Uncovered Interest Parity (UIP) condition approximation Until otherwise $e=\frac{\text{Domestic currency}}{\text{Foreign currency}}$.

The UIP condition can be written in the following form: Investing now in a domestic bond, must give the same return as investing in a foreign bond, which means, $1+i_t=e^{-1}_t\left(1+i^*_t\right)e^E_{t+1}$, implying

$$\frac{1+i_t}{1+i^*_t}=\frac{e^E_{t+1}}{e_{t}}=\frac{1}{1+\frac{e_{t}-e^E_{t+1}}{e^E_{t+1}}}$$ where $i_t$ is the domestic nominal interest rate at time t, $i^*_t$ is the foreign nominal interest rate at time t, $e^E_{t+1}$ is the expected exchange rate at time $t+1$.

Using the derivations presented here I get:
$$i_t-i^*_t=-\frac{e_{t}-e^E_{t+1}}{e^E_{t+1}}$$

However, if we define the $e=\frac{\text{Foreign currency}}{\text{Domestic currency}}$, we get $1+i_t=e_t\left(1+i^*_t\right)\left(e^E_{t+1}\right)^{-1}$, and using an analogous reasoning to the above:
$$i_t-i^*_t=-\frac{e^E_{t+1}-e_{t}}{e_{t}}$$

However, this wiki link gives(notice the missing minus sign):

$$i_t-i^*_t=\frac{e^E_{t+1}-e_{t}}{e_{t}}$$

It's not the first time I see this last(wiki) approximation being used.

What's the reasoning for this last approximation? are my approximations wrong? I would really like to know the reason for the minus signs to be missing.

Any help would be appreciated.

From the basic equation you get
\begin{eqnarray*}
\frac{e^E_{t+1}}{e_{t}} & = & \frac{1+i_t}{1+i^*_t} \\
\\
(1+i^*_t) \cdot e^E_{t+1} & = & (1+i_t) \cdot e_t \\
\\
e^E_{t+1} - e_t & = & i_t \cdot e_t - i^*_t \cdot e^E_{t+1}.
\end{eqnarray*}
Now comes the approximation. You divide either by $e_t$ or by $e_{t+1}^E$.

First method
$$
\frac{e^E_{t+1} - e_t}{e_t} = i_t - i^*_t \cdot \frac{e^E_{t+1}}{e_t} \approx
i_t - i^*_t.
$$
Second method
$$
\frac{e^E_{t+1} - e_t}{e^E_{t+1}} = i_t \cdot \frac{e_t}{e^E_{t+1}} - i^*_t  \approx
i_t - i^*_t.
$$
I would say there is no crucial difference, both of these just assume that the proportional change in the exchange rate multiplied by the interest rate is an order of magnitude lower than either the interest rate or the change in the exchange rate alone.


What is an economic crisis? I'm not sure if this is a silly question or overly semantic but I think the term "Crisis" is misunderstood (or at least misunderstood by myself). There are a number of ways the term is used, namely:

A Financial Crisis.
Sovereign Debt Crisis.
World Health Crisis.

These are all different scenarios but all have the word "crisis" attached to them.

Is there a unifying definition of what constitutes a economic crisis?

I don’t think crisis itself has set definition as opposed to recession, unless you attach some clarifier to it as the examples you mention. Generally the word is even in academic work thrown around quite loosely.

For example in Romer advanced macroeconomics in the subject index for word crisis you will find:

  crisis; see debt crisis; economic crisis of 2008-; recessions

Hence the word crisis is at least in macroeconomics used very loosely. However, when it comes to for example debt crisis or financial crisis you get more precise definition.

For example consider these definitions of the terms you mentioned (excluding health crisis since for that I think we have to go outside economics and also you tagged it with macro so I will focus on that) by leading authors:

Financial Crisis: according to the Mishkin and Eakins popular textbook financial markets and institutions a financial crisis is defined as “a major disruption in financial markets, characterized by sharp declines in asset prices and failures of many financial and non financial firms”.

Debt Crisis: which is also used as a synonym for fiscal crisis in Romer advanced macroeconomics it is argued that debt/fiscal crisis is a situation that involves “a sharp contraction in fiscal policy, a large decline in aggregate demand, major repercussions in capital and foreign-exchange markets, and perhaps a default on foreign debt.”

To further add to the confusion many authors have their own slightly different definitions of debt crisis. For example Reinhart and Rogoff in their now infamous research on debt crises define two kinds of debt crises external debt crisis defined as debt crisis that “involve outright default on a government’s external debt obligations—that is, a default on a payment to creditors of a loan issued under another country’s jurisdiction, typically (but not always) denominated in a foreign currency, and typically” and internal debt crisis defined as default that do not involve powerful external creditors.

If we look at the meaning of crisis as recession we find that they are defined as a drop in economic output. For example according to Blanchard et al macroeconomics an European perspective a recession is “a period of negative GDP growth. Usually refers to at least two consecutive quarters of negative GDP growth.”

Hence I don’t think there is any precise definition of crisis in macroeconomics and I would say in economics in general. A crisis is just used as an umbrella term for times of economic distress or fall in economic activity. Sometimes even within the field some authors might have different definitions.


Share of domestic and foreign monetary flows between sectors of a country In multiregional input-output analysis, monetary flows between or within economic sectors are given in a transaction matrix T. Both, the rows and the columns, have the same index (Region 1 Sector 1, Region 1 Sector 2, ... Region 2 Sector 1, Region 2 Sector 2, ... ). The diagonal of the T matrix contains the intra-sectoral monetary flows. Analagously, the diagonal of the "regional blocks" (all sectors from one region) contains the intra-regional monetary flows between sectors.
I figured that in the database with which I am working, the share of intra-regional monetary flows (all flows belong to region 1) compared to all monetary flows of a region (all flows from region 1 to region 2 or vice versa) are rather low. Let's say around 20% on average. To me (non-expert), this seems plausible. However, I would like to crosscheck my findings with "real world data".
Question: Is there an indicator which compares intra-regional monetary flows with total extra-regional monetary flows? ... in other words: How much monetary flows between sectors take place within a country and how much monetary flows between sectors of a country take place with sectors from other countries?
I came across "Trade (% of GDP)" (https://data.worldbank.org/indicator/NE.TRD.GNFS.ZS). However, 1) in input-output analysis I understand GDP to be given by the final demand vector/matrix and 2) the given figures of about 50 to 60% are completly different to my findings of around 20%.

The exports and imports in an input-output analysis should correspond to imports and exports as components of GDP so you can use those or some measures derived from them.
For example, trade as a $\%$ of GDP is the sum of exports plus imports over GDP (i.e. $\frac{E+M}{Y}$, where $E$ is export, $M$ import and $Y$ GDP/output).
Hence to calculate any comparable measure from input output table you would have to take the import and export variables from your study (your input-output table should have import row and export column), sum them together and then divide them by output.
You can also find data for $E$, $M$ and $Y$ separately and compare those to the measure you constructed. Based on the question I guess you came to $20\%$ by perhaps doing something like $\frac{E-M}{Y}$?
In addition input-output analysis is not necessary accurate and you should not be surprised to find discrepancies between what your input-output model shows and what holds in real world. Especially, if your input-output model has too many sectors small errors introduced by rounding and linearization can add up and produce results that will not be accurate.


Permanent Income Hypothesis - Changes in r I want to analyze the changes in the interest rate $r$ when dealing with the permanent income hypothesis. First of all,

$r$ is the interest rate $0<\beta<1$ is the discount factor, $Y_t$ is the labor income for period $t$, $A_t$ is the net asset wealth, $C_t$ is the consumption in the period $t$.

Now the Euler equation comes up :

$$u'(C_t)=\beta(1+r)u'(C_{t+1}) \ \  (*)$$

And If we assume $\beta(1+r)=1$ (We are going to be assuming this all the time when we are doing our analysis (closed economy)). Then:

Consumption in each period equals to each other (derived from the Euler equation) and it is equal to :

$$C = \frac{r}{1+r}(A_0+\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t )  \ \ (**)$$

I'm not gonna go through the details of how the equation $(**)$ is derived, since it will distort the scope of my question here.

Now Clearly It could be observed that when the interest rate $r$ increases, we use the $eq(*)$ to identify the substation effect and $eq(**)$ to identify the income effect. Substitution effect could be identified easily by using the fact that $u''()<0$

However when identifying the income effect there are two cases :

$(1)$ Consumer is saving
$(2)$ Consumer is borrowing

When looking at the case $(2)$ I could easily show that consumers that are borrowing will decrease their consumption for all periods if the interest rates increase (Intuition = formula)

However if Consumer is saving:

Then $A_0 >0$ So $$ A_0 \frac{r_2}{1+r_2} > A_0 \frac{r_1}{1+r_1}$$

Where $r_2$ is the new interest rate. However, We need to look at the change that occurs for $\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t$ when the interest rate increases. However this decreases with a higher $r$. Since the change in $A_0 \frac{r}{1+r}$ and $\sum_{t=0}^{\infty} (\frac{1}{1+r})^t Y_t$ act in opposite directions, I couldn't show whether $C$ would decrease or increase. (i.e I couldn't show that $intuition = formula$)

Any help?

If you impose the condition $ \beta (1 + r) = 1 $ and then move around $ r $, you're assuming that consumer preferences are changing. In other words, you have introduced a third effect: increased interest rates lead to consumers decrease their preference for consuming in the future. In other words, there is no substitution effect in your model, there is only an income effect. The result you derived for consumption is fine, and the reason you see an ambiguity is because the consumer not borrowing today doesn't mean the consumer will not borrow at some point in the future, and if the consumer does want to borrow, the higher interest rates will induce them to borrow and consume less. This happens when the consumer expects their income to rise in the future. To illustrate, assume that $ Y_t = Y (1 + \gamma)^t $ for some $ \gamma < r $ (which could change the NPG condition you used to derive this equation, but whatever) so we get

$$ C = \frac{r}{1+r} A_0 + \frac{r}{r- \gamma} Y $$

If $ \gamma \leq 0 $ and $ A_0 > 0 $, i.e if the consumer expects his future income to stay constant or decline, then the impact of a rise in $ r $ is unambiguous: it leads to higher consumption in all periods. However, if the consumer expects his income to rise, i.e $ \gamma > 0 $, then the two effects work in opposite directions: higher rates mean that the consumer borrows less, and this decreases $ C $. The net effect is ambiguous, and depends on the values of the variables in the model.


