Weak axiom of Revealed Preference application The following is a problem I am dealing with related to Weak Axiom of Revealed Preference. I have given my solution below to the situation. What I am not getting is how is WARP not violated?

A law firm looking to hire to fill three positions gets applications from Andrew, Barbara and Celia.

The law firm's set of alternatives is the set of possible hiring decisions:

$ X = \{ \phi, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,c\}, \{a,b,c\}  \} $

For any $Y \subset \{a,b,c\}$, define the power set of Y as

$ 2^{Y} \equiv \{Z | Z \subset Y \}$.

$ 2^{Y}$ is the set of hiring decisions that the firm can make when it receives applications from the lawyers in Y.

The law firm's budget sets $ B \in \mathcal{B} $ are the sets of hiring decisions it can make after receiving applications from some combination of Andrew, Barbara and Celia :

$\mathcal{B} = \{2^{Y} |Y \subset \{a, b, c \} \}$

1) When it receives applications from Andrew and Barbara, it will choose to hire Barbara (and not Andrew):-

$ C(2^{\{a,b\}}) = \{b\} $

2) When it receives applications from Barbara and Celia, it will choose to hire Celia (and not Barbara):

$ C(2^{\{b,c\}}) = \{c\} $

The following is the problem I am confused in :
Q) What restrictions does the weak axiom place on the firm's hiring
decision $C(2^{\{a,b,c\}})$ when it receives applications from Andrew, Barbara and Celia?

My Solution:

According to Mas-Colell et al (Definition 1.C.1) , the Weak Axiom of Revealed Preference says that if x is ever chosen when y is available then there can be no budget set containing both alternatives for which y is chosen and x is not.

So based on my understanding of WARP, in my situation above, when Andrew and Barbara apply the firm chooses Barbara , i.e.
$ Barbara \succsim_R Andrew$
and when Barbara and Celia apply, the firm chooses Celia i.e. :
$ Celia \succsim_R Barbara $

Here we see that since Barbara is not chosen over Celia , WARP is violated. Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set. So when Andrew , Barbara and Celia apply, and WARP violates the above relation given , the firm would hire only Andrew.

What I am not getting is how is WARP not violated?

Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set.

This is not true. Instead WARP would say that it is not possible for Andrew to be chosen whenever both Andrew and Barbara are in the choice set, which is different than your claim above.

Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should always be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.

Write out the two conditions explicitly:
$$
\begin{align}
C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&=\{b\}\tag{1}\\
C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&=\{c\}\tag{2}
\end{align}
$$
$(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only.
Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only.
These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$).

For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get
$$
C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}
$$
where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.


Statistical tests/models to use for Macroeconomic and Financial Research I am a student, new to research and have chosen a challenging field to perform my research in. While I have been reading about similar research's, as part of my literature review, I observed various researchers using various models in their study. I am having a tough time understanding why researchers use different models/tests and how they choose it, although its the same kind of research being performed.

My research is about the impact that macroeconomic variables have on Stock indexes. I have 6 indexes and 5 variables x 2 countries (Quarterly data). Each of the index/variables have about 60-70 observations.

What I am planning to find -

Which of the MV's could potentially explain the movements of the index prices. Which of the MV's impact the Stock prices.
Is there a variable/combination of variables that specifically create a downward/upward movement.

Should you have any suggestions of statistical tests/models for the above, I'd be pleased to learn and know more about them.

Thanks in advance.

One of the reasons why there are so many different models is that "having impact" is not that easy to define. It is not to test whether a variable has "any" effect on another variable, so researchers first have to come up with a model (based on theoretical considerations) that they can estimate and test. And of course the tests are different depending on the model they assume.

Basic model

One of the most basic (but powerful and widely used) concepts for these type of hypotheses is Granger-causality. Simply put, variable $A_t$ is said to Granger-cause variable $B_t$, if $A_t$ is useful in forecasting $B_t$. So you might try to find out which macro variables Granger-cause your indices. However to do formal tests you need to restrict the previous definition a bit.

Let us say, that variable $A_t$ Granger-causes variable $B_t$ if
$$E[B_t|B_{t-1}, B_{t-2}, ...] \neq [B_t|A_{t-1}, B_{t-1}, A_{t-2}, B_{t-2}, ...]$$
or in words, knowing the past values of $A_t$ helps us get a better prediction of the current value of $B_t$. The nice thing is, that the above hypothesis is easy to test: you just have to fit a linear model.

So if your index variable is $Y_i$, and your explanatory variables are $\{X_{1t}, \dots X_{mt}\}$, then you just have to estimate the simple linear model
$$ Y_t = \alpha + \sum_{s=1}^S \gamma_s Y_{t-s} \sum_{k=1}^m \beta_{ks} X_{k, t-s} + \epsilon_t $$
by OLS. Then $X_k$ Granger-causes $Y$ (in the expectation sense), if $\beta_{k1}, \dots \beta_{kS}$ are jointly significant.

Practically to test which explanatory variables Granger-cause $Y_t$ you have to do the following:

Run a linear regression with $Y_t$ as the dependent variable and some lags of $Y_t$ and the explanatory variables at the right hand side
Test whether all the lags of a given explanatory variable are jointly significant
Repeat it for each of your indices

By estimating these models you also get an answer to your second question (you just have to look at the coefficients).

Another way to go would be to estimate a VAR model where you include all of your indices and explanatory variables, and then see whether the lags of a given explanatory variable for a given index are jointly significant. This will have a slightly different interpretation: you would test whether your explanatory variables Granger-cause your indices even if you condition on all the indices. In your case I would be careful with it due to the relatively low number of observations.

Practical concerns

Deciding how many lags to include in the regression might be tricky. It depends on the amount of data you have and the model you have in mind. For quarterly data, you should probably include at least 4 lags.
As have panel data you could be inclined to include a country dummy to get fixed effects estimates. However because you have lagged dependent variables on the right hand side, your estimates would be biased if you did this. There are ways around it, but they are quite advanced.

If you have questions how to implement it in practice feel free to ask for details.


Negative Interest Rates and Pension Funds How do negative interest rates impact the future prospect and business model of (defined benefit) pension funds?
What could be winning investment strategies for pension funds in a negative interest rate environment?

This is two questions.

How do negative interest rates impact the future prospect and business model of (defined benefit) pension funds?

Defined benefit pension funds have a discount rate and assumed returns to determine the required contributions. The exact mechanism for determining these depends upon the jurisdiction, but they are normally related to market interest rates, possibly on an averaged fashion.  Lower discount rates - regardless of whether they are positive or negative - implies greater contributions needed for the same level of benefits. This makes defined benefits more “expensive.”

What could be winning investment strategies for pension funds in a negative interest rate environment?

This can only be answered as an opinion, which is not in line with guidelines.


How should I regress income inequality on economic growth? I would like to know which variables to use to regress income inequality on economic growth. For income inequality, I am thinking of using the Gini index, but don't know what measurement exactly.
For the growth should I use GDP per capita, GDP per capita growth, real GDP per capita, or just annual GDP values?

If your are interested in the statistic relationship between income inequality and economic growth, you should use GDP per capita growth since the others indicate level rather than growth.
More generally, the choice of the variable in your (and any other) regression depends on your economics model, otherwise you are just doing some linear projections.
A good reference you may start with this topic is "Banerjee, A. V., & Duflo, E. (2003). Inequality and growth: What can the data say?. Journal of economic growth, 8(3), 267-299."


What Constitutes The Size Of An Economy? My Geography textbook defines the size of an economy as:

  Size of economy is measured in terms of value of goods and services
  and grows over
      time. This is caused by increase in population, improvement in income levels,
      advancement of technology and other associated factors.

I found these statements to be a little iffy. I don't understand how things like advancement of technology could possibly make the economy larger in size. I do get that they could give the secondary or tertiary sector a larger chunk of the economy, but could it make the economy larger as a whole?

The Netherlands, for an example, has just as elaborate an economy as any other country; with possibly the same factors acting on it. However, even with all the advancements in technology and increased income levels, it is no where as gigantic and mammoth (we are talking size, remember?) an economy as India or China when it comes to sheer scale.

I thus concluded that this statement was erroneous. Am I right in propounding that the size of an economy has nothing (absolutely nothing) to do with the intricacies and coming-together of processes, and solely to do with the scale or largeness of it?

Your book is correct.

To quote it

  Size of economy is measured in terms of value of goods and services.

Suppose the Netherlands had its current population of 16 million people but only 14th century technology. I hope you'll agree that the value of whatever they produced would be significantly less than the value of their current production.

Another comparison:
The Indian economy is currently smaller (in terms of value) than the US economy
despite the difference in population. The reason for this is that the US has much more capital in advanced technology.

To address the question whether value is a good indication of size is good question:

There are several arguments for this. Imagine that a country produces something that it consumes very little of (e.g. cash crops). In this case they will not care at all about the weight, volume etc. of their production, they will want to know how much they can sell it for so they can import other things. I would say in such a case value is much more relevant then other measures.

Another example:

Imagine that the A pill immunizes you against malaria for a day and the B pill immunizes you for a year. Would you say a plant producing 100 A pills produces more than a plant producing a B pill?  I would argue in some sense it does, in some sense it does not. There is no absolutely clear method of comparison. For reasons of international trade, value is used. Is the word size confusing? Perhaps. But not overly so.


How likely are the Sonnenschein-Mantel-Debreu results? I have been reading about the SMD results and it's "damning" nature for GET. My understanding of the result is as follows: price changes not only cause a substitution effect but also change the wealth distribution of a society. The latter could, in principle, cause the excess demand function to have any shape (save for homogeneity of degree zero, Walras' Law, boundary conditions, continuity)

My reaction to this is that it seems enormously unlikely. While I'm sure it's possible, it seems to me that if the number of individuals and commodities gets sufficiently large, that "hardly any" economies would actually exhibit this weird behaviour. The wealth redistributions required to get the weird shape would be just too improbable. I am curious if there was some way of defining the space of economies as a measure space, and then saying that the set of economies with weird aggregate excess demand function is of "small" measure (and tending to zero with the number of individuals/commodities). Do mathematical economists agree with this intuition of mine?

The space of economies can certainly be modeled as a measure space. Perturbing an economy (the parameters of preferences and production functions) generally changes the excess demand function continuously, so if there is one economy with a "weird" excess demand function (however defined), then there is a positive measure of them (open ball around the one weird economy).

As the number of commodities grows, and preferences for each new commodity are randomly drawn from some space, then my intuition is that the probability that at least one commodity has upward sloping demand converges to 1. The probability that all commodities have "weird" demand likely converges to zero.

Empirically, upward sloping demand is rare (examples of inferior goods are rare).


