Can I normalise the Dixit-Stiglitz price index to 1? I've built a model that I am trying to solve in Matlab. The model has Dixit-Stiglitz preferences and $m$ goods, so admits a price index of the following form:

$ P = \bigg[ \sum_m p_m^{1-\varepsilon} \bigg]^{\frac{1}{1-\varepsilon}} $

I would like to normalise this to 1. That is, everything in my model would be in terms of the final goods basket. However, the goods prices are already determined by the following well-known equation

$ p_m = P^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon} $

where $I$ is total income (you can think of it as $\sum_m p_m Y_m$) and $Y_m$ is production of good $m$.

Furthermore, given a set of wages and rental rates (supplied as a "guess" for Matlab's fsolve), the FOCs already (over)identify the capital and labour levels, and so $Y_m$ is determined, pinning down $p_m$. It seems like I can't set the price index $P =1$?

What shall I do? Shall I include the $P-1=0$ equation in my Matlab solver (adding to the overidentification issues!)? Shall I calibrate $\varepsilon$ to ensure $P=1$? Or do you have any other advice?

Thank you so much!

To me it depends. If you are dealing with price indexes that are somehow distributed spatially, i.e. you have $P_i \ \text{for} \ i=\text{NY},...,\text{CH}$ it would make no sense to normalize only one of them. I say "only" because it would be a priori impossible to normalize them all since, say, transport costs would ineluctably force them to be different.

If you are dealing with a unique price-index, I see no reason of why you could not do so. Indeed, note that you have

$ p_m = P(p_m)^{1-\varepsilon} \bigg[ \frac{I}{Y_m} \bigg]^{1/\varepsilon}$

which underlies an under-identified system. So, yes you can include the $P-1=0$ equation in your solver.

Regarding the contradiction it creates with your (over-identifying) wages and rental rates, to me it looks like you should take things upside-down. Under Dixit-Stiglitz monopolistic market, firms set their price by assuming a constant elasticity of substitution (CES), $\varepsilon > 1$, and profit maximization leads to a constant mark-up on variable cost

$p_m = \frac{\varepsilon}{\varepsilon - 1} c_m(w_m)$

where $c_m(w_m)$ is the unitary labor cost of producer $m$, function of wages $w_m$. So there are a priori nothing to guess freely at that level, just use the relation you have between prices and wages.

Also, the level of capital, $K_m$, should be set before any king of GE-resolution. Indeed this is a "calibrating" characteristic and I do not see why -- at least given the content of your question -- you want its obtention to be melt with that of the general-equilibrium (GE) vector of prices.

I would not calibrate $\varepsilon$ if I were you, since there is a big narrative behind the choice of its value: this is somehow representative of the "market depth", see Rivera-Batiz (1988) for an intuition about the level of elasticity that might be chosen. Do you want your narrative to be chosen by your solver?

Finally, you say that we can think of $I_m$ as $\sum_m p_m Y_m$, which is (only) true at equilibrium... But how is your $I_m$ formed ? It looks like each consumer also possesses her own firm (given that $I$ is indexed by $m$)? Is it $I_m = w_m + K_mr_m$, where $r_m$ is the rate of return of the firm $m$ ?

Need more precisions from you to go further.


Why does marginal cost (derivative of total cost) differ from variable cost at each level? Why does the marginal cost equation (as the derivative of total cost equation) make predictions of variable costs that are very different from costs calculated using the Total Cost equation?

Marginal cost is simply the change in cost divided by the change in quantity.

MC = ΔC / ΔQ

However, marginal cost also can be computed using the derivative of the Total Cost function.

Suppose you have a short-term Total Cost equation for a production case in which no capital is used; labor is the only input.

TC = w * L

The production function is

Q = L^(1/3)   ... therefore
L = Q^3

And given that the w = 1, then

TC = Q^3

Therefore the Marginal Cost equation, as the derivative of the Total Cost equation, would be...

MC = 3Q^2

Of course, when Q = 0 then the TC equation and MC equation = 0.
But raise Q to 1, and TC is now 1. Therefore, MC = ΔC / ΔQ = 1.

But the MC equation gives MC = 3.

The difference grows as Q increases more. When Q = 2, the TC equation returns 8, a cost change of 7, while the MC equation returns 12.

I understand mathematically why these are different, but I don't understand why both are supposedly correct and useful in economics.

Marginal analysis says to only produce a quantity if the marginal cost is less than or equal to the price at that quantity. But in this example, if the price were \$2, someone using the TC equation would produce the first unit for a profit of \$1 while someone using the MC equation would not produce the first unit because the MC equation predicts a \$1 loss. It seems like at least one of these methods lacks practical value. So, could someone explain the difference?

For $Q_1 = 1, Q_2 =2$, you calculate

$$TC(Q_2) - TC(Q_1)=7$$
which is the additional (incremental) cost to produce the second unit of output, and then you compare this to

$$\frac {dTC}{dQ} |_{Q=Q_2} = 12$$

which is marginal cost when you are already at production level $Q_2$. From this observation alone, the two are not comparable.

As for practical value: assume marginal revenue is a straight line at $P=8$. What to do? The "marginal cost equals marginal revenue" leads to  $3Q^2 \leq 8 \implies Q^*= \sqrt {8/3} \approx 1.633$. What's the problem with that?

Well, the product we are examining may not be divisible so much as to make incremental cost adequately close to marginal cost. This not-sufficient divisibility  may force us to resort to integer-value calculus, and use the concept of incremental cost instead : produce the highest quantity for which incremental cost is lower or equal to incremental revenue - and we will obtain the correct answer $Q^* =2$.

Note: "sufficient divisibility" is not a strictly "physical constraint" but is judged also on economic considerations. If we are contemplating producing, say, cars to the range of hundreds of thousands, then, even if each car costs say 20 thousand euros, the pains and computational inconvenience of integer-valued calculus may not worth the small reduction in profits if (by using marginal calculus) we end up producing one unit more than optimal given indivisibility.


Where do Montenegro and Kosovo get their euros from? Both Montenegro and Kosovo have unilaterally (?) decided to use euros as their official currencies. I think that most of their capital exists only as records in bank registers, so there is not much problem with that, but what about physical notes and coins that are in circulation? Where do they get them from?

Countries can get foreign currency from tourists, exports or intergovernmental aid programs. (And several other ways, but these should suffice here.)


EU-Canada Comprehensive Economic and Trade Agreement? Once applied, CETA will offer EU firms more and better business opportunities in Canada and support jobs in the EU, it will remove customs duties, end restrictions on access to public contracts, open-up the services market, offer predictable conditions for investors, etc.

