Do banks really have people on the boards of corporations, and why? I read somewhere that in the United States, there are often people from banks on the boards of executives of corporations. Note, I don't mean ex-bankers who now work for that corporation, I mean people who are actually employed by the bank, and who are on the board in their capacity as a representative from the bank.

This surprised me very much, since the board of directors in principle represents the equity-holders, not the debt-holders of corporations.

How often does this really happen?

Why do banks do this, and why do corporations allow this?

What you describe is neither unusual nor pernicious, and occurs more commonly at smaller companies when the company has borrowed significantly from a single bank. Larger companies have direct access to capital markets (in addition to stock, they can issue bonds and commercial paper, which are both forms of debt), and are thus less reliant on their relationship with a single bank.

When companies are reliant on a relationship with a bank for access to debt, the bank will often seek a position on the board as a way of monitoring the activity of the company. From the bank's perspective, this is totally reasonable, but to understand why, you first have to understand a tiny bit of industrial organization theory.

In a nutshell, firms have what we refer to as "capital structures," which are a way of allocating risk and return.

At the top of capital structures is senior debt, which is first in line to be paid, but which receives the lowest (though a fixed and predictable) rate of return. If you hold senior debt, you want the company that issued the debt to take little risk, as you'll be paid the same amount so long as the company remains solvent. If the firm goes into bankruptcy, you'll still be paid first, but your money may be tied up for a time in bankruptcy proceedings, and you face having to reinvest the cash you get back, potentially at a lower rate of return. As a result, you want to avoid unnecessary risk.

At the bottom of capital structures is equity, which is high-risk and high-return. The equity portion gets paid the upside if a firm does well, and it bears the first loss if a firm performs poorly. If a company goes bankrupt, the equity is worthless. Equity investors have a lot of incentive to take on risk, because for each dollar invested, the downside is a maximum of a dollar (this is known as "limited liability"), but the upside is limitless.

In between these two parts of the capital structure, there are all kinds of hybrids of debt and equity, but we can set them aside for simplicity. Most important is that there is a fundamental conflict between debt holders, who want limited risk and who only have control of a firm once it is bankrupt, and equity holders, who favor risk and control the firm in the normal state of the world.

Once you understand this conflict, it's easy to see why a bank would ask for a seat on the board of a company when it holds that company's debt: by having a member on the board, it can ensure ongoing, regular access to the information it needs to be comfortable that the firm isn't taking on too much risk. If it sees that the firm is moving in a direction that it isn't comfortable with, it'll decline to roll over its debt, getting its money back and forcing the firm to find financing elsewhere.

So: is this unusual? No. Directors who don't directly "represent" equity holders aren't at all uncommon. In fact, the majority of directors at large firms in the US are "independent" directors who have no direct relationship to the firm, other than serving on the board. Many non-independent directors are "executive" directors, who are company management— a group that, like creditors, may have incentives that differ from those of shareholders. Other groups, like labor unions, are also sometimes represented on boards. So if in addition to equity holders, outsiders, management, and unions are all represented on boards at times, what is the specific objection to creditors being represented?

It's additionally worth noting that the only case where a creditor's vote matters is solely when the other directors are evenly split on an issue. If a particular matter clearly benefits creditors at the expense of equity holders, a significant fraction of equity holders would have to vote against their own interests for this to be the case.

Finally, to your question as to why regulators and corporations allow it. Simply put, regulators have no reason (from a market conduct, safety and soundness, consumer protection, or financial stability perspective— the general scope of regulators' mandates) to object to this sort of arrangement. For a regulator to intervene in this way would interfere with the company's freedom to contract however it chooses. Creditor board members are not a thing that banks do to corporations, who passively allow it; rather, the boards of directors have to vote to add the board member representing the creditor. So if one views positively the judgment of a board composed of equity holders, why should one question the judgment of that board when they vote to add a creditor as a member?

So in summary: banks as board members is a way of monitoring/limiting risk-taking by firms, firms "allow" this because they want to borrow money from the banks, whether it is socially optimal depends on whether firms should be taking more or less risk, and it's more about correcting information asymmetries than about asserting day-to-day control— boards of directors don't tend to have lots of measures passing or failing by a single vote.


When everyone sells at an event of a recession, who buys? I'm trying to walk through and understand the basic events of the great depression. I understand that at one point (Black Thursday) the news of a downfall of stock prices hit investors, so they obviously wanted to get rid of their shares.

Given the very high pace investors started to sell, there needs to be an equally big number of buyers, or -- more likely -- a not that big number of buyers, who bought that huge number of shares.

Is this a valid assumption?
If it is, I don't see who bought those shares.

the news of a downfall of stock prices hit investors, so they obviously
  wanted to get rid of their shares.

This is not the way you want to think about prices. Stock prices are determined via the equilibrium of demand and supply. It sounds like you are assuming that they drop from the sky and investors react, but they are determined via the actions of investors. What happens is that news of the fundamentals of the economy (Things like interest rates, employment numbers, job forecasts, etc.) hit, investors react, and then price falls.

Suppose there are two investors A and B and a single stock. Suppose that $t=0$ both investors 'subjectively value' the stock at 10, this valuation is determined via each investors subjective expectations of future earnings and future prices. So the price at $t=0$ is 10 and lets suppose that investor A owns the stock. At $t=1$ news about the fundamentals of the economy hits, each agent updates their subjective value of the stock. Suppose that the 'value' to agent A is 5 and to agent B is 6. Its clear that Agent A will trade the stock to agent B at a price between 5 and 6, hence price falls endogenously not exogenously.  As this example illustrates, trade occurs between pessimistic sellers and relatively optimistic buyers.


Value added by workers Is there any empirical evidence on the value added by workers?

That is, their "productivity" minus the wage payments, social security benefits etc. that are paid by the firm?

I'm eventually looking for cross-sector or time variation, but anything would be a good starting point right now.

Normally when we think about value added, we think of marginal revenue product of labor: how much more revenue does adding another worker add to the firm?

If you assume that the market is competitive, then the MRPL should be exactly equal to what you said: wage, benefits, etc. Otherwise another worker would be willing to work for an epsilon less.

In other words, the "value added" by your definition (their "productivity" minus payments) should be zero. From my experience, researchers measure worker's productivity by wages.

If you want to find where this does not hold, then I guess you'll need more detailed data. Perhaps comparing workers with same job description but different compensation.


Cobb-Douglas production function with expenditures rather than units I want to estimate a standard Cobb-Douglas production function of the form

$ Y=AK^{\alpha}L^{\beta}$. However, I only have data on labour expenditures, not units of labour. I don't have data on the price of labour $w$ either, so I can't back out the units of labour from expenditures. Is it okay to estimate the production function (in a GMM regression framework) using labour expenditures instead of units of labour?

Sure you can, just that your interpretation of your variables in your analysis changes however. In this case you are analyzing how investment in differing factors of production affect output.

I'd recommend that you may want to estimate a more flexible functional form like the Translog Production Function to check if your function is CES instead of just a simple cobb-douglas.


Why are we taking the logarithm for risk-averse decision-makers? I'm a mathematics student learning a bit of Game theory. Many examples are given within a very economic setting and up to know I could follow most of it because they were very basic and I learnt some basic economics in high school.

Now I've come across some examples where a player is risk-neutral whereas the other is risk-averse (e.g. insurance company vs. customer). Say we have outcomes $(A_i)_{i \in I}$ which occur with probabilities $(p_i)_{i \in I}$, and they have payoffs $(w_i)_{i \in I}$. It seems the expected payoff for risk-neutral players is simply

$$\sum_{i \in I} p_i w_i$$

whereas for risk-averse player we apply the natural logarithm and end up with

$$\sum_{i \in I} p_i \log w_i.$$

In the lecture notes I am reading, I have found no justification for this and I assume it is covered in other lectures in the standard curriculum of any economics student.

Question 1: Why are we taking the logarithm? (mathematically and economically)

Question 2: Are there more "measures" of risk-attitude? If so, how does the expected payoff change?

Let's step back. In general, utility functions are (monotone increasing) functions $f(w)$ that map wealth into utility. That is, the input $w$ is money and the output $f(w)$ is utility. If we have a distribution $\vec{p}$ over possible wealth outcomes $w$, then my expected utility is $\mathbb{E}f(w) = \sum_i p_i f(w_i)$.

Now, what is risk aversion? Intuitively, it means I prefer a certain outcome to an uncertain one even though they have the same expectation.

Thus, a risk-averse utility function is defined to be a concave utility function $f$. This is natural because of Jensen's inequality: if $f$ is concave, then for any distribution over outcomes of $w$, we have $\mathbb{E}f(w) \leq f(\mathbb{E} w)$. In other words, I prefer to get $\mathbb{E}w$ for certain rather than some uncertain amount of money with that expectation.

To connect back to your question (1), you are comparing the expected utility from two different utility functions: $f(w) = w$ and $f(w) = \log w$. The latter is just a very nice and handy concave (hence risk-averse) utility function. But it is only one example (albeit an important one) of a risk-averse utility functions.

And to connect back to your question (2), to get risk aversion, any concave function will do. But the "more" concave, the "more" risk averse. Meanwhile, risk-neutral functions should satisfy $\mathbb{E}f(w) = f(\mathbb{E} w)$, so that's any "affine" function of the form $f(w) = a\cdot w + b$. Finally, risk-seeking utility functions are convex (and less-often studied as they seem less natural).