Yield curve and cash rate (multiple choice) I'm working on this supposed to be a simple multiple choice question:
The yield curve indicates?

A. the relationship between existing short, medium and longer term interest rates, independent of expectations about future interest rates.

B. the relationship between the existing cash rate and the expected level of cash rate in the future

C. expectations about future levels of medium and longer term interest rates

D. among other things, expectations about the future levels of the cash rate and hence about future monetary policy

I believe all the answer could be correct, but I'm leaning toward A because the yield curve could be seen as a "snapshot" of short, med and long term rates. Hence, although the expectation of future cash rate or monetary policy does affect short and long term rate, the existing yield curve should already capture these expectations. But A could be wrong because of the word "independent" alone.

D)

A can be eliminated because the yield curve includes expectations of future rates (forward rates).

B is true but incomplete because it doesn't include monetary policy or current long term rates.

C is mostly true, but incomplete because it doesn't mention the expectations for future short term rates.  You could also make an argument that the curve does not include expectations for long term rates.  For example, if the curve goes out to 30 years, it does not give you the 6 month forward 30 year rate.  Therefore if your definition of "long term rates" is the 30 year rate, the curve does not provide expectations for


Intermediate macroeconomics: optimal bundle for quasilinear utility? How would I go about solving this question:

Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. find:

a. The maximum amount of leisure the house hold can have and still pay taxes
b. find the optimal bundle of consumption and leisure.

For part (a), I took the derivative of $U$ with respect to $L$ and set it to 0, giving me $l^{0.5}=0$ and $l=0$. Is this part correct? Intuitively, this feels very wrong to me.

I'm also stuck on part (b). I took the derivative of the utility with respect to $C$. From my understanding, MRS is the derivative of $U$ with respect to $L$ over the derivative of $U$ with respect to $C$. I ended up getting $1/(L^{0.5})$.

I know the optimal bundle is when the budget line is tangent to the indifference curve and I also know that the slope of the budget line would be $-W$. However, when I solve for $1/(L^{0.5})=0.5$, I end up getting $L=4$ and $C=-1$. It doesn't make sense to me that a consumer will be able to have negative consumption.

Would greatly appreciate it if someone could point out where I went wrong!

From the utility function it seems L is leisure.

a) I think you went about a) in a wrong way. For a, simply you want to work as little as possible but be able to pay 11 in taxes. So you will want to earn exactly 11USD and not more (earning more means working more, which you want to minimize). So at 0,50USD an hour you have to work 22 hours. Since h=24 the maximum leisure you can have is 24-22=2.

b) Where you are going wrong in your analytical approach for b) is not including the budget constraint in your problem (when taking derivatives), which you should as you explained for the graphic analysis.

For consumption you have expenditures - income: p*c=wage*labor - T (technically an inequality, but in this case you can assume it holds with equality). I would suggest using a langrangian with this constraint. You also have the constraint that Leisure=24-labor. This you can plug in directly and is advisable so that your problem only has two unknowns (c and labor).
Solving this problem now should do the trick.


Cross-section experiment with Differences-in-Diffrences estimation I am trying to answer the following question related to econometrics:
We have a sample consisting of a cross-section of individuals in 2016 plus another cross-section of different individuals in 2018. We believe that, in 2017, a natural experiment took place affecting individuals of a particular type (the ‘treatment group’) and not affecting individuals of other types (the ‘control group’). Both types were present in our two samples. We are thinking of running a differences-in-differences estimation.
I would like to know whether:

With the sample that we have, is this estimation feasible?
If the estimation is indeed feasible, then why so?

If possible, please provide a detailed answer, for me to go over.

Dif-in-dif (DiD) strategy relies on the identifying assumption of parallel trend. This essentially means that in the absence of the treatment, the control group and the supposedly treatment group would have evolved similarly (ideally both in the pre-treatment and post-treatment period). The information you provided did not mention anything specific to this underlying assumption. If the systemic difference you mentioned introduces some differences in levels between the control and treatment group but nothing else, i.e. they evolve similarly notwithstanding the difference in levels, then you can use the dif-in-dif. More generally, the dif-in-dif strategy assumes there is no other time-varying factor affecting the groups other than the introduction of the treatment.
There are other things that should also be taken into account when doing a dif-in-dif study. Is the assignment of the treatment affect the control group? Does the composition of the treatment and control groups change as a result of the treatment?
For a more comprehensive yet approachable analysis of the DiD strategy, you can have a look at chapter 5 of Mastering Metrics: The path from cause to effect by Angrist and Pischke (2014).


Ex post evaluation of conservation projects - accounting for time in cost calculations I'm looking at differences in the cost-effectiveness of two alternative conservation strategies (treatments). The analysis will compare benefit-cost ratios between the two treatments. Projects vary in age (number of years since undertaken eg. trees planted) within and between treatments, from 20 to 7 years. I have several projects as replicates for each of the two treatments. I have field data (birds) for my measure of effectiveness, measured in one year. I have calculated a total cost using input costs from an established pricing structure, giving a total cost for each project in dollars equivalent to the same year as the bird data. I do not have access to the original (actual) cost for the year undertaken. Note the costs were one-off in the year of implementation, not ongoing.
My question relates to the estimation of costs and the different ages of the projects. My concern is that similar projects will have the same cost, despite one being twice the age. Should the costs account for different ages of the projects, and if so how to approach it? The literature commonly focusses on calculating present value using discounting for projects to occur in the future; I can find less regarding past costs.

If for project $P_i$ the cost calculated was based on prices that held during the year that the project was executed, then certainly you should bring all costs on the same temporal base, in order to compare them.

This can be done in different ways, namely, the common base may be the year of the oldest project, or the year of the most current project, or something in between. Choose the one that will make more sense to the recipients of your study: this usually means bringing "past costs to the present", because this facilitates comprehension (people can better understand current costs and to what they amount in term of resources), and also, makes the adjusted figures more relevant for future projects.

In order to bring past costs to the present, you can use a price index like the Consumer Price Index, and the implied inflation. In general, costs in current dollars is just

$$\tilde C_{i,t|t+k} = C_{i,t}\cdot \prod_{j=1}^k(1+\pi_{t+j})$$

where $t$ is the actual year the project was undertaken, and $t+k$ is Now. $\pi_{t+j}$ is the price inflation between year $t+j$ and year $t+j-1$. Data on yearly inflation are usually available and easily accessible.

Another possible issue to consider is whether all these projects were undertaken in the same economy/country. If they are not, then you also face the task of incorporating in your calculations "purchasing-power-parity" considerations.


Literature on potential games Is there any comprehensive survey on potential games and their usage in game theory?

Potential games seminal paper is by Monderer and Shapley.

I could find this chapter of a book online, but I am not sure from which book it is coming from.

Your link contains chapter 2 of the following book:

Potential Game Theory - Applications in Radio Resource Allocation

Authors: Quang Duy Lã, Yong Huat Chew, Boon-Hee Soong
ISBN: 978-3-319-30867-8 (Print) 978-3-319-30869-2 (Online)

Perhaps Jason R. Marden * , Jeff S. Shamma , "Game Theory and Distributed Control", Chapter 16 in "Handbook of game theory", Volume 4, 2015 could helpful as well.


Homogenous of degree one in utility function. Question

    My solution is as follows. Please check my solution. If I make a mistake, please tell. I am really not sure about my solution. Thank you

U(x) is homogenous of degree one i.e. u(tx)=tu(x)

  Firstly I show that the indirect utility function is homogenous of degree one in m.

By the utility maximization,

V(p,m)=max u(x) subject to px$\le$ m

tv(p,m)=max tu(x) subject to px$\le$ m

Since u(tx)=tu(x),  tv(p,m)=max u(tx) subject to px$\le$ m

Then v(p,tm)=tv(p,m)

That is the indirect utility function is homogenous of degree one.

  I show that the expenditure function is homogenous of degree one in u by using previous result.

I know that

      v(p,m)=v(p, e(p,u))=u(x)

Since u(x) is homogenous of degree one and v(p,m) is homogenous of degree one in m, v(p, e(p,u)) have to be homogenous of degree one in e(p,u).

In other words, v(p, e(p,u(tx)))=v(p, e(p,tu(x)))=tv(p, e(p,u)) holds iff e(p,tu(x))=te(p,u(x))

i.e. The expensive function e(p,u) is homogenous of degree one in u.

    Now I will show that the marshallian demand x(p,m) is homogenous of degree one in m.

By Roy's identity,

$$\frac{\partial v(p,m)/\partial p}{\partial v(p,m)/\partial m}=x(p,m)$$

By the first result, since v(p,m) is homogenous of degree one in m, then x(p,m) is homogenous of degree one in m.

  now lets show that the hicksian demand is homogenous of degree one in u.

I know that

      x(p,m)= x(p,e(p,u))=h(p,u) ........(1)

x(p,tm)=tx(p,m)=tx(p,e(p,u))=x(p,te(p,u))

Since e(p,u) is homogenous of degree one by the second part,

x(p,te(p,u))=x(p,e(p,u(tx))=h(p,u(tx))=h(p,tu(x))=th(p,u(x)) must hold since the equality(1) exist.

That is the hicksian demand is homogenous of degree one in u.

The way you show that $v(p,m)$ is homogeneous of degree one in $m$ is correct, but the reason why this implies that, $e(p,u)$ is homogeneous of degree one in $u$, is not very precise in your argument. For example, duality tells us
$$v(p,e(p,u))=u,$$
where $u$ is just a target utility level, but should not be $u(x)$ as in your proof.

Here is one possible way to proceed: Since $v(p,m)$ is homogeneous of degree one in $m$, it can be written as
$$v(p,m)=mv(p,1)=m\tilde v(p).$$
Applying the equality $v(p,e(p,u))=u$ gives
$$e(p,u)=\frac{u}{\tilde v(p)},$$
which clearly implies that $e(p,u)$ is homogeneous of degree one in $u$. You can use a similar argument to prove homogeneity of the Hicksian demand.