Having in mind the existence of the North American Free Trade Agreement (NAFTA), does the enabling of CETA provide the US corporations (at least some of) the opportunities cevered in TTIP? For example, is it possiblede jure for a US corporation just to register in Canada and to access the EU market, achieveing in this way indirectly the aims of TTIP?

Update: reflecting the remarks in the comments, a more direct version of the question would be:

Is there a legal path through which CETA could act as a back door for TTIP?

I did not have either the time or the courage to plough through the text. The last consolidated text I could find stems from 2014 and has since been amended with an annex. Changes made are unclear to me.

The full text can be found here.

Excerpts that might be useful in this case are article 1.2 and 1.3 with respect to the scope of the application. Other excerpts might be the articles 17.2 and 17.3.

AS for the relevance of articles 1.2 and 1.3:
CETA applies to both the citizens and the territory of the agreeing countries. So, assuming USA corporations can have, and often do have, offices in Canada these companies can through these offices take advantage of the CETA agreement, assuming they keep to the regulations and requirements guaranteed under CETA. This means not everything from TTIP will be directly transposable. There are other treaties that cannot be overruled by CETA.

But in a way yes - a corporation holding office in the Canadian territories could in essence access the European markets under the same conditions. At least that is what I understood. I get back to the question when I'm through the 1598 pages of text and annexes (the latter being the greater part, but not less important).

Vague answer I know. I'm looking forward to a more in-depth conclusion because what you ask is something that a lot of people here in Europe are worrying about.


Can the Fed control the shape of the entire yield curve? Is it often taught that the Fed can only affect the short end of the (Treasury) yield curve, while the long end is governed by investors' inflation expectations. Could the Fed in principle control the long end as well, e.g. by buying long dated Treasuries. Could the Fed in theory set the whole yield curve to zero?

Yes, and they don't need any "unconventional" policy tools to do this, all they need to do is precommit to a future path of short-term nominal interest rates which they will maintain no matter what. The reason that in practice the Fed is unable to control the long end of the yield curve is because the Fed is not an institution which precommits to these kinds of policies - their policy decisions are always contingent on the information available at the date they take the decision, which means bond investors will try to forecast Fed funds rate movements not by looking at what the Fed says they will do in the future, but by forecasting the relevant macroeconomic variables that enter the Fed's decision process and thinking about how the Fed will react to what they expect will happen. Long-term bond yields reflect investors' future short-term nominal interest rate expectations, and by Fisher you may split that into inflation and real interest rates, so inflation expectations are a part of it, but they are not the whole story. Could the Fed set the entire yield curve to zero? Sure, if it could convince investors that it would maintain a zero interest rate policy for 30 years.

If government bonds bear no liquidity premium, then buying up lots of long-term debt and replacing it with short-term reserves will have no effect on long-term bond yields given the short-term nominal interest rate path the Fed intends to follow in the future. If long-term debt is important for some other purpose, for instance to meet some regulatory safety requirements, as collateral in secured lending, etc. then the Fed buying up long-term government bonds may create a shortage of long-term debt, increase the liquidity premium and force long-term bond yields down, but this policy is not optimal. When the Fed tries to reduce long rates, they intend this to have a stimulative effect on the economy today; reducing yields through this kind of mechanism would instead lead to inefficiency a la Friedman's rule.

In practice, a lot of the correlation between QE announcements and bond yield movements comes from what Bernanke called the forward guidance effect: investors interpret QE as a sign of the central bank's commitment to maintain lower policy rates in the future. According to Bernanke, this effect was the primary reason for the "taper tantrum" in 2013, and when the Fed clarified that "tapering" would not mean that the Fed would start raising policy rates sooner, bond yields came back down again.


Understanding interest rates When a central bank 'pays' interest rate X / year, what does it mean exactly? Is that pure money printing? Where does that money (interest rate) come from? does it increase the debt of the country?

A popular way to fight high inflation is to increase interest rate, but doesn't that mean printing more money -> more inflation?

Thank you!

When a central bank 'pays' interest rate X / year, what does it mean
  exactly?

It means that by lending to the central bank \$100/(1+X) at the beginning of the year, the central bank will pay \$100 at the end of that year.

  A popular way to fight high inflation is to increase interest rate,
  but doesn't that mean printing more money -> more inflation?

Not necessarily. Increasing the interest rate may attract cash flows from foreign lenders/investors. The effects of it on the exchange rate (appreciation of the local currency), currency reserves, and/or trade balance tend to alleviate the need to merely print more money in the local economy.


Dummy variable regressor OLS coefficient formula Consider the standard linear regression model: $y_i = \alpha + \beta D_i + e_i$ where the coefficients are defined by linear projections and $D_i$ is a dummy variable. In the population, the coefficients are given by:

$$\alpha = E[y_i \mid D_i =0] \ \text{and} \ \beta = E[y_i \mid D_i = 1] - E[y_i \mid D_i =0]$$

Using OLS to estimate the coefficients, we get:

$$\widehat{\alpha} = \overline{y}_{D_i=0} $$

$$\widehat{\beta} = \overline{y}_{D_i=1}-\overline{y}_{D_i=0} $$

In other words, $\widehat{\alpha}$ is just the sample mean of $y_i$ in the subsample with $D_i=0$ and $\widehat{\beta}$ is the difference in sample means of the two groups. The expressions seem very obvious because they are just sample versions of the population, but my question is, how can we arrive at the above coefficient estimates by using the standard OLS formulas? That is using:

$$\widehat{\alpha} = \overline{y} - \overline{D}\widehat{\beta} \ \ \text{and} \ \ \widehat{\beta} = \frac{\sum_{i=1}^{N}(D_i - \overline{D})(y_i - \overline{y})}{\sum_{i=1}^{N}(D_i - \overline{D})^2}$$

To start, let's derive $\hat{\beta}$. As you note:

$$\hat{\beta} = \frac{\sum(D_i - \bar{D})(y_i - \bar{y})}{\sum (D_i - \bar{D})^2} $$

The key to evaluating this is to recognise that $D_i$ is a binary variable that equals $0$ or $1$. Thus, we can use standard results on Bernoulli random variables.