Intuition behind Comparative Advantange I already read and ask NOT about pp 52-54, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw. I understand such numerical examples that corroborate Comparative Advantage, but how can I intuit comparative advantage? I wish to quash my need to revisit these numerical examples to recollect this idea. i start with definitions from pp 52 and 53 (supra):

  absolute advantage = the ability to produce a
  good using fewer inputs
  than another producer

  comparative advantage = the ability to produce
  a good at a lower
  opportunity cost than
  another producer

I read this by Prof Steven M. Suranovic and understand the gardening example, but still lack intuition.

  First, ... comparative advantage is clearly counter-intuitive. ... Secondly, the theory is easy to confuse with ... the theory of absolute advantage. The logic behind absolute advantage IS quite intuitive. This confusion between these two concepts leads many people to think that they understand comparative advantage when in fact, what they understand is absolute advantage.

  However, instead of assuming, as Adam Smith did, that England is more productive in producing one good and Portugal is more productive in the other; [David] Ricardo assumed that Portugal was more productive in both goods. Based on Smith's intuition, then, it would seem that trade could not be advantageous, at least for England.

A classic example is to consider the question

  Why doesn't Tiger Woods mow his own lawn?

Woods is an excellent golf player. But he also may be excellent at mowing. Still, if he devotes time to mow the lawn he sacrifices time playing golf, in which he is much more productive.

A gardener who is skilled at mowing lawns but not at playing golf sacrifices less in order to mow, so even though Tiger Woods may be better at it (i.e. has an absolute advantage), the gardener has a comparative advantage. Both benefit if the gardener is paid for the time that he mows the lawn by Tiger Woods who in the time he saves can earn more by concentrating on golf.


Fixed Effects Model Interpretation $$\ln(\text{pay})_{itj}= \alpha_{ij} + \beta\ln(\text{performance})_{jt} + \gamma X_{ijt} + T_t + ε_{ijt}$$
This fixed effects model was used to investigate the link between pay-performance throughout the corporate hierarchy. It controls for employee-firm match fixed effects ($\alpha_{ij}$). $i=$ employee, $j=$ firm, and $t=$ time.

How would the coefficients on performance from this table be interpreted.

You have a log-log model, so the coefficients on performance can be interpreted as the elasticity of your outcome (performnce) with respect to the regressor of interest (level in the corporate heirarchy).
This table only shows the coefficient of interest for four separate regressions. For each of these job levels (CEO, other board, white-collar, all other), a 1% increase in the 'total expected ex ante pay' leads to a x-% increase in 'total shareholder returns'.
For example, based on the results reported in this table, a 1% increase in total ex-ante CEO pay is associated with a 0.22% increase in total shareholder returns, a 1% increase in total ex-ante 'other board executive' pay is associated with a 0.21% increase in total shareholder returns, etc.
All four coeficients are statistically significant at the 1% level (which you can confirm by looking up the values of each coefficient's t-statistic, which is just the coefficient value divided by the standard error).


From single-consumer-multi-units to multiple-consumers-single-unit When I took a course in consumer theory, the economy always had a single consumer, represented by a monotone positive utility function $u(x,y)$ and an income $I$. Given prices $p_x$ and $p_y$, it is possible to calculate the demands of the consumer for products $x$ and $y$.

Now, I deal with a different type of economy: there are many consumers, each of them wants only a single unit of each product. Each consumer is represented by three positive values: $u_x$ (utility for having $x$), $u_y$ (utility for having $y$) and $u_{xy} \geq \max(u_x,u_y)$ (utility for having both $x$ and $y$). Given prices $p_x$ and $p_y$, each consumer buys either $x$ or $y$ or both, whichever gives the highest net utility (utility of product/s minus price). So it is possible to calculate the aggregate demands for $x$ and $y$.

MY QUESTION IS: Is there a natural/standard way to convert between these two types of economies?

I.e, given a utility function $u(x,y)$ and income $I$ for a single consumer, is it possible to construct a set of consumers with different $u_x$, $u_y$  and $u_{xy}$, such that the demand curves in both economies are the same?

The cumulative distribution function (CDF) describes the probability that a random variable X with a given probability distribution will be found to have a value less than or equal to x. That is: $$F_X(x) = P(X \leq x) $$

In this context, think of $X$ as the valuation of a particular unit demander. The demands are distributed with a certain probability distribution. So the the CDF tells you the probability of  a randomly drawn demander of having a valuation at least as large as $x$. If everyone demands either one unit if the price $p \leq X$ and 0 otherwise, then this also gives a demand function:

$$Q_d(p) = P(X \geq p) = 1 - P(X \leq p) $$

because at each price $p$ everyone with a valuation at least as high as $p$ will buy their one unit. But the fraction of people with a valuation at least as high as $p$ is one minus the fraction of people with valuations $p$ or lower.

Next, consider a single household with a demand function that just so happens to look like $1 - P(X \leq p) $. This household will have an identical demand for the good at all prices as the collective demand of the unit demanders above.

I believe this generalizes to the more complex case you present in the question but I haven't worked out the details.


What is time inconsistency in monetary policy? I am reading through a bit of policy literature and see authors referring often to the problem of time inconsistency when setting monetary policy.

Can anyone explain what is time inconsistency?

An example from Daniels and Vanhoose (1998):

  Rogoff’s paper makes one of the most fundamental points about policy coordination: In the presence of time inconsistencies, monetary policy coordination is not necessarily welfare improving.

In a nutshell, time inconsistency occurs whenever a policy maker can publicly commit to a certain policy action in a non-binding way. This is an issue in several settings (commitment vs. discretion settings and, as you see, policy coordination settings). You've run across the issue of time inconsistency in the context of policy coordination across interdependent economies. One can summarize the issue as follows: I promise in $t-1$ to deliver policy action A in period $t$ but instead deliver policy action B once we arrive in period $t$.

Consider, as does the paper you cite here, a simple two-country model where the policy actions in a domestic country impact the loss function of the foreign economy and vice versa. Under certain first order conditions (laid out in the very paper you quote) there are obvious gains available to the policy makers of these nations if they are able to arrive at a coordinated equilibrium rather than the Nash equilibrium. However, in a dynamic game with no commitment device aside from reputational concerns it might be the case that one country will promise to play the coordinated equilibrium and then deviate unilaterally to the cheater equilibrium. As you can see, time inconsistency in this context means (literally) for one to be inconsistent over time with promised and realized policy actions.

Further, see Luchonacho's references in his comment. Both links are quite helpful here and themselves could serve as an answer to your question.


If you simply "copy" the trades of a professional trader, but with less money, isn't that *guaranteed* profits? I keep coming back to this thought:
I just don't understand the complex "rules" or "patterns" of how the prices of Bitcoin (or any financial instrument) move up and down, so I will not "gamble" with my money to make money that way.
But then there are people who must know their stuff, because they make endless YouTube videos about it and seem to know so much. And they trade themselves.
If I simply copy their trades, except with much smaller sums, isn't that going to guarantee that I make profits perpetually? (Assuming that they will show me their trades.)
They are not going to suddenly make some crazy bet and lose all their money. If I just do what they do, every time, am I not going to keep making money, guaranteed? What am I missing? What is the flaw in this logic? It cannot be that simple...

Within finance, there is interest in attempting to replicate the positions of strongly performing funds. So your basic premise has some merit. However, most funds attempt to obscure what they are doing to foil copycats.
However, what you describe has risks.

The investors you are copying may lose money themselves. In particular, there is no reason that people making videos on spreading them online are telling the full truth about their performance. Securities laws generally prohibit giving unsolicited investment advice, and professional investors are generally aware of those laws.
If you get information with a lag, you are not going to replicate their results. In the worst case, they buy before the price rise, and then recommend others buy at the higher price (allowing them to sell). This is a common feature of fraudulent stock trading schemes in penny stocks.
Even if you can avoid the previous problems, a smaller investor may not be able to trade all the instruments a larger investor can, and face higher transaction costs. If positions are actively traded, higher transaction costs (as a percentage of the portfolio size) can mean the difference between a profit and a loss.


Demand estimation with a lagged dependent variable Suppose I have the following structural equation for demand estimation in time-series:

$$q_t=\beta_0+\beta_1\hat{p_t}+\beta_2incom_t+\beta_3q_{t-1}+\epsilon_t$$

Where $q_t$ stands for the quantity of the product, $\hat{p}$ stands for the instrumented price with supply shifters, $incom_t$ stands for income, $q_{t-1}$ stands for the lagged quantity and $\epsilon_t$ are the random residuals. All the betas are estimated coefficients -- say, through two-stage least squares. All variables are log-transformed too.

Suppose also that the lagged variable seems to solve the problem of autocorrelation between the residuals and its coefficient is also significant. Suppose also that the model passes through all the necessary tests for a viable two-stage least squares estimation -- the sargan test is ok, the instruments are strong in the first stage etc.

Knowing this, what are the possible problems of having a lagged variable in your estimation? Does it change any interpretation of the elasticity ($\beta_1$)? I understand that I could also have some sort of long-run inferred elasticity if I set $q_t=q_{t-1}$ after estimating the coefficients of the equation (Am I wrong?).