How to interpret the intersected information partition element of different actors? In this paper, it said that given $\omega$ is the current state of the world and $P_1(\omega)$ is the unique element from the information partition $\textit{P_1}$ of actor 1 in which the player 1 is informed that $P_1(\omega)$ contains $\omega$. According to the aforementioned paper (p.1237), it says that "To say that 1 knows that 2 knows (Event) E means that E includes all $P_2$ in the information partition $\textit{P_2}$ that intersect $P_1(\omega)$."
 I wonder why is the aforementioned intersection in this passage so important, since intersection here only mean that two information partition elements share some states.

I think there are two thing I need to understand: The interpretation of the shared states of two information partition elements from different people; the interpretation of the case when two informational partition elements intersect.

Also why is it seem to be inapplicable for the case when the partition elements mentioned are partially in $E$

I'm not an expert in epistemic game theory but let me see if I can help with an example.

Suppose the state space is $\Omega = \{1,2,3,4,5\}$, and there are two players with information partitions
$$
\mathcal{P}_1 = \left\{\{1,2,3\},\{4,5\} \right\} \\
\mathcal{P}_2 = \left\{\{1,2\},\{3,4\},\{5\}\right\}
$$
Suppose the true state is $\omega = 5$.
Let us consider the event $E = \{4,5\}$.

Player 1 "knows" $E$, since $P_1 = \{4,5\} \subset E$.

Player 2 "knows" $E$, since $P_2 = \{5\} \subset E$.

However, does Player 1 know that Player 2 knows $E?$ No.

What are the elements of $\mathcal{P}_2$ that intersect $P_1$? $\{3,4\}$ and $\{5\}$.

However, $\{3,4\} \not \subset E$, so Player 1 does not know that Player 2 knows $E$.

On the other hand, does Player 2 know that Player 1 knows $E$? Yes.

What are the elements of $\mathcal{P}_1$ that intersect $P_2$? $\{4,5\}$, and $\{4,5\} \subset E$, so Player 2 knows that Player 1 knows $E$.


Why there is no supply on monopoly markets? We learned in the class that monopoly markets do not have supply curve. Our textbook (Microeconomics and Behavior, page 388 from, R. Frank) says:

A monopolist has no supply curve

and it explains it by saying:

The reason is that the monopolist is not a price taker, which means there is no unique correspondence between price and marginal revenue when the market demand curve shifts. Thus, a given marginal revenue value for one demand curve can correspond to one price, while the same value of marginal revenue for second demand curve corresponds to a different price. As a result, it is possible to observe the monopolist producing $Q_1^*$ and selling at $P^*$ in one period, and then selling $Q_2^*$ at $P^*$ in another period.

I don't get this. We learned that in competitive markets demand can change from period to period even if the price stays the same (like when supply is just horizontal line). Also, why can't we just pick one point in time look at possible combinations between P and Q on monopoly market, and call those combinations supply? The book has some examples but they don't explain this clearly.

The argument is that the monopolist's decision is based on the demand curve (in effect matching marginal total revenue to marginal cost) so is not independent of the demand curve, and in that sense there is not a corresponding supply curve with price and quantity in equilibrium where the two curves cross; the monopolist equilibrium point is likely to be at a lower quantity and higher price.
The conventional demand curve illustrates the price against quantity.  If instead the demand curve illustrated marginal total spending against quantity, then where it crossed the supplier's marginal cost curve would give the monopolist equilibrium quantity (though not the equilibrium price), and in that sense the monopolist's marginal cost curve could the be seen as the supply curve.


Macroeconomic models with labor-replacing capital? I am reading article http://public.gettysburg.edu/~tlin/research/robots-main.pdf "A New Keynesian Model with Robots: Implications for Business Cycles and Monetary Policy" and I wonder how actual this article is, actually, in my opinion, this article will bring Nobel prize to its authors.

My question: is there research on macroeconomic models about labour-replacing capital, i.e. about robotics and artificial intelligence as the new factor? Google search gives no relevant results and as far as I can find, the mentioned article is really pioneering in this field. Or maybe there is ongoing research on these themes but it uses different terms and keywords?

Of course, capital is usually included in all the major macroeconomic models, but the key issue is that today the certain forms of capital can almost fully replace (e.g. see https://www.hlai-conf.org/ or http://people.idsia.ch/~juergen/) labour and it brings with itself new issues, including the necessity to model new types of taxes and also the universal basic income. Maybe such labour-replacing capital has already been modeled in traditional (e.g. DSGE) models?

The Grundrisse’s “fragment on the machines” expounds on the tendency to universal automation rather famously, without Marx having stripped his more idealistic hopes from his exposition, and without the deadening jargon of Capital’s chapters on the organic composition of capital. It isn’t perfect on the automation of science itself, but that just needs to be reread with science posited as a commodity, AI as fixed capital etc. Nick Dyer-Witheford works in this area last time I checked.

The assumption, or rather observation, of the centrality of a diminishing living component of labour is central to Marxist political economy. Normally it is posited as the OCC critique where transcending this process only exists as a negative, but the fragment on the machines pictures the real possibility of social mastering of social production. In “fragment” Marx goes a bit further than UBI and instead poses universal leisure (relief from labour) due to hyper abundance. This is viewed as a fundamentally political achievement and one not achieved by parliament, treasury or finance ministries.

As far as formalisations, OCC seems to be an accepted category amongst the analytical Marxists (Basu 2018 …Ground-rent… [working paper] https://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1242&context=econ_workingpaper). This means it ought to appear in heterodox Marxist economics, as opposed to just Marxist political economy.


Why lower the deposit rate if it is already negative? The European Central Bank (ECB) has been lowering the interest rate on its deposit facility, first to -0.1% in June 2014, then to -0.2% in September and eventually to -0.3% in December 2015.

But what difference does it make whether it is -0.1% or -0.3%, as long as it is negative? I would expect anyone to withdraw all their money immediately as soon as the interest rate gets negative, for you would always be better off just keeping it for yourself, even if the interest rate is just -0.0001%.

I do understand the intention of the ECB, but I do not understand why lowering the already negative interest rate further should make it more effective.

You are assuming that the supply of deposits is zero when the price (the rate) is zero but it definitely is not. There are several reasons for this.

While you can withdraw cash from the bank it is unwieldy, costly, and unsafe to have large amounts of cash lying around. Therefore, if there were no other alternatives, heavy spenders would still want some wealth in deposit system even if rates were quite negative. Consider, for example, that it is common for depositors to pay account fees that greatly exceed interest earnings.
These costs are actually very low as it goes. A rate of -0.1 percent a year is ten cents per \$100 of average balance, and since these costs are proportional to average balances, which themselves are small relative to total spending, the value of transaction services provided by these accounts is probably large relative to total negative interest rate costs.
Assuming non-bank storage costs are high relative to negative rates, it isn't the nominal negative rates (the advertised rate) that matters but rather the real rates (relative to inflation). Under these circumstances, you should be indifferent between 2 percent deposit rates under 4 percent inflation and -1 percent deposit rates under 1 percent inflation. So a rate can drop and the supply of funds can be unchanged if it is accompanied by a drop in inflation.

And, in the case of the EMU / ECB / Euro, that may well be what happened. Inflation fell for the five months after June 2014, which may explain why nominal rates had to fall: to keep the tightness of monetary policy unchanged or at least to keep real rates unchanged.


How can lending be feasible on blockchain? Given the fact that mostly all cryptocurrencies have a permanently set supply, they will invariably rise in value as cryptocurrency transactions rise. Even if cryptocurrency deflation cools off from where it is now, Crypto/fiat would still rise at a high rate due to population and economic growth.

So, I don't understand how loans would be feasible on blockchain.

Let's say I get a 1 BTC car loan at 1% interest that's amortized over 6 years, then the amount I would have to pay in dollars to cover the appreciation of BTC/USD along with the interest on the loan would probably be way above that on standard car loans.

To me, the only time it would be feasible to borrow BTC is when I am shorting BTC... Or am I missing something?

So, I don't understand how loans would be feasible on blockchain.

Lending and borrowing money via blockchain is similar but not equal to traditional methods.

The similarities:

Same parties: lender, borrower, broker
Same goals: those with capital wishing to increase their capital, those without wishing to increase their capital, a (optional) agent broker who receives a commission as part of their brokerage services.

The Differences are in how the process works:

I presume you are knowledgeable of traditional lending and borrowing mechanisms: banks, private loans, bond, equity, etc. So I'll cover the blockchain side of things.

You and I would sign up at a blockchain institution where the service of the firm is to match borrowers with lenders. I 'deposit' my blockchain assets and I either allow the firm to look for matches based on my preferences of interest rate, term, risk, and credit (in the abstract sense, not your FICO score); or pick and choose contracts to my liking.

The caveat for blockchains is that there is no known of recourse in case of default. The keystone basis of blockchain tech are the control and usage of private 'keys' (or password) matched to public keys that allow a user to control the 'data' associated with 'wallets'.

Unlike a traditional funding mechanism, if 'keys' are lost due to theft or malice (default), it is currently impossible to recover lost funds. No matter how powerful the judiciary and law enforcement system is in a given jurisdiction, they can't use a court order to brute force a solution to find the lost keys.

In a traditional lending environment, a creditor can go after assets that are tied to your identity and have monetary value (other forms of cash, real estate, vehicles, furniture, etc.) and liquidate them for their monetary worth.

Riskier? Definitely, but this doesn't mean that blockchain tech hasn't been used to facilitate lending arrangements with differing ranges of success.

  To me, the only time it would be feasible to borrow BTC is when I am shorting BTC... Or am I missing something?

You are, the key point here is if the loan is in BTC or fiat currencies (USD, Yuan, Ruble, Yen, etc.) BTC can be the mechanism of which the current market prices of BTC to X is transferred, but the loan would still be valued in USD.

So if you enter into a contract for 10,000USD at 1% over 10 years, (simple interest). Your USD obligations for the contract would be $100 a year, regardless of the USD to BTC rates.

Year 1: BTC/USD might be 1BTC/1USD. Your debt obligation is 100 BTC.
Year 2: BTC/USD might be 1BTC/10,000USD. Your debt obligation is .001 BTC.

The best way to approach the system is to first determine A: if the underlying monetary system is deflationary or inflationary. Deflation benefits the creditor, inflation credits the borrower. But with blockchain lending system: no one can force the borrower who is in default to return the Blockchain based assets. With traditional systems: you have law enforcement and judicial systems available.

Here's something I wrote for LinkedIn.


Explanation of differential equation for Price Elasticity of Demand I understand that Price Elasticity of Demand (PED) measures the percentage change in quantity demanded of a good with respect to a percentage change in its price.

However, I don't quite understand the intuition behind the formula for PED being:

$e_{{\langle p\rangle }}={\frac  {{\mathrm  {d}}Q/Q}{{\mathrm  {d}}P/P}}$

and not simply described by the following:

$e_{{\langle p\rangle }} = {\frac  {{\mathrm  {d}}Q}{{\mathrm  {d}}P}}$

This might be quite an elementary question, but I'm hoping for a bit of clarification.