To evaluate the numerator, note that if $D$ is Bernoulli distributed:

$$ \mathrm{Cov}[D, y] = \mathbb{E}[Dy] - \mathbb{E}[D]\mathbb{E}[y] = \mathrm{P(D=1)}\mathbb{E}[y|D=1] - \mathrm{P(D=1)}\mathbb{E}[y] $$

By the sample analogue of this:

$$ \frac{\sum(D_i - \bar{D})(y_i - \bar{y})}{n} = \bar{D}(\bar{y}_{D_i=1}-\bar{y})$$

To evaluate the denominator, note that if $D$ is a Bernouilli random variable:

$$\mathrm{Var[D]} = \mathrm{P(D=1)}[1-\mathrm{P(D=1)]}$$

Sample analogue:

$$\frac{\sum (D_i - \bar{D})^2}{n} = \bar{D}(1 - \bar{D})$$

We therefore see that:

$$\hat{\beta}=\frac{\bar{D}(\bar{y}_{D_i=1}-\bar{y})}{\bar{D}(1 - \bar{D})}
= \frac{\bar{y}_{D_i=1}-\bar{y}}{1 - \bar{D}}
\\ = \frac{\bar{y}_{D_i=1}-(\bar{y}_{D_i=1}\bar{D}+\bar{y}_{D_i=0}(1-\bar{D}))}{1 - \bar{D}}
\\ = \bar{y}_{D_i=1} - \bar{y}_{D_i=0}$$

which is what we wanted to show.

It is then trivial to derive:

$$\hat{\alpha} = \bar{y} - \bar{D}\hat{\beta} =\bar{y}_{D_i=1}\bar{D}+\bar{y}_{D_i=0}(1-\bar{D}) - \bar{D}(\bar{y}_{D_i=1} - \bar{y}_{D_i=0}) = \bar{y}_{D_i=0}$$


model design - fixed effects model for paired differences I have two panels. One panel that consists of an economic indicator variable across firms and years. The second panel consists of the same economic indicator variable across the same firms and years. However, the indicator variable in the second panel is calculated on the basis of different data than the one in the first panel, I call that data "adjusted". Hence, the values of the variable in the second panel data differ from that in the first panel. Put differently, I have two panels of paired observations of a single variable.

I now want to assess wether the indicators based on the adjusted data are significantly larger than the ones based on the unadjusted data.

Or simply: Are the values in one panel larger than the values in the other panel that describes an alternative set of values of the same variable for the same firms and years.

One issue to keep in mind is that the indicator variables are autocorrelated by construction, since they are computed on a rolling (overlapping) window basis. What I have come up with is as follows:

Step 1: I calculate the paired differences, i.e. $\delta_{i,t} = x_{i,t} - y_{i,t}$, where $x_{i,t}$ is the economic indicator based on adjusted data and $y_{i,t}$ is the economic indicator based on unadjusted data.

Step 2: I estimate a fixed effects model on a constant only, i.e. $\delta_{i,t} = \alpha + \gamma_{i} + \epsilon_{i,t}$, where $\alpha$ is the constant and $\gamma_{i}$ is a firm fixed effect.

I test the hypothesis that $\alpha > 0$ against the null hypothesis $\alpha = 0$ using the usual t-test. I employ robust standard errors to account for the autocorrelation that I expect in the $\delta_{i,t}$.

If the $\alpha$ is significantly larger than $0$, I assume that the $x_{i,t}$ is, on average, larger than $y_{i,t}$.

I am not an expert in panel regression, so my question is: Is this model design reasonable or flawed? Moreover, are there "better" ways to assess my question of interest?

--------------------------------------------------------------------------------

EDIT 1: What I mean by paired samples is that the observations are paired, i.e. they are somewhat linked because they refer to the same {i,t}. That is why I compute the differences of each "pair" (paired t-test).

EDIT 2: I actually analyse the quality of reported earnings. The one panel consists of quality metrics that are calculated based on actual reported earnings ($y_{i,t}$). The other panel consists of quality metrics that are based on "alternative" earnings values ($x_{i,t}$). What I mean by that is: For example, one could hypothetically imagine a firm in the US applying different accounting standards (e.g. German accounting standards), which leads to different values of reported earnings of the same firm in the same period. Now simply imagine that the second panel are earnings quality metrics that are based on earnings quantities reported under a set of different accounting regulations than the one in the first panel, but for the same years and firms!

EDIT 3: The panels consists of around 500 firms and 10 years each. I want to conduct the test for the entire sample. That is, I have 500 $\delta_{i,t}$ (500 firm-year differences).

The constant term in your final FE model has no specific meaning without further restrictions. For Stata, it is only chosen such that the (sample) mean of the estimated individual effects add up to 0. So your testing $\alpha>0$ is in fact based on
$$\frac{1}{n} \sum_{i=1}^n \hat\alpha_i,\quad \hat\alpha_i = \frac{1}{T_i} \sum_{t=1}^{T_i} (x_{it} - y_{it}),$$
where I am using the $T_i$ notation in order to allow for the panel data to be unbalanced.

For balanced panel data, the test is (almost) identical to one based on OLS if you use a cluster standard error for OLS because then
$$\frac{1}{n} \sum_{i=1}^n \hat\alpha_i = \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T (x_{it}-y_{it}) = \frac{1}{nT} \sum_{i=1}^n \sum_{t=1}^T (x_{it} - y_{it}),$$
where the LHS is a statistic from the FE regression and the RHS is the corresponding one from the OLS regression. OLS, FE and RE (and BE) will all give an identical intercept estimate, I think. Only standard errors will be different. If the panel data set is unbalanced, the method using the FE regression and that using the OLS regression differ in how the average differences ($\hat\alpha_i$) are weighted across different $i$.

The test you mention is about whether the mean of $x_{it}-y_{it}$ is bigger than 0 or not. This looks OK. What else? Which technique to use (OLS or FE or RE or BE) doesn't seem to be critical here. I would just use OLS + cluster se, as it is simple to both the researcher and the reader. Try:

xtset id year
gen d = x - y
reg d, vce(cl id)
xtreg d, fe
xtreg d, re
xtreg d, be


First order conditions in a Moral Hazard problem Consider a principal-agent problem where effort is observable. Let $\pi$ denote profit, let $w$ denote the wage to be paid to the agent, and let $f(\pi|e)$ denote the pdf of the profit, conditional on effort $e$. Further, $e \in \{e_L,e_H\}$. The problem for the principal is then as follows: For each level of effort, $e$: $$ \min_w\int w(\pi) f(\pi|e)\, d\pi$$ subject to $$\int v(w(\pi)) f(\pi|e)\,d\pi - g(e) \ge \overline u$$
(Here, $v$ is the utility function for the agent, $g$ is the cost function of effort and $\overline u$ is the outside option for the agent). Then, choose $e$ that gives maximum profit given the wage $w$, which has now been solved for at each effort level.

I'm not sure how to take the first order conditions in the minimisation problem above. If $w$ were the argument for the integral, it'd be manageable.

Notice that MWG choose a wage function $w(\pi)$ to maximize profit, not a wage level $w$. As footnote 6 on p.481 of MWG says:

  The first-order condition for $w(\pi)$ is derived by taking the derivative with respect to the manager's wage at each level of $\pi$ separately.