Does it change any interpretation of the elasticity ($β_1$)?

You walk into the firm where you work as an analyst, and the Sales Director calls and asks "I want to raise the price $10\%$ today. How will demand be affected in percentage terms?

Well, you don't expect income to have changed from one day to the next, and what was demanded yesterday -it was yesterday. So the only variable (for today) in the equation is price, and the best you can say is "I expect a $(10 \times \hat \beta_1) \% $ effect".

So it is the short term price elasticity of demand, irrespective of the fact that the level of demand is determined by other factors also, lagged demand included.

  I understand that I could also have some sort of long-run inferred
  elasticity if I set $q_t=q_{t−1}$ after estimating the coefficients of the
  equation.

Yes, either in an old fashioned "deterministic", $q_t=q_{t−1}$ way, or in a stochastic, $E(q_t) = E(q_{t−1})$ "mean-stationary" way.

Assume now that demand is already mean-stationary according to your analysis , and you get the same question from the Sales Director as before. Will your answer change compared to the previous one?

ANSWER

    The answer to the Sales Director should not change, because you can condition on the given and known past, for a more accurate (= focused on the specific situation) answer so you use $\hat E(q_t \mid p_t, I_t, q_{t-1}) = \hat \beta_0+ \hat \beta_1\hat{p_t}+\hat \beta_2I_t+\hat \beta_3q_{t-1}$ and not the unconditional (estimated) relation $\hat E(q_t) = \hat \beta_0+ \hat \beta_1E[\hat{p_t}]+\hat \beta_2E[I_t]+\hat \beta_3E[q_{t-1}]$,which would lead to the long-run (unconditional) elasticity.  But if he was to ask, say, "I want to introduce the product to a new market where the consumers are comparable in income to what we already have. What demand-responsiveness to price should I have in mind?", then the best you could do is provide the long-run elasticity.


How should I interpret (BTC/CNY) / (BTC/USD)? So, I've been following bitcoin for a number of years now and I decided to look at how it plays with the dollar and the Chinese yuan. I'm wondering if someone can help me interpret the graph below. I have some background in physics but not much in economics.

So I understand that the green line ("USD/CNY") is the value of the dollar compared to the yuan in the global foreign exchange market. When it goes up, that means the value of the dollar has gone up for those buying with yuan.

The purple line is "BTC/CNY" divided by "BTC/USD", and here's where I'm interested in your help. I have a couple thoughts:

The lines both have the same units
The purple line represents the same thing as the green line EXCEPT that it's mediated by the bitcoin market instead of the global forex market
The purple line compares the value of bitcoin for buyers with yuan to the value of bitcoin for buyers with dollars; When the purple line is above the green line, bitcoin has a higher value to buyers with yuan than buyers with dollars.

Are those reasonable interpretations? Thanks in advance!

Regarding your question before the update:

1) Yes, they have the same units ("how many units of CNY you need to buy one USD", either directly or through a bitcoin intermediary). That is the whole point of arbitrage.

2) It seems that "the global forex market"  and "the bitcoin market" seem separate to you. If by the former you mean big investment banks, then maybe you are right, as there seem to be no indication that big investment banks are systematically engaging in Bitcoin trading (at least I found no evidence of it; please correct me if I'm wrong). However, there are tons of bitcoin-oriented banks and brokers doing transactions on currency pairs, and I'm not sure these should not be counted as part of the global forex market.

Having said this, I think the simplest interpretation of the lines is that of a currency pair, and that of the currency pair being arbitraged through bitcoin.

3) The green line refers to how many units of CNY you need to buy one unit of USD. Or equivalently, how many units of CNY you get for one unit of USD. If the purple line is above (below) the green line, then you can get more (less) units of CNY by (1) selling USD through bitcoin and (2) buying CNY with those bitcoins. This means your interpretation in the update is correct. If you have USD, sell them to BTC, and buy CNY (purple line), and then exchange these directly to USD. As the green line is below the purple one, you need less CNY to buy one USD, which is the same to say that you get more USD for the same amount of CNY.

So, there seem to be arbitrage opportunities, at least in a zero transaction cost market.


Solving for Feedback Effects / Amplification Consider

$$f(x) + z = y \\
x = g(y)$$

I have a feedback effect in mind:

We Shock z by 1 percent
as a direct response, y increases
x increases with y through the second equation
Through f(x), y increases. Repeat 3.-4. until new equilibrium

I'm interested in solving the elasticity of $x,y$ w.r.t. $z$. I guess I need some sort of concavity requirement in $f \circ g$. How do I approach this?

We have an equilibrium given by
$$h_1(x,y)=f(x)+z-y=0,$$
$$h_2(x,y)=x-g(y)=0$$.
The implicit function theorem then says that (omitting the arguments):
$$\frac{\partial x}{\partial z}=\frac{-\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial z} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial z} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)}{\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)},\quad \frac{\partial y}{\partial z}=\frac{-\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial z} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial z}
 \end{matrix}\right)}{\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)}.$$

My source for this is Mathematical Methods and Models for Economists by Angel de la Fuente, although I don't have the book to hand now and can't remind myself of the intuition.

This implies
$$\frac{\partial x}{\partial z}=\frac{g'(y)}{1-f'(x) g'(y)}$$
$$\frac{\partial y}{\partial z}=\frac{1}{1-f'(x) g'(y)}.$$

For the implicit function theorem to hold and this solution to be valid, we need $1-f'(x) g'(y)\neq0$.

More generally, the way this works is as follows: you write down a system of equations whose roots characterize the equilibrium:

$$F_1(\mathbf{x};a)=0,F_2(\mathbf{x};a)=0,\ldots,F_n(\mathbf{x};a)=0$$

(where $a$ is the parameter of interest). From them, we construct the vector-valued function

$$\mathbf{F}(x)=[F_1(\mathbf{x};a),F_2(\mathbf{x};a),\ldots,F_n(\mathbf{x};a)]$$
which has the Jacobian matrix
$$\mathbf J = \frac{d\mathbf F}{d\mathbf x}
= \begin{bmatrix}
    \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_m}\\
    \vdots & \ddots & \vdots\\
    \dfrac{\partial F_n}{\partial x_1} & \cdots & \dfrac{\partial F_n}{\partial x_m} \end{bmatrix}.$$

To calculate the derivative of $x_i$ with respect to $a$, we construct the modified Jacobian in which we replace the $i^{\text{th}}$ column with derivative WRT $a$ instead of $x_i$. So, for $x_1$ this would look like
$$\mathbf{J}_{x_1}=\begin{bmatrix}
    \dfrac{\partial F_1}{\partial a} & \dfrac{\partial F_1}{\partial x_2} & \cdots & \dfrac{\partial F_1}{\partial x_m}\\
    \vdots & \vdots&\ddots & \vdots\\
    \dfrac{\partial F_n}{\partial a}&\dfrac{\partial F_n}{\partial x_2} & \cdots & \dfrac{\partial F_n}{\partial x_m} \end{bmatrix}.$$

The derivative of interest is then calculated as $$\frac{\partial x_i}{\partial a}=\frac{-\det\mathbf{J}_{x_i}}{\det\mathbf{J}}.$$

We need $\det\mathbf{J}\neq0$ for the implicit function theorem to be valid.


Effects of long run stability in U.S price level comparing to other countries on net export Suppose the United States experiences a long period of relatively stable prices while other countries experience long periods of inflation. How will this affect U.S. net exports?

I think that if U.S have long period of stable prices ,when U.S export some products abroad ,  it will be cheaper than the others.Therefor Export will increase  How can we predict  net export?

Net exports are exports less imports.

If other countries begin to experience inflation while US prices remain stable, then the dollar becomes more attractive to investors as its purchasing power has maintained more so than foreign currencies. The demand for dollars increases, which leads to an appreciation of the home currency. This leads to a decrease in net exports as US goods become pricier relative to foreign goods.

This theoretical relationship between the exchange rate and the relative purchasing power of two currencies is known as purchasing power parity.


Economic Confirmation Biases I'm not sure if this is the right place to post this, since my question is a bit more general to all of social sciences (indeed sciences in general), but I figured since Economics is a social science and since the question I have is somewhat economics related someone here might have an idea.  I am a statistician but work with a lot of economists who test various economic theories through observational data (not experimentation).  Some of the "studies" I've seen carried out recently concern me because it seems, no matter the findings our outcomes, one particular researcher tries to tie the specific results back to a pet theory of his.  In other words, the hypotheses are not clearly laid out in advance, but he looks at some data and if it shows 1 thing, he's say, Ah-ha, that is because my theory explains it.  But if the same data showed just the opposite finding, he's still say, ah-ha, this is because my theory explains it!  Specifically, let's say he compares 2 items for purchase that are identical in all physical aspects but one.  If the price of item A is higher than price B, he's explain it with this theory.  But if price of item A is lower than B, he'd simply twist his theory to to explain it as well.

Is there a name for this?  It sounds like confirmation bias a bit, but not quite.  I'm just curious if there is a name for this way of manipulating the findings to fit a narrative?  Any ideas would be greatly appreciated.

