may as well use soda. Alternatively, in the absence of money this economy is an exchange economy, and the only way to pay for pizza in that case is with soda.

When you move beyond two goods willingness to pay is often measured in money value. However, measuring willingness to pay in money value is just a convenient way of saying: a bundle of "everything else that is available".

In a two good economy,given the current amount of pizza that you have, an additional slice of pizza may thus be worth 5 cans of soda to you. In a multiple good economy it may be say 5\$ where the 5\$ stands for "all the other (combinations of) goods you could get for 5\$.


What's the role of initial endowments in an edgeworth box? I am trying to solve a problem but I don't see the role of initial endowments.
For instance, the question is about finding competitive equilibria.
But there is information about initial endowment.
How should I make use of it?

It's crucial to understand that only initial endowments and preferences matter. Everything else (utility functions, prices, equilibria, core etc) follows from them.

Your initial endowment shows how strong is your negotiation position.

Situation I. Imagine that the wealth is equally distributed (50/50) and the 1-st player slightly prefers good A to good B and the 2-nd player slightly (to the same extent) prefers good B to good A  than something like this happens.

i.e. you get extra 10 units of A and give 10 units of B (1 to 1 exchange).

Situation II. Now, consider that by the same initial endowment, the 2-nd player prefers B to A much more than you prefer A to B. Then, something like this happens:

i.e. by the same initial endowment and preferences you get more, now e.g. 35 units of extra giving away 20 units of B (better conditions of exchange as before).

Situation III. Now, imagine that you have only 10% of wealth. Then your negotiation position is weak and by the preferences as in situation I you get much less after exchange.

p.s. Calculations and plotting were made with pyEdgeworthBox package (I wrote it).


Why do you typically have to pay for use of the washing machines while other amenities are free? I'm looking at apartments and noticed that many offer amenities such as gyms, pools, saunas and hot tubs which are "free" to use by tenants (paid for by the condo fees). However, I noticed that while most of them also have laundry rooms, tenants must pay to use the machines (at the machine, on a per-wash basis).

The washing machines are an outlier in the sense that they are the only facility on the premises that cost extra to use; everything else (tennis court, barbecues, pool and gym) is free.

Why is it that washing machines seem to be treated differently?

I thought about it for a bit, here are a few of my hypotheses so far (but I don't have data to support any of it):

I can't imagine that washing machines cost much more to operate than a pool, gym, tennis court or sauna - maybe I'm wrong - but considering that the aforementioned are all included I don't think the landlords left the machines out of the included amenities because it would have been too expensive to include them. (Or too expensive to include them instead of some of the other aforementioned amenities)

Clearly, "because they can" is true, if they can make money off the machine they'll opt to do so, but they also could save money by not having a pool and gym to maintain yet they do it because they seem like good incentives to live there. Are the machines not included simply because they aren't very convincing incentives for potential customers? Either because

people underestimate the cost to use the machines (therefore would not see free use as a big incentive)
it's the norm here (North America) for machines to require payment for use and since it's not a "sexy" topic of conversation the laundry rooms are simply left out of sales discussions (making it hard to advertise it as an incentive). A landlord wouldn't gain much by giving the customer more than they expected in this case, so they don't.

I'm asking because, personally, I'd rather have free use of those machines than free access to a pool and sauna I'd never use and I'm wondering why no one seems to offer free washing machines for tenants.

As you said, you would prefer to have free washing machines instead of free swimming pool. What does this mean? That washing your clothes is more important than having a bath in a swimming pool. Usually, it applies to almost anyone. So you could stop using swimming pool, but not washing machines.

Also, it does not seem probable that many people will use that swimming pool at the same time. But even if they do, people will not complain about massification unless it is too much and permanent (both conditions are rare to be filled).

So:

1) They know that washing clothes is more important than having a bath in a swimming pool. And even more, they know that you cannot avoid to wash your clothes (but you can avoid having a bath).
2) They do not expect the swimming pool to be always full of people, so they do not expect many complaining about it.
3) Having a swimming pool is cool. Having a free swimming pool is very cool. Marketing.

In conclusion: they use the swimming pool as part of their marketing, while they make money from washing machines since you have to wash your clothes with no potential alternative (while having a bath in a swimming pool has several alternatives, so if you had to pay for it, maybe you would not use it).

I hope it helps.

EDIT:

For those who do not know what I am talking about, take a look of these two terms: "utility" and "elasticity" ;)


Proof that EV = CV when there is no income effect In every textbook it says that it is easy to see that with no income effect, the integral

$\int_{p^0_1}^{p^1_1} \! h(p,u_0) \mathrm{d}p_1. = \int_{p^0_1}^{p^1_1} \! h(p,u_1) \, \mathrm{d}p_1$

Could someone explain to me why?

This is because when the Marshallian demand $x_i(p,w)$ is constant in $w$ (i.e. no income effect), the Hicksian demand $h_i(p,u)$ is constant in $u$. To see this, we only need to apply the relation
$$h_i(p,u)=x_i(p,e(p,u)).$$
For $u^0\neq u^1$, we have
$$h_i(p,u^0)=x_i(p,e(p,u^0))=x_i(p,e(p,u^1))=h_i(p,u^1).$$

Therefore,
$$\int_{p^0_1}^{p^1_1} \! h_i(p,u^0) \mathop{dp_1} = \int_{p^0_1}^{p^1_1} \! h_i(p,u^1) \, \mathop{dp_1}.$$


Prove that variance of a portfolio cannot exceed variance of individual assets When reading on Markowitz's portfolio theory, I stumbled across the fact that in a market with two risky assets, if no short selling is not allowed, the variance of a portfolio consisting of the two assets cannot exceed the variances of the risky assets individually. That is:

$${\sigma _p}^2 \le \max \{ {\sigma _A}^2,{\sigma _B}^2\} $$
Where A and B are two different assets.

Could you kindly prove this statement, and possible provide some intuition for why this is the case?

Let $P = \alpha A + (1-\alpha) B$ where $A$ and $B$ are returns (random) from the two assets, and $P$ is their portfolio.
Variance of portfolio $P$ can therefore be written as

\begin{eqnarray*} \sigma^2_P & = & \alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\text{Cov}(A, B) \\ & \leq &\alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\sigma_A\sigma_B \\ & = &(\alpha\sigma_A + (1-\alpha)\sigma_B)^2 \\ & \leq & (\alpha \max(\sigma_A, \sigma_B) + (1-\alpha)\max(\sigma_A, \sigma_B))^2 \\ & = & \max(\sigma_A^2, \sigma_B^2)\end{eqnarray*}


Is the marginal cost the same for every firm in a perfectly competitive market? Spin-off from: market equilibrium quantity $\ne$ firm profit maximising quantity?

Consider a perfectly competitive market with equilibrium price $P_{eq}$ and quantity $Q_{eq}$ and firm with profit maximising quantity $Q_f$ as illustrated below:

Is the MC for each firm in the market the same?

Can we deduce such from any of the assumptions listed in the PC Wiki page?

No, the marginal cost curves are not necessarily the same for each firm in the market. However the values of marginal costs are.

To disprove the general claim that "The marginal cost curve of each firm in a competitive market is the same" we simply need to find one counter-example, such as the one given below:

Suppose there are two firms in the market and the market inverse demand function is given by $P=7 - q^m = 7 - (q_1 + q_2)$

Here $P$ represents the price, $q^m$ the market supply and $q_1, q_2$ represent the respective supplies of the firms 1 and 2.

Suppose the marginal cost curves for each firm are given by:

$MC_1 = 2q_1$,

$MC_2 = 4q_2$

Note that these are not the same! Since we are looking for a competitive equilibrium, the firms behave competitively and hence we must assume that they do not realize they have market power. Hence they each set:

$P = MC_i, \; i\in\{1,2\}$

From this, solving for quantity we have the respective Supply functions of each firm:

$S_1 = q_1 = 0.5P$

$S_2 = q_2 = 0.25P$

Inserting these two supply functions into the inverse demand function we have:

$P=4$, $q_1=2$, $q_2=1$.

Is this an equilibrium with perfect competition? Well we found it by combining Supply and Demand assuming perfectly competitive firms, so it must be. To verify:

$Market \; Demand = 7 - p = 3$

$Market \; Suppy=q_1 + q_2 = 3$

So the market is indeed in equilibrium. Are both firms setting $P=MC$ as they should in perfect competition?

$MC_1 = 2*q_1 = 4 = P$

$MC_2 = 4*q_2 = 4 = P$

So indeed we have a market equilibrium with perfect competition and different marginal cost curves. QED

Note however that the values of the marginal cost curves at the equilibrium quantities are the same for each firm (and must be the same for each firm in equilibrium).


ln(log) of energy consumption in a econometric analysis i'm applying a panel data analysis. I
 have four variables; real GDP, trade openness, energy price and energy consumption. I took ln (log)of trade openness,real GDP and energy price. but i cannot decide whether ln(log) of energy consumption should be taken or not. please inform me on this. Thank you

Note: Kg of oil equivalent per capita is used to measure energy consumption.

EN= energy consumption

Y=Economic growth

T= trade openness

EP=energy price

W= country fixed effect
EN=f(Y,T,EP,W)

Thank  you.

Prior to any trasnformation:

Energy consumption is a per capita variable, specifically "Kgr per capita"
Real GDP is a per capita variable
Trade Openness is a per capita variable
Energy price is a unit price of "crude oil"

I see two problems: Is "Energy Price" the "price per Kgr" -because usually oil prices are not quoted per Kilogram. But it should expressed here "per Kgr" so as to stand for the price of the dependent variable.

If it does, it is only natural to consider the logarithms of the variables, dependent variable included.

But one more thing that needs clarification is what $Y$ stands for. The authors write "$Y$(economic growth)". If this is the case then for this variable you don't just take the logarithm of real GDP but the first-difference of the logarithm  (which approximates the rate of growth).