With all that said, I would suggest you prove the original statement directly using the definitions of expenditure function and Hicksian demand. For instance,
\begin{align*}
e(p,\lambda u)&= \min p\cdot x ~~\text{ s.t. } u(x)\geq \lambda u\\
&=\lambda\min p\cdot\frac{1}{\lambda}x ~~\text{ s.t. }\frac{1}{\lambda}u(x)\geq u\\
&=\cdots
\end{align*}


Theories relevant to institutional quality and economic development I want to write a term paper on institutional quality and economic development. What will be its possible theoretical framework?

The following researchers have written a number of papers and books on this topic:

Doran Acemoglu
James Robinson

You may want to read their papers for a good start.


Are US-branded products manufactured in China considered Chinese? Many US-branded products like Apple's are manufactured in China. Are these products considered Chinese when they are imported back to US? And are they thus subject to tariffs?

Short answer, yes, they are considered Chinese imports. Your excellent question speaks to the flaw in using Gross Domestic Product as a measurement of the health of a nation's economy.

The flaw is that the simplistic assignment of "import" and "export" completely misses the reality of modern manufacture and trade, where parts come from multiple nations.

If trade numbers more accurately accounted for how products are made, it is possible that the United States would not have any trade deficit at all with China. The problem, in short, is that trade figures are currently calculated based on the assumption that each product has a single country of origin and that the declared value of that product goes to that country. Thus, every time an iPhone or an iPad rolls off the factory floors of Foxconn (Apple’s main contractor in China) and travels to the port of Long Beach, California, it is counted as an import from China.


Relation between foreign currency reserve and exchange rate and others What is the relationship between foreign currency reserve(USD) of a country with it's exchange rate (against USD), and also with export and import strength ?

And if it's too complicated to explain, which books I should head over to read more to know it better ?

Quick and easy way is to think of a foreign currency (let's call this USD) as another good with its own demand and supply.

Imports and exports

When a country imports a lot (assuming it imports from the US or the invoice is in USD), the importers will need to pay for these goods in USD. To do this, they go to a bank (roughly speaking) and buy USD, pay for USD with the local currency. This drives up the demand of USD, and hence the price of USD goes up... the local currency depreciates against the USD.

When a country exports a lot, the reverse happens. Importers in other countries will need to pay local currency, so they sell the USD they have in order to get the local currency. The supply of the USD goes up (or you can think of it as demand of USD goes down)... price of USD goes down, and the local currency appreciates against the USD.

FX reserve

Now to go back to your original question... FX reserve is just a place for a country to hold foreign currencies (since USD can't be used locally). It's like a warehouse for USD, controlled by the central bank.

When a country can export a lot there will be a lot of people who wants to sell USD. The rapid appreciation of the local currency could affect firms (profits, etc.) so the central bank might "intervene" to stabilize the rate in the short run. They do this by buying up USD... in a sense create additional demand for USD so the price of USD doesn't fall down too fast. They then put this USD they bought in the reserve.

This is why the change in FX reserve could be used as a proxy to see how much central bank has intervened in the FX market.

In buying USD, they could print new local currency and buy USD outright, resulting in higher supply of local currency and higher inflation, or they could "sterilize" the action by selling bonds just as they would do when they want to reduce money supply by the same amount that they used to buy USD.

On the other hand, if the local currency is depreciating too fast (price of USD going up), the central bank could help by selling USD in the reserve to increase USD supply.

The problem, though, is that the reserve could run out. This is what happened in the Asian Financial Crisis in 1997.

Hope this helps!


Where does the Euler equation get its name from? The Euler equation, specifying the dynamical constraint for optimality of an intertemporal optimization problem, is widely used in modern economics. But where does it get its name from? What did Euler have to do with it?

It appears, like a lot of other math economics uses, this one was lifted from fluid dynamics. From Wikipedia:

  The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Academie des Sciences de Berlin in 1757 (in this article Euler actually published only the general form of the continuity equation and the momentum equation;[3] the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down.


Is inflation a lack of production? If every person in America was given a million dollars what fundamentally changes that causes prices to rise? Intuitively I was thinking that this is simply a production problem. With more money there is more effective demand but without a rise in productivity prices will simply shift to a new equilibrium.

I had this thought and I was wondering if it's a good way to understand inflation.

Yes, this appears to be correct. This intuition is fundamentally Keynesian.

Consider the AS-AD curve.  During the vertical portion, additional demand increases prices while supply remains roughly fixed. As a result, prices increase dramatically.

https://en.wikipedia.org/wiki/AD%E2%80%93AS_model


How are GDP rates calculated? We often hear that the GDP grew 2 % the past quarter, or something like that.

But we are also told the GDP grows annually about 2-3 % every year.

How can both be true? If the GDP grows 2 %  in just one quarter, surely annual growth rate should be closer to 8-10 %?

This question overlaps with the question: link to question on quarterly/annual GDP.

Since you are referring to news commentary, I just want to add that the standard in the United States is to quote annualised quarterly GDP figures (e.g. 2.0% annualised roughly equals 0.5% on the quarter), while European countries (e.g., Germany) often quote the non-annualised quarterly growth rate. (These are the conventions used by the national statistical agencies when quoting growth rates, and the business press usually follows the local convention when reporting on the results.) This difference sometimes can cause confusion in international comparisons.


Computation of Different Axiomatic Barganing Solutions I need help with the following question.

Take a two-agent bargaining problem. d=(0,0). Pick a strictly concave function for the bargaining frontier. Fix the intersection of the frontier with the y-axis to be at 1. Vary the intersection of the frontier with the x-axis between 0.1 and 10 with increments of 0.1. Graph the points proposed by the following solution concepts.

So y^2=1-x^2/0.1 can be the function. Does anyone know how can it be computed in matlab or a similar program?

The answer depends on the "following solution concepts", which you did not specify. I guess the Nash solution and the Kalai-Smorodinsky solution will be among them.

The frontier functions you seem to propose are of the form $y_n(x)=\sqrt{1-\frac{100x^2}{n^2}}$, where $n\in\{1,2,\ldots,100\}$.

For the Nash solution you maximize $xy_n(x)$ and after substituting for $y_n(x)$ and setting the derivative to zero you will find $x_n=\frac{n}{\sqrt{200}}$ and $y_n=\frac{1}{\sqrt{2}}$. For the KS solution you set $y_n(x)=\frac{10x}{n}$ and after substituting you will again find $x_n=\frac{n}{\sqrt{200}}$ and $y_n=\frac{1}{\sqrt{2}}$. Graphing these solutions is easy.

Remark:
That the solutions coincide is not a coincidence. The different frontier functions are constructed in such a way that they are all just horizontally "compressed" or "stretched" versions of the frontier function $y_{10}(x)=\sqrt{1-x^2}$. This means that they are really all equivalent in the sense that they result from each other after some rescaling of $x$ (the utility of agent 1). But the frontier function $y_{10}$ just describes the quarter unit circle in the positive orthant, which is symmetric w.r.t. the agents. In symmetric feasible sets the Nash and the KM solutions are known to coincide.


Does the Lucas (1978) asset pricing model feature complete markets? The Lucas (1978) asset pricing model seems to be one of the workhorse models in finance / asset pricing models. It also seems to be the case that the environment, with claims to $n$ (exogenous) productive units traded, features complete markets. I have also seen another version of this model with the agent trading a single stock and a single bond that the instructor told me had complete markets.

To me, this is not obvious at all -- the agent does not have access to Arrow securities for each state, or something of that sort. In fact, in the original paper, the states are continuous, and this makes it even harder for me think about market completeness.

Can anyone tell me why (or perhaps why not) this model features market completeness?

Any help would be greatly appreciated (or pointers to relevant links would work). Thanks!

Edit #1: ----------------------------------------

Just to clarify, while I understand that in equilibrium the net holdings (of all the identical agents) must be zero, my concern was that having access to AD securities vs. not having access may have asset pricing implications. What I mean by this is that although the outcome in terms of asset holdings must (almost by construction) be equal in any case, but perhaps the presence of AD securities may distort the prices of other assets (e.g. equity claims "trees").

Intuitively, perhaps the price of trees must be different in the presence of AD securities to offset the agent's desire to hold the AD securities in equilibrium. My next logical step in determining whether or not this is the case, was to think about whether the markets were complete or not in the first place -- if markets were already complete, then the additional AD securities are "redundant" is some sense, and prices should not change. If markets are incomplete, my initial intuition was that this may possibly have consequences (not in asset positions, but in prices).

To me, this is not obvious at all, and something I would really like to see a proof of.

It seems to me that we might as well say that markets are complete. It seems to me to be somewhat inconsequential since this is a representative agent model and that market clearing requires that the only assets that the agent may hold in positive net supply are the $n$ trees. The prices of all Arrow-Debreu securities may be inferred from shadow prices. Suppose the optimization problem is rewritten such that the agent may invest in the $n$ trees plus the set of all Arrow-Debreu securities. (There will be redundancies---I'm hand-waving a little.) The market clearing conditions say that, in equilibrium, the agent must have a unit holding in each of the $n$ trees and zero holdings in the rest of the assets and the price of each of these securities in equilibrium must be set to support these holdings. Since market completeness is a matter of whether or not the agent may trade in all of these securities, we may interpret the situation a two different ways: either the agent can invest in the Arrow-Debreu securities (as well as the trees) but the AD securities must be in zero net supply or the agent cannot invest in the AD securities (only in the trees). In either case, the result is the same.

On the other hand, if you had a model with more than one agent and they had different endowment processes, then the question of market completeness would matter for equilibrium outcomes.

Reference

Take a look at the notes here.

  We have used the equilibrium of the model to determine a single price,
  the price of a tree. With trading in trees, markets are complete.
  However we can introduce additional (redundant) assets and use the
  same approach to determine their prices.

Additional Note:

Your question is somewhat related to the question asked here: "Complete Markets in Continuous Time". See the solutions given there.

In addition, you can also find a discussion in these notes by Markus Brunnermeier:

  If the horizon is infinite, the number of events is also infinite.
  Does that imply that we need an infinite number of assets to make the
  market complete?

  List item

  Do we need assets with all possible times to maturity and events to have a complete market?