Consider a demand equation,
$$Q=\alpha\ +\beta\ P$$

If you were to just use,

$$e_{{\langle p\rangle }} = {\frac  {{\mathrm  {d}}Q}{{\mathrm  {d}}P}}$$

Then the result is only the differentiation with respect to $P$. This doesn't give you the information that you are looking for because you haven't specified the point at which you are measuring elasticity. To do this the second portion of the equation is necessary. Suppose you want to know the elasticity at $P=\$8$. By plugging $\$8$ into the demand equation you can calculate the quantity demanded at $P=\$8$.

Personally, I have to think of this in two parts, (1) the differentiation with respect to $P$ (2) multiplied by the "spot" of the demand curve that I am examining, ${\frac PQ}$.

$e_{{\langle p\rangle }}={\frac  {{\mathrm  {d}}Q}{{\mathrm  {d}}P}}* {\frac PQ}$

edit

The number that you calculate is the percent change of a 1% increase in price. For example, if $e=-0.3$, you can say that a 1% increase in price should cause a 0.3% drop in the quantity demanded.


Why do we need the specific utility assumption? In many microeconomic models, we read the assumption that agents have CARA normal utility preferences. We need the assumption of the utility to do pricing, but should someone do differently?

Namely, is there a case where we do not need utility functions to make pricing? I think this has to do with the probability measure, if it is unique or not.

One of the most important consideration for using CARA utility is tractability. CARA utility and Gaussian errors yield a certainty equivalent completely described by a simple function of the mean and variance of the Gaussian distribution..

It also turns out that maximizing expected utility in this setup is equivalent to maximizing the certainty equivalent - which is a huge advantage in settings where we want some sort of quasi linearity in wealth.


What are good books on economic world history? I am trying to find a book that covers human history with a special focus on economics. It should answer questions like:

What economic systems prevailed during which eras?
How did these systems work in detail?
How did economic organization evolve (Not just since the industrial revolution)?
Were there civilizations which particularily focused on economic development?

It should NOT be a history of economic thought, like e.g. Niall Kishtainy's "A little history of economics".

Rondo Cameron's A Concise Economic History of the World may be helpful. I read an early edition many years ago and it certainly is primarily economic history rather than history of economic thought. One limitation however, identified in this review, is that despite its title it focuses mainly on Europe.


Simple equilibrium in monopoly question Given an inverse demand function for hangars: $$P = 3 - \frac{Q}{16,000}$$ and constant marginal cost of $1$, what is the equilibrium price and quantity of hangars in a monopoly?

In a perfectly competitive market we need $P = MC$, which gives us an equilibrium price of $1$ and quantity of $32,000$. However, in a monopoly we need $MC = MR$, and in this case that equates simply to price = $1$. Then, the demand gives us, again, an equilibrium quantity of $32,000$. Am I making some mistake somewhere, or are the equilibrium prices and quantities in a monopoly and competitive market in this example equal simply by coincidence?

The monopolist solves,

$\max \pi=R(Q)-C(Q)=P(Q)Q-C(Q)$

The general solution is (if the problem is globally concave)

$MR(Q^*)=P'(Q^*)Q^*+P(Q^*)=C'(Q^*)$.

Since $P'(Q)<0$, notice that $MR(Q)<P(Q)$ and, therefore, $Q^*<Q^{C}$.

In this example, we have $P(Q)=3-\frac{Q}{16000}$ and $C(Q)=Q$. Hence,

$MR(Q)=-\frac{1}{160000}Q+(3-\frac{Q}{16000})=1=MC(Q)$.

Rewritting, we get
$3-\frac{Q}{8000}=1$ or $Q=16000$. To compute the price simply substitute in the demand function and get $P(16000)=3-1=2>1$.


Disruptions in Server / Data Center Market With the advent of cloud computing, data storage and scaling has become extremely cheap when compared to the strictly physical server alternative (which require a dedicated facility and substantial fixed/variable costs). Implicitly, we could view cloud computing as having comparatively big ecological advantages over dedicated servers. From this perspective, companies with server/data center needs will have huge incentives to choose cloud computing.

My question is, is the verdict really this simple? Wouldn't some companies be reluctant to use cloud computing for security / conflict of interest reasons? Instead, might they prefer to be the sole custodians of their data? For example, investment banks, hedge funds, ect?

Are there any other compelling reasons that might hold the server / data center industry away from adopting the cloud?

Institutions like banks for many years now keep  back-up archives of their data in physical locations owned and operated by other legal entities. This is a standard case of outsourcing to specialists a requirement of a business process that is not a directly value-added activity. So the "sole custodian of one's own data" has been proven in the market not to be strong enough against cost-reduction considerations (or against price/value optimization if you wish).

Cloud computing is comparatively still new. The cost incentives to adopt it are indeed very strong. In practice, I believe we will observe, as in the past with other innovations, a certain degree of selectiveness: as the risks and rewards of cloud-computing will become more well understood (and measured), companies may select certain parts of their activities to keep off the-cloud, while enjoying the cost reductions for all the rest.

This by the way, can be squarely put in the framework of the, dear to us economists, marginal optimization - it is not a "yes-or-no" decision but a process where each company will adopt the Cloud up to the degree where marginal benefit equals marginal cost.


I need to prove how an increase in output p increases profit-max. Can someone help to understand why IFT implies that z is a unique maximizing point? MWG 5C6 asks: "Suppose a concave prod function f(z) with inputs $(z_1,...,z_L-1)$ and also that $\partial f(z))/\partial z_l \geqslant 0$ for all l and $z\geqslant0$ and that $D^2f(z)$ is negative definite at all z. Use FOC and Implicit Function Theorem (IFT) to prove that an increase in output price increases profit-max level. I did undertand the most part of Solutions, although it´s not clear why does the Implict Function Theorem implies that z is a unique maximizing point that satisfies $p\bigtriangledown f(z)-w=0$.

The uniqueness of z comes from the ND of $D^2f$ I believe. Let $z^*(p)$ be the solution of the problem at price $p$. Define the function, $G(p,z):=p \nabla f(z)-w$. From uniqueness we have $z=z^*(p)$ if and only if $G(p,z)=0$. That is, $G(p,z)=0$ is an implicit equation which tells us how $z^*(p)$ changes when $p$ does. The IFT gives us a way to find the derivative of this function
$$\frac{\partial z^*_i}{\partial p}=\frac{-\frac{\partial G}{\partial p}}{\frac{\partial G}{\partial z_i}}=-\frac{1}{p}H^{-1}\nabla f(z)$$
Where $H$ is the hessian of $f$.
So $\sum_i\frac{\partial f}{\partial z_i}\frac{\partial z^*_i}{\partial p}>0$ as $-H^{-1}$ is positive def.
The derivaitve of the profit function wrt to $p$ is $\frac{d}{dp}pf(z^*(p))=f(z(p))+p\sum_i\frac{\partial f}{\partial z_i}\frac{\partial z^*_i}{\partial p}$


Optimal decision for perfect substitutes utility function ? Given $u(x_1,x_2)=4x_1+14x_2$ and $m=\frac{1}{2}x_1+\frac{3}{2}x_2$, I shall choose the optimal decision among:
$a)(2m,\frac{2m}{3})$
$b)(2m,0)$
$c)(\frac{m}{2},0)$
$d)(0,\frac{2m}{3})$
The correct answer is $(d)$

But not sure how to find this, what I did:
$$m=\frac{1}{2}x_1+\frac{3}{2}x_2 \Leftrightarrow x_2=\frac{2m}{3}-\frac{1}{3}x_1 $$
And the marginal rate of substitution is $-\frac{2}{7}$,so we have :

Obviously $a)$ isn't the right one and $c)$ doesn't seem to be the best one... but really don't know 'how to think', some explaination would be great.
Thanks in advance

Utility maximization problem is:

$\max\limits_{(x_1, x_2)\in\mathbb{R}^2_+} 4x_1 + 14x_2 \\ \text{s.t.} \ \ m = \frac{1}{2}x_1 + \frac{3}{2}x_2$

This problem can be converted into an optimization problem involving just one variable $x_1$:

$\max\limits_{x_1} 4x_1 + 14\left[\frac{2}{3}\left(m - \frac{1}{2}x_1\right)\right] \\ \text{s.t.} \ \ 0 \leq x_1 \leq 2m$

Differentiating the objective $4x_1 + 14\left[\frac{2}{3}\left(m - \frac{1}{2}x_1\right)\right]$ with respect to $x_1$, we get $\left(4- \frac{14}{3}\right) = \frac{-2}{3} < 0$. So we observe that the objective is decreasing in $x_1$. Therefore optimal choice must be the smallest $x_1$ we can have i.e. $x_1 = 0$ and hence $x_2 = \frac{2m}{3}$.


Value of the ruble in year 1825 Would anyone know how to find out the value of the Russian ruble in the year 1825, in today's (2016) US dollars?

This website (in Russian) states that, in 1825,
you could sell 1 Imperial Ruble and get 0.72 US dollars. I don't know Russian so I cannot tell the source of that data.

Those 0.72 US dollars in 1825 are worth US$15.49 in 2016, according to this website.

Therefore, 1 Imperial Ruble of 1825 was equivalent to US$15.59 in 2016.

Additionally, this website has information of prices of different goods and wages for Russia since the 16th century.


Deadweight Loss and Excise Taxes I am currently reading through the 5th Edition of Economics, by Paul Krugman and Robin Wells. Right now I am learning about excise taxes and how their different aspects (government revenue, deadweight loss, reduction in consumer/producer surplus) can be represented on a supply and demand diagram.

In relation to the diagram featured in Figure 7-8 below, the book explains that “a portion of the loss to producers and consumers from the tax is not offset by a gain to the government—specifically, the two triangles B and F. The deadweight loss caused by the tax is equal to the combined area of these two triangles.”

This explanation makes a lot of sense, and is in keeping with what I have learned about deadweight loss up to this point. However, in the 'Check Your Understanding' section of this particular chapter (featured below), an answer to one of the questions seems to completely contradict this explanation. Allow me to elaborate.

Question 1f is the one that I am having trouble with.

The described supply and demand schedule and the excise tax imposed generates the supply and demand diagram featured below, where the yellow triangle represents the deadweight loss created by the imposition of the excise tax:

To clarify some of the components of this diagram, the 0.40 USD excise tax drives a wedge between the price paid by consumers and the price received by producers, equal to the value of the tax. This reduces the quantity transacted from 4 units at the equilibrium level to 2 units (represented by the vertical dashed line).

Based on the explanation of deadweight loss which I earlier cited, it seems to me that the deadweight loss here should be equal to 0.40 USD:

However, in the answers section (section S-7 of the linked PDF), it is said that the deadweight loss is actually 0.20 USD:

Though I can see the logic of subtracting the government revenue and the total surplus after the tax from the total surplus before the tax to arrive at the deadweight loss, the answer provided seems to contradict the explanation given earlier — either the deadweight loss is equal to 0.20 USD, or the deadweight loss is equal to the area of the shaded triangle (0.40 USD). From my perspective, it does not seem that these two assertions can be simultaneously true.