Confirmation bias is somewhat different to the behavior you are describing. This expression describes the attitude of a researcher who interprets ambiguous data as confirming his initial hypothesis. For instance, researchers who believe that death penalty has a deterrent effect on crime would interpret any additional dataset as confirming this theory, whereas researchers who believe the opposite would interpret the same evidence as indicating that the effect is null.

The behavior that you describe might be called "post hoc analysis" or "post hoc theorizing". Scientific standards in experimental economics have been  improving recently to mitigate this bias (among other mistakes/misbehavior). For instance, many researchers who do randomized controlled trials in development economics now pre-register their analysis plan (and even sometimes a theory with its predictions) before collecting the data: by doing so, they commit to a list of statistical tests that they will run, which prevents them from changing their theory/hypotheses (and even the question of interest) after looking at the data. Of course, as @denesp mentions, this cannot generally be done with observational data, except maybe in situations where the researchers need to go through an approval procedure to get access to the data, in which case a pre-registration of the analysis plan might be imposed as part of the application.


What does steady state mean? I am reading a book on business cycle models, and it keep using the word "steady state". It never defined what that actually means.

Obviously it is the idea that some key variables reach constant levels ... but WHAT key variables? It doesn't explain that.

For example, at one point it says "zero inflation steady state". Uhm, sorry, but what? Zero inflation steady state? Of course inflation is zero in steady state ... otherwise price levels would not be constant, hence not in steady state. In fact, if inflation was ever non-zero, we can NOT be in steady state since then price levels must be changing by definition.

So what am I supposed to make of this? What is steady state??

A system is said to be in steady state if certain variables do not change over time (and where "certain variables" depend on the context and ought to have been clearly specified by the writer).

Say we define a system to be in steady state if inflation does not change over time. Then if inflation is constant at 2%, we say that the system is in steady state even though prices are changing. (This is contrary to what you've suggested in your post.)

(The term steady state was, like the term equilibrium, appropriated from the sciences for use in economics. However, although the term equilibrium is often ill-defined, the term steady state is usually amenable to precise definition, provided the writer makes an effort.)


Identification with BBL In the last few years, the estimator proposed by Bajari, Benkard, and Levin ('07) for dynamic games has been gaining popularity. It is relatively straight forward and is one of the only viable options for estimating dynamic games with both continuous state space and continuous decision variables. I have heard from a few people, though, concerns about what it is really identifying (possibly not the structural parameters it's supposed to be identifying).

My question is three-fold. 1) what are the specific concerns about identification with BBL, 2) when do they (not) matter, and 3) is there a way of getting around the identification issues without having to, say, approximate the state/actions as discrete.

After searching for a while, this is the best answer I can so far come up with.

1) A formalized argument for why identification could break down under BBL is from Srisuma ('13). He gives two specific examples in the online appendix  where identification is lost because of using additive rather than multiplicative perturbations to construct off-equilibrium value functions (as was suggested in the original BBL paper). This is indicative of a broader issue with BBL that there might be off-equilibrium parameters that satisfy the BBL minimum-distance estimator.

2) The two examples given in the appendix are both quite basic and standard (single-agent and Cournot). This suggests that the phenomenon could be a problem in many/most applications.

3) Be creative with policy perturbations. Although Srisuma doesn't show the benefit of multiplicative over additive perturbations in general, the couple examples given show that multiplicative perturbations could improve the estimator. Formalizing optimal perturbations seems a good place for further research.


Euler equation in Continuous time VS Discrete time I have seen the euler equation in discrete time for the baseline neoclassical growth model written as:
$$\frac{U'(c_{t+1})}{U'(c_{t})}=\frac{1}{\beta(1+r)}$$
however I have also seen the euler equation for the continous time equivalent written as:
$$\frac{U''(c(t))}{U'(c(t))}\dot{c}(t)=r-f'(k)$$
Ignoring the right hand sides of these equations (which vary based on model set up) and exclusively looking at the left hand side I see two very different formulations of the same equation.
My question is: "How does the Euler equation in Continous time represent the same thing conceptually when compared to its discrete time counter part?"

One way to see (intuitively) the connection between the left hand sides is to write the discrete case as:
$$
\frac{u'(c(t + \tau))}{u'(c(t))},
$$
for $\tau = 1$. Now if we generalise this to a setting where $\tau$ is now a variable in $\mathbb{R}$, this becomes a function of $\tau$. Taking the derivative with respect to $\tau$ and evaluating at $\tau = 0$, gives:
$$
\frac{u''(c(t))}{u'(c(t))} \dot c(t).
$$
which is the left hand side for the continuous case. As such the left hand side for the continuous case can be seen as the "derivative" of the discrete setting (when allowing the time interval to go to zero).


Plotting the isoquant curves I have the following equation
$$Q_1(K, L) = AK^α L^β, A, α, β > 0$$
I'm asked to derive the isoquant curve which yields the following diagram:

Could anyone please explain to me how the form of the graph was found?

I have 5 unknowns in this equation and I just can figure out how would it be possible to plot the graphs.

Thank you very much in advance for your help!

Each curve shows the rate at which K can be substituted for L, or vice versa, while keeping output constant. The Marginal Rate of Technical Substitution (MRTS) equals the absolute value of the slope. The MRTS tells us how much one input a firm can sacrifice while still mainting a certain output level. Then the MRTS is equal to the following ratio: $$\frac{\text{Marginal Productivity of Capital } (MP_{K})}{\text{Marginal Productivity of Labour } (MP_{L})}$$

Then the substitution of Labour (L) for Capital (K) is given by:
$$ MRTS_{K,L} = -dL/dK = MP_{K}/MP_{L}$$

Now, note that in your equation $\alpha$, $\beta$ and $A$ are exogenous parameters (i.e. given to you). What your equation will then defined is, given these parameters, how your inputs (K and L) result in some amount of output given your production function.

You can find the isoquant curve that yields that diagram by varying the fixed level of output and by "playing with the fixed parameters".


Question about constant relative risk aversion The question:

Consider a person with constant relative risk aversion $p$.

(a) Suppose the person has wealth of $100,000$ and faces a gamble in which he wins or loses $x$ with equal probabilities. Calculate the amount he would pay to avoid the gamble, for various values of p (say, between $0.5$ and $40$), and for some values of $x$. For large gambles, do large values of p seem reasonable? What about small gambles?

(b) Suppose $p > 1$ and the person has wealth $w$. Suppose he is offered a gamble in which he loses x or wins y with equal probabilities. Show that he will reject the gamble no matter how large $y$ is if $p >= (log(0.5)+log(1-x/w))/log(1-x/w)$.

I'm not sure where to start with this. Am I solving for the risk premium and multiplying by $w$?

I know that for someone with CRRA utility $u(w)= (1/(1-p))w^(1-p)$ and that an individual will pay $\pi(w)$ to avoid the gamble if $u((1-\pi)w)=E[u(1+\epsilon \tilde)w)]$. But I'm not sure how to apply this information to solve the question.

Whenever you need to make someone indifferent between $x$ and $y$, it means that

$$U(x) = U(y)$$

a:

Denote amount he would pay by $z$. Paying $z$ to avoid the lottery gives him "certain utility" $u(100.000 - z)$. With Von Neumann-Morgenstern utility as given,
we can denote the lottery utilities as
$$prob*U(10.000 + x) + (1-Prob)*U(10.000-x)$$ (Understand why we can do that!)

b

Without uncertainty, he will have $u(w)$. Then, you need to compute the utility of the lottery as the function of $p, x,y$.

That is, you need to solve

$$U(100000) = prob*U(w+y) + (1-prob)*U(w-x)$$

Now, I'm not sure whether you were meant to say loses x or wins y, I suppose it is loses x or wins y. Anyhow, the value of the lottery is on the right hand side. Can it ever dominate the left hand side, given the additional information for $p$ that you were told in that subquestion?


Income effect $-\frac{\partial x_i}{\partial m} x_i$ or $\frac{\partial x_i}{\partial m}x_i$? Recall that the slutsky equation is:

$$\frac{\partial x_i}{\partial p_i}=\frac{\partial h_i}{\partial p_i}-\frac{\partial x_i}{\partial m}x_i$$

I know $\frac{\partial h_i}{\partial p_i}$ defined as the substitution effect, the second part of the equation is the income effect.

My Question: is the income effect defined as  $-\frac{\partial x_i}{\partial m}x_i$ or $\frac{\partial x_i}{\partial m}x_i$ (is the negative sign a defining characteristic?)

The income effect is defined as $\mathbf{-\frac{\partial x_i}{\partial m}x_i}$.

Let $x_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{\partial x_i}{\partial m} > 0\right)$. Even if there was no substitution effect $\left(\frac{\partial h_i}{\partial p_i} = 0\right)$ from an increase in own-price, the amount of $x_i$ consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is negative, which necessitates the negative sign since $x_i$ is non-negative. Analogous logic holds if $x_i$ is an inferior good instead.

This interpretation is consistent with Nicholson & Snyder (p. 156).


Is China at the golden savings rate s? If so, what is it? Consider Solow growth model. Let $s$ denote savings rate.

Suppose we estimate China invests 40% of it GDP rather than consuming. Assuming China is dynamically inefficient because they are a developing nation, how can I estimate what it's $s_{gold}$ is? Furthermore, how do I decide whether China is at this $s_{gold}$ versus some other $s$?