In other words, you'd treat the wage function as a variable, e.g. $w(\pi)=z$, and solve
\begin{equation}
\max_{z} \int [\pi-z]f(\pi|e)\mathrm d\pi +\gamma \left[\int v(z)f(\pi|e)\mathrm d\pi-g(e)-\overline u\right],
\end{equation}
which will yield the FOC
\begin{equation}
-f(\pi|e)+\gamma v'(z)f(\pi|e)=0.
\end{equation}
Substituting $z$ with $w(\pi)$, you'd have condition (14.B.3) in MWG.


Where do I find the US GDP growth on FRED? I have never used FRED before. I need to locate the annual growth rate in Real GDP to do some data analysis for my intro macro class. The TA and the Prof were utterly unhelpful and I'm frankly lost.

Specifically, I think I am supposed to use this site:
https://research.stlouisfed.org/fred2/series/A191RL1Q225SBEA

If you click on Graph tab, I can adjust the data present. However, there are a bunch of options and I can't tell which one I need.

MY QUESTION

(1) Am I looking in the right spot on FRED for the aforementioned information I'm interested in?
(2) If I am in the right spot, which version of the data do I want? Do I want "Percent change from preceding period" or "Percent change from year ago" for example. Also, what aggregation method.

I googled this terms to no avail.

With the link you posted you get quarterly data, so to get the yearly growth rate you need to change the frequency to annual. There are different aggregation methods but if it's not specified in the question you were asked I don't see how one should be more right than another.


Link goods to sector (UCC to NAICS) Universal Classification Code (UCCs) are six digit codes that identify items or groups of items. Here are some UCCs:

190903 Food and non-alcoholic beverages at restaurants, cafes, fast food places on trips
190904 Food and beverages purchased and prepared by CU on trips
200900 Alcoholic beverages at restaurants, cafes, bars on trips 210110 Rent of dwelling
210210 Lodging away from home on trips
210310 Housing for someone at school
210901 Ground rent - owned home

They are for example being used in the consumption expenditure survey to identify consumption items.

Is there any way (e.g. existing bridge data base) to link these UCC to the sector of production? For example, NAICS?

Unfortunately, I'm not aware of any publicly-available concordance, and I can promise you that there's no clean mapping between UCC and NAICS. Obviously, some categories will map well at low granularity (UCC 190903 will definitely map to 722, for example).

While I'm sure it's fine for whatever you're doing if multiple UCC codes map to a single NAICS code, the big problem is that some UCC codes will map to multiple NAICS codes. For example, 190113, "lunch at vending machines and mobile vendors" will map to some combination of NAICS 454210, "vending machine operators" and NAICS 722330, "mobile food services".

The way this is usually handled is through some form of allocation— you assume, for example, that the CES population you're observing has exactly the same distribution of expenditures as the overall economy.


Understanding GDP Chart When I Google "GDP of 1950", I get ~$300 Billion.

Looking at the graph @ https://fred.stlouisfed.org/series/GDP . I see

Q1 280 billion
Q2 290 billion
Q3 308 billion
Q4 319 billion

My questions:

Does last quarter value represents the GDP of the year? I thought we generally add all 4 quarters to get the annual number in finance?
Does GDP reset to 0 every year? So Jan 1st 12:00am, value starts at 0?
Is GDP culimnative? So the GDP created Q2 1950 is actually ~10 billion, and we just add that to the value of the previous quarter (to get ~290 billion)?
If the GDP value is culminative,  and you only add to the value of last quarter, the only reason why you can have a lower overall GDP is export is higher? The GDP equation is (GDP = C + I + G + (X – M)). The only subtraction I see in the equation is export. Or can  consumer spending, business investment, and goverment spending be negative???

What you are seeing in the data provided in the google search is an annual estimate  of GDP.  In the FRED data download is GDP reported on a quarterly basis.

Does last quarter value represents the GDP of the year? I thought we generally add all 4 quarters to get the annual number in finance?

Add up the all the quarters nominal estimates and divide the total by four to get the annualized numbers. (but when it reals you do not add them up, as real terms are not additive)

Is GDP culimnative? So the GDP created Q2 1950 is actually ~10 billion, and we just add that to the value of the previous quarter (to get ~290 billion)?

Nominal GDP estimate is the market value of the goods and services produced by labor and property located in the United States for that particular time period (quarter).  The estimate is not a running total from previous periods.

Now if you are asking if the nation's economy grow or contract compared to the previous period measured?   Then the GDP growth rate tells you how fast a county's economy is growing. It compares real GDP from one quarter to the next.

new: 2. Does GDP reset to 0 every year? So Jan 1st 12:00am, value starts at 0?

It depends on the country.  In the US, the source data for GDP is collected every month and the GDP is estimated at every quarter  once all final estimates are collected for each quarter in the year then the annual estimate is published (note there are revisions to the data and on a quarterly basis revisions are published 3 times).

https://www.bea.gov/resources/learning-center/what-to-know-gdp

If the GDP value is culminative, and you only add to the value of last quarter, the only reason why you can have a lower overall GDP is export is higher? The GDP equation is (GDP = C + I + G + (X – M)). The only subtraction I see in the equation is export. Or can consumer spending, business investment, and goverment spending be negative???

It would be really bad if GDP in nominal or real terms are negative.  However, GDP growth (percentage change)  can be negative but the interpretation is different as the growth rate tells you how fast a county's economy is growing.
The (X – M) term is defined as “net exports” .  So in the case where imports M is greater than exports X then net exports would be negative.      The other components would typically not have a negative values unless something catastrophic happens.


Teaching changes in "demand" v. "quantity demanded" I am always struggling with teaching this one.

The terminology of standard intro textbooks is unfortunate, although it's so pervasive that we probably have to live with it.

By a "Demand", intro textbooks really mean a "Demand function" (but talk about "Demand" either for concision, or because they think the word "function" will scare intro students).
Having shied away from calling a function a function, intro textbooks then cannot speak of "the demand at price $P$" (just like you would speak of the value of $f(\cdot)$ at $x$), and need to introduce the awkward terminology "quantity demanded at price $p$".

That's an ok compromise, until you reach the usual section on "changes in the quantity demanded v. changes in demand".

There, students are often told that we should only talk of a "change in the quantity demanded" if the demand function $D$ is fixed, and that such "changes in the quantity demanded" must be the consequence of a change in price only (typically following a change in supply).

Then how can I speak of the comparison between $D(P) > \tilde{D}(\tilde{P})$ without being terribly confusing to my students?

If I say,

  1) "Following a change in demand from $D$ to $\tilde{D}$, the equilibrium price changed from
  $P$ to $\tilde{P}$ and the quantity demanded therefore changed from $D(P)$ to
  $\tilde{D}(\tilde{P})$",

I think I am guaranteed to confuse the hell out of them, because the textbook essentially tells them that "we should never talk of a 'change in quantity demanded' unless the demand function remains unchanged".