I hope that I have provided enough information for readers to be able to figure out whether I have made a mistake here, or whether this is actually an error in the textbook.

The problem is that your supply-demand graph is wrong. You assume in your graph that the demand and supply are linear functions but in this case they are step functions.

For example, in your graph there would be 2.5 demand at price $\$0.55$ but the problem clearly states that the units are sold at discrete quantities of 1. So the drawing is incorrectly modeling the situation as there should be a steps in increases in demand not a simple linear function. The same holds for supply.

If you draw your supply and demand correctly you will see that in your case deadweight loss (DWL) is not a triangle but just two squares with areas 0.10 respectively which combine into a rectangle with area 0.2 and hence total DWL will be also $\$0.2$.

This is how your problem would look visualized (pardon the bad drawing):

Where the shaded area is the DWL. The two methods will always match but you have to use the same demand and supply functions in both problems if you assume different supply and demand of course results will change (in your original drawing also consumer surplus after tax would be wrong by the way as in your original drawing it would be area of two trapezoids both having areas of 0.15, and the pre tax consumer surplus in your picture would also not be 1.2).


Any major theory/model that considers return due to idiosyncratic risk? Are there any classical, major theories/models that consider positive return due to idiosyncratic risk? For example, CAPM only considers return due to systemic risk but not idiosyncratic risk. If there are no major theories/models that fit the criterion, are there perhaps unorthodox ones worth mentioning? I would be grateful for references.

(I am a novice in asset pricing, so pardon if the question is too basic.)

There are no major models that come to the conclusion that idiosyncratic risk can drive positive returns. The reason is that idiosyncratic risk is diversifiable. The models, however, do in fact consider idiosyncractic risk. The theories don't forget or ignore them. It just turns out that this risk doesn't matter.

In this sense, CAPM actually does consider idiosyncratic risk. In this model, if you are perfectly rational, you will diversify your portfolio to minimize risk. This leads you to hold the whole market portfolio, in which case idiosyncratic risks are diversified away in your portfolio and therefore no longer matter. In simple terms, market risk then is not diversifiable, because once you hold the market portfolio, there's no way to diversify further. What drives returns are risk, but rational investors can remove the idiosyncratic risk through diversification. Hence, what matters for returns in the CAPM therefore is just the market risk (which you refered to as systemic risk, but that term actually means something else). That's why under CAPM, the returns are driven by the asset correlation with the market.

EDIT:

The CAPM model does not put any explicit assumptions on the number of assets available, hence the model does not require an infinite number of assets. However, the derivation is based on the decision to add an asset to an already well-diversified portfolio, i.e. adding an asset to the market portfolio.

Nevertheless, if only a handful of assets exist, then it might not be possible to diversify away all idiosyncratic risk. In that case, the CAPM does not work, because it implicity assumes enough assets exist, since it assumes a well-diversified portfolio exists to which you can add an asset. In such a case idiosyncratic risk may become relevant.

Such considerations are unlikely to be relevant in practice though. It is difficult to put a number on how many assets are necessary for idiosyncratic risk to be diversified away completely. Often, a number that is mentioned is 30-40 assets. The number required in practice is usually very realistic and far below infinity. The number of assets required for idiosyncratic risk to be completely diversified away becomes larger if assets are more correlated with each other or very volatile.

Lastly, an asset with such high idiosyncratic risk that no investor would want to add it to their portfolio cannot exist in equilibrium, because supply of the asset could never equal demand.


How does burning a huge amount of cash money affect the monetary supply? I am not an economist by default. In other words, I am a newbie here.

Assume there is an extremely rich man who withdraws an extremely huge amount of cash money. He then burns all of it.

How does this action affect the monetary supply?

It would seem logical that this would result in reverse-inflation or lower prices, but this is actually not the case (at least over time).

The banking system has a "supply and demand" market for reserves.  When say 1 million dollar is withdrawn and not redeposited, the banking system in the aggregate will bid more for current reserves to replace the ones that were lost.  The Fed does not like to see changes in the rates the bankers lend each other reserves and if it thinks this exchange rate is higher than the targeted Fed Funds rate, it will sell REPOs to primary dealers in an attempt to move this rate down.

So in the end, the amount of money will roughly be the same (as near as I can tell).  What will change is that the Fed will have new assets it didn't have before.  These assets in turn will accrue interest, and profits from the Fed get turned over to congress.  So in a round about way, burning money is a transfer of wealth from the individual to the public.

In the same way, counterfeiting does not actually increase the money supply in the US over time.  New fake dollars are deposited, drive down reserve rates which results in the Fed selling assets to absorb the new money.  Less Fed assets = less taxpayer money.  So counterfeiting hurts, but just not in the way most suspect.


K in the Cobb Douglas function I'm using the Cobb Douglas production function for a Mathematics investigation into how optimisation works in Economics. The assumption is that the firm require's only one type of capital, that is, a coffee machine. Keeping this in mind, would it be reasonable to take K as the number of coffee machines  used, instead of hours?
Thanks, and cheers!

There is no "reasonable" choice here.

One can choose inches over centimeters, gallons of water over kilograms of water, etc. This depends on how you want measure the input quantities. No choice is wrong so long as it is consistent with your other modelling choices.


If we estimated the value of all the assets in the world and compared it with the estimated total wealth of individuals, would the results be similar? Say we estimated the value of all assets (money, stocks, bonds, property, etc.) in the world.

Next we estimate the wealth of all the individuals in the world.

Would the two resulting values be similar or substantially different? If they would be substantially different, then why?

I am happy for people answering the question to use the definition of wealth they feel is most appropriate.

As currently written, the question asks whether the value of all assets equals the wealth of individuals. This is trivially not the case, since assets can be liabilities of another entity.

As an example, imagine an economy in which one individual owns both an industrial firm and a bank. The bank can make a loan to the firm, and both entities’ balance sheets grow - the bank gets a loan asset, the firm gets a new deposit at the bank. This means that the total value of all assets increases by double the amount of the loan. However, the act of taking out a loan and not spending it does not increase the value of a corporation, and so individual owning the two banks has no increase in wealth.


How do we estimate discount factors? We often talk about time preferences in intertemporal utility maximization problems.

Usually we see some variant of:

$$\max_{c_t}\sum_{t=0}^\infty\beta^tu(c_t)$$
subject to:
$$\sum_{t=0}^\infty e_t=\sum_{t=0}^\infty p_tc_t$$

where $e_t$ is some endowment process over a consumers life time and spending is on a specific consumer good.

I have been thinking how exactly we estimate what $\beta$ is.

Our usual demand analysis methods only consist of estimating a expenditure function and then recovering underlying preference parameters. Here its tricky though since we are interested in what a discounting looks like.

how do we approach this problem?

There are multiple ways how the discount factor can be estimated. I dont think its possible to make exhaustive review of all of them (within format of this site at least), but one that nicely matches your question would be through estimating the Euler equations.

Following Attanasio & Browning (2009) an Euler equation for general asset would be given by:

$$ E \left[ \frac{c_{t+1}^*}{c_t^*} \right]^{-\gamma}(1+r_{t+1}) \beta = 1 \implies  \left( \frac{c_{t+1}^*}{c_t^*} \right)^{-\gamma}(1+r_{t+1}) \beta = \varepsilon_{t-1}$$

where $c$ is consumption $\gamma$ the coefficient of relative risk aversion, $r$ real interest rate and $\beta$ the discount factor. Authors show that assuming that measurement error is log normal with homoskedastic variance across households and independent then in terms of observed consumption it can be proven it will be equal to $e^{\gamma^2 v}$ where $v$ is the variance of measurement error.

I will skip all derivations, since they are nicely laid out in the above-mentioned paper itself and for the sake of brevity, but with additional assumptions that the Euler equation will also hold in future $c_{t+2}$ you can derive an estimator based on the following equations:

$$ u_{t+1}^1 = \left( \frac{c_{t+1}}{c_t} \right)^{-\gamma} (1+r_{t+1}) \beta - e^{\gamma^2v}$$

$$ u_{t+2}^2 = \left( \frac{c_{t+1}}{c_t} \right)^{-\gamma} (1+r_{t+1})(1+r_{t+2})\beta^2 - e^{\gamma^2v}$$

where parameters $\beta$, $\gamma$, $v$ are overidentified so we can estimate them using non-linear general method of moments (GMM). Authors call this estimator GMM-LN and they also derive version of it which require a bit less strict assumptions.

However, the above estimator is not the only way how to estimate discount factors, other ways exist including estimating it experimentally such as for example in Benzion, Rapoport,  Yagil (1989).


Pascali (2015) Natural Experiment In The Wind of Change: Maritime Technology, Trade and
Economic Development (2015), Pascali estimates various effects on freight rates (i.e. the cost of transport) on outcomes such as trade volumes and economic development. To do this, he exploits a kind of natural experiment that results from the fact that different shipping routes benefit in different ways due to the preexisting wind patterns (i.e. those with wind flows against the route direction).

I guess this controls for demand for shipping, so that freight rates are exogenously affected by transport times. But I don't understand why this is needed to estimate the effects of shipping technology. Isn't the invention of the steamship itself an experiment that allows you to see the effects of shipping technology on trade volume? Why is this additional variation due to asymmetric benefits needed?

A typical regression equation used by him is the following:

The issue is that the invention of the steamship is not exogenous to economic development. Assuming that trade exerts positive effects on development, it could be that past trade favored economic development and as result the invention of the steamship. By contrast, wind flows are exogenous.


Monopoly second degree discrimination - two part tariff In a Monopoly with second degree discrimination - two part tariff will the consumer with higher demand always pay a higher entrance fee and lower unit price than the cunsumer with lower demand or is this assumption not correct?

That is if in optimum High demand chooses offer (T1,P1) and Low demand chooses (T2,P2), do (T1 > T2) and (P1 < P2) always hold?

I'm inclined to say "yes" to $T_1>T_2$ and no to $P_1<P_2$.

For simplicity, suppose the monopolist's marginal cost is constant. Then consumer surplus is the area under the (inverse) demand curve and above the marginal cost. In a two-part tariff scheme, the monopolist would charge entrance fee equal to consumer surplus and unit price equal to marginal cost.

If, in your language, consumers with "high demand" are understood as those who are willing to pay more for each unit of the good than those consumers with "low demand", then high demand consumers generate higher consumer surplus than the low demand ones.

Suppose monopolist can distinguish between the two types and offer different two-part tariff bundles to each (e.g. a theme park company operating in US and India charging different prices in the two countries). Then it will be the case that the monopolist charges a higher entrance fee to the high type (because they generate higher consumer surplus) but the same unit price for both types (because this is related to the marginal cost of production, which is the same regardless of the consumer types).


Why does BOJ unchanged rate lead to Nikkei and Topix declines? Why does the BOJ's decision to leave rates unchanged lead to a stronger yen and declines in the Nikkei and Topix? Thanks in advance!