  No. Dynamic completion.

  Arrow (1953) and Guesnerie and Jaffray (1974)

The conclusion, which is in the slide titled "How many assets to complete market?", is that, in discrete time, it depends on the "branching number" (see the slides) and that, in continuous time, you can get complete markets simply with a risk-free bond and $n$ independent risky assets where $n$ is the number of independent Brownian motion underlying the economy.


Why are break-even and shut down points based on average cost? I understand the basic idea of break-even and shut-down points, where break-even is the price at which revenue covers all economic costs, and is located where the marginal cost equals to average cost. And that shut down is the price where revenue just covers variable costs (it is where marginal cost equals average variable cost).

My question, why are taking in to account average cost and average variable cost, rather than just total cost and just variable cost? Why is shut down point where revenue equals average variable cost rather than just variable cost or total cost, and why is break even where marginal cost equals average total cost, not just total cost or variable cost?

Thank you

When you are finding the break even point, you are looking for a price-point.

The price is essentially the average (and marginal) revenue. Suppose price of good $x$ is $p$.

Revenue = $p * x$.

Average Revenue = Revenue / $x$ = $p$.

So what you are really doing when looking for p is to set Average Revenue = Average Cost.

At that point, you will also have the fact that Revenue  = Cost. So your logic carries through. You still break even at the point where revenues exactly cover costs.

To see this, note Average Costs = Costs / $x$.

So Average Costs = Average Revenue $\iff$ Costs / $x$ = Revenue / $x$  $\iff$  Costs = Revenue

The interpretation when focusing on average costs is only slightly different. When talking about averages, we're just talking about prices and costs per unit. So you break even at the point where your revenue per unit equals your cost per unit. It is necessarily implied that you also break even where your total revenue equals your total cost.


Is there a pull of skilled workforce from rural to urban areas? Is there a pull of skilled workforce from rural to urban areas, i.e. is this scientifically described?

Especially in the European Union.

It somewhat depends what you mean by that. The process is more complicated with migrants to cities also acquiring skills/education there, e.g.

  Another important piece of work related to education-based motivation is Lucas (2004), who constructs a two-sector model where the urban sector pays a higher wage due to its high-skilled jobs which are not available to rural workers. Unskilled rural agents migrate to cities to accumulate human capital and invest in skills. Households’ preferences are isoelastic and they supply one unit of time endowment to either work or invest in human capital. Rural production uses labor and land; it has diminishing returns because land is fixed in quantity. The urban production technology is linear and requires only skilled labor input. Human capital investment can only take place in cities and labor is free to move between sectors. Rural–urban migration takes place once and for all with the growth rate of production of the urban (rural) sector changes from zero (one) to a constant positive level (zero).

This is somewhat of a no-brainer, you don't expect PhDs in AI to just pop up in rural areas and then migrate to cities...

There's a monster EU document (some 350 pages) published in 2018 on migration patterns inside the Union.  As far as I can tell from that (its summary is poor), already [mostly the younger] skilled workers prefer to move to another EU country (with better wages, less corruption, etc.)

For healthcare workers, which the EU study spends a lot of time on, within a given EU country, there's indeed a more acute shortage in the rural areas (p. 266). However across EU countries healthcare workers may migrate from an urban area e.g. in Greece to a rural area in the UK, all because of pay, prospects etc. (p. 269)


Where can I find County level income data for college graduates? Does anyone know where I can find data regarding the median income of college graduates by county in the US? Can't seem to find it searching through the census website.

The goal here is to obtain an opportunity cost basis to explore geographic wage differences in one field. I will be taking a ratio of field wage to this median income data. It would be more meaningful if I could find estimates of median income by educational attainment, rather than just general population data.

Try looking at the Current Population Survey. Using a tool like DataFerret you can find education (PEEDUCA) and income (HUFAMINC), along with the county code. That gives you the micro data, finding median income by education level can then be done in your statistics package of choice.


Rentier capitalism vs investment capitalism? A company has backers who put in money seeking a return from either share-price growth through company growth, and yearly rent in the form of dividend payments. Debt is clearly the case where share price growth is 0 and only the yearly rent is paid, (Socialism is 1/population for both + a wage). Why do stockholders get both the right to make money from the investment, and the right to extract rent at the same time? How does having both rights at the same time impact the economy? Is there any economic thinker that considers banning dividend payments on shares from a legal perspective?

A share is an entitlement to the ownership of a company. Such ownership brings several benefits like voting rights and dividends, and the owner is free to sell such ownership whenever s/he wants, making a capital gain (or a loss) in the process.

Why is this the case? Well, it was like that with the first publicly listed stock-owned company (the Dutch East India Company). In terms of incentives, it fosters the allocation of (financial) capital to new ventures by giving those taking the risk to benefit from it. It has proven a highly successful legal arrangement to foster businesses and investment.

Another common legal arrangement is that of non-profit corporation. These organisations are usually started by a group of people, which contribute capital to it (albeit this is not represented in the form of shares). Sometimes they transfer the ownership to a foundation. In any case, these companies cannot distribute profits to the owners.

Finally, as you say an alternative legal arrangement would be one were ownership of shares do not lead to income via dividends, but such shares still allow for capital gains. In fact, some shares pay no dividend at all (as a matter of fact, the taxonomy of shares is quite diverse). As this article explains:

  investors who buy stocks that do not pay dividends prefer to see these companies reinvest their earnings to fund expansion and other projects which they hope will yield greater returns via rising stock price.

So in principle you could set up a company that does not issue dividends in order to foster its growth. Naturally, this could be changed in the future through a motion in an annual general meeting of shareholders. There are also plenty of companies who decide not to pay dividends in order to foster expansion (e.g. Apple until a few years ago). Yet, many of these companies, and particularly in the last decades, have decided to execute share buybacks rather than foster expansion and investment. This is, the company buys their own shares with its profits, which fosters price growth and hence expands capital gains. This is part of a wider short-termist trend that some have denominated "Quarterly Capitalism".

There is very interesting new research about the consequences of this "quarterly capitalism" on investment and growth. Check here, here, and here. The latter concludes:

  We show that capital no longer flows more to the industries with the best growth opportunities because, since the middle of the 1990s, firms in high q industries increasingly repurchase shares rather than raise more funding from the capital markets.

This evidence gives a rather counterintuitive result (at least from the normative meaning of the words), which is that a rentier capitalism might be better for investment and growth than an investment captialism.


Why protectionism is bad? can you please explain from the economic point of view, why protectionism is bad? If sometimes it is sensible to protect some industries to avoid unemployment, loses of some professionals, death of the industry and decreasing independence? What do you think? tnx

In theory, it is bad because its inefficient. You protect your inefficient industries, I protect mine, we all end up consuming less. We would all be better off with totally free trade...

But of course, there are many complications to this. It might be that an industry needs time to become efficient and you want to protect it in the meanwhile. Or it might be that the other country is not more efficient, but is just dumping their products o your economy until your firms close, after which they will raise their prices and extract a handsome surplus...


Nash Equilibrium in a bargaining game I've got a question to ask about the Nash Demand game from my assignment.

  Sarah and Ruth find \$100 on the ground and decide to split it between them in the following manner. Each individual simultaneously and independently selects how much of the \$100 she wants to keep. Denote this value by $xi$. If $+≤100$, then each player receives the amount she desires. If $+>100$, they forfeit the \$100, and each player receives \$0. Find all of the Nash equilibria. Explain how you come to your conclusion (i.e. how you know that your answer is exhaustive.)

I think that I've solved the Pure Strategy Nash Equilibrium for this game which is basically $+=100$. But I'm wondering whether there are any Mixed Strategy Nash Equilibrium so if anyone could help me out that I would be very appreciative! :)

Nash equilibrium is a pair of strategies. So your answer needs to give two strategies. The informal definition is "pair of strategies that are best responses to each other".

So take an arbitrary strategy; let's say each player chooses 50. Can anyone gain by playing 51 or 49? No, so (50, 50) is a NE. Like your equation says, any two strategies that add up to 100 make up a PSNE. Because there can be no profitable deviation for any of the two player (the proof is trivial; show that playing less gives less, and that playing more, gives less for any pair of strategies adding up to 100)

Now about MSNE. A good way to start thinking about it is to take an arbitrary strategy and play around with it. You will find no MSNE this way.

The other way to look for them is to find a point where two strategies make players indifferent between their options and are also best responses. I could only find this in the symmetric case.

There are some MSNE if you play strategies $( a_1, a_2 )$ where $a_1+a_2 = 100$ and your mixing probabilities are $P(a_1) = a_1/a_2$ (assuming $a_1 < a_2)$. Example here would be playing $(10, 90)$ with probabilities $(11.111, 88.888)$. This makes the opponent indifferent between $10$ and $90$, and as such he could play the mirror strategy, with both being indifferent to any mixing and both being a best response to each other.

I did not find any other case of MSNE, even after some work.


Is worsening wealth inequality intrinsic to Capitalism? I am but a humble lay person, but I have a question.

I've been thinking about wealth inequality and the structure of capitalism. It seems to me that the pay rate for workers accumulates in a linear way. That is, during year 1 you make X, at the end of year 2 you've made 2X, at year 3 you've accumulated 3X and so on. You may get raises (although probably not so much in our current economic conditions of a labor oversupply) or switch jobs, but if you stayed at the same firm, and conditions didn't change much, then your savings accumulate in a linear way.

But for the business who hires you, they start out with some money, then they use that money to buy raw materials and pay you to convert them into a product, then they take your finished product and sell it for more money than they started with. Let's say they start off with Y, then at the end of a year they have 2Y. But this is an exponential growth: they then start over the process using their 2Y, and can hire twice as many people and buy twice as much raw materials and sell twice as much, to end up with 4Y at the end.

So while worker worth goes up in a linear way, 1X, 2X, 3X, 4X...the employer's worth goes up in an exponential way, 1Y, 2Y, 4Y, 8Y...So in that case, isn't wealth inequality fundamentally built right into capitalism, and therefore we should expect it to pretty much always get worse as time goes on, unless measures are taken to combat or reverse it after the fact? Am I wrong in my analysis here?