At the same time, is there any other ways to formulate 1)?

Should I use a different word and speak, e.g., of a "move" or a "movement" in quantity demanded from $D(P)$ to $\tilde{D}(\tilde{P})$  to avoid any confusion with a "change in quantity demanded".
Or is it better to be explicit about the ambiguity?
Or is it better to try and avoid the issue as much as possible?

What is your experience teaching this? What are your tricks to get around this terminological difficulty? (Or, maybe, why do you think it is not a terminological difficulty at all?)

I think the terminology below is pretty unambiguous and also common:

  a shift in the demand curve

to refer to movements from $D$ to $\tilde D$ (using shift is quite more specific than using change, as it more clearly refer to a change in the whole demand schedule).

  a movement along the demand curve

to refer to changes in $D(P)$ induced by movements in the supply curve (this I normally do it by showing the "dynamics" in the graph, from the original equilibrium to the new equilibrium along the demand curve).

  a change in the quantity demanded

to compare different $D(P)$ in the horizontal axis. A mention to the concept of equilibrium can be useful here too. For example, a change in the final/equilibrium quantity. Here the demanded adjective is in a sense redundant, as in the equilibrium the quantity demanded and supplied is the same, but if you are being very forma as in $D(P)=S(P)$ then its fine. This terminology is different from the first one (shift in the demand curve) in the sense that the latter is a ceteris paribus exercise, whereas the former is not (unless supply is perfectly in/elastic).

Regarding your question about whether to use the textbook's proposed terminology, I would not advice so. First of all, it is not clear what the authors propose to use to refer to changes in $D(P)$ induced by "changes in the demand [curve]". Why to have a separate notation for such changes and those induced by changes in the supply? That is very arbitrary.

Additionally, as their restrictive terminology is probably not the one you are used to (judging by question and comments), forcing you to adopt such incomplete terminology (which is likely not to come to you naturally) might confuse your students because you might not always consistently use it. Provided you want to keep using that textbook, I would tell them something like

  beware of this very restrictive termonology issue, which is uncommon in other books. More common instead is this terminology, and what I normally use is this. In any case, all this is not very relevant for understanding. In your answers, always explain what you mean, ideally with the aid of annotated graphs. Correct answers (mostly) do not depend on terminology.


Deriving the golden-rule savings rate in a Solow Model Consider an economy described by the per worker production function function:
 $$y = f(k) = 2k^\frac{1}{2}$$   and a depreciation rate $δ$ of $.05, 5%$.

Considering what we know to be true of the golden rule level of capital, find the golden rule level of capital $k_{gold}$ and output $y_{gold}$ using the above information.

Suppose there is no population growth rate $n = 0$.  Given the Solow model equation $(\Delta k = sf(k) – \delta k)$, and the answers for $k^*$ and $y^*$ you got in part A, what savings rate must this country have to put us at the golden rule level of capital?

Here is what I did:

  In part A, I took the derivatives of the depreciation rate, and the
  per worker production function, and got that $K =.05$ just by the
  simple power rule. To find the output I plugged $K=.05$ back into the
  per worker production function to get $2(.05)^{1/2}$ and came up with
  an output of $.447$

  Part B: Since the equation I need now is $sf(k) = \delta k$ which
  using what I know, $s\times.447 =.05 \times .05$ Solving for $s$ I get
  that the savings rate is $0.556$ %.

However, this is not correct. Please help me find the correct solution method and correct solution.

The "golden rule" is the level at which steady-state consumption is at a maximum, given the parameters of the model. Steady state consumption is

$$c^* = (1-s^*)\cdot f[k^*(s^*)] = f[k^*(s^*)]  - s^*f[k^*(s^*)] \tag{1}$$

where $0<s^*<1$

We also have that, at the steady state (for constant capital)

$$s^*f[k^*(s^*)] = \delta k^*(s^*) \tag{2}$$

Inserting $(2)$ in $(1)$,

$$c^* =  f[k^*(s^*)]  - \delta k^*(s^*) \tag{3}$$

We want to maximize steady-state consumption, so we take the first derivative and set it equal to zero,

$$\frac {\partial c^*}{\partial s^*} = f'[k^*(s^*)]\cdot \frac {{\rm d}k^*}{{\rm d}s^*} - \delta \frac {{\rm d}k^*}{{\rm d}s^*} =0 $$

$$\implies \big(f'[k^*(s^*)]\ - \delta \big)\frac {{\rm d}k^*}{{\rm d}s^*} =0 \implies f'[k^*(s^*)] = \delta \tag{4} $$

From the production function we have

$$f'(k) = \frac 1{\sqrt {k}} = \delta \implies k^* = \frac 1 {\delta^2}$$

The rest are evident.


Market Behavior under expected and unexpected news I found the following text in the paper "Real-time Price Discovery in Foreign Exchange"

  The general pattern is one of very quick exchange rate conditional mean adjustment, characterized by a jump immediately following the announcement, and little movement thereafter. Favorable U.S. “growth news” tends to produce dollar appreciation, and conversely. [...] [A] one standard deviation U.S. payroll employment surprise tends to appreciate (if positive) or depreciate (if negative) the dollar against the DM by 0.16 %.17 This is a sizeable move [...]

implying that this news are only available to the market once the announcement is made by the US authorities. Other similar announcements are considered: Retail Sales, Industrial Production, Consumer Credit, etc.

I am having trouble to understand why this is an acceptable explanation. From my point of view, the markets must "feel" the increase (decrease) and adjust accordingly before the announcement. Any price change after an announcement seem as irrational or herding behavior to me. On the other hand, I can think of examples in which a real surprise causes movements in the market: changes in Central Bank policy, social unrest, accidents, etc.

Am I missing something?

Short answer: the market has a forecast of the government numbers but the market considers the actual government numbers to contain information not available to the market. Therefore when unemployment  is lower than the market forecasts this indicates that on average things will be better than than the market forecast before the release. This is not to say that the forecast has no use, nor that the government number is perfect. Instead, it merely requires that there be some information in the release not in the forecast such that differences from the forecast are informative.


Phenomenon of equal prices without price collusion Some time ago, I came across a term describing a specific phenomenon observed in free markets. I'm now looking for this term.

When comparing prices of certain staples in some countries, one quickly notices that their prices are exactly the same in different supermarkets and discounters. For example, in Germany, the cheapest price for 1 kg of sugar in any supermarket or discounter is 65 ¢. I have never seen a different price for this for years. The exact brand of sugar is different for different supermarkets or discounters, but the cheapest offer is always 1 kg for 65 ¢. It's the same for eggs, milk, flour, yeast, salt, curd, and many other staples.