Furthermore, how does one determine this in general? Like are there distinct characteristic patterns that allow me to understand these rates?

Let our production function be $Y_t = (A_tL_t)^{1-\alpha} K_t^{\alpha}$.

I am asking how one calculates $s$ and $s_{gold}$ in practice, as in, if I had to find realistic numbers for these how would I do it?

To partially answer this question, in Advanced Macroeconomics by Romer, question 1.5(c) asks

  "What saving rate is needed to yield the golden-rule capital stock?"

Answering this tells us what $s_{gold}$ is.

But before we can find this we must find the golden rule level of capital stock $(k_{gold})$.  Which we shall derive with the following equations: The intensive form of the production function, the dynamics of capital in the Solow Model, and consumption per unit of effective labour

\begin{cases}
Y_t = (A_t L_t)^{1-\alpha} K_t^{\alpha} = A_t L_t \left( \frac{K_t}{A_t L_t} \right)^{\alpha} = A L k^{\alpha} \\
\Rightarrow Y/AL = y = k^{\alpha} \\
\dot{k} = s y - (n+g+\delta) k \\
c = (1-s)y \\
\end{cases}

Under the balanced growth path, $\dot{k}=0$ so

$$ \displaystyle s k^{* \alpha} = (n+g+\delta)k^* \Rightarrow k^* = \left( \frac{n+g+\delta}{s} \right)^{\frac{1}{\alpha-1}} $$

We maximize $c^*$ with respect to $k^*$ to implicitly define the golden-rule level of capital per unit of effective labour

$$ \displaystyle \frac{\partial }{\partial k^*} c^* = \frac{\partial }{\partial k^*} (1-s)k^{* \alpha} = (1-s)(\alpha k^{* \alpha - 1}) $$

$$ \displaystyle = \alpha (1 - s) \left( \frac{n+g+\delta}{s} \right)^{\frac{\alpha-1}{\alpha-1}} $$

Next replace $s$ with $(n+g+\delta)k^{* 1-\alpha}$

$$ \displaystyle = \alpha (1 - (n+g+\delta)k^{* 1-\alpha}) \left( \frac{n+g+\delta}{(n+g+\delta)k^{* 1-\alpha}} \right) $$

$$ \displaystyle  0 = \alpha (k^{* (\alpha-1)} - (n+g-\delta)) \Rightarrow k^*_{gold} = \left( \frac{\alpha}{n+g+\delta} \right)^{1/(1-\alpha)} $$

We can finally compute, $s_{gold}$

$$ \displaystyle s_{gold} = (n+g+\delta) k_{gold}^{*1-\alpha} = (n+g+\delta) \left( \frac{\alpha}{n+g+\delta}\right)^{(1-\alpha)/(1-\alpha)} = \alpha $$

So for a Cobb-Douglas production function the golden-rule saving rate should be whatever $\alpha$ is.


Why do increase in firms profit margins cause cost push inflation? I read somewhere that this was the case. I get what profit margins and cost-push inflation are, but don't see how increase in marginal profit causes cost-push inflation? Also, how does increase in money supply cause cost-push inflation?

You can think on prices like $p = (1+m)w$, which means that prices are the value of inputs ($w$) multiplied by the margin ($1+m$) of the firms. So it is clear that raising firms margin would increase prices. This is cost-push inflation because the margin are part of the firm´s costs, it is a payment to a factor, like capital or entrepreneurship.

A raise in money supply don´t cause cost-push inflation directly, but causes demand-push inflation, which can increase also increase the prices of productive factors and inputs to production (w) as well, if firms can pass this increases in their cost to produce to their price, then you have a channel of cost-push inflation.


Measure combining growth and distribution of income A frequently used measure of economic progress is GDP growth. (There are alternative economic indicators as well, such as the Human Development Index.) But there are problems with GDP growth. For example population growth also causes the GDP to grow, because more people produce and consume more goods. You could correct for this by usind GDP/capita growth instead.

But GDP/capita growth still does not take into account the distribution of wealth. Perhaps 90% of the population are doing worse than last year but 10% are doing so much better that on average there is still growth. You could argue that this is still 'better' overall, but I would argue that in a democratic setting most people would probably not vote for an economic policy that clearly and openly supports such an income shift. (If you are thinking of cases where this has happened and will happen: I am afraid I don't care about the political side of this argument. You are welcome to post about it over at politics SE.)

Are there indicators that capture both distribution and growth of income?
 A simple example with its own faults would be median GDP/capita growth. Something better (and slightly more complicated) could probably be constructed using the GINI index.
 Alternatively: Are there arguments against measuring something other than aggregate or average GDP?

One concept fairly established in economics is the idea of generalized Lorenz dominance.

The Lorenz curve (https://en.wikipedia.org/wiki/Lorenz_curve) plots percentiles of the population on the x axis and the cumulative percentiles of income on the Y axis. If the point (30,10) is on the curve, then this means that the bottom 30% of the population have 10% of the income of the population.

This allows us to compare the inequality of two countries by comparing how low or high their Lorenz curve is. If curve A is everywhere above or equal to curve B, then A is weakly less unequal than B. However, even the poorest person in B may still earn more than A.

The generalized curve multiplies the Y axis with the average income of society. Now we can not only compare A and B in terms of inequality, but also by how well individuals are off in these societies. Curve A being everywhere above or equal to curve B means that every percentile of society A is weakly better off than every matching percentile in society B.

Shorrocks showed that under certain conditions a parameterized family of social welfare functions is higher in society A relative to society B for all parameters if and only if the generalized Lorenz curve of A is higher than B at all points.

Shorrocks, Anthony F. "Ranking income distributions." Economica (1983): 3-17.


Why is the marginal utility of money assumed to be constant in Marshallian Theory of Consumer Behaviour While studying the Marshallian Theory of Consumer Behaviour, I came across the assumption that the marginal utility of money is assumed to be constant. Can someone please explain why is this so?

Marshallian Theory is notoriously about computing consumer surplus (and then welfare changes).

But it would be nonsensical to perform such calculations if at the level of one individual, the unit of valuation of surplus, i.e. the marginal utility of money (or the utility of one extra euro), were changing before and after changes in, say, prices. Also, what sense would it make to compute surplus (pleonastically) over many individuals if each of them had a different marginal utility of money?

In these two "within-individual" and "between-individual" cases, not having a constant marginal utility of money would be like summing objects with different units... Hence this assumption.


What exactly is "quantity of money per unit of output"? When making the case about inflation being monetary phenomena, Milton Friedman shows a graph where "quantity of money per unit of output" move together with CPI. What is "quantity of money per unit of output"? Is it real GDP?
Update:
Link to presentation

It is the quantity of money, measured by money supply of 'near money', divided by the real GDP. You can see that from reproduction of those tables in the:
The Collected Works of Milton Friedman, compiled and edited by Robert Leeson and Charles G. Palm.
Here is a link to chapter from that book which reproduces the lecture and the figure from the lecture. You can see the picture also below (the note under the table makes it explicitly clear Friedman is talking about quantity of money per real GDP.


What is the point of all the models in an economics degree? I am a student of Economics doing my masters degree and I have to say I am a bit tired of all the models.

On the one side, there is these models where I can see how they are derived, which always seems to be a mixture of using the Lagrangean with some - often rather unconvincing - assumptions. But I kind of fail to see the deeper meaning. For example how do New Keynesian (or RBC) models help me understand the real world?

How would they be used in practice - is it reasonable to assume they could forecast inflation, growth or interest? It seems like these things are inherently impossible to predict.
Do they help me understand how they work? The understanding I have about them comes more from non-mandatory applied courses and my personal readings about past financial crisis etc.
But probably most importantly, do I believe anything that comes out of them which I did not assume already before? Certainly not.

Or the Solow model, how is the fact that some countries seem to have followed a path that vaguely resembles that assumed by the Solow model a justification for its validity? Drawing a simplified picture of a human doesn’t help me understand why they became sick.
There is extreme cases like Barro's model of Ricardian Equivalence which seems like either the world's most boring circus trick or an attempt to fool policy-makers without basic maths skills.

On the other side there is a few cases where I would say my understanding improved with models in a way it would not have without, e.g. the notion of externalities, Pigouvian taxation or comparative advantage. But I have to say - given the countless pages of Lagrangeans I have set up, hours I have spend maximising profit functions and arguments I had in my head trying to find ways to justify some micro axioms - this is not a lot.

Don't get me wrong, I am fascinated by the study of the Economy and everything it relates to, I just don't see how a lot of what we learn is actually a helpful tool. I am a good student and neither looking for an easy way out, nor do I want to rant about studying. I am just looking for some motivation where this all leads. Can somebody explain to me what the deeper meaning of economic modelling is and why we give it this much attention?

And yes I know all models are wrong, but how are they useful?

Models are more than just math. Model is a simplification of a reality that allows you to study the underlying mechanisms. Models do not need to be mathematical. Many people actually create models without even realizing it. For example, if someone says "minimum wage will not lead to unemployment" that person is actually having a model even if they never present any mathematics. Such person definitely does not take into consideration full reality - every choice of every single individual, movement of every atom and so on.