While the owners of a successful business will generally experience an exponential growth compared to their employees, there are some factors to consider:

I find it unrealistic for any business to maintain a growth rate of 100% over many years. (As per the example) You can only sell twice as much if you can find twice as many buyers.
As a business grows, it is likely to start hiring more people and paying them better as they need better skills.
The business also takes the risk. The worker is generally guaranteed his X, but stockholders may have to expect the occasional decline in profit, or even a loss. The company may even go out of business in which case they cannot recover much, if any, of their investment. Those who take the risks, get the reward. Businesses failing also mean that bad investors can often lose fortunes, it's not always a good investment.
There is nothing stopping the employee from owning part of the company by investing in some shares. Then he too may experience this exponential growth. He must also keep in mind, however, that he must now also accept the risks of such an investment.


How should the general public evaluate economic arguments? This is a follow-up question to Why do economists disagree so much?

I know little about economics - just enough to know that it's a complex field I don't understand. However I see economic arguments in the media quite often. Since I don't understand economics, how should I make sense of the economics politicians might argue?

For example, in the Scottish independence debate, there are people who say Scotland is economically better off independent and people who say the reverse. Clearly they can't both be right. Ubiquitous's answer in the linked question indicates that this difference in opinion is because we aren't sure what will happen after Scottish independence, and the two camps make different assumptions as to what is more likely. Well and good, but that doesn't tell me how to vote (assuming economics is my main concern).

Hence the question: given that I can't evaluate economic arguments myself, how can I tell who to vote for? Is there a way to tell if a certain view is more fringe than the other? Can I just count the number of economists with a certain view? Can I go by fame? Popularity? Or just flip a coin?

For most purposes, economics is not effectively understood in 10-second segments or catchy phrases used by politicians.

Generally, a willingness to openly acknowledge that something is complex, and to acknowledge merit to empirical evidence and/or ethical argumentation on multiple sides of a debate, is a good indication that what follows has at least some merit. Presentation of an issue which is preoccupied with fun-sounding insults, or promoting black/white thinking on a question, is unlikely to be based on quality analysis.

Differentiating between acknowledgment of complexity, as opposed to obfuscation, is also important. If complexity is acknowledged and some other analysis claims "ah HA, you admit you don't know, stooopid, because you said it's complex", then don't pay any attention to the people producing the second 'analysis', unless wanting to understand the nature of their manipulation.

Another strong indicator for a non-expert to guess at the quality of arguments is the transparency in who funds the research. If there is no disclosure of the funding, the research output should be treated as worthless unless you have specific abilities to understand the analytical methods and data inputs. An independent researcher may explicitly declare that no external funding was provided, as compared with someone paid to produce a pre-determined result who may not disclose anything. Disclosing funding sources at least allows a non-expert to observe that the conclusion does or does not align with whoever paid for it. This can be gamed, but a couple minutes of online research should most often be sufficient to determine if it's likely to be BS -- compare a) a web search ending with unnamed corporations or funding organizations with no detail about who/what/when/why of some front organization, and no ability to contact any person from the front organization, as opposed to b) a web search that leads you to a named organization registered in a specific jurisdiction with a named board of directors and an explicit mission statement.

Also, this is fundamentally not an economics question. The thematic area is more "how can a non-expert evaluate any expert claim?"


Today's value of the Louisiana Purchase? A video I watched claimed (at 6:50) that while only \$0.25B had been paid for the Louisiana Purchase, the current value was \$1200B (all in 2017 dollars).

I couldn't find any references, but assuming the latter number is correct, what might it refer to?

The current value of the land?

The estimated value of the land if all structures and roads were removed?

(It seems to me that these numbers would be vastly different, and comparing empty land to one with infrastructure is comparing apples and oranges)

The video has a transcript with the references. The \$0.25B figure is obtained from here (after adjusting for inflation). Unfortunately, the author does not provide a source for the \$1.2B figure.

However, there are estimates of the value of land elsewhere. For example, here. Their estimates on a map:

These estimates consider the value of land only, removing the value of structures on that land (which, as you say, makes a big difference; see paper). They use hedonic regression methods to separate land and building values (check from page 6 of document).

I leave to you the task to compute the precise value of the Louisiana Purchase. (you could ask them to send you the data on the map).


How does the Federal Reserve unwind the asset purchasing scheme announced in response to the 2020 Coronavirus Pandemic? On Monday, 06 April, 2020 the Financial Times reported that the Federal Reserve balance sheet could increase to $9 trillion.  This is partly due to the myriad of initiatives, some new, to protect the economy of the United States during the 2020 Coronavirus Pandemic.

A lot of online commentary is alarmist and centres on the threat of hyperinflation and many of the news articles don't inform the reader how the Federal Reserve purchases assets.. This puts the average citizen at a disadvantage when understanding the economic impact.

My question is multi-faceted (but singlular) and aiming to focus purely on the economics.

How does the Federal Reserve unwind the positions on their books?  Do financial services have to purchase the assets back at a later date and does the Federal Reserve charge a premium for the service?
Can a position of this size (60% of national output) actually be unwound?

If this question is similar to the approaches of other Central Banks then please tag accordingly if appropriate.

The unwind depends upon what the central bank did.

If the central bank did a repurchase agreement (“repo”) or lent against assets for a fixed time, the agreement automatically unwinds at the term of the deal (which is short). They would need to enter into new deals to keep their balance sheet size unchanged.
If they bought the asset outright, it will either mature, or the central bank can sell ahead of maturity.

There are no “charging a premium” involved; the cost of the operation is embedded in the lending cost (or purchase/sales price of an asset).

There was a very large debate about unwinding central bank balance sheets in the 2010-2016 (?) period. The Fed was eventually able to slowly reduce the size of its balance sheet in the last cycle, and there is no reason to believe that this time is different. There is no sign that an “unwind” was necessary, nor are the effects of unwinding a deep concern.


Why is wage-induced inflation a bad thing? More than once, I have heard on the news how the US Federal Reserve would raise interest rates to counter the risk of inflation, even when they indicate that the potential inflation would be caused by "wage pressure".

I can understand why the wealthy disapprove of inflation -- it causes the value of their wealth to decrease -- but for the poor, this is not a significant concern.  And if (for example) the price of products are 50% due to labor, doubling the salary of that labor would only increase their cost by 50%, while the workers who were paid that salary could buy twice as many products. Mathematically speaking, then, I would expect prices to asymptotically approach some new value at a rate of e^(-k/t), not spiral into hyperinflation.

What am I missing here?  Is inflation due to wage pressure really a bad thing for the general population, or is this a case of the Federal Reserve bowing to the will of those few who would benefit the most?

Wage induced inflation can create inflation spirals, which can lead to hyperinflation. I think everyone agrees hyperinflation is a bad thing for everyone and not just the rich. Furthermore, inflation can be a significant problem for the poor as well.

Such a case of inflation spirals was even threatening post WW2 Germany.

The spiral happens because, if workers demand higher wages that firms cannot pay, then firms may increase prices in order to make up for those losses. This causes inflation, which reduces workers' real wages, so they ask for another wage increase. Firms not in a position to deny or to afford these wages will have to comply and increase prices again. This can go on and on and easily spiral out of control.

In principle it is possible to have higher inflation and higher wage increases. This was the case in Italy before the introduction of the Euro.

However this wage induced inflation also affects non-wage earners, such as those living on welfare benefits, the unemployed living on savings and retirees.

As for whether fiscal stimulus is better put in the hands of the poor, this is a different topic related to fiscal policy and unrelated to inflation (i.e. monetary policy).


Relation between Nash equilibrium , subgame perfect equilibrium, backward induction, perfect Bayesian equilibrium My claims are

1- Backward induction solution is Nash equilibrium solution.

2- Not all Nash equilibria are sequentially rational

3- All Backward induction solutions are sequentially rational

4- SPNE solutions are Nash equilibria

5- SPNE solutions are sequentially rational if game has at least one
proper sub game. If only has improper sub game then it may not be sequentially rational.

6- PBE solutions are Nash equilibrium and SPNE

7- PBE solutions are sequentially rational no matter what.

8- WPBE solutions are Nash equilibria. And sequentially rational no matter what.

Are these true? What do you think? Anything want to add?

1- Backward induction solution is Nash equilibrium solution. TRUE

2- Not all Nash equilibria are sequentially rational. TRUE; NE allows for sequentially irrational behavior off equilibrium path.

3- All Backward induction solutions are sequentially rational. TRUE

4- SPNE solutions are Nash equilibria. TRUE

5- SPNE solutions are sequentially rational if game has at least one proper sub game. FALSE (Counterexample below: the game has one proper subgame; $(XC,L)$ is an SPNE but involves a sequentially irrational strategy $L$ by player 2). If only has improper sub game then it may not be sequentially rational. TRUE (Treat the subgame after $X$ as a game of its own.)

6- PBE solutions are Nash equilibrium and SPNE. UNCERTAIN / TRUE (in most cases); it actually depends on how exactly you define PBE (there are many definitions). See Watson (2017) for more details.

7- PBE solutions are sequentially rational no matter what. TRUE

8- WPBE solutions are Nash equilibria. And sequentially rational no matter what. TRUE


Econometric analysis of mergers from a company's point of view I am learning about financial analysis of mergers and acquisitions: most of the models are based on accounting principles and simple assumptions.
I wanted to delve deeper into more statistical/ econometric methods to analyse mergers from a company's point of view: how would profitability be affected, synergies, debt etc., what variables are usually taken into account etc. But I have no idea of how that is done (although I have studied econometrics before, I haven't studied it in this context).
I read an article by Christopher Rybak and Loren Smith titled "the Use of Econometrics in Merger Reviews" but this article is not from the company's point of view, it is from the regulator's point of view instead, focusing on the merger's impact on market competitiveness, anti-trust etc. They say that in this case, three things are usually used: Demand Estimation and Merger Simulation, Event Studies and Price-Concentration Studies and then goes on to explain them.
I wanted to know if similar to the regulator's point of view, there are also commonly used practices to analyse mergers from the company's point if view, and where can I learn about them

A good textbook for this sort of empirical research is Introductory Econometrics for Finance by Chris Brooks. That textbook helped me a lot when I was writing my thesis, which was actually on M&A (if they improve financial ratios for firms). I know writing a thesis is not exactly the same as actual research but reading your question I think you could find it useful.