This price does not change as its in every competitor's interest to keep it exactly where it is:

If a market participant increases the price to make more money off the specific product, many people will avoid the supermarket or discounter because their prices are perceived to be higher in general as the staple in question serves as a price comparison between shops for many people.
If a market participant decreases the price to attract more customers, any competitor will decrease the price of their equivalent product to not appear to have higher prices than the participant that decreased their price.

This phenomenon results in supermarkets and discounters limiting the amount of a product customers can buy rather than increasing prices when running short of resupply or taking losses.

Note that I'm not looking for the term "price collusion". The term describes the exact phenomenon described above and does not suggest that price collusion is happening.

I'm not 100% sure this term exists in English. I only know that it exists in German because I read about this phenomenon on the German version of Wikipedia, but no one I asked in real life can tell me what it is. I guess that it exists in English, too, as it is an economic phenomenon not limited to any specific geographic region. Now I need the English term (if it exists) for usage in an English document.

I know you are at pains to say you are not looking for 'collusion' as an answer, but what you describe sounds very much like tacit collusion. Here's a quote from Competition Policy by Massimo Motta (pp138–140):

  I briefly characterise the concept of collusion from the point of view of industrial economics. Note that here I will not use the term "collusion" as a synonym for "collusive agreement that should be outlawed". Indeed, one of the main themes of this chapter is that whereas in economic theory collusion is defined as a market outcome (i.e., "high prices", in a sense to be specified further below), anti-trust authorities and judges should consider as illegal only practices where firms explicitly coordinate their actions to achieve a collusive outcome.

  [...]
  In economics, collusion is a situation where firms' prices are higher than some competitive benchmark. [...] In any case, in economics collusion coincides
  with an outcome (high-enough price), and not with the specific form through
  which that outcome is attained.

  [...] It is important to stress that [colluding firms need] not talk to each other, neither directly nor through intermediaries: collusive prices will arise through purely non-co-operative behaviour of the sellers. In other words, if detection of deviations is rapid, and if (market) punishments of deviations are likely and credible, then tacit collusion can arise: firms do not necessarily have to talk to each other, let alone agree on complicated schemes, for a collusive outcome to be sustainable. All that is needed is the awareness that a deviation will be identified, and that a "punishment" will follow.

The 'punishment' that Motta refers to is exactly the idea that, if one firm cuts its price to break the tacitly collusive status quo, other firms will 'punish' it by responding with a price cut of their own.

Sometimes, tacit collusion is known by the alternative name conscious parallelism to reflect the idea that firms are consciously mimicking each others' strategies. Particularly among legal scholars, the term conscious parallelism is used to mean coordination of prices without an agreement, whereas the term collusion is reserved specifically for the case in which an agreement has taken place. Among economists, as Motta explains above, both practices are known as (respectively, tacit and explicit) collusion and the conscious parallelism terminology is less common.


Is there any scientific proof that 2%-3% target inflation rate is ideal? There are a couple of central banks (RBA, Federal reserve for example) that use inflation rate as a key indicator to regulate their official cash rates.

Those target inflation rates usually range from 2%-3%.

This is taken from federal reserve's official website:

  Over time, a higher inflation rate would reduce the public's ability to make accurate longer-term economic and financial decisions. On the other hand, a lower inflation rate would be associated with an elevated probability of falling into deflation

World has changed in the past 10 years or so:

World's prevailing overnight cash rates have plunged from 3%-20% in the 20th century to 0%-3% in the past 5 years.

So is there any scientific proof that 2% inflation rate (or 2%-3% in RBA's case), is still the optimum target in today's low interest rate regime?

Science doesn't work on the basis of "proof". Proof is something that belongs only in the realms of mathematics and philosophical logic. Whereas science works on the basis of the best-available explanation that fits the available evidence.

That's as true of physics as it is of economics.

The challenge in economics is that it's much harder to do experiments to gather additional evidence, particularly in macro-economics. So decisions have to be made on the basis of quite thin evidence. Nevertheless, the principle is the same: we look for the best-available explanation that fits the available evidence.

The point of a 2-3% inflation target was that it was neither too high nor too low. And that's based on the harm we see being done during periods of deflation (such as the Great Depression) and during periods of high inflation (the oil crises in the early and late 1970s).

And because it's a target, rather than a cap or a floor, that means that it's applicable in a period of high inflation and high interest rates; and it's applicable in a period of near-zero inflation and near-zero interest rates. The evidence that supported this target pre-2007, is still the evidence that supports it now.


How can the bond market be overvalued when the value of a bond is a function of interest rates? I seem to have a major misconception in my understanding of the bond market and how interest rates and demand and supply influence valuation levels. I hope someone will be able to clarify.

My thinking is as follows. If the fair value of a bond is the present value of the cash flows resulting from coupons and principal repayment, discounted at an interest rate that depends on a policy interest rates, how can the bond market be overvalued, given a specific set of interest rate levels?

I understand the idea of overvaluation in equity markets, where demand and supply freely determine prices, and demand may be artificially high in expectation of fiscal stimulus or because investors buy into unrealistic success narratives of specific companies. But in bond markets, the situation is a little more complex because both the policy interest rate and demand and supply feature in the valuation mechanism.

So my questions are the following:

How to interest rate levels on the one hand and demand and supply on the other hand interact to determine the market price of bonds?
Can the bond market be overvalued or can bond market valuations be inferred perfectly from policy interest rates?
How can it be that bond markets are referred to in the media as overvalued when valuations depend on interest rates? Do they mean that market participants are assuming inappropriately low spreads to discount bond-related cashflows in valuation?

Thank you in advance for any clarifications you may be able to provide!

How can the bond market be overvalued?

To value a bond, you need, as you mention, to discount future cash flows. The cash flows (or coupons) are fixed, but the discount rate is not, and every investor may have a different view on interest rates and hence price bonds differently.

Therefore, someone thinking that interest rates will remain at rock-bottom levels will find different prices for bonds from someone who thinks that interest rates will rise sharply in the near future (the first investor will find higher prices).

So, in your case, someone thinking that the bond market is overvalued simply has a different view on interest rates than the consensus on the market. To be more precise, they expect interest rates to increase faster than the market consensus.

Bond Prices and interest rates

Taking into account what we have said above, you can view the bond market as follows: every investor forms a view on discount rates, for a given bond. Doing so, they actually compute the yield, that is the discount rate that makes the NPV of the cash flows of the bond equal to 0.

However, investors disagree on where the yield should be. They will thus trade the bond between them until they all agree.