Did real interest rates decrease during the Great Recession Assuming the stock market/housing market crash decreased consumer wealth, decreased business confidence, and decreased expected appreciation rates on houses (thus increasing perceived user cost of residential capital), the IS curve shifted left at the onset of the Great Recession (in the IS-LM model):

But this indicates that real interest rates would have fallen.  Given that there was a liquidity crisis at the onset of the recession, I am guessing this did not happen.  Did it?  And if not, how could a liquidity crisis be accounted for in the IS-LM model?

The Great Recession precipitated in the financial sector, as a liquidity crisis, as I'm sure you know. It didn't precipitate in the real sector. I would represent that in the theoretical framework as a decrease in the stock of money:

That is because there was a lack of liquidity. The evidence is that interest rates soared. That is, the inter-bank lending rate soared, along with the increase in the perceived riskiness of financial instruments. The above shift correlates to a leftward shift of the LM curve.

The constraint on credit had real effects—a recession. The traditional way to smooth a recession is through monetary policy, but the Fed reached the lower limit when lowering the federal funds rate—a so-called "liquidity trap". So further, non-traditional methods were needed to address the recession (such as quantitative easing, where risky securities were purchased by the Fed in an attempt to lower those interest rates, and they also sought to lower forward interest rates).


Why aren't house prices included in CPI? Swedens central bank recently lowered their prime rate to -0.25% from -0.10% in an effort to avoid deflation.

At the same time, house and apartment prices are higher than ever, up 11% last 12 months, and up hundreds of percents the last 5-10 years. Measures are taken to avoid a housing bubble, like regulations on mortgage payments and so on, these are having an effect but the effects of a rapid rise in rate would be disastrous.

As I've understood, and I guess this might differ from country to country, the measure of inflation, CPI, includes prices for services and goods, but not for houses and apartments, why is that? In Sweden, rent for apartments and the bank rate are included in CPI, but the consumer isn't paying rate, they pay rate * house prices.

If house prices were included in CPI, we would definitely have inflation rather than deflation, so the right action would be to increase the prime rate, which would also lower house prices.

The CPI stands for a Consumer Price Index. As in the price of things that are consumed (at a particular moment in time). Real estate prices are not the price of something consumed because they contain the value of current housing consumption but also the capitalized value of future housing consumption. As such, including house prices would make the CPI a mixture of consumption at different times, and therefore unsuitable for comparing the price of consumption bundles at distinct times. Instead, they use a purer measure of the price of housing consumption: rents.

Rents reflect the price of consuming a flow of real estate services at a moment in time. Of course, many homes are owned by the occupants and not rented. Therefore, the calculators of national accounts generate something called "owner occupied rents", which is an attempt to calculate what the rents would be on homes that are occupied by their owners. This measure has problems, but for many purposes is quite adequate (Crone, Nakamura, Voith (2004)).


How to Graph Short-Run Average Cost? I've learnt to roughly draw graphs of various functions like isoquants of Cobb Douglas function, i.e., $k=√q/L$. Here first derivative is negative so it's downward sloping and second derivative is positive so convex to origin.
Now if Short-Run cost function is $C = (w/k)q^2 + (rk)$ then average cost is $AVC= (w/k)q +(rk)/q$.
First derivative is $(w/k)-(rk)/q^2$ but how do I know if it's positive or negative?

$\newcommand{\fone}{\color{red}{f_1(q)}}$
$\newcommand{\ftwo}{\color{blue}{f_2(q)}}$

For the sake of simplicity, call

$$
f(q) = \frac{w}{k}q +  \frac{rk}{q} = rk\left(\underbrace{\frac{1}{q}}_{\fone} + \underbrace{\frac{w}{rk^2}q}_{\ftwo} \right) = rk (\fone + \ftwo)
$$

where I have factored $rk$ out of the expression. Now you want to understand each term separately:

  $\fone = 1/q$

This term is drops as $q$ increases, and diverges when $q$ is small.

  $\ftwo = \alpha q$, with $\alpha = w/rk^2$

This is a linear term with slope $\alpha$: it is small for small $q$ and large for large $q$.

  Combined

In this particular case the function one of the terms grows while the other shrinks. So in extreme cases only one matter. The question is where is the point in which one becomes more relevant than the other.

If you notice above I always use the expressions small and large, but these are relative words. You can actually find a value $q^*$ at which these two terms are equal, and this defines in which each term dominates.

So, if $q < q^*$ this is what I mean by small $q$ and therefore $\fone$ dominates. If, on the other hand $q > q^*$ $f$ will be dominated by $\ftwo$. To find $q^*$ we make

\begin{eqnarray}
\fone &=& \ftwo \\
\frac{1}{q} &=& \alpha q \\
q^* &=& \alpha^{-1/2}
\end{eqnarray}

With this in mind, below there's a graph for $\alpha = 1$


Mixed Nash equilibrium I have game table posted below:

$$\begin{matrix} &\#2 \\ \#1 &
\begin{array}{c|c|c|c}
&D &E &F \\ \hline
A &4,4 &6,6 &2,6 \\ \hline
B &6,4 &2,2 &0,4 \\ \hline
C &2,6 &8,4 &0,0
\end{array}
\end{matrix}
$$

Are there any Nash equilibrium where the agent 1 mixes between

(i) A & B

(ii) A and C

(iii) B & c?

What I did is that

For example I did first one

I calculate its expected utilities

$$u1(A, p2)= 4p_D+ 6 p_E+ 2+(1-p_D-p_E)$$

$$u1(B,p)=6p_D+2p_E$$

Since first payoff is higher than the second payoff, there is no such a mixed Nash equilibrium.

I think this approach to such a question seems wrong. Therefore I ask this question. What is your idea about the solution?

(i) $A$ & $B$

If player 1 play $A$ with probability $p$ and $B$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+4(1-p) = 4$
$E$ is $6p + 2(1-p) = 4p + 2$
$F$ is $6p + 4(1-p) = 2p + 4$

Since payoff from playing $F$ is more than the payoff from playing any other strategy for player 2, he will always choose $F$ in response to any mixed strategy of player 1 where 1 play $A$ with probability $p$ and $B$ with probability $(1-p)$. Therefore, $F$ is the best response of player 2 if 1 play a mixed strategy $A$ with probability $p$ and $B$ with probability $(1-p)$. However, given the strategy $F$ of player 2, player 1's best response is to play a pure strategy $A$. Therefore, there is no Nash equilibrium in which player 1 mixes between $A$ & $B$.

(ii) $A$ & $C$

If player 1 play $A$ with probability $p$ and $C$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+6(1-p) = 6 - 2p$
$E$ is $6p + 4(1-p) = 2p + 4$
$F$ is $6p + 0(1-p) = 6p$

Since payoff from playing $E$ is more than the payoff from playing $F$, he will never choose $F$ in response to any mixed strategy of player 1 where 1 play $A$ with probability $p$ and $C$ with probability $(1-p)$. Therefore, $F$ is never a best response of player 2 if 1 play a mixed strategy $A$ with probability $p$ and $C$ with probability $(1-p)$. Now let us consider the following cases for $p$ :

$0 < p < 0.5$ In this case player 2 has a unique best response and that is to play $D$, in response to which player 1 would like to play a pure strategy $B$, and not a mix of $A$ and $C$.
$0.5 < p < 1$ In this case player 2 has a unique best response and that is to play $E$, in response to which player 1 would like to play a pure strategy $C$, , and not a mix of $A$ and $C$.
$p = 0.5$ In this case player 2 is indifferent between playing $D$ and $E$, and if 2 mixes between them appropriately i.e. by choosing to play $D$ with probability $0.5$ and $E$ with probability $0.5$, it will leave player 1 with two best responses $A$ and $C$ and therefore, 1 can now choose to play $A$ and $C$ with equal probability. So, we've found a mixed strategy Nash Equilibrium in this case: Player 1 choose to play $A$ with probability 0.5 and $C$ with probability 0.5, and player 2 choose to play $D$ with probability 0.5 and $E$ with probability 0.5.

(iii) $B$ & $C$

If player 1 play $B$ with probability $p$ and $C$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+6(1-p) = 6 - 2p$
$E$ is $2p + 4(1-p) = 4 - 2p$
$F$ is $4p + 0(1-p) = 4p$

Since payoff from playing $D$ is more than the payoff from playing any other strategy for player 2, he will always choose $D$ in response to any mixed strategy of player 1 where 1 play $B$ with probability $p$ and $C$ with probability $(1-p)$. Therefore, $D$ is the best response of player 2 if 1 play a mixed strategy $B$ with probability $p$ and $C$ with probability $(1-p)$. However, given the strategy $D$ of player 2, player 1's best response is to play a pure strategy $B$. Therefore, there is no Nash equilibrium in which player 1 mixes between $B$ & $C$.


Current Value Hamiltonian VS Present Value Hamiltonian in Economics I've been looking at a number of optimal control problems and have been wondering under what conditions one should use the current value Hamiltonian over the present value Hamiltonian.
Does it depend on the specific result we seek to derive? (pick whichever you want) or does it depend on the nature of the problem?