The economics of continually selling a product for under its market value? What are the economic principles behind a company continually selling a product for well under its market value. Continually here expressing that it does not appear to be some short term method to bankrupt competitors.

I am talking about companies that sell products that can be routinely resold on the second hand market for many times their initial cost.

I recently watched a documentary on Nike, where it describes their humongous Limited Edition department and how they sell shoes for a few hundred dollars a pop to client base willing to wait in line for weeks and pay thousands of dollars for any one of their limited editions shoes.

It describes Nike underselling their shoes by 1000% in both quantity and price. But Nike is hardly the only example, a few years ago, anyone who wanted to buy a Nintendo Wii could be certain of making a profit reselling it.

1st of all, you should distinguish cases from the Nike case and others, if any:

Since Nike sells only a tiny portion of its shoes "irrationally" it cannot be seen as market value - equilibrium price, but rather as marketing expenses

Think that simply generating a rare nike products market adds value to the brand and motivates consumers to prefer nike

Regarding any other case: they can either be "dumping" - that is selling at a loss in order to throw your competitors, selling above your *average * cost, meaning making  a profit, maybe fightinv for market share, or being cross funded by different activity, to hurt competitors in the subject market


Are homothetic preferences monotonic? I'm trying to understand intuitively what a homothetic preference is, and I am still not quite there. I understand the definition, that a homothetic preference implies that the slope of the indifference curve remains constant along any ray from origin.

But intuitively, like in real life, how would we know that a preference is not homothetic? If its not strictly monotonically increasing (i.e. you don't prefer more of a product to less of it always), does it mean that the preference is not homothetic? Can someone give an example of what exactly is not a homothetic preference so I understand what is?

Are homothetic preferences strictly monotonically increasing?

Homotheticity requires that

$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$

This is not defined over the "increasing" part of strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds.

Are nomothetic preferences weakly monotonic?

However, the "strict" part (in both jointly) is required, i.e. $U(\alpha x, \alpha y) \neq U(x,y)$ for $\alpha \neq 0$.. You cannot have homothetic and weakly increasing or weakly decreasing preferences.

Proof outline To prove that, we assume otherwise. There is some homothetic $U$ with a strictly increasing part and a flat part. You will see that the $\gamma$ corresponding to the increasing part can potentially be any number (it depends on the functional form). However, the $\gamma$ corresponding to the flat part needs to be 0.[1] Hence, there cannot be a $\gamma$ that holds for the utility function as a whole, for any general increasing part.

[1]: Unless of course, $U(x,y) = 0$ at the flat part. However, as utility functions are immune to monotone transformations, we can ignore such cases.


Nash Equilibrium and Dominant Strategy If I have a game that goes as follow:

Player 1 is the row player and player 2 is the column player. I think that the Nash Equilibria should be (10, 5) and (5, 10), since neither of the player has incentive to unilaterally deviate given the other's strategy. But then the dominant strategy, I thought, for both player is to always invest. Does this mean that the stable outcome is actually (5, 5) and that it is inconsistent with the NEs, which should be stable? (I only work with pure strategy here)

You're on the right track here. You need to check every outcome for its potential to be a NE. You're correct in stating that outcomes (5,10) and (10,5) are NEs however you didn't identify that (5,5) is also an NE.

(5,5)
If player one deviates he receives a payout of 5.
If player two deviates he receives a payout of 5.

Therefore no player has an incentive to unilaterally deviate and (5,5) is a Nash equilibrium.

You're also correct that (5,5) is the dominant strategy solution. Just keep in mind that the dominant strategy solution will always be a Nash equilibrium but not all Nash equilibria are dominant strategy solutions.


Two-player first-price auction with resale Exercise 2.5 in Vijay Krishna's Auction Theory (slightly modified):

  Consider a two-bidder ﬁrst-price auction in which bidders’ values
  are distributed according to $F$. Let $\beta$ be the symmetric equilibrium, i.e., $$\beta(x)=E[Y\mid Y<x],$$ where $Y$ is drawn from $F$.

  Now suppose that after the auction is over, both the losing and winning bids are publicly announced. In addition, there is the possibility of postauction resale: The winner of the auction may, if he so wishes, offer the object to the other bidder at a ﬁxed “take-it-or-leave-it” price of $p$. If the other bidder agrees, then the object changes hands, and the losing bidder pays the winning bidder $p$. Otherwise, the object stays with the winning bidder, and no money changes hands. The
  possibility of postauction resale in this manner is commonly known to
  both bidders prior to participating in the auction. Show that $\beta$ remains an
  equilibrium even if resale is allowed. In particular, show that a bidder with
  value $x$ cannot gain by bidding an amount $b>\beta(x)$ even when he has the
  option of reselling the object to the other bidder.

Call the two bidders Jane and Mike. Given Mike's bid, and assuming that Mike follows the equilibrium strategy $\beta$, Jane can determine Mike's exact value. This implies that if Jane wins the item but realizes that she has lower value than Mike, she will resell it to Mike at exactly Mike's value.

We should calculate Jane's expected payoff when bidding $\beta(z)$ with $z\geq x$ when her value is $x$, and show that the expected payoff is maximized when $z=x$.

Jane wins with probability $F(z)$. If she loses, her payoff is $0$. If she wins and her value is higher than Mike's value, her payoff is $x-\beta(z)$. If she wins and her value is lower than Mike's value, $Y$, her payoff is $y-\beta(z)$. As a result, her expected payoff is $$-F(z)\beta(z)+F(x)x+(F(z)-F(x))E[Y\mid x<Y<z].$$

The term $F(x)x$ does not depend on $z$, so we are left to maximize $$-F(z)\beta(z)+(F(z)-F(x))E[Y\mid x<Y<z].$$

How do we maximize this?

Let's look at this with backward induction. Let $v_{i}$ be my valuation and $v_{-i}$ be your valuation. Suppose I've won the item. Then $v_{i} \geq v_{-i}$. If I sell you the item at price $v_{-i}$, then my utility is $v_{-i} - b_{i} \leq v_{i} - b_{i}$, where $b_{i}$ is my bid according to the symmetric equilibrium bidding strategy (which may not necessarily be $\beta$). Notice that I do not gain $v_{i}$ by selling you the item, as I do not possess it.

We see that I will never have incentive to sell you the item. So I should bid as if I'm not going to sell you the item. We have an equilibrium bidding strategy for this anyways, given by the first price auction.


Trying to apply in practice the theory of finding the optimal price in a cournot competition I recently received my bachelors degree in economics. For fun I wanted to try to apply some micro-economic theory of finding the optimal price in a cournot competition.

I wanted to do this for a business my friend has. He's a goldsmith and has his own established goldsmith store in a city with a population of approximate 100.000 people. There are two more goldsmiths with their own store in this city.

My plan would be to only look at one product, suppose a simple and plain gold ring. Finding the marginal cost function for this should be easy. However, I struggle with thinking about how I would find the marginal revenue function to eventually find MR=MC. For the marginal revenue function I need to know the demand curve. In order to find the demand curve I need to know the the sum of utilities of individuals (for simplicity assuming that consumers are homogeneous), right? How could I estimate this utility of an individual?

Furthermore, I would believe that this market could be described as a cournot competetion (n=3) where we could assume, for simplicity, that each goldsmith would have the same marginal costs, since they all buy gold from the same supplier and the costs of effort and production wouldn't differ that much from each other.

So mainly I struggle with finding or estimating the utility function. Since a simplistic utility function could be U(p)=X-p, where X is the willingness to pay and p the price of a golden ring. But how do I find the willingness to pay of an individual. Is it perhaps possible to estimate deduce this from my friends previous years revenues or profits?

Last question. This optimal price would only be optimal in a cournot competition when the other two goldsmiths would also use the same price? Assuming that these firms are symmetric.

If you have critique on my process of thought of how to approach this, please do and correct me if I mentioned something wrong about the micro-economic theory I had in mind for this application.

Thank you for your attention :)

Suppose you are willing to believe that the demand curve has a constant elasticity $\epsilon$. Then the optimal price $P^*$ is given by
$$ P^* = \frac{|\epsilon| \hspace{0.1em}c}{|\epsilon|-1}$$
where $c$ is the marginal cost of producing gold rings (assumed constant). So if your friend can estimate the elasticity $\epsilon$ (e.g. by randomly varying the price and seeing what happens to the number of rings sold), they should be able to estimate the optimal price.
To illustrate, suppose that your friend thinks that increasing the price by 1% will decrease demand by 2%. Then $\epsilon \approx -2$, i.e. $| \epsilon | \approx 2$. The formula then proposes charging $2c/(2 - 1) = 2c$; that is, a 100% markup on the constant marginal cost.
As an aside, note that this formula holds independently of the market structure. So you don't need to worry about whether the market is described, say, by the Cournot model (except insofar as this bears on your estimate of $ \epsilon$).


What is the implication of perfectly reversible capital? I am reading the lecture note in labor economics by Acemoglu and Autor. In the chapter of basic search model, they assume capital is perfectly reversible, and argue that  "Perfect reversibility implies that $w$ does not depend on the firm's choice of capital ... Suppose $k$ is not perfectly reversible then suppose that the worker captures a fraction $\beta$ all the output in bargaining. Then the wage depends on the capital stock of the firm, $w(k)=\beta A f(k)$ ;$A f^{\prime}(k) =\frac{r+\delta}{1-\beta}$ capital accumulation is distorted." (page 241.)
I don't understand how is the reversibility of capital related to the determination of wage?

I think I get the answer but I am not 100% sure. Any comment would be welcome.
I realized that this is related to the search-matching nature of the model (comparing to competitive labor market), so that if capital is not perfectly reversible the worker can bargain with the firm. Note that even with perfect reversibility, given the match-specific surplus, the wage is still determined by a Nash Bargaining. However the firm's choice of capital would be independent to this Nash Bargaining and be efficient ($f^{\prime}(k)=r+\delta$).