It doesn't in general.

It did in this specific case because the Nikkei and Topix had earlier priced in the possibility that the Bank of Japan would do something other than leaving rates unchanged this time. Now that that possibility has gone, the market no longer needs to price in the possibility, and so has corrected.

That's not just specific to this case: that's true of all market responses when an unknown event becomes known.


Simultaneous Equations Models in R I use the systemfit package to evaluate Simultaneous Equations Models (for example 2SLS). But I do not understand how to predict endogenous variables using the new values of exogenous variables using this package.

Therefore, I would like to know how to make prediction (including prediction intervals) using models of simultaneous equations in the "systemfit" package or in some other way (maybe someone knows other packages for this).

I am also interested in how I can calculate the elastic coefficients of models.

I will be grateful to the literature on the topic of modeling simultaneous equations in R.

If you search for “simultaneous equations” inside the Econometrics Task View web page you can find two results: the systemfit and the bimets packages.
As you stated, the systemfit package is a powerful tool for econometric estimation of simultaneous systems of linear and nonlinear equations, but it only provides fitting procedures, thus it cannot be used in your example in order to work out a forecast.
On the other hand, the bimets package implements, among others, simulation and forecasting procedures, and multiplier analysis; therefore you can easily calculate elasticities.
Take a look also at the Rstudio blog post about SEM in R:
https://rviews.rstudio.com/2021/01/22/sem-time-series-modeling/
An example of a simple Klein model in R follows:
#load library
library(bimets)

#define the Klein model
kleinModelDef <- "
MODEL

COMMENT> Modified Klein Model 1 of the U.S. Economy with PDL,
COMMENT> autocorrelation on errors, restrictions and conditional equation evaluations

COMMENT> Consumption with autocorrelation on errors
BEHAVIORAL> cn
TSRANGE 1923 1 1940 1
EQ> cn =  a1 + a2*p + a3*TSLAG(p,1) + a4*(wp+wg)
COEFF> a1 a2 a3 a4
ERROR> AUTO(2)

COMMENT> Investment with restrictions
BEHAVIORAL> i
TSRANGE 1923 1 1940 1
EQ> i = b1 + b2*p + b3*TSLAG(p,1) + b4*TSLAG(k,1)
COEFF> b1 b2 b3 b4
RESTRICT> b2 + b3 = 1

COMMENT> Demand for Labor with PDL
BEHAVIORAL> wp
TSRANGE 1923 1 1940 1
EQ> wp = c1 + c2*(y+t-wg) + c3*TSLAG(y+t-wg,1) + c4*time
COEFF> c1 c2 c3 c4
PDL> c3 1 2

COMMENT> Gross National Product
IDENTITY> y
EQ> y = cn + i + g - t

COMMENT> Profits
IDENTITY> p
EQ> p = y - (wp+wg)

COMMENT> Capital Stock with IF switches
IDENTITY> k
EQ> k = TSLAG(k,1) + i
IF> i > 0
IDENTITY> k
EQ> k = TSLAG(k,1)
IF> i <= 0

END
"

#load the model
kleinModel <- LOAD_MODEL(modelText = kleinModelDef)

#define data
kleinModelData <- list(
  cn  =TIMESERIES(39.8,41.9,45,49.2,50.6,52.6,55.1,56.2,57.3,57.8,
                  55,50.9,45.6,46.5,48.7,51.3,57.7,58.7,57.5,61.6,65,69.7,
                  START=c(1920,1),FREQ=1),
  g   =TIMESERIES(4.6,6.6,6.1,5.7,6.6,6.5,6.6,7.6,7.9,8.1,9.4,10.7,
                  10.2,9.3,10,10.5,10.3,11,13,14.4,15.4,22.3,
                  START=c(1920,1),FREQ=1),
  i   =TIMESERIES(2.7,-.2,1.9,5.2,3,5.1,5.6,4.2,3,5.1,1,-3.4,-6.2,
                  -5.1,-3,-1.3,2.1,2,-1.9,1.3,3.3,4.9,
                  START=c(1920,1),FREQ=1),
  k   =TIMESERIES(182.8,182.6,184.5,189.7,192.7,197.8,203.4,207.6,
                  210.6,215.7,216.7,213.3,207.1,202,199,197.7,199.8,
                  201.8,199.9,201.2,204.5,209.4,
                  START=c(1920,1),FREQ=1),
  p   =TIMESERIES(12.7,12.4,16.9,18.4,19.4,20.1,19.6,19.8,21.1,21.7,
                  15.6,11.4,7,11.2,12.3,14,17.6,17.3,15.3,19,21.1,23.5,
                  START=c(1920,1),FREQ=1),
  wp  =TIMESERIES(28.8,25.5,29.3,34.1,33.9,35.4,37.4,37.9,39.2,41.3,
                  37.9,34.5,29,28.5,30.6,33.2,36.8,41,38.2,41.6,45,53.3,
                  START=c(1920,1),FREQ=1),
  y   =TIMESERIES(43.7,40.6,49.1,55.4,56.4,58.7,60.3,61.3,64,67,57.7,
                  50.7,41.3,45.3,48.9,53.3,61.8,65,61.2,68.4,74.1,85.3,
                  START=c(1920,1),FREQ=1),
  t   =TIMESERIES(3.4,7.7,3.9,4.7,3.8,5.5,7,6.7,4.2,4,7.7,7.5,8.3,5.4,
                  6.8,7.2,8.3,6.7,7.4,8.9,9.6,11.6,
                  START=c(1920,1),FREQ=1),
  time=TIMESERIES(NA,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,
                  1,2,3,4,5,6,7,8,9,10,
                  START=c(1920,1),FREQ=1),
  wg  =TIMESERIES(2.2,2.7,2.9,2.9,3.1,3.2,3.3,3.6,3.7,4,4.2,4.8,
                  5.3,5.6,6,6.1,7.4,6.7,7.7,7.8,8,8.5,
                  START=c(1920,1),FREQ=1)
);

#load time series into the model object
kleinModel <- LOAD_MODEL_DATA(kleinModel,kleinModelData)
## Load model data "kleinModelData" into model "kleinModelDef"...
## ...LOAD MODEL DATA OK

#estimate the model
kleinModel <- ESTIMATE(kleinModel)

#In order to forecast the model up to 1944,
#we need to extend exogenous variables up to 1944
kleinModel$modelData <- within(kleinModel$modelData,{
    wg    = TSEXTEND(wg,  UPTO=c(1944,1),EXTMODE='CONSTANT')
    t     = TSEXTEND(t,   UPTO=c(1944,1),EXTMODE='LINEAR')
    g     = TSEXTEND(g,   UPTO=c(1944,1),EXTMODE='CONSTANT')
    k     = TSEXTEND(k,   UPTO=c(1944,1),EXTMODE='LINEAR')
    time  = TSEXTEND(time,UPTO=c(1944,1),EXTMODE='LINEAR')
  })

#forecast model
kleinModel <- SIMULATE(kleinModel
                      ,simType='FORECAST'
                      ,TSRANGE=c(1941,1,1944,1)
                      ,simConvergence=0.00001
                      ,simIterLimit=100
                      ,quietly=TRUE
  )

#get forecasted GNP
TABIT(kleinModel$simulation$y)
##
##       DATE, PER, kleinModel$simulation$y
##
##       1941, 1  ,  125.3
##       1942, 1  ,  172.5
##       1943, 1  ,  185.6
##       1944, 1  ,  141.1


Data on Directed Technical Change I'm interested in technologies that are directed towards improving productivity of workers with high human capital.

What kind of data could one use to look at directed technical change, as in Howard-Howitt-Aghion, or Acemoglu (2002)?

My understanding is that the literature has so far used three kinds of empirical analysis of the directed technical change.

Illustration based on the relationship between the relative supply of skills and the college premium as in Acemoglu (2002).
Quantitative Analysis to shed light on the theory as in Acemoglu, Gancia, and Zilibotti (2015). They match key moments and simulate their model.
Estimate elasticities of substitution between more and less educated workers. Behar (2009) use four types of data to estimate those elasticities, namely (i) cross sections, (ii) a long range panel of 40 observations,(iii) time series for Brazil and (iv) an annual panel for Latin America.


Local non-satiation in economics I am having trouble completely understanding the mathematical definition of non-satiation. I have stated the definition from Wikipedia below. It would be great if someone can graphically explain.

  Formally, if $X$ is the consumption set, then for any $x \in X$ and every $\varepsilon > 0$, there exists $y \in X$ such that $||y-x|| \leq \varepsilon$ and $y$ is preferred to $x$.

The Wikipedia article shows it graphically.  But here, you're leaving out the important bit:  "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.

So how does this relate? Let's go bit-by-bit. Refer to the graph on the Wikipedia page as needed.

-"for any bundle of goods" means "choose any bundle $x \in X$, where X is the box in the Wikipedia graph.

-"arbitrarily close" means "choose any $\epsilon > 0, \epsilon \in \mathbb{R}$".  So, pick any non-zero real number as small or as large as you like.

-"there is always another bundle of goods arbitrarily close" means "given $x$ and $\epsilon$, you can always find a bundle $y \in X$ that is within a distance $\epsilon$ from $x%$." This is expressed as $||y-x|| \le \epsilon$.

-"is preferred to it" means that for any utility function $U(x)$, we have $U(y) > U(x)$.

Putting this all together, "non-satiation" means that (without a budget constraint) an agent can "never be satisfied" because no matter what bundle of goods they have ($x$) there is always a $y$ that yields greater utility. "Local" means that $y$ and $x$ can be very similar (ie arbitrarily close to each other in $X$).


Derive demand function from utility Never encountered such a problem as I am new.
$$U(x_1,x_2)=(a\ln(x_1)+b\ln(x_2))^n$$
and  $a,b,n>0$ with income $w>0$ and prices $p_1,p_2>0$.

Find the demand function.

Attempt
I am thinking that I am able to use that $h(u(x))$ for a strictly increasing function represents the same præferences. I.e I can just work on $a\ln(x_1)+b\ln(x_2)$. I am also thinking about using the $MRS$ but not sure on how to continue.

Firstly find the marginal utility of $x_1$ and $x_2$. Hint: Use the chain rule.
$\frac{\partial U}{\partial x_1} = \frac{na(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$
$\frac{\partial U}{\partial x_1} = \frac{nb(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$
The consumer maximises their utility when $\frac{MU_{x_1}}{P_{x_1}}= \frac{MU_{x_2}}{P_{x_2}}$.
From that we simply need to solve for $x_1$ and $x_2$.
Plug each expression into the budget constraint which in this case is $w =P_{x_1}x_1 + P_{x_2}x_2$. Then solve for $x_1$ and $x_2$, giving each demand function.


Why's it always better to choose a lump sum than periodic payments over time? Gill, Madura. Personal Finance, 4th Canadian Edition 2019. p. 56.