Hence, whether you use mathematics or not you are using some model anytime you are describing any mechanism, such as x causes y, or there is such and such relationship between x and y because of this or that. Thinking otherwise is just self-deception as no human mind can take into account full reality when examining any issue. You have to simplify whether you do it with math or without. However, I get that at the heart of your question is why economists use mathematical models, but I wanted to highlight that there is no way of escaping models.

Mathematical modelling has advantages over plain text.

Dani Rodrick put it best when he said:

  We need the math to make sure that we think straight--to ensure that our conclusions follow from our premises and that we haven't left loose ends hanging in our argument. In other words, we use math not because we are smart, but because we are not smart enough.

  We are just smart enough to recognize that we are not smart enough. And this recognition, I tell our students, will set them apart from a lot of people out there with very strong opinions about what to do about poverty and underdevelopment.

Hence the reason why economists use mathematical models is that they are tremendously helpful in disciplining your thinking and by forcing you to say exactly what you mean. Written text is imprecise, words can have double meaning, people can easily cover up unsound logic by good use of literary devices or appealing to your emotions all of which can cloud your judgement.

This makes text  harder to actually analyze analytically. For example, even in moral philosophy - the least mathematical field I can think of, to fully analyze argument made in text and determine whether it makes sense you have to often resort into translating it from plain English into symbolic logic - which is form of math. Mathematics itself is just a language based on logic, that's also why in mathematics we often talk of equations as sentences or statements.

When you reduce model to set of mathematical expression you are forcing yourself to make your own thinking bare. You cannot easily conceal faulty logic anymore. You can argue that math can be used to 'fool policy-makers' as you put it but you cant easily fool other scholars with math while with text you can easily fool both policy makers and scholars.  My argument here is not that the math is bullet proof, as argued by Romer some scholars can try to mislead with using "mathiness", but lying with math is much much more harder than lying with text.

Models actually do help predict what will happen. First, do not confuse prediction with forecasting. For example New Keynesian model predicts that if there will be sudden shock to demand wage rigidity will lead to higher than natural level of unemployment - that is a prediction but its different from forecasting as the prediction is based on the ceteris paribus conditions while the word is constantly changing.

Predictions that model make can be useful in itself for several reasons. Predictions actually allow us to test whether one theory is superior than another. In fact the only way of testing a theory is to test its prediction against empirical observations. If someone has elaborate theory written in text that says that minimum wage does not affect employment and other person will make equally compelling case in text against that how will you decide which person is right?

One way how to do that is to turn the first person text into a mathematical model. The person says that minimum wage does not affect unemployment? Okay that means he postulates the following relationship:

$$ U(w_{min}) = a + bw_{min}$$ where $b=0$

Now converting the thinking of that person into mathematical model actually allows us to test it - we can based on the model above construct regression of unemployment on minimum wage:

$$U = \beta_0 + \beta_1 w_{min} +\epsilon $$

and test whether the first person was correct by testing the null hypothesis of $\beta_1=0$ against alternative hypothesis $\beta_1 \neq 0$.

Predictions also allow us to do counterfactual analysis. We can ask ourselves what would happen if in the presence of wage rigidities government increases spending during the recession. New Keynesian models provide you with framework to think analytically about such questions and fully explore all their logical conclusions.

Does this mean that math is the only way of studying economics? No. Of course, every individual is different. Some people are visual learners, some people are auditory learners.  Some students respond better to narration than to math. Moreover, as you yourself admitted math helped you to understand some problems so it worked for you well at least a couple of times. While you might have learned better most concepts by reading about them  other student might have completely opposite experience.

Moreover, depending on your career choices you might end up not using most of the math in real life. However, a master degree especially if its master of science and not just master of arts is often a stepping stone (at least in the Europe) towards PhD or more academic career in policy institutes or government institutions where you will need the math if not directly then at least indirectly to understand new advances presented in academic journals. For example, even routine psychologists must have some statistical understanding if they want to keep up to date with new techniques to help their patients as in order to see if some technique is better than other some testing has to be done. Otherwise you will be just at mercy of trusting what authors say in conclusions, or some intermediary like academicians who like to blog for wide audience etc.

Furthermore, you should not interpret anything I said above as saying math is the only way of doing economics or that narrative analysis is useless. Sometimes narrative analysis can be more nuanced, and with some problems we might not even yet discovered ways how to model them mathematically in satisfactory way. However, models and especially mathematical models are incredibly important tool in any economist's toolkit.


Negative expenditures in public budgets I need to better understand the concept of negative expenditure in UK public spending. The definition says

  Receipts classified as negative public expenditure receipts are
  netted off, and hence reduce levels of total public spending. The most
  common examples are payments for goods and services. However negative
  PE receipts also include: royalties, income from insurance payments,
  guarantee fees and income from rent of buildings (but not land).

More on page 11 of this paper.

For example in the 2015 UK Budget's Total Managed Expenditure is £765bn, where the HM Treasury spending is -£43bn, because HM Treasury earned more than it spent in that period, and the sum of all other departments is 765+43=£808bn. My question is, what do we say is the "size" of the government budget? The government physically spent more than 765, they just did not financed all of it from tax revenue, that is all.

The problem becomes much more exposed if we look at one of the state agencies the Agriculture & Horticulture Development Board. It receives only £1.2M from the government budget but then it "earns" another £57.4M from Levies that enters the budget on the spending side as -57.4 and the total expenditures I see are over £66.3M. But in their annual report they reffer to it as

  The total 2014/15 income was £66.746

Apparently Total Income and Total Managed Expenditure are two very different things. What we say is the "size" of the budget? If we nett off the negative expenditures then the budget of this agency is only £1.2M but that is misleading, isn't it? What is the correct terminology?

There are several issues here, but the three most significant are:

The National Accounts treatment of the public sector, which is set out in international standards and controlled by the Office for National Statistics.  This is what you might naturally look to if you wanted to compare the size of the UK's public sector with those of other countries, or its impact on GDP or other macroeconomic indicators.  Examples of ONS data include https://www.ons.gov.uk/economy/governmentpublicsectorandtaxes and https://www.ons.gov.uk/economy/governmentpublicsectorandtaxes/publicspending/datasets/esatable2mainaggregatesofgeneralgovernment
Her Majesty's Treasury's control of public expenditure for budget management purposes, which includes passing on money voted by Parliament.  The Treasury has a natural tendency to try to minimise this, and so often encourages parts of government to obtain resources in other ways such as levies and fees, and sometimes allows these to be deducted from the control totals it applies in a way not deemed economically meaningful by the Office for National Statistics. Total Managed Expenditure is part of these control totals.  Sources of information include https://www.gov.uk/government/statistics/public-expenditure-statistical-analyses-2016
Individual parts of government managing their own budgets, primarily ensuring that money coming in  balances money going out.  This can include receiving money from or giving money to other parts of government, which as internal transfers would not count as public expenditure from an economic point of view, but do matter to an individual body

The Treasury paper you link to is largely trying to describe some of the differences of treatment between (1) and (2), explaining phases in its analyses such as "Receipts treated as negative DEL [departmental expenditure] but revenue in National Accounts".  Meanwhile, in the case of the Agriculture and Horticulture Development Board, it gets its money from the industry either in the forms of levies or commercial income and is an example of (3); in this case the Treasury interest is largely in ensuring the board does not become a burden on public funds and that it operates efficiently, while the ONS is interested because it how it is funded (it is similar to a tax as the levies are compulsory, but it is different in that the board uses this money solely to provide a explicit service to the levy payers)

If you are looking to measure the size of the UK public sector as a whole, you may find the ONS figures in (1) most useful, though they may not give the detail you can find from (2) or (3)


How can energy consumption be an indicator of per capita availability of capital? "An important indicator of low capital per head available in developing countries is the consumption of energy". I could not decipher the concept.

In economics, capital refers to things that are used to produce other things.

So, capital includes tangible things like a delivery truck, an office building, or a factory. These are things which often require energy to run.

Hence, if a developing country has low capital per head, then it probably also has few delivery trucks, office buildings, and factories. And hence, it probably also has lower energy consumption.


Computing Subgame-Perfect Equilibrium Suppose that two players play each other for two periods. In the first period they play the
first game below, and in the second period they play the second game below. There is no
discounting between periods. Players observe the action their opponent took in the first
period before choosing their second period actions.
\begin{bmatrix}(2,2)&(-10,x)\\(y,0)&(0,0)\end{bmatrix}
\begin{bmatrix}(8,4)&(0,0)\\(0,0)&(4,8)\end{bmatrix}
(a) For x ≤ 2 and y ≤ 6, find a subgame perfect equilibrium in which player 1 receives a
payoff of 10.
(b) For x = 5 and y = 3 find a subgame perfect equilibrium in which player 2 receives a
payoff of 10.
(c) For x = y = 4, show that there is no subgame perfect equilibrium in which (U,L) is
played in the first period.
I'm totally lost as how to find subgame perfect equilibria. Aren't these games simultaneous games? Why would the players consider the choices made in the first game to play the second game?