There is no clear right and wrong about this, it's just a matter of convenience. The current-value Hamiltonian is likely to be more convenient when the objective function includes a discount factor. Following Chiang (1), suppose the problem is:
$\qquad$Maximise $V = \int_0^T G(t,y,u)e^{-\rho t}$
$\qquad$subject to $\dot y=f(t,y,u)$
$\qquad$and boundary conditions
The standard (present value) Hamiltonian is:
$\qquad H=G(t,y,u)e^{-\rho t} + \lambda f(t,y,u)$
If we proceed from this Hamiltonian, the co-state equation (one of the first-order conditions) is:
$\qquad \dot \lambda = -\frac{\partial H}{\partial y}= -\frac{\partial [G(.)e^{-\rho t}]}{\partial y}-\lambda\frac{\partial f}{\partial y}$
While it is possible to obtain a solution this way, the discount factor complicates the derivatives and can make interpretation more challenging.
Suppose instead we use the current-value Hamiltonian:
$\qquad H_c = G(t,y,u) + mf(t,y,u)$
where $m$ is a current-value Lagrange multiplier defined by $m=\lambda e^{\rho t}$.  The co-state equation is then:
$\qquad \dot m -\rho m = -\frac{\partial H_c}{\partial y} = -\frac{\partial G}{\partial y} - m\frac{\partial f}{\partial y}$
This is simpler because it contains no discount term.
Reference

Chiang A C (1992) Elements of Dynamic Optimization pp 210 ff


Show that First order conditions are necessary and sufficient for utility maximization I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order conditions are necessary and sufficient for an interior solution to the utility maximization problem
**My solution **
First step : I maximize utility subject to the given budget set
Second step: I calculate first order conditions
Third step : I use hessian matrix.
If the determinant of this hessian matrix H is negative, then I can say that the FOCs are necessary is sufficient. (Is this true?)
My solution is like that
where $$u_{11}={\partial^2 u(x_1, u_2)\over \partial x_1^2}$$
$$u_{22}={\partial^2 u(x_1, u_2)\over \partial x_2^2}$$
$$u_{12}={\partial^2 u(x_1, u_2)\over \partial x_1 \partial x_2}$$
$$u_{21}={\partial^2 u(x_1, u_2)\over \partial x_1 \partial x_2}$$
I don’t know this answer is enough and correct. Because this solution seems not to be sufficient. Please share me your ideas about my solution.
Thanks a lot.

Consider the following situation :
Preferences of the consumer are represented by the utility function $u(x_1, x_2) = x_1 + x_2$. Note that preferences are strictly monotonic and convex. Now let income be any $y > 2$. So the consumer's problem is :
\begin{eqnarray*} \max_{x_1, x_2} & x_1 + x_2 \\ \text{s.t.} &  \ \ 2\sqrt{x_1} + x_2 \leq y \\ & x_1\geq 0, \ x_2 \geq 0 \end{eqnarray*}
Now let us set up the Lagrangian :
\begin{eqnarray*} \mathcal{L}(x_1, x_2) = x_1+x_2 - \lambda(2\sqrt{x_1} + x_2 - y) +\mu_1x_1 + \mu_2x_2 \end{eqnarray*}
Here are the first order necessary conditions:
\begin{eqnarray*} \dfrac{\partial\mathcal{L}}{\partial x_1} & = & 1 - \dfrac{\lambda}{\sqrt{x_1}} + \mu_1= 0 \\ \dfrac{\partial\mathcal{L}}{\partial x_2} & = & 1 - \lambda + \mu_2= 0 \\ \dfrac{\partial\mathcal{L}}{\partial \lambda} & = & y -2\sqrt{x_1} - x_2  \geq 0, \ \lambda\geq 0, \ \lambda(2\sqrt{x_1} + x_2 - y) = 0 \\ \dfrac{\partial\mathcal{L}}{\partial \mu_1} & = & x_1  \geq 0, \ \mu_1\geq 0, \ \mu_1x_1 = 0 \\ \dfrac{\partial\mathcal{L}}{\partial \mu_2} & = & x_2  \geq 0, \ \mu_2\geq 0, \ \mu_2x_2 = 0 \end{eqnarray*}

Check that one of the solutions to the above set of conditions is $x_1^* = 1, \ x_2^* = y -2, \ \lambda^*=1, \ \mu_1^* = 0, \ \mu_2^* = 0$,

But it is not the solution to the utility maximization problem stated above. This is because utility from $(x_1^*, x_2^*) = (1, y-2)$, i.e. $u(x_1^*, x_2^*) = 1 + y - 2= y-1$, is less than the utility from another affordable consumption bundle $(x^{**}_1, x^{**}_2) = (0, y)$ as $u(x_1^{**}, x_2^{**}) = 0 + y = y$ and $2\sqrt{x_1^{**}} + x_2^{**} =y$.

Therefore, first order conditions are not sufficient for utility maximization.


What does the Fed mean by "smoothening money market functioning"? In many recent interviews, Fed Chairman Jerome Powell has repeatedly mentioned that ensuring the smooth functioning of money markets is a key goal of the Fed's recent monetary policy actions. I want to know how exactly do their policy actions achieve this. Money markets are the markets that deal with short-term borrowing and lending in the economy and instruments traded in these markets include short-term treasury bills, commercial paper and mortgage- and asset-backed securities. What I interpret from what he said is that the price mechanism in these markets is not functioning optimally (i.e. some form of market failure) and that government intervention is required to restore proper market functioning. From the policy actions listed in this article, it seems that the most relevant ones would be the lowering of interest rates and the increase in asset purchases. In essence, I would like to know what problems in the money market is the chairman referring to and why the Fed's policy actions would resolve this problem.

Money markets are where financial institutions and corporates access short-term funding.
One money market where the Fed has intervened directly is the overnight repo market where overnight loans ("repurchase agreements") are made in exchange for high quality collateral such as Treasuries.
It is arguably the most important money market in the world, with volume---the demand of the financial system for overnight cash---up to 1 trillion USD.

The overnight repo rate is determined by supply and demand for overnight cash.

  ...what [potential] problems in the money market is the chairman referring to...

If the spread between overnight repo rate and the Fed's policy rate (fed funds rate)
is too wide, it could be symptomatic of:

The Fed's monetary policy is not being effectively transmitted.
Shortage of cash in the financial system.

Typically, the spread is in single-digits basis points.
This slightly higher rate of interest in the repo market is normally sufficient to  induce (big) banks to lend out their cash reserves held at the Fed.

As mentioned by @Henry, there was a September 2019 episode where the overnight repo rate shot up to 10%, or about 800 basis points over the then-policy rate.

This points to severe imbalance of supply and demand for overnight cash.

  ...why the Fed's policy actions would resolve this problem...

After the September 2019 spike, the Fed responded by simply flooding the market with cash, increasing supply to meet funding demand.
The NY Fed started making overnight loans in the repo market, later expanding into two-week loans, as well as purchasing short-dated Treasuries.

(The fact that such measures were effective indicates that the spike was a liquidity event, rather than credit event. The unwillingness of banks to lend was not due to concerns of counterparty credit-worthiness.
There are several possible reasons that bank were unwilling to lend. One is that reserves held at the Fed is allocated to satisfy capital requirements, as mentioned by @Henry. There was also a concurrent unwinding of the Fed's QE balance sheet, which takes cash away from banks.)

The Fed is currently maintaining its repo facilities, By doing so, the Fed signals to the market that it is willing to step in as, to mis-use terminology somewhat, "lender of last resort".

In another money market, the purchase of short-term commercial paper by the Fed has a similar effect, on funding for businesses instead of banks.
It is meant to ensure that demand of short-term credit for businesses is met.

  ...it seems that the most relevant ones would be the lowering of interest
  rates and the increase in asset purchases.

I would say those measures constitute a different aspect of monetary policy. (Whether purchasing short-dated securities is the same as QE is a different question.)


Even if immigrants take USA jobs, don't they consume proportionally to maintain supply-demand equilibrium? I heard many times politicians argue that immigration (in USA in particular, but applies to anywhere) are taking "real" Americans jobs. Then I tough, but they also are new consumers on the market, so they create supplies too. If immigrants also consume goods in a similar quantity and type as Americans, that will produce new more demand and thus the extra supply the jobs produce will be "justified", "balanced" or "matched".

Obviously, this economical situation is EXTREMELY complex with millions of factors: how productive/educated both populations are, average age of immigration, in which sectors will immigrants work and if they will overload some part of the economy with too much supply... There are millions more factors.

The only thing I care is if my argument disproves the IMO fallacious argument

  "Immigrants will take our jobs"

Tell me if this in more correct:

  "immigration will prompt the creation of more jobs and the new
  population demand will (hopefully, if everything goes well, which would
  not) balance with the new supply, without taking jobs in
  massive proportions, thought some people may loose jobs to more
  skilled immigrants."

The only argument that I think can be made is that immigrants often don't consume similarly to Americans. Large amounts of salaries are sent home from the US by immigrants which means that money is now out of circulation.

Additionally, coming from poorer countries with a lower standard of living it can be argued that immigrants will choose to live a lower quality of life in the US which again could cause a decrease in Economic activity in the U.S.

Lastly, for illegal immigrants wage/labor restrictions don't always apply to them meaning that U.S workers who are protected by these restrictions now have to compete with workers who do not have those restrictions which can drive down U.S employment and wages for workers across the board.

With that said, I think immigration as a general rule increases economic growth in the long run.


In March 2020, why did stock indices of Asian countries that curbed COVID19 still plummet? By March 1 2020, both HK and Japan restrained and quelled COVID19, compared to  France, USA, UK. Don't hesitate to discuss other countries unscathed by COVID-19.

I cite some of  better comments from r/investing and r/stocks.

Because all their customers are in countries that didn't curb it.

Everyone else follows America. US equities are half the value of all equities in the world.

I would guess that Asia is an export economy largely to the rest of the world so tends to correlate with the rest of the worlds performance. However, the P/E ratios are likely lower than what you'd find in the developed world so there might be some opportunity for good gains when the broader economy picks up.

Because there was a gigantic flight to the US dollar in March. Pretty much every single asset across the board (yes, including safe havens like gold and US treasuries) fell versus the USD.


Data on Argentina car industry I am working on a paper for university about the car industry in Argentina.
I found a lot of data, but I cannot find any data about the turnover, profits and part of the GDP of this industry (Neither for each one one the companies) between 2014 and at least 2000 maybe more if possible.

The national office of statistics (INDEC) does not seems to have it, neither the ADEFA (Association of car constructors).

Does anyone have a good data website about this industry in Argentina?