Let's take a numerical example :

Investor A owns a Bond, which has a face value of 100, a coupon of 1 and will mature in one year.
Investor A thinks that the yield of this bond should be 2%
Investor B thinks that the yield of this bond should be 0.5%

The value of this Bond to Investor A is : $(100 + 1) / 1.02 = 99.02$
The value of this Bond to Investor B is : $(100 + 1) / 1.005 = 100.50$

Therefore investor A and Investor B will both improve their current state by trading the bond anywhere between \$99 and \$100.5, ie. Investor A should sell the Bond to Investor B.

If the market price of the bond is \$100, then Investor A thinks that the bond is overvalued, and Investor B thinks that the bond is undervalued.


Who is Jacob? And where did he say this? In Karl Marx "Das Kapital", in Ch I.1 we find this quote

  "Jacob questions whether gold has ever been paid for at its full
  value."

Is Jacob Vanderlint? Where in his book “Money Answers all Things” is that claim about gold's value?

As we discussed in the comments, in  William Jacob's An Historical Inquiry Into the Production, and Consumption of the Precious Metals, on page 101 it says

  It is probable that in all ages those metals have cost more in their
  production than their value ever repaid.

This is the likely source of the quote.


Regression Line Method Suppose we are given a set of observations on income $X_{i}$ and consumption $Y_{i}$ and we plot all the $(X_{i},Y_{i})$ on a graph. We want to draw a sample regression function as close as possible to the existing yet unknown population regression function. Why is it considered wrong to take $\sum \hat u_{i}$ as small as possible to draw our regression function?

My study book says that this method assigns "equal weights" to the $\hat u_{i}$ which is not accurate. However, I also agree with the least order square method.

Anyone have a specific example to show me that the normal summation method leads to bad estimation?

Thought about a very simple example: Suppose we have two distinct points $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$. The best regression line for these two points would be the line that passes through both points. However, any line passing through $(\frac{X_{1}+X_{2}}{2},\frac{Y_{1}+Y_{2}}{2})$ would let $\hat u_{1}+\hat u_{2}=0$, just like the perfect regression line.


Matching unpopularity It seems that in economics, matching techniques (such as pscore, covariate matching, regression adjustment) are not as popular as selection-on-unobservables techniques (such as DiD, IV).
Indeed I recently had a paper rejected by one referee on the basis that "no-one uses matching these days".
If that's true then, why? In some case it's not possible to come up with a natural experiment or perfect instrument to tease out treatment effects, and so matching strikes me as attractive.

It's not true that "no-one uses matching these days". It's a bit too definitive. For example, see Bilicka (2019) who recently published in the American Economic Review. I am sure it's possible to find other recent examples.
It's true however that the matching estimator is much less popular than the DiD or IV. This is because the results are not always consistent and robust across estimators. They sometimes differ depending on the estimator used (propensity score matching, PSM, or nearest neighbor matching, NNM) and the way in which the treatment and control groups are matched. It is also sometimes difficult to obtain matched samples that are as comparable as those in Bilicka's paper (see her Table 1).
As an example, there is a well know controversy in the Journal of Econometrics, involving Rajeev Dehejia, Jeffrey Smith and Petra Todd.

Smith, Jeffrey and Petra Todd. 2005. "Does Matching Overcome LaLonde's Critique of Nonexperimental Estimators?" Journal of Econometrics 125(1-2):305-353.
Dehejia response (2005)
Smith, Jeffrey and Petra Todd. 2005. "Rejoinder." Journal of Econometrics 125(1-2):365-375.
Dehejia Final Thoughts

My recommendations would be

to be very specific about how you match your treatment and control groups.
to provide lots of robustness checks to show that your results are not sensitive to a particular estimator, for example, PSM or NNM with Euclidean or Mahalanobis distances.

Good luck with your resubmission!


How to deal with challenging exam questions and truly understand concepts? In my university, as well as many others around the world, undergraduate Economics exams can often be quite challenging – necessarily so, given that this is higher education. My question is – how can I meet this challenge?

A common source of frustration after exams is that “these questions were so different from what we did in class!” Over the past few semesters, I’ve come to realise that a difficult Economics exam question is difficult because it challenges the assumptions with which we learnt the central concepts and/or introduces external conditions which makes it difficult to directly apply how we learnt it in class.

For instance, just as a very simple example, in intermediate microeconomics, while we may have learnt how to solve for the optimal basket for a consumer who has a Cobb-Douglas utility function, in exams we could be “suddenly” thrown a scenario where the consumer’s utility function is now a minimum (min) function when we have not seen a min function before. Given the necessary scaffolding so that we know how a min function works, we are expected to solve for the optimal basket.

Another example: in introductory Econometrics, the topic on "estimation" focused on estimating mean, variance etc...(typical stuff). In exams, we were asked to estimate a cumulative distribution function i.e. P(X<=x). While we had learnt distribution functions quite extensively, we were never asked to estimate it in class.

In some ways the above examples are over-simplified, but the point is to illustrate a few ways “typical” questions could be tweaked to be challenging.

So my question is – how do I study Economics more skilfully so that I can be able to face up to these sorts of questions during the exam? Alternatively, how do I truly understand the concepts so that I can answer the question no matter how it’s tweaked – in a realistic amount of time under exam conditions. I can’t possibly compile a list of all permutations and combinations of how the question might deviate from the typical examples – that would be too inefficient, especially since the “assumption challenging condition” could come totally unexpectedly. Students are often advised to “truly understand” the concepts – but I have not heard enough on how this can be achieved. How can I do this?

I'm saying this from the perspective of a guy who was a student and a TA.

Understanding the material

Saying from the perspective of a TA, as you have hinted, the point of learning economics is not to memorize ways to drive equilibrium outcome for Cobb-Douglas utility function. The point, rather, is to understand how the demand function is derived (starting from the consumer's maximizing her utility). Often, in the process, simple utility functions have many simplifying assumptions involved (continuouity, differentiability, etc.) and recognizing this will help you become more aware of other things... a good example is when you have linear or quasilinear utility function so that you need to consider endpoints as well.

If you truly understand the material, then you have no need to memorize things. You can derive them all in the exam room.

Your question, of course, is how to do that. I feel that if you work on the example questions and focus not just on the mechanics (take derivative of this, solve for $p_x$, plug into demand, etc.) but rather truly think about what the economic agent (consumer, producer) is actually doing... what they want to achieve, and what their constraints are... what would the market demand look like... etc., that might help.

One more technique I use if I have some spare time is to tweak the questions myself. Instead of having this utility function, what would happen if the utility function looks like that instead? Sometimes you work it out and check with friends. Sometimes it doesn't work out (price becomes negative!) and you could figure out why the question you set up yourself doesn't make sense.

Exam format

This leads me to my second point, and where I might feel a bit different from you (and that you didn't ask).

As a student, I often find exams too long. There are too many questions and the time is too short. The only thing, I felt, that the exam is testing is whether the student could memorize that, when met with this kind of problem, the demand function is in the form of this and that. Many of my friends resort to memorizing shortcuts.