  True or False? It is always better to choose a lump sum than periodic payments over time.

p. 519

  T

Doesn't the answer depend on the amount of lump sum vs. periodic payments and inflation?

Or are the authors assuming that if lump sum = P, then you get n periodic payments of $\frac{P}{n}$ (for a total of $n × \frac{P}{n}$)? Then inflation will cause each P/n to be worth less.

The answer depends on the assumed interest rate of the periodic payments, compared to the expected return of alternative investments.

To be viable as a 'good' choice, the period payments must total to an amount that is equal to or greater than what could be achieved by taking the lump sum and doing something with it. In practice, this means that a period payment must total to a larger amount than the lump sum payment, with the difference increasing as the total amount of time of payments does.

For instance, the very large lotteries in the United States offer a lump sum amount that is somewhere on the order of 60% of the 'advertised' amount, which is what would be earned over 30 years of periodic payments. This figure is calculated assuming a 5% year-over-year increase of the payment amount.

Whether one is better than the other depends on what would be done with the lump sum money in the meantime. If one could invest the money or start a business with an expected rate of return greater than the one used to generate the periodic payment amount, then it is a bad idea to take the periodic payment; conversely, if the rate used in the periodic payment calculation is greater than what one would expect to earn themselves, then it would be a bad idea to take the lump sum.

In the real world, since the periodic payment calculation is usually made with a conservative, guaranteed rate, the lump sum would be better. If the periodic payment calculation is made with an assumed rate of 0% (ie, the total of all payments is equal to the lump sum amount), the lump sum will always be superior, assuming it can be used to earn any money at all. It sounds like the author is implicitly assuming one of these two scenarios.


Bertrand game with delivery risk and side payments Consider three agents $A_i$, who engage in a Bertrand game. All agents have perfect knowledge on all parameters and the distribution $F()$.

$A_1$ moves first and selects price $0\leq p_1\in \mathbb{R}$. $A_2$ moves second and selects price $0\leq p_2\in \mathbb{R}$. $A_3$ moves third and selects $0\leq q_1\in \mathbb{R}$ and $0\leq q_2\in \mathbb{R}$, that is, the quantities that it buys from $A_1$ and $A_2$, which it sells at market clearing price.
$A_1$ tries to maximize $  q_1 p_1$.
$A_2$ tries to maximize $ q_2 p_2$.
$A_3$ tries to maximize $$((A - q_1 - q_2) (q_1+q_2) - q_1 p_1 - q_2 p_2) (1-F(-q_1 p_1))+
( (A - q_2) q_2- q_2 p_2+B) F(- q_1 p_1) $$, with $F$ being a continuous cumulative distribution function. $A_3$'s profit function captures that buying from $A_2$ comes at no risk, while $A_1$ might be unable to deliver. However, the higher $A_1$'s profits the smaller that risk and the higher the chance to capture the benefits/losses $B$.

Note, all agents have perfect knowledge on all parameters and the distribution $F()$.
Does $A_3$'s profit increase in $B$? How to prove this?
(If $F$ is uniform, $A_3$ will only buy from one of the agents and it is easy to show that the answer is yes. If $p_1$ and $p_2$, then it is easy to show that the answer is yes.)
In the general case, which is outlined above, my numerical studies suggest that the answer is yes, but I struggle to write down the analytical proof.

I am not going to give a solution to you question (as I am not able too). However, let me try to indicate the difficulty of obtaining one if you cannot find a closed form solution for $q_1$ and $q_2$ in the last stage of the game.
Let $\Pi(q_1, q_2, p_1, p_2, B)$ be the objective function that we want to maximize. Assuming we can totally differentiate this, we get:
$$
d\Pi = \Pi_{q_1} dq_1 + \Pi_{q_2} dq_2 + \Pi_{p_1} dp_1 + \Pi_{p_2} dp_2 + \Pi_B dB. \tag{1}
$$
Assume the first order conditions hold:
$$
\Pi_{q_1} = 0\\
\Pi_{q_2} = 0 \tag{foc-stage3}
$$
This then simplifies $(1)$ to:
$$
d\Pi = \Pi_{p_1} dp_1 + \Pi_{p_2} dp_2 + \Pi_B dB \tag{1.a}
$$
The problem here is that we can not drop the $dp_1$ and $dp_2$ terms as they are (in general) both going to be a function of $B$.
Assume that the first order conditions $\rm{(foc-stage3)}$ can be solved to give a solution $q_2(p_1, p_2, B)$ and $q_1(p_1, p_2, B)$. Then in stage 2 of the game, Firm 2 maximizes:
$$
p_2 q_2(p_1, p_2, B)
$$
Again, assume that the first order conditions hold:
$$
q_2 + p_2 \frac{\partial q_2}{\partial p_2} = 0 \tag{foc-stage2}
$$
We can totally differentiate this with respect to $p_1$ and $p_2$ and $B$ (notice that $q_2$ is a function of all three these variables). This gives:
$$
\left(2\frac{\partial q_2}{\partial p_2} + p_2 \frac{\partial^2 q_2}{\partial p_2\partial p_2}\right) dp_2 = -\left(\frac{\partial q_2}{\partial p_1} + p_2 \frac{\partial^2 q_2}{\partial p_2 \partial p_1}\right) dp_1 - \left(\frac{\partial q_2}{\partial B} + p_2 \frac{\partial^2 q_2}{\partial p_2 \partial B}\right) dB \tag{2}
$$
Let $p_2(p_1, B)$ be the optimal solution of firm 2 (by solving $({\rm foc-stage2})$. Then in stage 1, Firm 1 maximizes
$$
p_1 q_1(p_1, p_2(p_1, B), B)
$$
So:
$$
q_1 + p_1 \frac{\partial q_1}{\partial p_1} + p_1 \frac{\partial q_1}{\partial p_2} \frac{\partial p_2}{\partial p_1}= 0 \tag{foc-stage1}
$$
Assuming again that this can be totally differentiated with respect to $p_1$ and $B$, we obtain:
$$
\begin{align*}
\left(\frac{\partial q_1}{\partial p_1} + \frac{\partial q_1}{\partial p_1} + p_1 \frac{\partial^2 q_1}{\partial p_1 \partial p_1} + \frac{\partial^2 q_1}{\partial p_2\partial p_1}\frac{\partial p_2}{\partial p_1} + \frac{\partial q_1}{\partial p_2}\frac{\partial^2 p_2}{\partial p_1\partial p_1}\right) dp_1\\
=- \left(\frac{\partial q_1}{\partial B} + p_1 \frac{\partial^2 q}{\partial p_1 \partial B} + \frac{\partial^2 q_1}{\partial p_2 \partial B} \frac{\partial p_2}{\partial p_1} + \frac{\partial q_1}{\partial p_2} \frac{\partial^2 p_2}{\partial p_1 \partial B}\right) dB \tag{3}
\end{align*}
$$
Then we can use $(3)$ to substitute for $d p_1$ in $(2)$. Then $(2)$ and $(3)$ give $dp_1$ and $dp_2$ in terms of only  $dB$. This can be substituted into $(1.a)$ to give the total effect of $B$ on $\Pi$.


What really is inflation? I understand that you measure inflation by first measuring what the average person buys. Then you compare the total cost of these goods in year x with year y to get the inflation (from x to y).

The problem: Not only the cost of goods, but also the quantity that people buy changes all the time. Things go out of fashion or are replaced by different (better?) things. When you calculate inflation, you implicitly assume that the amount of goods people buy is constant (during the measurement period, say one year).

I do not doubt that this is reasonable from the practical perspective. But I doubt that today's inflation measurement approximates a well-defined concept.

Assume you know for every day/month/year how much people buy from each product and what it costs, what would be the formula for inflation? How do you incorporate the fact that people might by apples at 1\$ in May and oranges at 2\$ in June, i.e. that costs and quantities change all the time?

This is a really big topic and you're essentially questioning why don't we calculate price levels using Paasche's indexes instead of Laspeyre's. Well, one "simple" answer is that's it's very expensive to maintain a detailed account of what a representative consumer basket should be.

There's people trying alternatives to this, like the Billion Prices Project, but their estimated inflation is very close to the one we estimate through traditional, constant consumer basket method.


What aspects of macroeconomics should I address for convincing an investor to invest in a certain country? I have a task at my internship which is to write 2 pages about the economic conditions in my country. My question is what I could include in my research. Surely, I'll address inflation, currency fluctuations since they are very unstabe. What else can I add? Where can I get reports on a certain country, in this case is Egypt.

As Brian said, international organisations are a good place to start. For example, the World Bank publishes periodic economic outlook updates on each country. For Egypt, see here. There is a pdf link to the right of the page. They covers aspects like GDP growth, inflation, financial and fiscal accounts, and also some prospects on ongoing, pro-business reforms.

Another source is the African Economic Outlook reports. Here is the one for Egypt. This is a bit more broad, including analysis of debt, revenues and spending, monetary policy, governance, natural resources management, and the social sphere (poverty, social protection, gender equality). See also the website of the IMF dedicated to Egypt.

Finally, having worked in the financial sector for a few years, I know that most of the investment banks produce periodic reports on every country their clients might have investment interests, reports that usually include a lot of forecasts. Albeit most of these reports are not available to the public, some googling still yield some results. For example, here is a very comprehensive analysis for Egypt from Deutsche Bank, published two months ago.  Other asset management companies also produce such reports (e.g here).


Has the world become poorer? This question is not backed by serious economic research but mainly by a very simple attempt to look at how the world economy has changed.

I am asking whether and why the world has, over the past 50 or so years, become less productive (not used in the strict economic sense here) and therefore people have less money to do stuff/buy things.

Several reasons why I believe this to be the case (I am open to be corrected):

Real wages in developed economies have not risen for a long time
Financial crisis now been going on for many years (almost 10)
Unemployment in the EU is very high
Tax rates are ever-increasing (it seems) relative to median wages
More and more people are reliant on welfare (from what I gather, unfortunately I don't have a source at this point
Growth in developing countries (China) has decreased
National debts are ever-increasing

This has left me with the feeling that in many situations, there is simply 'less' money to go around and be spent. I can observe this, for example, by the seemingly constant need of companies and countries (even where I live, in wealthy Switzerland) to save money and reduce expenses (I don't know if it only seems this way because of some biases or it is actually the case).

Again, not sure if this is actually happening or if it only seems so to me (Also apologies for not being able to express this in very 'economy' terms).

Several reasons that I could explain to myself why it would seem like we have less to spend and are less productive than we used to be:

Rising inequality (the rich keep getting more so the majority has less)
Less incentive for technological progress (apart from in IT). In many industries, there is no reason to keep making everything better, only to make it cheaper and more profitable. We have most of what we need and don't see a point in improving it (cupboards, beds, kitchen utensils etc.)
Lethargy of people. Many people are happy with what they have and do not want to work more/harder
We have most of what we need, and the things that are left to buy are luxury items (which we maybe didn't think about before)

Again sorry for the very vague way of asking this question without any evidence, but if someone could help me make it more concrete I would be very open to that.