The ADEFA website does have information on sales, disaggregated by company, market (internal/export), and for very year since 1959. For example:

This data is here. Disaggregation by company and market is here.

ADEFA does not provide data on profits thought. As you can see from the latter link, all companies are multinationals with subsidiaries in Argentina (e.g. Toyota Argentina S.A. where "S.A." stands for Sociedad Anónima, meaning PLC). As such, they provide annual statements to the local tax authorities, including profits. They should also publish annual reports to their investors (e.g. Toyota 2014 here; see profits in page 8). And official government organisations also make these statement public (e.g. here). In other words, all the data must be available online. You just need to google something like "name of company memoria anual 20XX", where memoria anual stands for annual financial report, for year 20XX.

Whether the data is systematically available free of charge, I doubt it. Normally consultancy companies, investment banks and other financial firms produce compilations and analysis of this data for their clients (know this from first hand, when worked at Deutsche Bank). Unless you have access to one of those, you will have to compile the data yourself. At least there are just a few car companies, so this should not take you long.

PS: beware of over/under reported utilities due to transfer pricing though.


When a stock market crashes, how does money just disappear? I looked it up online (here), but I find it hard to believe that money literally "disappears".  For example, if I buy 10 stocks of a toaster company for \$100 per share, and then the stock value drops to \$10, I've lost \$900, but the entities I bought from have gained \$900 collectively.  I understand that money is supposed to be an abstract concept, but we should be able to trace where the dollars go.

So how does money just disappear when the stock market crashes?

Stocks are not money.

The valuation of a company - the market captialisation - is the number of shares multiplied by the share price.

The share price is the price people are willing to trade at right now. It does NOT mean all the shares have been traded at that price.

If a company issues 1 million shares, at a starting price of £10, the market cap is £10 million. They may only sell a few of these shares, so much less than £10 million actually changes hands. If the price rises to £20, only a few of the shares changed hands at that price - but the market cap is reflected in all the shares, and is now £20 million. People have commented on the flaws in this methodology.

The same happens in a crash. Only a fraction of the shares actually change hands, but the media report on the change in the market cap.

If you want an analogy, consider a house burning down. You buy the house for £200,000. Then it sadly burns down (and without insurance). It is now worth £0. However, no money has been created or destroyed; it's a loss of value of an asset.


Perfect competition: Finding short run equilibrium price? I am trying to self teach myself some Economics, I am using an old textbook given to me by a friend, which does not contain an answer key. I have run into the following problem and I was wondering if anyone could guide me (I don't want the answers, I just want to understand the method).

The firms in a perfectly competitive industry have a cost function given by:

$c(w_{1},w_{2},Y)= Y^{2} (w_{1}, w_{2})^{\frac{1}{2}} + 8$

Where $(w_{1},w_{2})=(4,25)$

The market demand in this industry is $D(p)=40-p$

The number of firms in this industry are $30$, I'm not sure if this ties in with this half of the question or not.

Anyways, what I have done so far is computed the total cost function:

$TC= 10Y^{2} +8$

From this point, how can I proceed onwards to find the short run equilibrium price?

I would appreciate it if anyone could offer some advice.

Thanks.

These kinds of questions depend heavily on the market structure that you assume. Most importantly, when firms have (some degree of) market power, they usually take the demand function as given and try to solve something of the kind

$$ \max_p D(p)\cdot (p-c(D(p)))$$

However here, when there is no market power, firms take the price as given and solve for the optimal quantity of production $q$:

$$ \max_q q\cdot (p-c(Q))$$

Then, we get the supply of any single firm, given the price $S(p)$ from solving this problem. To clear the perfectly competitive market, we remember that there are $30$ firms and solve:

$$p: D(p) - 30\cdot S(p) = 0$$


Is there an economic interpretation of the envelope theorem? I wonder if envelope theorem has also some hidden economic interpretation. For example, Lagrangian multiplier in economics can have interpretation of 'shadow price' which is useful economic concept.
The envelope theorem seems like something that should have some deeper economic interpretation as it is about examining how function adjusts when parameters change given that we are at some optimum. The envelope theorem says that only the direct effects of a change in an exogenous variable need be considered, even though the exogenous variable may enter the maximum value function indirectly as part of the solution to the endogenous choice variables. But I am not sure if this makes sense. Is there any economic concept attached to the envelope theorem?

First, let's consider a standard "economic interpretation" of the envelope theorem for unconstrained maximization:
if $x^*(b)$ solves $\max_x f(b,x)$, then $\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial bk},$ i.e., the total derivative of $f$ w.r.t. $b_k$ equals the partial derivative of $f$ w.r.t. $b_k$.
The economic interpretation (in the given problem): Let $\pi^*(p,w) = p f(x^∗) − w·x^*$ be the maximized value of profits given output price $p$ and input price vector $w$. Then the $i$'th input demand function is $x^*_i(·) = −\frac{\partial \pi^*(·,·)}{\partial w_i}$, known as Hotelling's Lemma, after Harold Hotelling, (1895–1973). This derivative is negative: if the price of the input increases, then the firm's maximal profit decreases.

Second, let's consider a standard "economic interpretation" of the envelope theorem for constrained maximization:
if $x^*(b), \lambda^*$ solves $\max_x f(b,x)$ s.t. $h^j(b,x) = 0$, $j= 1, ...m,$ then
$$\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial b_k} + \sum_{j=1}^m \lambda_j^*(b)\frac{\partial h^j(b,x^*(b))}{\partial b_k}, $$
i.e., the total derivative of $f$ w.r.t. $b_k$ equals the partial derivative of $f$ w.r.t. $b_k$, plus the $\lambda^*$-weighted sum of the partial derivatives of the $h^j$'s w.r.t. $b_k$.
The economic interpretation (in the given problem): Let $\hat c(\bar q, p, w) = w·\hat x$ be the minimized level of costs given prices $(p,w)$ and output level $\bar q$. Then the $i$'th conditional input demand function is $\hat x_i(·) = −\frac{\partial \hat c(·,·,.)}{\partial w_i}$,  known as Shepard's Lemma, after Ronald Shephard  (1912-1982). The partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods.


Converting per capita growth rates for subgroups into contribution to total per capita growth rate You are given total debt for the second quarter of 2003 and the second quarter of 2004 for the overall population and four age groups. The four age groups collectively account for the entire population and their debt holdings account for overall total debt.

Quarter, Age_Group, Population, Total_Debt, Per_Capita_Debt
Q2 2003,0,1000,10000,10
Q2 2003,1,250,3000,12
Q2 2003,2,400,5000,12.5
Q2 2003,3,250,1500,6
Q2 2003,4,100,500,5

Q2 2004,0,1300,15000,11.5
Q2 2004,1,380,4000,10.5
Q2 2004,2,450,7000,15.6
Q2 2004,3,270,2500,9.3
Q2 2004,4,200,1500,7.5

The per capita growth rate for age group 0, representative of the total per capita growth rate is 15 percent. But how does one convert the individual per capita growth rates to the contribution to the total rate? Perhaps weight by the population shares? Taking a weighted average of the per capita growth rates yields approximately 6 percent. Dividing the change in per capita debt for each of the groups over the per capita debt level for the population doesn't work as it would when using total debt. The result should be r1 + r2 +r3 + r4 = r0 Where rn is the growth rate for age group n.

The problem is that you've got two inconsistent definitions of growth rate.

One is driven by per capita debt across the whole population.

The other by per capita debt within each subgroup.

These are inconsistent, because different subgroups have different baseline debts in Q2, and different population sizes. Because both of those things are different, and both are divisors in the calculation of your statistic of interest, then simply re-weighting by population size can't fix the problem.

So you can calculate a meaningful contribution of each to overall growth rate, so that together they satisfy  r1 + r2 +r3 + r4 = r0. But then the individual r1 to r4 won't mean much within their own groups - only in terms of their contribution to the whole. Or you can calculate growth rates for each sub group, as you've done in your question, but then they won't mean much in the overall context.


Resources to improve my economic intuition I am trying to improve my economic logic skills and ability to think about day-to-day issues in the systematic way economics and decisions science approaches scarcity or optimization.

A simple example to clarify, I know that MC=MR however the intuition as to why MC$<$MR is not optimal is something which educators seem to overlook to explain.

I am looking for any resources online or textbooks, workbooks, non-fiction, etc that will help me sharpen these skills.

Let me know if I can further improve my question, apologies in advance as this is my first post.

While textbooks are the best way to learn the material (MC=MR etc), here are some suggestions for improving your intuitive understanding of economics.

Books

The Undercover Economist by Tim Harford

The Armchair Economist by Steven E. Landesburg

Blogs

Marginal Revolution

Noahpinion

Conversable Economist

The Enlightened Economist (Great for book recommendations)

Podcasts

Econtalk with Russ Roberts

FT Alphachat

This is just a start. Hopefully these will send you to other excellent sources.


Long Run Equilibrium of Oligopolies The long run equilibrium of a perfectly competitive market is well established. My question is - are the concepts of a long run equilibrium in a perfect competition extendable (analogous or otherwise) to an oligopoly, specifically considering the the Cournot model and the Bertrand model.

Suppose I want to find the equilibrium number of firms $ N $ in a Cournot model, for example - will the approach be similar to that of a perfect competition?

The 'long run' assumption is not about whether the firms already on the market are price takers (perfect competition) or oligopolists but whether entry to the market is free. If entry to the market is free then in the long run profits tends toward zero, as a profitable market makes it tempting for more firms to enter.

If you have special asymmetric conditions where some entrants have different cost functions than the new ones the zero profit condition may not hold even in the perfect competition case.


When action and strategy differ in game theory It looks like in a static, simultaneous move game, action is used interchangeably with strategy.

But when do they differ, say in a more complicated game structure? I am trying to understand the difference between the two and try to keep in mind that I should, in general, NOT equate these two.