I am not sure that I am able to fully answer your problem, but I can give you a short answer to your questions. "Aren't these games simultaneous games? Why would the players consider the choices made in the first game to play the second game?"
I would refer to collusion and repeated games. Every stage game is simultaneous, but the fact that they are played sequentially makes them a super-game. In super-games the outcome of the first stage game influences the second, because in this way every player can introduce a punishment at the second stage if the other player did not cooperate. So pretty much like in collusion games, what matters are the threats one player can pose to the other through the second stage to obtain collaboration at the first stage.
Having said that, I'd suggest some easy readings:
Introduction: Cabral "introduction to industrial organization" chapter on game theory and collusion"
Game theory: Chapter 14 and 15 Osborne "Introduction to game theory"
Regards.


what was the world's wealth in 2015? I was reading an article from 2015 stating that 1% of the world has half of the world wealth. They have several figures such as the world's billionaries have around 7 trillions dollars or the global world wealth pyramid which shows a percent of population by percent of the total wealth etc. but I can't see anywhere a figure of the total wealth of the world. Which is it?

Your article probably resulted from an Oxfam campaign

If you believe table 2 of a January 2015 Oxfam report then about $\$250$ trillion. The campaign spliced together some information from Forbes on the richest $x$ individuals and a series of Credit Suisse reports - see the bottom of page 94

The analysis has faced criticism: it concentrates on individual assets and liabilities, e.g. a US doctor whose student debt exceeds personal financial assets counts as poorer than a subsistence farmer in Africa, and uses a very large amount of estimation, so other reasonable calculations could be very different


Bayesian game and the set of types In a finite Bayesian game, in most textbooks, a type of player $i$ is defined as $\theta_i\in\Theta_i$. Little is said about the "nature" of the set of types.

For example, could we have a two-element set $\Theta_i=\{(a,b),(c,d)\}$? If I am not mistaken this means that $\Theta_i$ is a subset of $\mathbb{R^2}$, right?

For the common prior then, we could have, for example, $\mathbb{P}((a,b))=p$ and $\mathbb{P}((c,d))=1-p$. Both probabilities describe the joint probability distribution that Nature assigns the types to the players.

Is this allowed in the definition of a Bayesian game?

I'd appreciate any help. Thank you.

Sure.

In the two type case there is an alternate solution that is frequently used. If $a \neq c$ and $b \neq d$, that is the types are really different in both attributes, then you can define a function $f$ for which
$$
b = f(a), \hskip 20pt d = f(c).
$$
Whatever you need the second attribute for, this way it becomes a function of the first one, and the values of the first attribute completely define the type space.

Example:
 In Spence's model of job market signaling, people have a type that shows how much they are worth to a company, how hard working they are. It is assumed that the cost associated with getting a degree is a function of this type as well, and harder workers get a degree more easily. One way to show this would be to include both worth to the employer and cost of getting a degree in the type attributes, but Spence only defined the types and gave some functions that mapped to these values.


Currency substitution after withdrawal from a currency union? Currency substitution is the situation where a country uses a foreign currency.  For example, Montenegro and Kosovo unilaterally use the euro, El Salvador and Panama unilaterally use the dollar.  In the scenario that a country would be forced to leave an existing currency union (if at all possible), what would be the advantages and disadvantages should this country unilaterally decide to keep using the larger currency?

Summary:
Using a national currency gives you full control over monetary policy while using a larger foreign currency provides stability.

Detailed answer:
If a country has its own currency then the government, or to be more precise the central bank, can intervene and change exchange rates. This might be necessary if the countries products are not competitive on the international markets on the current exchange rates. An example:

Cost of a foreign laptop: 200 USD. Cost of a domestic laptop: 100 Peruvian Nuevo Sol (PEN). The domestic laptop can only be sold if the USD/PEN exchange rate is over 1/2.

In theory the government of a country using a foreign currency could launch policies that would simply reduce all salaries and rents, but this is politically and psychologically undesirable (downward rigidity of wages).

Unless big shocks (drastic events or drastic new information) occur such interventions would not be necessary as prices, wages and exchange rates slowly adjust thanks to the free market.

So a benefit of having a national currency is that it gives a country full control over its monetary policy and this makes such interventions possible. A disadvantage would be that a small nation's currency may be less stable than a large ones, for an example see this question. Stability (or at least predictability) of exchange rates is important for foreign investments and as they have a direct effect on the prices of imported goods and on the profit of export firms it is also important for the citizens of the country.

In the circumstances described there may also be a secondary effect:
A government in control of its monetary policy and facing extremely serious problems may start printing extreme amounts of cash because it turns up indirectly as income in the governments balance sheet (this is known as seigniorage). This can cause undesirable rates of inflation. Most economies avoid this problem by delegating money printing to a central bank which is independent from the government (to some extent). The mandate of the central bank is not to provide the government with income but to stabilize prices so it will probably try not to cause hyperinflation. This however is ultimately a political decision and in drastic times it may be overruled.


Can game theory be used to predict economic specialization (division of labor)? There is a hint of a connection to game theory in Adam Smith's Wealth of Nations. For example, he says:

  It is not from the benevolence of the butcher, the brewer, or the
  baker that we expect our dinner, but from their regard to their own
  interest.

We can put his general argument into a payoff matrix, like shown below:

There are two Nash Equilibria here: {DS,DS} and {S,S}.

Obviously, if they both specialize, they are both much better off. However, each of them not specializing is also an equilibrium.

Is there any way that Game Theory predicts the emergence of specialization in a population of such players?

I think you're over-analyzing here. Game theory is not just about maximizing your own pay-off, but about doing the best you can given the behavior of others. What Smith describes is just that people do what they're good at to make a living. That is, the way I read it, Smith claims that people specialize independent of what others do.

That being said if the emergence of specialization was to be predicted with game theory I'd look into evolutionary game theory. I'm not sure if you'll find anything though.


Geographic mobility of workers within Germany, as compared to within the US? Martin Feldstein (1997, p. 36) claims that workers within the US are more geographically mobile than say within Germany. I had known that US workers were mobile. But one sentence of his seems a bit hyperbolic:

  While Americans don't hesitate to move from Ohio and Massachusetts to Arizona and California, Germans are loathe to leave one part of Germany for another.

Boston to say San Francisco (about 5,000km) is about seven times the distance from Hamburg to Munich (about 700km). It would seem surprising that Americans don't hesitate to move from one end of their country to another, while Germans are loathe to do so.

Of course there are many ways this could be measured. But here is my attempt at a question that is in principle answerable:

Each year, what percentages of US and German workers relocate to a different state (within the same country) for work? Is the figure for the US greater than that for Germany?

P.S. Feldstein was writing in 1997 but I imagine the numbers do not differ very greatly between now and then--correct me if I'm wrong.

Here is a figure documenting annual (I think) internal migration rates in Germany:

Internal Migration in Germany, 1995-2010: New Insights into East-West Migration and Re-urbanisation  (Sander (2014))

This table show 5 year migration rates in many countries, including the USA:

Cross-national comparisons of internal migration: An update on global patterns and trends (United Nations Department of Economic and Social Affairs (2013))

Depending on methodology, annual and 5-year rates can't exactly be compared because some people will move to another area and move back within five years, so asking people if they live in the same state as 5 years ago won't give the same result as asking someone if they lived in a different state last year for five years. That said, if we can make some rough comparisons if we treat the five year rate as roughly comparable to five times the annual rate. The USA rate is about 9 percent every 5 years or 1.8 percent per year. This would be higher (but not much higher) than the West-East, East-East, and East-West German internal migration rates but quite a bit lower than the internal migration rate within West Germany. It might be that this converts the 5-year rates to annual ones that are too low, but the sizes seem roughly comparable. This is just two sources but they provide little or no evidence for Feldstein's claim that the German rates are far lower than in the US.


How might the Swiss currency floor be unwound? The Swiss National Bank has been enforcing a currency floor since 2011, not allowing the EUR/CHF rate to drop below 1.20 by buying unlimited amounts of foreign currency as needed with newly created CHF.

I'm having trouble finding exact amounts, but it seems from their balance sheet that their foreign currency reserves have increased by about CHF200bn since 2011.

How will they unwind this situation? They state that they still view CHF as overvalued at the 1.20 rate, so presumably they would let it drop some way from that before they start selling the reserves, leaving them with a gain in CHF. Will they just leave the money base permanently contracted to account for this? What long-term impacts will this have?

Central banks have a lot of leeway from a balance sheet perspective to do these things. They can simply stop the practice of buying foreign currency when the CHF appreciates which would let it appreciate beyond 1.2. If it keeps appreciating they will book losses, but unless they are enormous, these typically don't make a dent in a central bank's balance sheet.

If it depreciates, they can use the foreign currency to buy CHF, or not. If they do not use it, then they book a profit, as the comments suggest.

In many cases, the legal implication of large loses or gains in the CB's balance sheet is a transfer to or from the country's treasury. Here's a recent example from Mexico Bloomberg Article


Inflation without Increase in Money Supply? Is it possible to have inflation without increase in the supply of money?

For example, if workers demanded higher wages, and goods went up in price, but the money supply in circulation did not increase. Or is this an impossible paradox?

It all depends on what you mean by inflation and by money supply. Technical questions and answers need specific definitions, otherwise everyone ends up talking at cross-purposes.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation? Yes: if the velocity of circulation of money increases, and the amount of goods and services available to buy does not increase by as much.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation? Yes: if the amount of goods and services available to buy, decreases, so that there's more money chasing fewer goods.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation, and with no decrease in the amount of goods and services available to buy? Yes, if the demand curve changes so that the same amount of money is now used to buy a smaller quantity of stuff at higher prices.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation, and with no change in the amount of goods and services bought? No, because the velocity of circulation is by definition total transaction value divided by the amount of money in circulation, so if velocity, quantity and money supply are constant, then prices must be too, because total transaction value equals prices times quantity.