Is corporate tax an argument in equal income tax (from labour and capital) debate and does the return on capital depend on rate of the corporate tax? In our country we have 25% personal income tax on labour, 15% corporate tax on corporations and 10% personal income tax on income from capital (e.g. dividends). Pundits in our country are saying that we have flat tax rate across all types of income, because - they are saying - 15% corporate tax and 10% capital income tax amounts to the 25% tax on the income from the capital. Are their analysis correct?

This question leads to the following question - does return in capital (after taxes) depneds on the corporate tax rate? What empirical analysis and models are saying about this? From the one hand - it seems that corporate tax have impact on the gains after corporate tax from the other hand - maybe free market reprices the capital in such manner that the return on capital (after corporate tax) does not depend on the corporate tax rate?

Does the tax system of our country is just, normal, appropriate and similar to for the developed Western democracy country?

I feel that to speak about flat (equal) taxation we should have 25% tax on all the types of income and we should not take into account the corporate tax when considering personal income tax. How this analysis proceeds in Western democracies?

Or should we speak about price/earning share ratio as the most sensible measure of return on capital - does this ratio depend on the corporate tax?

There are too many questions in your post, and I advise to edit out most of them, especially those that are not answerable (like "is the system just" etc).

For just the first question the answer is No, you don't have the same tax rate on income from labor and from capital.

Let $\pi$ be profits before tax, $\tau_{\pi}$ the tax rate on profits, $\tau_d$ that tax rate on dividends. Then capital owners will receive (as a percentage of profits)

$$(1-\tau_d)(1-\tau_{\pi}) =  1-\tau_d-\tau_{\pi}+\tau_d\tau_{\pi}$$

since the dividend tax is applied to profits after tax. So the total tax rate on capital is

$$\tau_c = \tau_d+\tau_{\pi}-\tau_d\tau_{\pi}$$

In your case $\tau_d\tau_{\pi} = 0.10\times 0.15 = 0.015$

So $\tau_c = 0.235$ while tax on labor income is $0.25$. This, in general, is not considered a negligible difference.


Original Work for Pareto Refinement of NE under Strategic Form Just looking to see if anyone has any knowledge of original work for pareto refinement of NE under strategic form. I want to give credit to the refinement in a paper I am writing. I pulled the refinement from Nachbar's notes on "Topics in Advanced Game Theory", but obviously that is not a reliable source to cite.

I think you're referring to the criterion of payoff dominance in equilibrium selection. This idea can be attributed to Harsanyi and Selten (1988):

John C. Harsanyi and Reinhard Selten. 1988. A General Theory of Equilibrium Selection in Games, MIT Press.


Find a Perfect Bayesian Equilibrium with a game that includes Nature I want to determine all pure-strategy Perfect Bayesian Equilibria for this task, but I cannot get very far. Side note: First number is payoff for A, second number payoff for player B.
For the last stage, I know that upper A will always choose to exit, and the bottom A will always choose to pass.
But I don't know how to proceed from there? I have ascribed beliefs for player B being in the upper and bottom node (beta and 1-beta, respectively), but am not sure what to do with it.
I tried assuming that Nature goes 'up' and in that case beta=1. Then B would choose pass because it has a greater payoff than exiting. That's where I get confused.

You've rightly identified A's optimal actions at his last decision nodes (shown as red arrows in the figure below). Next, you need to

conjecture a pair of actions at A's initial nodes,
use those actions to determine B's belief at her information set,
find B's best response(s) to that belief, and
check whether A's initial actions are optimal with respect to B's best response(s).

If the answer at step 4 is "yes", then you have a PBE; if not, repeat the steps with a different conjecture.
In your particular example, there is no pure strategy PBE. I will nevertheless go through an example just to demonstrate the above approach:

Suppose A chooses "pass" at both initial information sets (blue arrows).
Using Bayes' rule, B's belief must be $0.9$ at the top node and $0.1$ at the bottom node.
Based on this belief, B's expected payoff is $1$ from choosing "exit", and $0.9(3)+0.1(-5)=2.2$ from choosing "pass". So her best response is to choose "pass" (green arrows).
Given that B would choose "pass", A's best response after Nature chooses the top path would be "exit" (purple arrow), since $4>3$.

Thus we do not have a PBE involving A choosing "pass" at both initial nodes. You can go through the rest of A's possible action combinations at his initial nodes to verify that there is no PBE with pure strategies.


Creating non-linear cost functions From what I understand, the basic cost function looks like
$$C(x_i)=\sum_{i=1}^{n}w_ix_i$$

What I'm wondering is whether or not its possible to create a non-linear cost function which adjusts based on the bundle consumed.

Example:

Lets say we have a discrete case where our  bundle price changes based on the combination of goods in a non-intuitive way. as seen in the table below.
$$\begin{array}{|c|c|c|c|}
\hline Bundle\ Price&x_1&x_2&x_3
\\\hline 5&1&1&0
\\\hline 10&2&1&0
\\\hline 15&2&2&0
\\\hline 25&2&3&1
\\\hline
\end{array}$$

We see such pricing methods employed in a number of commercial businesses.

How does one model such a case1?

1. I would appreciate an answer that is more theoretical and does not rely upon regression , however if that is the only way to get an estimate i'll take it.

When you say non-linear cost function, I assume you aren't referring to the firm's production having non-linear costs, but judging by your example, you rather mean the firm's output prices having weird optimal bundle pricing.

From the abstract of the linked article (Hanson and Martin, 1990):

  " Bundle pricing is a widespread phenomenon. However, despite its importance as a pricing tool, surprisingly little is known about how to find optimal bundle prices. "

So this may be a good place for you to start. For an article that is a little more recent/available online, you could look at extensions of the original work in this area, such as bundle pricing for homogeneous items (Grigoriev et.al., no date), or this article correlated valuations for the items (Chen and Ni, 2017).


In fisher equation, Should I use "expected inflation" not "actual inflation"? In fisher equation,
 real interest rate = nominal interest rate - expected inflation rate

During calculating real interest rate, I've question that whether I can use actual inflation rate instead of expected inflation rate.

so, is it make sense?
 real interest rate = nominal interest rate - actual inflation rate

The Fisher equation as $$r=i-\pi$$ is little more than an approximation to the definition of the real interest rate $$r=\dfrac{1+i}{1+\pi} -1$$ where $r$ is the real interest rate, $i$ is the nominal interest rate and $\pi$ is inflation (the proportionate change in prices). You can complicate things by "coninuous compounding" but it makes little actual impact.

Looking backwards you should look at actual interest and actual inflation at the time of the loan. Looking forward, if you do not know what inflation or interest rates will be then you can only use estimated values and forecaster often prefer to estimate inflation and real interest rates, turning the Fisher equation round to give $$i = r+ \pi$$


What term/theory describes more efficient utilization of a resource? I may be asking the wrong question here, but I wonder if there is a term/theory that describes/studies the evolution of an economy that continually improves the utilization and allocation of certain resource? The resource here can be different things, e.g. physical goods, human labor, financial capital. In fact, I am more interested in the former two than the latter.

For example, living space used to be mostly utilized by the its owner or long-term occupier, and many were under-utilized in terms of both the utility of having people living in them and the monetary value extracted from them. Now some significant portion of these spaces is better utilized by services like Airbnb. A similar case is car-sharing services like Uber and Lyft, where both cars and human labor are arguably better utilized and allocated.

To clarify, I am not specifically interested in the so-called 'share economy' concept though the two examples above are closely related to it. Actually it'd be better if you can identify other examples in different contexts. Pointers to any related term/concept/theory is welcome and appreciated.

At the abstract level, this phenomenon is covered by the concept of "Technological Change", which, in Economics, does not have a strict/narrow "Engineering" interpretation: here, "Technology" is not only "how things are done" but also "how things are organized", and includes also new business models like the one you mentioned (see also this post).

So the phenomenon is already embedded in the logic of economic models.

Stochastic Frontier Analysis (SFA) and Data Envelopment Analysis (DEA), are two sub-fields focusing on modeling and measuring the degree of efficiency (under various interpretations of the term), in the use of factors of production at the micro-level.

It is my impression though that since the examples you describe are relatively new, we still do not have enough data available to subject them to econometric analysis, so I cannot say if there are already papers that examine them (in a quantitative way).


Value of Used Goods in GDP Why do sale of used goods do not add to income of a country in a particular year? If you think of income as the inflow in your bank account. All outflows are just expenditures. So I think of income as how much stuff can someone buy in a particulat year. Lets say person A bought a football for 50 dollars. Someone's income is increased by 50 dollars. So currently total income of society is 50 dollars in the current year. Now the guy who bought it sells it for 30. His income in this year is 30. All outflow is expenditure, in other words you are making use of your income. It does not really reduce income. So total income of society in this year should be 50 + 30 = 80? What is wrong with this logic?

As @FooBar mentions in a comment, with "Income" we want to measure what we produce (production being the whole targetfocus of economic activity, with consumption being the ultimate purpose).

Measuring Income as

  "If you think of income as the inflow in your bank account."

may appear an adequate approach to measure it for some situations: if I work and my employer pays me by bank-transfers, all my income is the sum of these inflows (I am sure that the OP meant to exclude other kinds of inflows, like for example from one of my accounts to another).

So, If I sell something that belongs to me, doesn't that also increase my Income, and so National Income?
No, it restructures your wealth in terms of liquidity: your non-liquid wealth decreased while your liquid wealth increased. You did not produce anything, you just liquidated part of your wealth.

Buying a football, is buying a durable consumer good. You don't get to eat the football, or burn it at a temple (at least in normal usage), you make use of its bouncing services to entertain yourself. So while you hold on to the football and you use it, it is a utility-enhancing capital good, and so part of your wealth.

The very fact that you were able to re-sell it, reveals that this durable/utility enhancing capital good was not fully depreciated, it still had some potential flow of services in it -this is why the other person paid you for it.

So it is selling of a capital good after this good has become wealth, and so it does not reflect production.

Mind you  that the "Used goods Sector" does create Income: it offers services around the used goods: the good "Used football in my closet" is not the same good as "Used football in a shelf for sale". But the Income they generate is the difference between how much they paid me for the football, and how much they will sell it... which naturally leads us to the concept of "Value Added" which is the one we should make certain to understand accurately, in order to navigate in quantitative economics.


Real-life applications of repeated games theory What are some scenarios in which the theory of repeated games have been applied?