Even when I think about the normal form game, prisoners' dilemma, it is not all clear why in this case, action is equivalent to strategy and many authors choose to use two terms interchangeably.

When criminal is called upon to act, she can either cooperate or defect. But the strategy set is

$S=S_1$x$S_2=\{(U,U),(U,D),(D,U),(D,D)\}$

Here, how do I see that criminal 1's strategy is equal to her action which is $S_1\{U,D\}$?

And when do these two notions depart from each other?

If you happen to know that the "normal form" is sometimes also referred to as the strategic form, you'd most likely not be surprised that it doesn't help you distinguish between the notions of action and strategy. The normal form is precisely used to represent strategies (not actions) in a game. To appreciate the difference between strategies and actions, it's probably best to consider another form of game representation: the extensive form.

Suppose 2 players move sequentially, and that Player 2 observes Player 1's choice before making his decision. The scenario can be represented using a game tree as follows:

Here we can call $\{A,B\}$ Player 2's actions (actually the correct jargon here is behavior strategies, but I sense that this level of nuance is unnecessary at this point). However, they are not his strategies, for strategies in this case is a "full contingent plan" that specifies an action at each information set the player moves. Since Player 2 moves at two information sets (one after $L$ and the other after $R$), so one of his strategies would be

  "choose A if Player 1 chooses L and choose A if Player 1 chooses R", or simply denoted as AA.

So, Player 2 has four (pure) strategies: $\{AA, AB, BA, BB\}$. In comparison, since Player 1 only moves at one information set (at the beginning), her actions and strategies are identical: $\{L,R\}$.

To represent this game using the normal/strategic form, we'd have

It's not a 2-by-2 game, but 2-by-4, precisely because strategies and actions are not the same.

Lastly, a note on Prisoners' Dilemma (PD) and other similar one-shot simultaneous-move games. If you consider the extensive form of PD, you'd see that each player moves at only one information set, and so their set of actions and set of strategies coincide (just like Player 1 in the above example). This is why you can use the two terms interchangeably in those games.


Why was the term 'Economic Rent' chosen? How does it relate to the ordinary meaning of 'rent'? Pindyck, Rubinfeld. Microeconomics (2017 9 ed). pp. 314-315.

Etymonline substantiates that 'rent'  originally didn't mean the microeconomics meaning overhead: no surprise, as microeconomics is a recent discipline. So please see the titled question. Why worsen the ambiguity and polysemy of 'rent', rather than using some other word?
Is Economic Rent related to the mainstream meaning of "payment for use of property"?

Note that the Etymoline page you link traces the origin of the word rent as meaning, more generally, "income revenue". Here's their definition (my emphasis):

  rent (n.1) "payment for use of property," mid-12c., a legal sense, originally "income, revenue" (late Old English), from Old French rente "payment due; profit, income," from Vulgar Latin *rendita, noun use of fem. past participle of *rendere "to render" (see render (v.)).

This meaning (income revenue) is quite close to the use of the term rent by economists—i.e., an income arising from private control of some scarce factor of production.

Indeed, if we consult the Oxford English Dictionary (which I usually take as the definitive reference on the language), here's what we find in the first entry:

  1. b. Revenue, income. Also fig. Obsolete.

The OED has examples of the word being used in this sense as late as 1783. Meanwhile, the first recorded usage of "rent" in the sense meant by economists is 1662. So economists didn't simply invent a new meaning for the word rent; they adopted the word because, at the time early economists were writing, it meant exactly what they wanted to say.

The more modern meaning of "payment for use of property" is essentially a special case of the older, more general meaning of the word rent. Indeed, such a payment is an expense for the payer but an income for the recipient.

If a recipient owns an apartment and earns an income ("rent") from letting it out to a tenant then s/he is earning rent both in the common modern sense of the word, and in the sense described by Pindyck and Rubinfeld. So the usage is clearly related. It's just that this situation (someone earning an income because they own a scarce factor of production) is somewhat more general than the simple contexts in which the word rent finds every day usage.


Advice for finding literature for thesis I have to write my bachelor thesis in the coming months and I was planning on replicating one of the leading papers of the last years in the field of macro economics. Problem is that I am insufficiently acquainted with the literature and my supervisor does not want to help me in any way with regards to the topic/research question. My hope is that the community could advise me on what the most important/influential papers have been the last 5 or so years in the field. Any help is appreciated.

Your supervisor might want for you to come up with a research question yourself, maybe because that's part of the assignment. At the thesis, you should be able to demonstrate independent thinking rather than being told what to do for an assignment, and your supervisor might want you to do some more research on your own. However, it is also part of his/her job to give you guidance on the relevant fields where s/he considers your skills fit best.

Since you want to research on macro, check which papers were part of the recommended readings for that subject and ask your supervisor if s/he would recommend you to start there.

Another way to get ideas is to scout the papers in economics search engines. The IDEAS/RePEc can be a good starting point. You can look for topics and sort by citations.
Elsevier also has a list of most cited work in economics since 2013. Keep in mind that importance in academic terms is usually often measured by number of citations.

If you want a more mainstream reading, I'm sure a google search will also help you find articles like this list of 2017's most important researches.


Piketty's explanation of elasticity of substitution (from his book Capital in the 21st century) I have some trouble following the explanation of the elasticity of substitution between capital and labor and its implications on p189.
Take this part:

The relevant question is whether the elasticity of substitution
between labor and capital is greater or less than one. If the
elasticity lies between zero and one, then an increase in the
capital/income ratio β leads to a decrease in the marginal
productivity of capital large enough that the capital share α = r × β
decreases (assuming that the return on capital is determined by its
marginal productivity).

Specifically, when he talks about that elasticity, does he mean something like (dL/L)/(dK/K) with L for labor and K for capital? Assuming that, how exactly does a higher elasticity translate in a lower fall in marginal product of capital?
Thanks

The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here):

Take a CES production function
$$Y=F(K,L)=(a⋅K^{\frac{\sigma-1}{\sigma}}+(1−a)⋅L^{\frac{\sigma-1}{\sigma}})^{\frac{\sigma}{1-\sigma}}$$

Me: $\sigma$ is the elasticity of substitution. They have a small typo here with the outer exponent of $\frac{\sigma-1}{\sigma}$ instead of $\frac{\sigma}{1-\sigma}$, but I'm pretty sure that is wrong so I corrected it here. It doesn't change the rest of what I'm doing.
Back to Piketty and Zucman:

The rate of return is given by $$r = F_{K} = a \cdot  \beta^{\frac{-1}{\sigma}}$$ with $\beta = K/Y$.
The capital share is given by $$ \alpha = r\cdot\beta = a\cdot\beta^{\frac{\sigma-1}{\sigma}}$$

Me: Now take the partial derivative of $\alpha$ with respect to to $\beta$
$$ \frac{\partial \alpha}{\partial \beta} = a \cdot (1-1/\sigma) \cdot \beta ^{-1/\sigma}$$
By assumption $a$ is always positive. Because $K$ and $Y$ are always positive so is $\beta ^{-1/\sigma}$. That means the sign of $\alpha$ is the sign of $1-1/\sigma$. For $0<\sigma<1$ you can see that $1-1/\sigma < 0$ and therefore $$\frac{\partial \alpha}{\partial \beta}  < 0$$.


Indifference curves and preferences? I am going through some micro concepts and I am confused, is there a difference between deriving preferences through indifference curves and actual preferences of the consumers?

The question I was going through was:

Jim's utility function is $U(x; y) = xy$. Jerry's utility function is $U(x; y) = 1,000xy + 2, 000.$ Tammy's utility function is $U(x; y) = xy(1-xy)$. Oral's utility function is $-1/(10+xy)$. Billy's utility function is $U(x; y) = x/y$. Pat's utility function is $U(x; y) = -xy$.

(a) No two of these people have the same preferences.

(b) They all have the same preferences except for Billy.

(c) Jim, Jerry, and Pat all have the same indifference curves, but Jerry and Oral are the only ones with the same preferences as Jim.

(d) Jim, Tammy, and Oral all have the same preferences.

(e) There is no truth in any of the above statements.

The answer apparently is C, which implies that even though Jim, Jerry and Pat have the same indifference curves, The preferences of Pat and Jim are different, while those of Oral and Jim are the same even though they have different indifference curves. How is that possible?

A nerd loves both mathematics and physics equally, and hates sports and drinking beer equally.

A football player hates both mathematics and physics equally, and loves sports and drinking beer equally.

The nerd and the football player have the same indifference curves, but not the same preferences.

The point is this: Both agree on the "grouping" of things. But they disagree in the ranking of the groups.

Your Example

If you look at Jim and Pat, they have exactly inverse utility functions and hence inverse ranking in their preferences.

That is, if for Jim $A > B = C = D > E$, Pat will have $A < B = C = D < E$. Both agree on the three groups, but disagree on the order. Jerry's just a monotone transformation (poor Jerry).


Self policing incentive in Goods and Services Tax Amongst several advantages of GST (Goods and Services Tax) I read one as that it prevents tax evasion through self policing incentive. But self policing incentive is already provided in VAT (Value Added Tax). So what's new in GST that improves the current self policing incentive system?

Self-policing incentive, in theory, works similarly for both Value Added Tax (VAT) and Goods and Services Tax (GST). The improvement, however, could come from the reduction in the number of instances where suppliers can circumvent the self-policing system, to evade taxes.

VAT is collected by every supplier in the supply chain and paid by every buyer in that supply chain. At every incidence of tax there is an opportunity for tax fraud. One example of VAT fraud is: the supplier provides an invalid VAT number on the invoice and collects the appropriate amount of VAT from the buyer and instead of transferring the collected tax to the government, the supplier pockets it. Of course, such fraud can be detected by the customs department and when audited the supplier may claim that there was a printing error in the VAT number on the invoice. But if suppliers are not audited (perhaps because the cost outweighs the benefit) then they can plausibly pocket all the tax they collected from the buyer. It is also plausible that such a fraud can be carried out by many suppliers, if not every supplier, in the supply chain. Hence, even though there is self-policing mechanism fraud can still be committed by one or more suppliers in the supply chain under the VAT regime. In contrast, under the GST regime there is only one supplier in the supply chain who can theoretically commit that tax fraud. Since there are less instances under GST to evade taxes it is plausible that the amount evaded will be less and that it would be easier to audit.