I guess what I want to say is do people have less money and less to spend than they used to, and if so, why?

The short answer is No.

Every single year, except 2009, for the past 55 years of continuously recorded economic history, the world has been getting richer. The -2.1% global recession in 2009 was made up in 2010 with 4.1% growth. I was just working with the World Bank's World Development Indicators, which track global GDP growth, and I double checked.

We measure prosperity and growth by the quantities of goods produced. Money is irrelevant. Every year we, humans, produce more and more goods and services and that makes us richer. Not every country grows steadily though. Some countries are stagnant, like Greece, and some grow remarkably rapidly over very long periods of time, like China. Moreover, there is increasing inequality withing many countries (measured by GINI coefficient). So it is very natural for many to feel like everything around them either stands still or degrades. But on average, the world keeps getting richer.

If you are in Switzerland, you may have felt the shock from recent currency appreciation. Sharp currency appreciations hammer exporters and industries reliant on tourism, which make up a large portion of Swiss economy.


What is the effect of expansionary fiscal policy in case liquidity trap situation? I've read that liquidity trap means interest rate is at its minimum and increase in real money stock will not lead to fall in interest rate because people will be demanding whatever the amount is being supplied at that minimum interest rate. Now in books they say that in this situation when government expenditure rises leading to a rise in income and hence a rise in money demand would not lead to a change in interest rate.

But according to me when the demand for money goes up it shifts the money demand curve to the right and above. Given the money supply and fixed price, this should lead to a rise in interest rate.
Same confusion arises when we reduce the real money supply under the liquidity trap situation. Book says that interest rate will not go down when money supply goes up but it says nothing about money supply going down.

If I've understood correctly (it's not entirely clear from the question), you're asking about a policy of fiscal expansion in a depressionary environment, accompanied by a monetary expansion.

It is conceivable that an increase in money supply, coupled with an increase in money demand, would lead to unchanged interest rates, if those two opposing forces balanced out.


Wealth Distribution of the Super-Rich I am looking for a mathematical model which can accurately forecast what percentile ($1-p$) of a population a member with wealth $w$ would be in, based on Gini coefficient ($G$). I am planning to apply it mainly only to the top end of wealth.

I have tried to use the Pareto distribution so far. What I am unsure of there, however, is what to do with the parameter $x_{min}$ as described in Wikipedia's article (https://en.wikipedia.org/wiki/Pareto_distribution):

p = $(x_{min}/w)^P$

where $P$, the Pareto parameter, can be expressed in terms of $G$ by a known equation.

Wikipedia describes $x_{min}$ as the (necessarily positive) minimum possible value of $x$. How does one apply this to a real-world population? I can't imagine setting the value to $1 or less would allow for accurate predictions of top-end wealth.

The other problem is, this $x_{min}$ seems to me to behave like an average rather than a minimum value. For example, in a maximally unequal society, $P=1$ so $p = x_{min}/w$. This predicts that 10% of the population have wealth ten times $x_{min}$, 1% have wealth a hundred times $x_{min}$, and so forth. But while this is a pretty unequal society by real-world standards, it is definitely not the most unequal possible - imagine a system where 10% had a wealth a hundred times the "minimum" (or, say, the average of the remaining 90%), 1% had a wealth $100^2$ times the minimum, and so forth. The most unequal system would really be everyone having "minimum" wealth except one individual. But the inequality here seems to be capped much before that.

Can anyone help with my model?

Some thoughts:

accurate prediction seems ambitious without empirical data;
many people have negative or zero net wealth;
a continuous distribution could lead to oddities;
the world may have a finite total wealth and so a finite average wealth thus excluding some distribution shapes such as Pareto distribution with $\alpha \le 1$

But you could certainly develop a model, even if it would be wrong.  For example, let's try:

A Pareto distribution with shape parameter $\alpha$ and scaling parameter $x_{\min}$
The application of the Pareto 80-20 principle, so $\alpha=\log_4(5) \approx 1.160964$
A total global wealth of perhaps of the order of $\$ 250$ trillion and a global population of about $7.4$ billion, giving an average wealth $\mu$ of about $\$33784$ and $x_{\min}=\mu\frac{\alpha-1}{\alpha}$ would be about $\$4684$ (looks rather high to me, but $x_{\min}$ is really a scaling factor and should not be taken too seriously)
A Gini coefficient of $\frac{1}{2\alpha-1}$ would be about $0.76$
The top $1\%$ (so $74$ million people) would have over $x_{\min} / 0.01^{1/\alpha}$ each, so here about $\$250,000$
a person with wealth $w$ would be at the $\left(\frac{x_{\min}}{w}\right)^{\alpha}$ quantile measured from the top

Since this probably does not fit reality, you can then start adjusting it if you want: for example you can get closer to a Gini coefficient of $1$ if you reduce $\alpha$ towards $1$ from above, and this would also lower $x_{\min}$.   If you use a Gini coefficient of $G$, that would make $\alpha=\frac{G+1}{2G}$ and $x_{\min}=\mu\frac{1-G}{1+G}$ so the person with wealth $w$ could be at the $\left(\frac{\mu(1-G)}{w(1+G)}\right)^{(G+1)/(2G)}$ quantile, measured from the top.

Or you could decide to give a certain share of the population a wealth of zero.  If the rest keep the same $\alpha$ on a (now zero-inflated) Pareto distribution this will increase $x_{\min}$ for the rest while also increasing the Gini coefficient overall.

Or you could look for empirical information on the actual wealth distribution and then try to fit a curve to that.  Or perhaps look at global income distribution instead, which has not had a particularly simple function in the past.


Real wage stagnation and union membership In the United States, wages in real terms have seen very little change over approximately five decades:

(source: http://www.pewresearch.org/fact-tank/2014/10/09/for-most-workers-real-wages-have-barely-budged-for-decades/ )

The reasons for this are many and varied.

Over approximately the same time period, union membership has fallen dramatically. BLS data since 1983 shows the following chart

(source: data roundup via https://www.bls.gov/news.release/union2.nr0.htm with the same chart in a more accessible article at https://www.bloomberg.com/news/articles/2018-01-19/union-membership-rate-in-u-s-held-at-record-low-of-10-7-in-17 ).

The reasons for this too are many and varied.

I think the causal relationship between the two is weak, but I'm wondering if there has been any research on whether or not there is a relationship, and to what degree.

Repeated, in question form: What is the relationship, if any, between declining union membership and real wage stagnation? Thank you.

Edit: UK data.

The UK has a report on union membership:  https://www.gov.uk/government/statistics/trade-union-statistics-2016 . One table lists "Trade union membership, UK, 1892 to 2015-16" in absolute terms; the series counting percent of employees only starts in 1989. But it shows that absolute membership peaked in 1979 and has been declining since.

In contrast with the United States, ever since the (absolute) union membership peak in 1979, real wages have increased in the UK (actually shown is since 1980), although they did take a dip after 2008.

(source: "Real Wages and Living Standards in the UK" )

My understanding is that theories and empirical methodologies demonstrating the relationship between declining union membership and wages risk overestimating the causality of the relationship.

Anyone who argues that businesses do NOT, in a relevant share of cases, endeavour to suppress union formation, activities or recruitment, whether individually or collectively, is either ignorant or a political shill. (Things are not nice 100% of the time in the other direction either.) This should be recognized, in order to proceed with rational discussion of the economics of the issue.

However, factors like increased international competition in lower-skilled jobs from a very large international labour pool (which is much more accessible to producers than in 1950 or 1980) can be one reason that the causality of declining union membership can be overestimated.

Also, mechanisms of tacit collusion between businesses can act to hold down wages as well, for example with various agreements which prevent an employee from working for a competitor for some number of years after they leave (e.g., Google and Apple could both force employees to sign such an agreement as a condition of employment, thus holding down wages for both firms, which both account for a substantial share of employment in their specific areas of activity). This could be another explanation for some portion of wage stagnation that would be difficult to frame as resulting from declining union membership per se.

Between 1) international competition and 2) a proliferation of methods to suppress wages unrelated to union formation or membership per se, it should suffice to say that significant effort for robustness must be demonstrated before any empirical work can be taken seriously when discussing causality between declining union membership and 50 years of stagnant average wages.

The risk, then, is that an inadequate methodology which involves a) under-specification of a model, and/or which b) includes inadequate robustness checks, could indicate that declining union membership caused stagnant wages, when theoretically this could be a false positive when other factors explain both the declining union membership and also stagnant average wages.


Public Goods - Voluntary provision I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell (2006), page 452-453. e have a two-commodity world, in which there are
${n_h}$ agents (households): commodity 1 is a pure public good and commodity 2 is
purely private.

Each agent has an exogenously given income ${y^h}$ , denominated in units of the
private good 2. We imagine that the public good is to be financed voluntarily,
each household makes a contribution ${z^h}$ which leaves

$$x_2^h = {y^h} - {z^h}$$ of the private good available for h’s own consumption
and
$${x_1} = \phi (\bar z + {y^h} - x_2^h)$$    where z is the total input of the good 2 used in the production process. Each agent realises that the total output of the public good depends upon his
or her own contribution and upon that made by others

The expression (13.30)

The expression (13.31)

Did he come to expression (13.33) by applying the chain rule as

$$\eqalign{
  & z = U({x_1},x_2^h)  \cr
  & {x_1} = g({x_2})  \cr
  & {{dz} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}}{{d{x_2}} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}} \cr} $$

since ${x_1}$ is a function of $x_2^h$ ?

Given

$$\max_{x_2^h} \quad U^h(\phi(\bar z + y^h - x_2^h), x_2^h)$$

You have to find FOCs by taking the chain rule twice for the first argument. Denote the argument for function $\phi$ as $(\cdot)$.

$$\frac{\partial U^h}{\partial x_2^h} = \frac{\partial U^h}{\partial \phi} \cdot \frac{\partial \phi}{\partial x_2^h}$$
$$\frac{\partial \phi}{\partial x_2^h} = \frac{\partial \phi}{\partial (\cdot)} \cdot \frac{\partial (\cdot)}{\partial x_2^h}$$
$$\implies \frac{\partial U^h}{\partial x_2^h} = \frac{\partial U^h}{\partial \phi} \cdot \frac{\partial \phi}{\partial (\cdot)} \cdot \frac{\partial (\cdot)}{\partial x_2^h}$$

Thus, letting $U^h_{1,2}$ represent the partial derivative with respect to the first and second arguments of $U^h$ respectively:

$$\begin{align}
\frac{\partial U^h}{\partial x_2^h} & = U_1^h(\phi(\bar z + y^h - x_2^h), x_2^h) \cdot \phi_z(\bar z + y^h - x_2^h) \cdot -1 \\
& + U_2^h(\phi(\bar z + y^h - x_2^h), x_2^h) \cdot 1
\end{align}$$

The second argument in $U$ has no $\phi$, so only chain rule should be applied there.