I am looking, for example, for scenarios in which a government, a firm or a person accepted a decision which relied upon a result such as the Folk theorem.

I believe insights from repeated game theory have been used in designing the assignment scheme for security patrols in many occasions.

In this matter, the main research projects with close connections to applications that I know of are

(a) PROTECT for the US Coast Guard

(b) TRUSTS for the Los Angeles Sheriff's Department

(c) IRIS for the Federal Air Marshal's Service

(d) LAX for the Los Angeles Airport Police

(e) GUARDS for the Transportation Security Administration

which all revolve around the teamcore research group at the University of Southern California.

Examples of papers involving members of the group and repeated games (usually repeated Stackelberg security games) include:

Defender Strategies In Domains Involving Frequent Adversary Interaction
Learning Bounded Rationality Models of the Adversary in
Repeated Stackelberg Security Games
Conducting Longitudinal Experiments with Behavioral
Models in Repeated Stackelberg Security Games on
Amazon Mechanical Turk
A Game of Thrones": When Human Behavior Models Compete in Repeated Stackelberg Security Games


Why is the rising price of a commodity inflationary, while the rising price of money is disinflationary? Everybody buys oil and everybody borrows money. If the price of oil goes up, we call that inflation and the central bank uses monetary policy (rising interest rates) to reduce inflation. Why is the price of money (i.e. interest rates) different from the price of any other commodity in regards to its impact on inflation? Does not a rising price of oil in itself cool the economy in the same way as a rising price of money?  If we need to spend more to drive our cars, then we can spend less on televisions?

If everybody borrows money, where can you borrow money from?

In what unit do you denote the price of oil, chocolate, or cows? Well, in units of money. So calling interest rates the price of money doesn't quite make much sense.

You can think about it as the price of money today in terms of future money though. So an increase in interest rates makes money today more expensive in terms of future money, which is the same thing as making future money cheaper in terms of money today. And if future money gets cheaper, more people might "buy" future money (usually, we call that saving).

The more future money people buy with money today, the less money they have left for oil, chocolate, and cows. But we already have all the oil, chocolate, and cows around. So that people still buy these things, oil, chocolate, and cows have to get cheaper and the price level for goods and services measured in money today goes down. Less inflation.


Introductory or Primer text on Economics? I've been meaning to learn more about economics, and am looking for an introductory text or a primer on the topic.

I've already taken a subject on this back in high school, but I guess what I'm looking for is more on breadth. Something that'll introduce me to the things that I should know if I am to read an article on the news concerning updates on the world's economy and the local market.

I just want to get into being updated on such things. And thought of asking because maybe you guys who're more experienced can point me to the right direction instead of me reading about on different things, when in fact there are better references.

I'm a computer engineering undergrad, and my reason for wanting to learn about this is to be updated about the economy (it interests me), and because I want to get into stocks and forex eventually. I've already been reading books and articles on stocks, but I guess I'm just looking for something comprehensive and structured.

So a recommended text on stocks or forex would be nice too! Something on terms and the likes I guess :D same for the references I'm looking for concerning economics. Thanks a lot for the help!

My interest on the topic also concerns my intention to eventually focus on a field regarding energy consumption, the smart grid and the likes, and environmental sustainability.

So being able to understand and gain a working knowledge about economics seems like something that I should know.

If you were to read only introductory book in econonomics:
Economics: The User's Guide by Ha-Joon Chang.
It explains all basics concepts in economics, and the differences between the main school of thoughts. It is quite easy to read, it is not a textbook.

For an overview of financial markets:
Capital Ideas, and Capital Ideas Evolving by Peter L. Bernstein

For your interest in electricity:
The Grid: A Journey Through the Heart of Our Electrified World by Phillip F. Schewe, very interesting on how the grid came to be.

Smart Grid: Integrating Renewable, Distributed & Efficient Energy
by Fereidoon P. Sioshansi (Editor), Sioshansi is a reference in Smart Grid and the paradigm change from consumer to prosumer.

Renewable Economics: How to compare the costs of different technologies:
Energy Technologies and Economics


Optimal spending over several periods with log utility and uncertain lifetime If someone has probability p(n) of being alive after n periods and p(n) is known with p(n) = 0 for n >= m, and if he has log utility of consumption, and his utilities are additive over time, and interest rates and inflation are zero, can one derive the fraction of wealth he should consume in each period? I think that the amount of wealth he sets aside for a period is proportional to the probability he will be alive then but have not proven it.

I have solved the two-period case. If he starts with one unit of wealth, and he has probability p of living to a second period, and no probability of living after that, and he has log utility of spending, then his total utility is

U(x) = log(x) + p*log(1-x)

U'(x) = 1/x - p/(1-x)

Setting U'(x) = 0 gives

p*x = 1-x

So

x = 1/(1+p)

which makes sense. If p = 1 he is guaranteed to be alive in the 2nd time period, and he splits consumption equally between the two time periods. If p = 0 he spends all his wealth in the first period.

Hint
Generalizing from your 2-period exercise, let $N=m-1$ be the last possible period of being alive. The problem to solve is
\begin{equation}
\max_{\{c_n\}_{n=0}^N}\;u(c_0)+p(1)u(c_1)+\cdots+p(N)u(c_N)
=\max_{\{c_n\}_{n=0}^N}\;\sum_{n=0}^Np(n)\log(c_n)
\end{equation}
subject to a lifetime budget constraint: $c_0++c_1+\cdots+c_N=y$ where $y$ is the lifetime income.


Is there any economic meaning to log returns? I recently had a discussion with a friend regarding log returns and any economic meaning that could have as a measure in the sense of "assessing the benefit received for compensating work done in a previous time period."

He rather suggested that this was a mathematical construct beneficial in constructing stories such as historical var for the assessment of risk (owing to their additive nature.. $\log(a/b)+\log(b/a)$ causing no information loss).

Is there any economic meaning that could be ascribed to log or arithmetic returns?

Take the 1st order Taylor series the function $log(x)$. around one:
$$ log(x) \approx log(1) + \frac{x-1}{1} = x-1$$

Therefore, for a random variable r that is close to zero:
$$ log(1+r) \approx r$$

Log returns are typically something like $log(P_{new}/P_{old})$:
$$\Delta log(P_t) = log(P_{new}/P_{old}) = log((P_{old} + \Delta P)/P_{old}) = log(1 + \Delta P/P_{old}) \approx \Delta P/P_{old}$$
But $\Delta P/P_{old}$ is just the percent change in $P_{old}$. We've therefore shown that we can approximate the percent change in a variable $X$ with $\Delta log(X)$.

Why bother? Mostly because log differences can be summed but percent changes cannot be:

  Using logs, or summarizing changes in terms of continuous compounding,
  has a number of advantages over looking at simple percent changes. For
  example, if your portfolio goes up by 50% (say from \$100 to \$150) and
  then declines by 50% (say from \$150 to \$75), you’re not back where you
  started. If you calculate your average percentage return (in this
  case, 0%), that’s not a particularly useful summary of the fact that
  you actually ended up 25% below where you started. By contrast, if
  your portfolio goes up in logarithmic terms by 0.5, and then falls in
  logarithmic terms by 0.5, you are exactly back where you started. The
  average log return on your portfolio is exactly the same number as the
  change in log price between the time you bought it and the time you
  sold it, divided by the number of years that you held it.

Jame Hamilton: Use of logarithms in economics


Independence axiom of lottery when $\alpha \ge 1$ When studying preference over lotteries we learned the independence axiom which goes like this:

The preference relation $\succsim$ on the space of simple lotteries $\mathscr{L}$ satisfies the independence axiom if for all $L$，$L^\prime$，$L^{\prime \prime} \in \mathscr{L}$ and $\alpha \in (0,1)$ we have
$$L \succsim L^\prime \iff \alpha L + (1 - \alpha) L^{\prime \prime} \succsim \alpha L^{\prime} + (1 - \alpha) L^{\prime \prime} $$

What if $\alpha \ge 1$?  I guess that will be a quick extension for the case of $\alpha \in (0,1)$ but I don't know how to show it.

To understand why $\alpha$ must be constrained in $(0,1)$, one has to contemplate the meaning of the expression

$$\alpha L$$

when $L$ is a "lottery". How is a lottery denoted mathematically? Authors do not agree on that: for example, the way Jahle and Reny define a lottery (a "gamble" in their terminology), a lottery can be written as a vector whose elements are bivariate vectors themselves:

$$L=\{(p_1,w_1),...,(p_n, w_n)\}$$

where $p_i$ are probabilities, and $w_i$ are quantitative outcomes.

But MasColell, Whinston and Green, define a lottery as a vector containing only the probabilities:

$$L=\{p_1,...,p_n\}$$

and so the "Lottery Space" is a vector space, including vectors each containing only probabilities.

But in both cases, the authors make clear that an expression like $\alpha L$, when the time comes to translate it into a mathematical operation, denotes a multiplication of only the probabilities linked with $L$ by another probability, $\alpha$. This is implementing the reduction of compound lotteries to simple ones. JR describe it as "the decision maker cares only about effective probabilities of each outcome". MWG call it the "consequentialist premise", that allows them to work only with simple lotteries.

So it is not valid to consider $\alpha$ outside $(0,1)$, because it is defined as a probability (the open bounds are used in order to avoid triviality in the statement of the Independence axiom).

Also, the above imply that $\alpha$ does not directly interact with the quantitative outcomes of the lottery/gamble...

...which points to a (perhaps interesting, perhaps not) research direction: What can we say (if anything), if we start tweaking the outcomes linked with a lottery? Will "attitude towards risk" get in the way and prevent us from drawing any general conclusions? Or not?


Big research goals in economics? Are there any big, overarching research goals in economics?
Examples from other fields that come to my mind:

Mathematics: Millenium problems, Hilbert's problems, Collatz conjecture, ...
Computer science: Proof of P =?= NP, quest for an exascale system
Biology / Medicine: Cure disease, cancer, stop ageing

Probably, economics as a (social) science is inherently not the right field for such grandiose goals, but is there anything comparable?

Yes there are. Two big ones are:

Why are some nations rich and why are some nations poor?
What are the causes of large scale involuntary unemployment of resources and individuals (I.e. a depression).