Cournot Model Monopoly example Good day

In my Microeconomics course, we are handling the 4 imperfect competition models.

Currently we are discussing the Cournot Model, however I am unsure about a certain example. This is given as a lecture example (all information provided).

  Let us first assume that we start with a single firm (monopoly).

  This monopoly is the owner of a costless spring (and sells the water).

  I.e., MC = 0

  Start with our demand function:

Q = 120 – P
Now, determine the profit maximizing price/output combination.
MR = MC
MC = 0
Q = 120 – P (Demand function)
P = -Q + 120
Thus, MR = -2Q + 120
Thus -2Q + 120 = 0
Thus Q = 60
P = 60
Profits = 3600

I am familiar with the concept of calculating each firm's demand function, what confuses me is, in this example, we have a a single firm (monopoly) and when deriving its mariginal revenue function, one should end up with the following:

P = -Q + 120

d/dQ = P' = -1 (using chainrule for -Q => -1 and dQ of constant = 0)

thus MR = P' = -1

but in the example given, we arrive at:

MR = -2Q + 120

What am I missing or is this example incomplete (not all information given)?

The example you gave is correct.

What you are missing is that you took the derivative of the demand function and not of the revenue function. What you should have done is:

$Revenue = R = Price*Quantity = (-Q + 120) * Q = -Q^2 + 120Q$

$Marginal \; Revenue = MR=\frac{dR}{dQ} = \frac{d(P*Q)}{dQ} = 120 - 2Q$

What you did was:

$MR = \frac{dP}{dQ}=-1$.

Hence, you took the derivative of the Price and not of Revenue to get marginal revenue, which is wrong.


Why do anti-trust regulators in China need to approve NXP's takeover? Qualcomm is currently awaiting Chinese regulatory approval for its acquisition of rival NXP.

Both companies aren't based in China. I'd understand that vehicles like tariffs can be used by countries to exercise control over certain trade inbalances, but how could government block a takeover taking place outside of its sovereignity? Are anti-trust laws of international nature, in some case?

Is the rationale behind this case the same as US technology companies operating in the European Union, with regards to anti-monopoly policy? No long ago, I recall Google being fined by the European Commission on monopoly charges with regards to some of its services in the EU. The case of NXP's takeover seems of an entirely different nature.

Generally, antitrust regulators have authority over any company doing business inside the zone of its control, hence why the EU antitrust regulators hit Google in 2017, or the EU's recent examination of US company Apple's attempted acquisition of another US based technology firm, Shazam.

Of course, for these companies, there's always the option of simply pulling out of those markets (not selling any goods or services, or holding any capital inside their borders) which dodges their authority. But given the size of these markets (US, China and the EU particularly), that's simply not a feasible option.


Does cornering a market has any "bad" economic consequence? (Not sure If this question is suited for here or it should be asked at money or law SE)
As the title reads, Does any specific economic inefficiency arises when someone corners or tries to corner a market e.g. a commodity like silver?
My limited knowledge as a economics undergraduate student tells me that if the cornering leads to higher demand, then in a perfect market that would lead to a increase in supply (and the price could go up or down or stay flat depending on the industry costs).
For cases when perfect competition is not present or the supply is already limited (like precious metals) I can't be sure what could go wrong other than that the party who cornered the market can establish a monopoly on supply now.

You already have your answer in the last sentence. A monopoly can be established and that leads to well known "bad" economic consequences, i.e. economic inefficiencies.


Quarterly industry-level productivity growth? Are there any quarterly data on industry-level productivity growth (i.e., by NAICS code) in the United States? I've looked around for a while but haven't been able to find anything. BLS publishes aggregate productivity growth quarterly, so I find it hard to believe that there isn't a breakdown by industry.

Here is exactly what you are looking for, at the finest level of NAICS disagregation.

Example for aircraft manufacturing:

Example for Software publishers:


Questions about the gold standard and exchange rates So I'm a total economics noob and I'm trying to understand some of the consequences of using vs. not using a gold standard listed on Brad DeLong's blog here: https://www.bradford-delong.com/why-not-the-gold-standard-talking-points-on-the-likely-consequences-of-re-establishment-of-a-gold-st.html. The first point DeLong mentions is that with a gold standard an economy somewhat loses it's ability to stimulate growth by increasing the money supply. He gives the following example:

  ...in the spring of 1995 the dollar weakened against the yen. Under a gold standard, such a decline in the dollar would not have been allowed: instead the Federal Reserve would have raised interest rates considerably in order to keep the value of the dollar fixed at its gold parity, and a recession would probably have followed.

It's been a while since I've taken Econ 101 so bear with me. I'm trying to work through this example with some simple test numbers to understand whats going on. Say that the U.S. is on the gold standard, and a \$1 bank note can be exchanged at any time for 1 unit of gold. In DeLong's example the dollar weakened against the yen, so let's say the exchange rate between dollar and yen was previously \$1/¥5, and now it's \$1/¥3.

I have a few, potentially silly questions. First, why does the change in exchange rate change the "gold parity" of the dollar? If a \$1 bank note still gets you one unit of gold (since the gold standard ensures it), won't you still be able to get one unit of gold, assuming you only have U.S. dollars? Maybe this assumption is where my confusion lies? My other thought was maybe that if the dollar weakens against the yen, people with yen will be able to get more gold, which may be problematic for the U.S's gold reserves, but how does increasing interest rates address this problem? As I recall, raising interest rates reduces the amount of money circulating in the economy. But again, \$1=1 unit of gold always, so how does this change things? Sorry if this is somewhat incoherent -- any help explaining DeLong's reasoning (ideally in very, very simple terms with not a lot of jargon) would be greatly appreciated. Thanks!

I would interpret the “Gold Standard” as being all (major) countries each pegging their currency to gold, or at least to a currency pegged to gold (e.g., the U.S. dollar in the Bretton Woods system). A single country pegging its currency to gold would make gold investors happy, but it would do very little otherwise, since the currency would just float like gold prices already do.

This means that the exchange rates cannot appreciably move (there is usually a small trading band around parities).

This is why that DeLong says that the decline “... would not be allowed.” That is, if tge currency was weakening within its small trading band, the central bank needs to defend the parity: typically raising interest rates. If that does not help, the government is forced to launch fiscal austerity, which shrinks the economy. This reduces imports, which will hopefully take the pressure off the exchange rate.


The monopoly markup with inelastic demand? This question is bothering me a lot... I think the answer must be really simple, but I haven't been able to answer it.

In a monopoly, $p=\frac{c}{1-1/|\epsilon|}$, where $p$ is price of the good, and $c$ is the constant marginal cost, and $|\epsilon|$ is the absolute value of price elasticity of demand.

Why is this equality not valid for inelastic demand $|\epsilon|<1$ ? If $|\epsilon|<1$, then we get a negative price... Where in the deduction of this formula, from the profit maximization firm's problem, do we have to assume that $|\epsilon|>1$?

Edit: So for the FOC we need $|\epsilon|>1$, but where in the description of the profit maximization firm's problem do we need implicitely the condition?

The expression $p=\frac{c}{1-1/\epsilon}$ is derived on the assumption that an interior solution is feasible, i.e. FOC is set to equal zero. But inelastic demand curve may not yield an interior solution at all, i.e. it may be impossible to set FOC equal to zero.

Perhaps this example will help. Consider a demand curve that features constant price elasticity: $q=p^{-\epsilon}$, or its inverse $p=q^{-1/\epsilon}$, where $\epsilon$ captures the price elasticity of demand. We set $\epsilon<1$, so that the demand is inelastic everywhere.

Assuming constant MC at $c$, let's maximize the profit of a monopolist facing this demand:
\begin{equation}
\max_q\; pq-cq=q^{1-1/\epsilon}-cq.
\end{equation}
The FOC is
\begin{equation}
\left(1-\frac1\epsilon\right)q^{-1/\epsilon}-c<0.
\end{equation}
Note that since $\epsilon<1$, the entire LHS is negative, and therefore we cannot have an interior solution.

On a more intuitive level, when a monopolist is operating at a point on the demand curve that is inelastic, it could always increase profit by raising price (or lowering quantity):

since demand is inelastic, 1% increase in price will cause a less than 1% reduction in quantity, so total revenue will go up;
with less quantity, total cost will also go down.

With more revenue and less cost, the monopolist will simply not be content at operating on the inelastic segment of the demand.


"Competitive equilibrium" vs. "General equilibrium" What is the difference between the term "competitive equilibrium" and the term "general equilibrium"? Here in econ.SE, they are two different tags, so there is probably a difference between them, but I don't understand what exactly.

Competitive Equilibrium

A competitive equilibrium ("Walrasian Equilibrium")'s defining characteristic is that it's competitive. It's about an equilibrium in which market forces (say, consumers, firms)' supply and demand responds to prices, and prices respond to supply and demand, and no Pareto-improving trade possibility remains in the end.

To be technical, consider a market with $I$ agents, $J$ goods, and initial endowments $E$ of shape $I\times J$. Also, each agent has rational preferences over the of goods.

A CE consists of a ($J\times 1$) vector of prices $P$ and a ($I\times j$) vector of allocations $X_i$ with two characteristics:

Feasibility: $\sum_i J_i \geq \sum_i X_i \, \forall j\in J$
Rationality: $X_i \succsim_i \, H \forall H: HP \leq E_iP, \forall i$

General Equilibrium

The general equilibrium's defining characteristic is that it is an equilibrium on more than one market; as opposed to the partial equilibrium in which we hold at least one price fixed and analyze the response of other markets/prices only.