Sequential Price Competition for Perfect Complements There are two goods, $1$ and $2$ produced by two firms at zero marginal costs. The goods are perfect complements. The demand for each goods is: $Q_1=Q_2=a-(p_1+p_2)$. The prices are set sequentially, with the leader firm ($1$) credibly committing to a price which the follower takes as given. Determine the equilibrium prices, quantities and profits.

I understand that in standard Stackelberg games of prices, the follower has a second mover advantage. For the above question, I found that the reaction curves turned out to be downward sloping, i.e. prices are strategic substitutes. $(p^*_1,p^*_2)=(\frac{a}{2},\frac{a}{4})$.
Would the second mover advantage still hold? Evaluating the leader's profit function at Bertrand prices would show that the leader has an incentive to set a price above $p^B_1$ since prices are strategic substitutes and goods are perfect complements (evaluating $\frac{\partial\pi_1^L}{\partial p_1}$ at bertrand prices).
Is this the correct method to evaluate second mover/first mover advantage?

Just looking at the setup, everything is symmetric. This means that if they were to move simultaneously then we should have $p_1 = p_2$, and the payoffs for the two firms the same.

According to your calculation, $(p^*_1,p^*_2)=(\frac{a}{2},\frac{a}{4})$. Since $Q_1 = Q_2 \Rightarrow \pi_1 > \pi_2$. So it's quite clear that in this case, the first mover has the advantage.

Intuitively, you can figure out which mover has the advantage by simply asking, would the first mover get higher payoff if he/she moves second?

Mathematically, if strategies are strategic substitute, then you can show that the first mover will have the advantage. Reference here.

Added later: one interesting way (I think) to look at this is this is just like Cournot competition, just with price and quantity switched.


Burden of debt; Internal or external financing? Q1) How does the burden of an external public debt differ from that of an internal public debt?
Q2) Are there sny conditions under which neither external  nor internal debt is a burden to a nation?

Q1 : If you define external debt as "forex debt" context.
If the debt repayment amount for every period are greater than a country import/export amount, then it will influence the exchange rate.  Subsequently affect goods price.

For internal debt (e.g. money borrow within the country, using various financial tools or creative derivative ).
On positive side, if the debt spending bring in long term production or/and cost saving benefit, it will be reduce the debt burden, increase income. As it suppose to get the debt payoff from the positive benefit.

On negative side, internal debt can be wasted, or lead to speculative activities. In such case, the problem can be many folds, depends on the magnitude of of the debt. For "smaller scale", creditor or so call "investor" lost their money. E.g. Enron, Worldcomm scandal.
For the worst snowball case, the effect is much dire and difficult to assess. E.g. the worst private sector debt, you can check out example of subprime mortgage crisis. For country level, you can check out the Greek government-debt crisis.

Q2 : Under perfect world production context (no production of unwanted goods, goods/finance speculation/corruption), as long as the production income are more than the debt, there will be no ill effect from borrowing.

Note : In each economic question, context is always required, no matter how you apply famous/infamous economist that apply math formula. Because human society keep adapting and exploiting the system.


Market baskets: cross-border comparisons The concept of a market or consumer basket is quite intuitive. I assume a market basket for a given country tells for which goods an average person spends how much of her money.

To be able to compare market baskets between different countries (and cultures) their (groups of) items must be chosen quite generically (such that they make sense everywhere in the world). The official German market basket as found here includes the following groups of items:

  01 Food and non-alcoholic beverages

  02 Tabacco and alcoholic beverages

  03 Clothing and shoes

  04 Housing and energy

  05 Furniture

  06 Health

  07 Transportation

  08 Telecommunication

  09 Leisure, culture

  10 Education

  11 Hotel, restaurants

  12 Other

I wonder if this market basket makes sense everywhere in the world, or if there are significantly different ones around. Maybe there is an international standard? If so: Where can I find a good overview and comparison of this market basket all over the world?

It's very likely that that these major categories in the hierarchy of goods in the market basket differ somewhat by country.  The sub-categories within these major groups are even more likely to differ across countries, as the market basket is supposed to reflect the purchases of a typical consumer, and typical purchases are obtained from consumption surveys of the population.

In the U.S., the eight major categories used for calculating the rate of inflation (Consumer Price Index) are:

Food and Beverages
Housing (includes furnishings)
Apparel
Transportation
Medical
Recreation
Education and Communication
Other Goods and Services

In Canada, the major groups are:

Food and non-alcoholic beverages
Shelter
Household operations, furnishings and equipment
Clothing and footwear
Transportation
Health and personal care
Recreation, education and reading
Alcoholic beverages, tobacco products and recreational cannabis

Sources:

U.S. CPI Handbook of Methods (https://www.bls.gov/opub/hom/pdf/cpihom.pdf)
The Canadian Consumer Price Index Reference Paper (https://www150.statcan.gc.ca/n1/en/pub/62-553-x/62-553-x2019001-eng.pdf)


How do you determine the strict core in a matching game? So I have a game in which home owners prefer certain other's homes.  The owners are A, B, C, and D with a, b, c, and d referring to their respective homes.  They each prefer homes according to the ranking

A: bdac
B: cadb
C: dbca
D: acbd

I've read an article by Shapley about an algorithm to find one assignment in the strict core, but I'm asked to "determine the strict core" which I assume means finding all assignments in the strict core.  How does one do that?

In case it's of any use, I'll describe the algorithm in the Shapley article.  Make a graph of people's top preferences and select "top cycles", then delete the people who were in top cycles and do it again.  Repeat the process until everyone has occurred in a top cycle and every cycle represents a permutation of people and houses that will be in the strict core.

When preferences are strict, the strong core is unique and the top-trading cycle procedure (TTCP) is the unique mechanism yielding the strong core allocation. Here are the highlights.

Lemma 1: Let $x$ be the allocation generated by the Top-Trading Cycle Procedure. Then $x$ is in the strong core.

Lemma 2: Let $P$ be the set of players, $\Omega = \{ \omega^{i} = i : i \in P \}$ be the set of initial allocations, and $\succ^{i}$ be the set of strict preferences. We have $(P, \Omega, (\succ^{i})_{i=1}^{|P|})$ as the instance of the house allocation problem. The strong core is unique.

Proof: Suppose to the contrary that there exist two allocations $x, y$ in the strong core and suppose $x$ is the allocation returned by the TTCP. As $x \neq y$, $S = \{ i : x_{i} \neq y_{i} \}$ has at least two elements. Observe that if there is a player $i \in S$ such that $x_{i} \succ^{i} y_{i}$, then $x_{i}$ can form a weakly improving coalition with the other players in $x_{i}$'s cycle whose allocations are the same in $x$ and $y$. As preferences are strict, every player in $S$ strictly prefers either $x$ or $y$. We construct a weakly improving coalition to block $y$.

Let $i \in S$ such that $y_{i} \succ^{i} x_{i}$. Let $i$ be matched in iteration $k$ of the TTCP. Since $y_{i} \succ^{i} x_{i}$, we have that player $i$ prefers some other house $j$ to $x_{i}$ and that $j$ was assigned in a previous iteration by the fact that $x$ belongs to the strong core. And so using the allocation $y$ displaces the player who received house $j$ under the TTCP. As preferences are strict, each player who received a house when $j$ was allocated under the TTCP can form a weakly improving coalition to block $y$. And so the fact that $S \neq \emptyset$ implies that $y$ is not in the strong core. QED.

I actually have a blog entry going into more detail about this: https://michaellevet.wordpress.com/2015/06/01/algorithmic-game-theory-house-allocation-problem/


Real Positive Eigenvalue, but Stable Dynamics UPDATE
I was not thinking straight anymore and got totally confused after working hours on my equations. The point is, I have an unstable system, but I force it on the stable path. After realizing that crucial point everything made perfect sense.

Problem

(I fixed some major errors)

I'm having a two dimensional system with dynamics
\begin{align}
\dot{k} &= \frac{1}{1+\frac{1}{\rho}} - \frac{1}{1+\lambda}\\
\dot{\lambda} &= \rho\lambda - \frac{1}{k}.
\end{align}
Where $k\in[0,2]$ is the state, and $\lambda$ the costate. There is a (symmetric) fixed point $E(\tilde{k},\tilde{\lambda})$ at $\tilde{\lambda} = \frac{1}{k\rho}$ which yields $\tilde{k}=1$. The Jacobian at the fixed point is given by
\begin{align}
J_E = \begin{bmatrix}0,& \frac{1}{\left(\frac{1}{\rho}+1\right)^2}\\
1,& \rho\end{bmatrix}_E
\end{align}
I get two eigenvalues which are opposite in sign
\begin{align}
(\mu_1,\mu_2)=\left(\frac{p(p + \sqrt{p^2 + 2p + 5}+ 1}{2(p + 1)},\frac{p(p - \sqrt{p^2 + 2p + 5}+ 1}{2(p + 1)}\right)
\end{align}
where $\rho\in\mathbb{R}_{++}$ (time preference rate). The first eigenvalue is always positive ($\mu_1>0$) and the second one is always negative ($\mu_2<0$). So it is an unstable saddle,right? Nonetheless the system is stable. How is that possible, cause I used to think that it must be unstable?  The image shows the evolution of the state and the control which is a function of the costate. ($\rho = 0.05$)

For reference I add figures with $\rho=2,5$.
The first one seems to converge to a different fixed point $k<1$ (I didn't solve for that one, cause I deal with a symmetric situation; I think there are three in total).

And the second picture shows a strange attractor? Which I actually quite like cause of its chaotic stability. For $k\in[0,3]$ the policy function $\tau_1(k)$ is quite weird.

If you have are trying to discretize the continuous time model
$$
\dot{\textbf{x}} = A\textbf{x},
$$
then in discrete time you will have
$$
\textbf{x}_{t+1} = B \textbf{x}_t
$$
but $A\neq B$, since $A$ describes the change in $\textbf{x}$ while $B$ describes the next value of $\textbf{x}$, not just the change. However
$$
\Delta\textbf{x}_t = \textbf{x}_{t+1} - \textbf{x}_t = B \textbf{x}_t - \textbf{x}_t = (B-I) \textbf{x}_t
$$
This new matrix $B-I$ would correspond to $A$.

(From this you can also see why the stability criterion is different.)

Given your equations
\begin{eqnarray*}
\dot{k} & = &  \frac{1}{1+\frac{1}{\rho}} - \frac{1}{1+\lambda} \\
\\
\dot{\lambda} & = &  \rho \lambda - \frac{1}{k}
\end{eqnarray*}
the equations for the discrete modell would be
\begin{eqnarray*}
k_{t+1} & = &  k_t + \frac{1}{1+\frac{1}{\rho}} - \frac{1}{1+\lambda_t} \\
\\
\lambda_{t+1} & = &  \lambda_t + \rho \lambda_t - \frac{1}{k_t}.
\end{eqnarray*}
You will have to calculate the Jacobian of this system to determine stability.