Basic New Keynesian Model - Price and Wage Level after shock I'm currently solving a new Keynesian model with government spending. It's the basic model version, with constant returns to scale on the only production factor work.

Everything has worked out fine in terms of solving the model with Dynare, the only thing that I don't understand is the output regarding P(Price level, not inflation) and W(Wages, not in real terms).

After the gov. spending shock hits, all variables move back into their steady state i.e. the impulse response functions return to zero as time goes on. This also is the case for >real

The only variables that do not return to their steady state are P and W(nominal). They both spike upwards as the shock hits and then return to a state even higher than that, as the shock fades out.

Why is this the case, what is the economic intuition behind it?

Try to give a look at what happens to inflation's IRF. If it stays positive for the whole horizon of the IRF then simply prices have increased over time at the inflation rate. I guess that any non-degenerate price level (nominal!) is compatible with such model structure, as its system is written down in growth rates, as that's what loglinearised variables are.


Explaining why Hicksian demand is more inelastic to intermediate micro student How would you best explain to a student in intermediate micro class that the Hicksian demand for a normal good is more inelastic than the Walrasian demand without heavy differential calculus?

When I mean inelastic, I mean specifically being less responsive to price changes, a picture similar to MWG Figure 3.I.3.

Here's a "no maths" explanation (including the inferior goods case, because I think it helps to understand what's going on):

Suppose we have a normal good, $x$, and we increase its price. Marshallian demand decreases thanks to two effects (i) consumers substitute away from $x$ towards cheaper alternatives; (ii) because prices are higher, consumers can afford less stuff, so it's as if their income were lower. Both of these effects point towards lower demand for $X$.

Now consider Hicksian demand, which shows the effect of a price change after we compensate consumers to eliminate the income effect. Instead of having two effects (income and substitution) pointing in the direction of lower demand, now there is only one (substitution). So Hicksian demand changes less with prices.

Note that on p83 of MWG, the authors note that the figure is drawn for the case of normal goods.

But let's look at the case of an inferior good. By definition, when the price of an inferior good increases Marshallian demand tends to fall due to the substitution effect, but increase due to the income effect. In other words, there is one negative effect and one positive effect on demand. These two effects cancel each other out to some extent, so that the overall effect of price on Marshallian demand is muted.

Hicksian demand eliminates the (positive) income effect, so  that the only remaining effect is the unambiguously negative substitution effect. Thus, Hicksian demand for an inferior good is more sensitive to price that Marshallian demand.


Why is the risk premium always positive for risk averse individuals? I think this has to do with the definition of concavity and the fact that a risk averse person has a concave utility function, but I'm not sure how that helps.

Suppose that the vector $W=\left(w_1,w_2,\dots,w_n\right)$ represents wealth in $n$ possible states. In addition, assume the probability of each state occurring is represented by the vector $\pi=\left(\pi_1,\pi_2,\dots,\pi_n\right)$. We can express this as the simple gamble:

$$g = \left(\pi_1\circ w_1,\pi_2\circ w_2, \dots, \pi_n\circ w_n\right)$$

The expected value of the simple gamble $g$ is:

$$\mathbb{E}[g]=\sum_{i=1}^{n}\pi_iw_i$$

Suppose a consumer has a sub-utility function $u(w)$. Further, suppose that the consumers preferences over gambles are such that they can be represented by a von Neumann Morgenstern utility function (has the expected utility property). The utility of the gamble can be expressed as the expected utility across all states:

$$u(g)=\sum_{i=1}^{n}\pi_iu(w_i)$$

Now, if we assume that the sub-utility function $u(w)$ is strictly concave (risk averse consumer), by the concavity of functions it must be the case that:

$$u\left(\mathbb{E}[g]\right) > u(g)$$

This implies that the consumer would prefer to receive the expected value of the gamble with certainty rather than the gamble itself. A question we could ask; how much would I need to give to the consumer such that they are indifferent between choosing the gamble and receiving some amount of wealth with certainty (Certainty Equivalent)? The certainty equivalent $CE$ must satisfy the following:

$$u\left(CE\right) = u(g) \implies u\left(\mathbb{E}[g]\right) > u\left(CE\right) \implies \mathbb{E}[g] > CE$$

Since $u(w)$ is increasing in wealth, there exists a $P>0$ such that the following will hold:

$$u\left(\mathbb{E}[g]-P\right)= u\left(CE\right) \implies \mathbb{E}[g] -P= CE$$

$P$ represents the risk premium. It is the amount you would be willing to pay to get rid of uncertainty (receive the certainty equivalent). The fact that the risk premium is positive does arise from the fact that we assume the sub-utility function is strictly concave.


Could the upper quintile pay the annual federal personal income tax for all other quintiles from their annual income? I am wondering if, in theory, the upper quintile (20% of earners in the U.S.)  could pay the annual federal tax bill for all quintiles from their annual income?
I have been looking at 2013 CBO data  here  that shows quintile income as a percentage of total income and quintile taxes paid as a percentage of total taxes paid. What's necessary is the amount of total income and total federal taxes paid - does anyone know these values?

Here is the latest report of the Congressional Budget Office on federal taxes by households. The main result you are looking for is this one (page 31):

The "Federal Taxes" row shows that the upper quintile tax bill is higher than the tax bill of the other four quintiles combined. (the data are averages but each quintile has same number of households so those numbers are enough to conclude this).

The figure below show this more clearly, in percentage terms:

Check here for older reports.

PS: amazingly, the top quintile gets more government transfers than the poorest quintile, which get the lowest. An entirely regressive system!


How to find change in the optimal choice with a utility function in general form? Suppose the utility function is represented as $U(x_1,x_2;I)$, where $I$ is the level of information the consumer possesses.

How to find the change in the optimal choice of $x_1$ as price of $x_1$ changes with and also with respect to changes in Information?

While I understand the procedure with a Cobb Douglas Utility function, I don't know how to work with general form.

My take:

The consumer maximises $U(x_1,x_2;I)$ subject to a budget constraint $px \leq Y$ where $p$ and $x$ are vectors and $Y$ is income.

The optimal choice can be represented as $x_1^*(p,w;I)$ - Marshallian Demand. Now differentiating this wrt $p$ and $I$ would be the solution.

Is my understanding correct?

It depends what you know about your utility function and the "space" of goods $x_1$ and $x_2$.

You ask in your question what the change of optimal choice is with respect to the parameter information.
This analysis is called comparative statics, and you can use a number of mathematical tools to achieve this. The most common are

The Implicit Function Theorem

If the space of goods in convex and everything nicely differentiable, quasi-concave etc. etc., then this is easiest. I will not discuss the theorem, you should work through it yourself (good references are in De La Fuente's book or in MWG).

First, what is the optimal choice for the actor? Since everything is differentiable, AND quasi-concavity implies that the first-order conditions characterize the optimum (this is important), you can derive the conditions for the optimal choice by the first order equations
$$ D_{x_1,x_2,I}U(x_1,x_2;I) = \mathbf{0}$$
where $D$ is simply the differential operator aka in your case the vector of $\frac{\partial U}{\partial x_i}$.

This characterizes the optimal choice. It is not yet "solved" as the form of $x_i=\ldots$, but this is often neither possible nor necessary. For example, maybe you don't really know what $U$ is except that it has certain properties, so you can never solve for $x$ explicitly. That is why we use the "implicit" function theorem.

Applying the Implicit Function Theorem allows you to calculate
$$\frac{\partial x_i}{\partial I}$$
simply by using the "ratios" of first-order conditions you derived above.

Again, you need to carefully check some properties of the functions, so I will give you the formulas but urge you to check the literature.

Monotone comparative statics

As it turns out, we can do comparative statics in an even more general world than a convex space of goods with differentiable utility.

Suppose we are only interested in "directions" of choice. That is, we care only about monotonicity. For example, the IFT does not work for discrete goods ("1 car", "2 cars", "3 cars") since the space is not convex. But the ideas should work similarly!

Doing optimization on these non-convex, and yet ordered sets it called lattice optimization.

Now, we merely need to assume that $X$, the space of goods, is a lattice - a set where the possible combinations of $x_1$ and $x_2$ have a certain order, and that the information values $I$ constitute a partially ordered set.

Now, if we assume that the utility function is at least supermodular (think similar to concavity in this more general space), then we can use a similar result to the implicit function theorem, called Topkis's theorem to establish how $x_i$ changes as response to $I$.

But in fact, this is not even the most general case. Remember the difference of concavity and quasi-concavity? Well! Here, too, we can invoke quasi-supermodularity. This brings us in the word of Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics”.

In particular, they claim that this form of comparative statics is the most general, and in a sense, necessary and sufficient from of "optimal choice" and "comparative statics".
I found that a very good reference is the book by Xavier Vives.

Well, you asked for generality, so there you go.

Edit: If you are an undergrad, then perhaps just file this response away and I will say that, yes, your understanding of the problem also works. You can differentiate the demand function to see how it changes with respect to the parameter ;-)