So, what's the difference?

It's about the emphasis. Any GE is a CE, but not any CE is necessarily a GE. We could look at a CE of consumer goods, but not analyze labor supply (i.e., not even model labor). Would this then really be a general equilibrium? Mostly, it is about the field.

In macroeconomics, it is a standard procedure to shut off several general equilibrium response mechanisms (for example, fix labor supply) and first look at a toy model in which only one margin is allowed to vary, and understand the mechanism - the partial equilibrium. Then, after that was understood, understand to which extent the mechanism is valid and relevant once one allows other margins to adjust. Therefore, here, it is important to be able to emphasize the difference - i.e. partial versus general equilibrium.

In other fields, one may not care as much about "completely different" markets. The distinction is for example not as important in many IO topics, and hence many there use the notion of the competitive equilibrium instead.


Why don't governments exploit negative interest rates more than they currently do? At the moment many governments, such as Finland's, are able to borrow money at negative interest rates. My question is: why don't they borrow as much money as possible while interest rates are negative?

The answers I have heard so far don't make sense.

  Governments avoid borrowing too much money because that would lead to spending more money

This doesn't make sense: they could borrow money and not spend it and just take the profit from the negative interest rate.

  Governments avoid borrowing too much money because they want to minimize the risk that a bank fails and loses their money

This doesn't make sense: they could take the money out as cash, pay people to guard it, and still make a profit from the negative interest rate even if the money was just sitting in a safe.

  Governments avoid borrowing too much money because there are limits to how much they can take out in loans in they won't to keep a healthy distance to those limits.

This doesn't make sense: if they borrowed money at a negative interest rate now, and just saved the money, they would be in a better position in the future compared to not taking the loan. (In all cases except when interest rates go even more negative in the future, in which case it might be beneficial to take the loans when the rates are lower - however, no-one is arguing that governments would be speculating like this).

(The arguments you listed are questionable, so I will not discuss them.)

One thing to keep in mind is that we need to distinguish between the central bank and the fiscal authority (e.g., the U.S. Treasury). This is doubly true in the euro area. I will put aside what the central bank is doing, to match the spirit of the question.

If we are in a place that has limits on debt issuance - e.g., the US debt ceiling, or countries in the Maastricht Treaty - there are limits on debt that need to be respected. (Why these limits should be implied when it is profitable to borrow is a mystery to me, but that is a different question.)

If we put those legal limits aside, there is a very practical issue: when the Treasury issues bonds (which is a form of borrowing), it is removing money from the “private sector” (or more accurately, the “non-government sector”). Under normal circumstances, borrowing is aligned (roughly) with deficit spending, and the deficit spending sends money to the private sector

As a simple example, imagine a government runs a deficit of \$100 million. The deficit results in sending \$100 million in extra payments to whomever, and that will roughly coincide with what it needs to borrow.

If the government just issues bonds, there is no source of money available to buy them. (The central bank can operate to make it possible, but this is limited.) The government will need to do something with what it borrows, and that would most likely be buying financial assets from the private sector.

(Since this explanation was not adequate, I will explain further. If the government issues \$100 million in bonds, private sector entities need \$100 million in balances at the central bank to pay for them. However, banks need to keep balance levels near minimums to meet liquidity needs. If they are already at the minimum, there is no capacity to buy the bonds at auction. The only way the bonds are sold is via some arm of government to transfer \$100 million to banks to allow the auction to proceed. If the central bank is not able to do it, the only course of action is for the Treasury to buy something worth \$100 million from somebody in the private sector.)

It is entirely possible that the securities it buys are worthless, which poses a risk to the government. Also, it will hand windfall gains to people selling assets, which raises political issues.

They could borrow to fund investment, which is exactly what many people have argued for governments to do. However, the capacity to so is limited by useful projects to invest in, and any projects raise other political issues.


Relationship between default risk and value of bond In particular Mortgage-backed securities. I've read from a book that underestimating the risk of default is tantamount to overestimating the value of the bond, giving rise to a bubble in MBS. I'm unclear of how that works.
My guess is that since the probability of default risk is low, thus banks give out lower coupon rates. Investors buy MBS up because they are safer, demand for MBS increase, price of these MBS pushed up.

[...] underestimating the risk of default is tantamount to overestimating the value of the bond.

The reason behind this is simply due to how bonds are priced.

$$
P = \sum_{n=1}^N \frac{C + M[{I_{n=N}}]}{(1+i)^n}
$$

Where $i$ is the contractual interest rate, $C$ is the coupon payment, $N$ is the number of payments, $I_{n=N}$ is an indicator function taking the value of $1$ when $n=N$ is $true$ or $0$ otherwise, $P$ is the market price of the bond, $M$ is the value of the bond at maturity.

When $i$ goes up, $P$ goes down and vice versa, see

$\rightarrow$ if $i$ is underestimated, $P$ will be overestimated. Indeed, it is $i$ which, via a premium, internalizes the default risk consideration and is an increasing quantity of it.


Would an $8 National Minimum Wage Work in the US? For those who advocate a \$15 minimum wage, would the benefits espoused also come from an \$8 minimum?

The reason I ask, is that it’s often said that the minimum wage needs to be updated as it was legislated years ago, and hasn’t kept up with inflation. But, given the fact that the wage was last increased in 2009, the rate of inflation is only 14.1%.

Thus we see a marked difference between the ostensible goal of modernizing the wage (\$8) and the \$15 wage frequently talked about.

Are the problems reformers seek to address only addressable via a \$15 minimum, or is it simply that the wage hasn’t kept up with inflation, in which case \$8.26 would suffice?

The labor market is missing one of the criteria of a free market, the ability of participants to decline a transaction if they only receive disadvantageous offers.

This is not a problem as long as there are more open jobs than applicants; market forces provide sensible price points (different for different skill sets) here. If there are more applicants than jobs however, prices fall quickly as there is no supply elasticity, and the gradient is basically given by the cost of the commute.

The two major ideas to counteract this market failure are a minimum wage, and a Universal Basic Income (UBI).

The minimum wage approach tries to change as little as possible. Competition for low-skilled jobs is still fierce, but at least the holders of these jobs are able to live in houses, get proper nutrition and possibly also have free time in which to improve their skill set, which will enable them to enter a different market segment, reducing pressure at the bottom. Upward mobility is greater when there is a bigger surplus in workers' wages after cost of living, because it allows for better learning conditions.

From a government perspective, we want everyone to be upwardly mobile, as we can expect that low-skilled building and manufacturing jobs to become even more scarce in the future, while at the same time, we need lots of specialists (to the extent that the US still has special visa for skilled workers).

The \$15 figure is pretty much arbitrary, but a full time (40h/wk) worker should come out at roughly \$28,000 before taxes, which is not great, but should be sufficient for rent, transportation, food and further education; at the same time the number isn't so high that it leads to competition with skilled jobs.

Going with \$8.26 would still avoid the "race to the bottom" in wages, but leaves workers nothing above subsistence level, diminishing their education opportunities and thus worsening the situation as more low-skilled workers enter the market than leave it, so unemployment remains long term and workers remain low-skilled until retirement, which is not sustainable especially in light of technological advance.

For completeness, the UBI approach simply corrects the missing part of the free market by allowing everyone to reject bad offers. This would be easier to police than a minimum wage as it removes the requirement to look at individual contracts, and just goes with equilibrium prices.


Capital flight and interest rates I am currently reviewing some stuff on capital flight and self-fulfilling crises.

In this scenario, investors generally think that a low availability of reserves could imply that the central bank may not be able to hold a fixed exchange rate. Consequently, investors quickly convert their domestic assets into foreign assets.

Given that this implies that the money supply decreases (as currency in circulation reduces), shouldn't this automatically increase interest rates? (Under IS-LM framework, a decrease in money supply increases interest rates). As a result, domestic assets should automatically become more valuable, thus fixing the problem naturally. Is there a problem in this line of reasoning?

It is very difficult for interest rates to cover the losses caused by a sharp devaluation. For example, if you believe that a currency is going to drop to 75% of its current value within 3 months month due to a devaluation, you need an annual interest rate of about 216% ($3.16^{\frac{1}{4}} \approx 1.33 \approx \frac{1}{0.75}$) to cover that loss.

Since interest rates of that magnitude would cause deep stress for domestic borrowers, we can reach a point where speculators believe that the central bank will not raise interest rates by enough to cover the losses (relative to a foreign currency) created by a devaluation.


S=I in a closed economy In a closed economy, the national income identities imply that $$S=I$$

However, the intuition is not clear to me. The saving is the part of national disposable income that is not consumed. $I$ is the sum of residential and nonresidential investment. When firms invest, they generally borrow to finance that investment. However, why is it the case that this would necessarily equal aggregate savings? Individuals might save, but firms (or households) need not borrow. Does $i$ include idle funds as well?

This is a macroeconomic tautology, and like many others, the words often do not have their commonly expected meaning.  In particular saving and investment are not directly about money but about stuff measured in money terms

Suppose you had an economy based purely on grains of wheat.  The real activities are:

Plant wheat grains, grow them and harvest them, getting more grains as a result
Consume wheat as food
Store unconsumed wheat for future planting or consumption

In this case, macroeconomic saving is the amount that production exceeds consumption while macroeconomic investment is either increasing stored stocks or the planting of grains for future production.  (If you were wondering, grains which rot after having been purchase by consumers still count as consumption, while grains which rot while in producers' storage count as negative investment.)  So in this case, both saving and investment  measure the same thing, seen from different perspectives.

Clearly you could measure this particular economy purely in grains of wheat, but if you want to do so in money terms then you would need to know the price of wheat, where say a million grains are worth $\$5$.  This would then allow you to incorporate the production and consumption of other stuff into your calculations, such as barley where say a million grains are worth $\$3$ and yields are different, or beer produced from barley and bread from wheat, or anything else.  But saving representing unconsumed production, and investment representing the other non-consumption uses, would remain equal, almost by definition


Why is it said that Sweden has a regressive income tax system? Quoting from Wikipedia

  Sweden has a progressive income tax, the rates for 2014 are as follows:

  0% from 0 kr to 18,800 kr (~0 – 2,690 USD)
  Circa 31% (ca. 7% county and 24% municipality tax): from 18,800 kr to 433,900 kr (~2,690 – 62,140 USD)
  31% + 20%: from 433,900 kr to 615,700 kr (~62,140 – 88,180 USD)
  31% + 25%: above 615,700 kr (88,180 USD and up)[4]

Why is it said that it is regressive?

E.g.

This post from Washington Post

Excellent question. You can understand this by reading the original study referred in the article. I will try to provide a short explanation here.

Basically the study argues that if you take into account all sorts of taxes like taxes on properties, consumption, dividends, etc. the European countries tax their rich residents relatively less than the poor residents, while the US does the opposite. This is possible because there are other significant taxes besides the income tax. As the study states:

  The main reason for this is the relatively low reliance of the
  American fiscal system on sales taxes, which constitute a much higher
  share of revenues in other countries (even when including state and
  local taxes in the US case, as the figures above do). General sales
  taxes place a disproportionate share of the tax burden on the lower
  end of the income distribution because expenditures are higher as a
  share of income as we move down the distribution.

To gain an exact understanding of the key claim in the WP article we have to understand both the Lorenz Curve and the Kakwani index. The Lorenz Curve has a pretty good Wikipedia page, I recommend browsing it. The Kakwani index is chiefly used to calculate payments for health services.

  The Kakwani index is defined as twice the area
  between a payments’ concentration curve and the Lorenz curve and is
  calculated as, $\pi_K = C - G$ , where $C$ is the health payments’
  concentration index and $G$ is the Gini coefficient of the ATP variable.
  The value of $\pi_K$ ranges from –2 to 1. A negative number indicates
  regressivity; $L_H(p)$ lies inside $L(p)$. A positive number
  indicates progressivity; $L_H(p)$ lies outside $L(p)$. In the case of
  proportionality, the concentration lies on top of the Lorenz curve and
  the index is zero.

Source

The difference in the Kakwani index is small though, the range of the graph in the article is in the hundredths.


What is the evidence that econometrics has empirical value? Economies are extremely complex systems with many variables, not to mention the fact that they emerge from the interactions of complex beings. I agree that economies have certain underlying principles, but I remain skeptical of the overall value of econometrics as a science.

A truly useful econometrics would be a valuable tool in predicting the future behavior of the economy, particularly in predicting shocks like the recent financial crisis and Great Recession. But it did not happen.

If econometrics can't predict the future, how do we know it even effectively describes the past? What is the evidence that it has empirical value?

Clarification

I regret any confusion this may have caused, but let me clarify what I'm looking for here. I regard this as an epistemological question. A better way to phrase it might be, "Can econometrics conform to the scientific method?" or just "Is it science?"

This is, at least in theory, a specific, answerable question, even if it is, as Alecos Papadopoulos noted in his answer, a matter of degree. (A view I share.)

This does not mean that specific examples of econometrics' successes and failures are not relevant.

To (slightly) paraphrase the OP:

  Economies (Human Bodies) are extremely complex systems with many variables, not to mention the fact that they emerge from the
  interactions of complex
  beings (factors). I agree that economies (human bodies) have certain underlying principles, but I remain skeptical of the
  overall value of econometrics (medicine) as a science.

  A truly useful econometrics (medical science) would be a
  valuable tool in predicting the future behavior of the economy
  (human body), particularly in predicting shocks like the recent financial crisis and Great Recession (severe illnesses or near
  death). But it did not happen.

  If econometrics (medical science) can't predict the future, how
  do we know it even effectively describes the past? What is the
  evidence that it has empirical value?

Comment: There is a Present that needs to be dealt with, just like with human bodies.

To (slightly) paraphrase Milton Friedman:

  I don't care how I predict, as long as I predict adequately.

Comment: But what do we do when our predictions are not adequate ?

Econometrics is based on Mathematical Statistics on the one hand, and on the assumptions made by Economic Theory on the other, which in turn are based on (imperfect) empirical observation, and are then led to their logical conclusions. In other words, Econometrics uses rigorous mathematics, induction, deduction and all the words dear to an epistemologist. Its epistemological foundations are solid as a rock.

What hangs on the balance is the observed validity of its abstractions. So the question is not whether Econometrics is a science, in the sense of whether it follows the scientific method or not. It does, fully.  The question is whether its results are useful.

But what are the criteria in order to determine "useful"? Is it just adequate predictions? That would be a matter of disagreement between humans.

And is it a matter of a "yes/no" answer?  Or is it a matter of degree to which it is useful? In which case, we have to somehow measure this degree (after we have agreed on the criteria), which brings us back to where we have to collect, analyze, assess, and debate the evidence.


European Commission getting rid of excess carbon emission permits According to this Economist article,

For a long time after it was launched in 2005, the [European Trading System] barely functioned; a glut of allowances (which give the holder the right to emit an amount of greenhouse gases) kept prices close to zero. But after the European Commission sucked out excess permits in 2019, the market began to thrive.

the European Commission got rid of, so-called, excess permits.
What were these excess permits of carbon emissions?
Is it the amount of permits that are still up for sale (or to be distributed for free); and hence, aren't used?
And how did the European Comission get rid of them? Is it by decreasing the cap of overall supply of permits within the EU?
This article of the European Environment Agency designates the surplus of permits as a market stability reserve.
The Swedish Energy Agency, describes the market stability reserve as, a rule-based mechanism that enables the delivery of allowances to respond to changes in demand, thus maintaining the balance of the EU ETS.
It says

If the surplus of allowances on the market does not decrease to less than 400 million but if the price of an emission allowance for more than six consecutive months is three times higher than the average price over the last two years, up to 100 million allowances are injected into the market through an increase of auction volumes with units from the reserve.

Hence the number of allowances is dynamic, set according to the condition of the market. But isn't this ultimately limited by the overall EU cap that is decided politically? Also, what is the unit of an allowance?

What were these excess permits of carbon emissions? Is it the amount of permits that are
still up for sale (or to be distributed for free); and hence, aren't used?
Page 92 of the ETS handbook states that since 2009, because of the economic crisis of 2008, a higher than expected imports of international carbon credits and a large uptake of renewables led to a significant surplus of allowances in the EU ETS.
So, yes, excess permits are the ones that aren't used by the holder in the near future. According to the ETS handbook, page 92, in early 2012 there were 1 billions surplus allowances. And 2 billions by the start of 2013.
From the handbook:

The presence of the surplus and resulting low price may discourage participants in the
EU ETS from taking actions now to reduce emissions, which may result in a deviation from the most
cost-efficient path to long-term emission reduction goals. The large surplus has led to persistently low
carbon prices in the EU ETS, reducing the price incentive for low-carbon investments.

One mechanism of tackling this surplus in allowances is the creation of a market stability reserve (MSR).
And how did the European Comission get rid of them? Is it by decreasing the cap of overall supply of permits within the EU?
The MSR acts to combat this imbalance of allowances. From the handbook, page 95:

Using the total number of allowances in the market as an indicator, market imbalances due to
unexpected shocks that impact demand, such as the economic crisis, can be addressed. This allows
the EU ETS to maintain its objective to reduce emissions in a cost-effective and economically efficient
manner, even under unexpected circumstances.

Hence, this way, each year the number of allowances injected into the market by auction will take the market demand into account.
So, yes, the amount of overall supply of permits, by auction, will be decreased if there is an excess. This will act as a carbon reduction incentive, by keeping prices competitive.
From this diagram of the ETS handbook, company B, with a shortage of allowances, has two options. Either buy from company A's surplus allowances or buy them at an ETS auction.

Also, what is the unit of an allowance?
The website of the European Commission states that each allowance gives the holder the right to emit one tonne of carbon dioxide (CO2), or
the equivalent amount of other powerful greenhouse gases, nitrous oxide (N2O) and perfluorocarbons (PFCs).
But isn't this ultimately limited by the overall EU cap that is decided politically?
Yes, indeed the total amount of allowances injected into the market by the ETS is determined politically. This overall cap will decrease accordingly, to realize the European Green Deal; no net emissions of green house gasses by 2050.


Growth rate of variables on a balanced growth path (BGP) Normally, in the very basic endogeneous model of growth. It is said that all variables grow at a constant rate.
Let's take the basic capital accumulation ;

$$\dot{K}=Y-C$$

If I write this in following way ;

$$g_{K}=\frac{\dot{K}}{K}=\frac{Y}{K}-\frac{C}{K}$$

Normally, at balanced growth path (BGP), $g_{K}$ grows at a constant value (an arbitrary constant)

In this case, this means that variation of $g_{K}$ according to time at BGP should be equal to zero.

So ;

$$\frac{dg_{K}^{BGP}}{dt}=\left(g_{Y}-g_{K}\right)-\left(g_{C}-g_{K}\right)=0$$

which implies at BGP ;

$$g_{Y}=g_{K}=g_{C}$$

Is this reasoning correct ?

Because in fact, it seems me weird to differentiate a growth rate $g_{K}$ according to time.

The concept of "balanced growth path" in economics incorporates three characteristics at the same time (related to the main macroeconomic aggregates):

1) Growth rates are constant  (reflecting a notion of equilibrium/stability)

2) Growth rates are strictly positive  (otherwise the economy would eventually vanish)

3) Growth rates are equal to each other (hence there is "balance" and no macroeconomic aggregate becomes negligible relative to another)

To obtain the first characteristic you have to impose that the derivative of the growth rate with respect to time is zero. Nothing weird about that, just a valid mathematical operation.