I would have much prefer exams you have... something that tests the understanding of the material, rather than your memory of the material. Of course, with this kind of problem, there needs to be enough time so that students could carefully go through and really think about the problems.


Simulating macro / benchmark data Question

Are there any simulated datasets that have been designed specifically to represent macroeconomic data? In particular, are there any such datasets that can be used in benchmark studies?

Background

To give an analogy, in optimization, one may be interested in assessing the quality of a given algorithm. To do this, there are a number of test functions for optimization, such as the well-known banana function, that can be used to evaluate the performance of an algorithm; for example, by analysing if and how fast the algorithm can find a global minimum.

I am interested in working in this controlled experimental setting using simulated data that has a macroeconomic interpretation. With data simulated from a known data generating process (this would be my control), it would be possible to investigate, say, the performance of algorithms designed to detect structural breaks and to assess techniques/models used for forecasting.

I would greatly appreciate if someone could help: either by pointing me to some simulated datasets that can be used for the above purposes or by providing some guidance on how to simulate data that carries a macroeconomic interpretation.

It's possible for me to simulate a variety of ARMA processes (or VARMA model), but I am really interested in something that goes beyond that; the simulated data ought to have similar properties to observed macroeconomic data. Obviously, I am trying to avoid using actual data because my control (of knowing the data generating process) would be lost.

Update

For one of the purposes I had in mind, a quick read of Castle, Doornik, and Hendry (2013) suggests that it is, perhaps, not that difficult. Their "experimental design" is based on the following equations
$$
y_{t} = \beta_{0} + \gamma y_{t-1} + \beta_{1}x_{1,t} + \cdots + \beta_{10} x_{10,t} + \epsilon_{t}\\
x_{i,t} = \rho x_{i,t-1} + v_{i,t}, v_{i,t} \sim IN[0,1], i=1,\ldots,10,\\
\epsilon_{t} \sim IN[0,1], t=1,\ldots ,10.
$$
along with some further qualifications. So, it would appear that for one of my examples (the case of evaluating structural break algorithms), relatively simple (although by no means, "not tricky") DGPs are all that's required.

Reference: Castle, Doornik, and Hendry (2013) Model Selection when there are Multiple Breaks

I think yes. Macroeconomists usually build DSGE models to predict short-run fluctuations. Calibrating the data is the part of the process. After using numerical methods to run the models, researchers check the moments of the simulated values. This is a way to examine whether the simulated data represents real life outcomes.

However, every economic situation differs so you have to choose wisely the simulated data depending on your research every time. I mean the growth was huge in the 1950s in the US, but today the main objective is to kill the deflation monster.

Here's the benchmark model by Hansen and Wright (1992):

$\sigma$ denotes standard deviation, $Y,C,I$ and $L$ are macroeconomic variables.


Effect on GINI of eliminating poverty in China China eliminated extreme urban poverty this year and is on schedule to eliminate it in the rest of the country by the end of 2020.

How do I calculate the approximate impact of this change for China's 2021 GINI?

Such calculation is extremely difficult, because all the factors involved in shaping the personal distribution of income: relative change in return to different skills/job (think e.g. on skill-biased technical change), demographical/educational composition change (e.g. many new educated young into the labour market), taxation/redustribution policies (if less poverty, less income support welfare needed?), variations in the functional distribution of income (ownership of financial assets?), etc. It also depends on whether you are measuring inequality at the individual or household level, and which equivalence scale you use in the latter. Not to mention that you need to somehow differentiate between urban and rural sectors to incorporate your change.

I can think of three options you can follow to get what you want:

Microsimulation: this requires to have a general equilibrium model where a population of (commonly) households are simulated, usually to analyse policy changes. For the case of China, an example is here. This is the best approach in terms of using economic theory, but it is very time-consuming.
Regression: here you estimate a regression to evaluate determinants of nation-wide inequality. Included in these determinants should be the rate of urban and rural poverty. Forecasting them (and other regressors) you can forecast future Gini. This method is a highly reduced-form approach and might be highly inaccurate or inconsistent (due to endogeneity, etc). Still, to get an idea, see this paper.
Percentile income growth: as suggested by ahorn, you could go atheoretical and simply predict income growth of each percentile. Then you can easily compute Gini. The plot of these percentile income growth are commonly known as "growth incidence curve". For an example on China see here (Figure 4).


Historical returns for popular American funds over 20-30 years? I have seen charts like this one  showing the return for the past 100+ years cited as evidence why one should invest in stocks for their retirement. I don't disagree with that, but realistically, nobody invests for 110 years; my guess is that they actually invest for 20-30 years on average (for those who actually have a retirement account).

Also, it's not advised to try to play the market by investing in individual stocks, but rather mutual or index funds, so nobody can actually invest in "the stock market", but rather some subset of stocks.

So if I were making a case for someone to invest their retirement money in the stock market on a long-term basis, I would want to show them historical returns for popular consumer-grade funds over 20-30 year periods. The Dow over 110 years doesn't really tell me what an average investor's returns might look like (I understand that past performance is not indicative of future performance, but this is the whole reasoning behind looking at a chart of historical returns).

Has anyone crunched the numbers on this and published the charts and figures?

Edit: The wikipedia article notes that "[t]he first retail index fund, First Index Investment Trust, was formed in 1976...", so it looks like mutual fund investing by the average Joe could only have been done since then. Which makes the 100-year charts even more irrelevant.

There are a few important things you need to consider when planning your retirement.

1. Risk tolerance.

You cannot invest only in stocks, whether indices or individual stocks. Even the most risk loving investors will get stressed out when their lifetime savings drop 10% over a week amidst some market volatility. It is usually best to put a portion of the money into bonds or other fixed income assets. They usually have much lower yields than stocks so it will bring down the yield of the entire portfolio. How much to put into bonds and how much into stocks depends on your risk preference and how old you are. The older you get the more you should allocate into fixed income assets.

2. Inflation.

Inflation will eat up a lot of your savings over the course of your life. Not much can be done about it. Stocks are usually more resistant to inflation so having a decent position in stocks helps.

3. Hidden costs of index funds/ETFs

ETF and index providers sometimes try to hide some costs like annual fees, commissions for good performance, dividends etc. It is very important to look thoroughly into the costs of the fund you are going to invest in.

So if you build a balanced portfolio of stocks (index funds) and bonds, and the broker charges are either very low or free, then your annual returns on average should be around 3-5% depending on inflation. 4% is the rough estimate of returns you can get on a portfolio that is mostly composed of index funds. There are tons of literature out there, but most legitimate sources converge on the number 4%.

Here is an article from a good practical blog about retirement that I came across recently: http://www.mrmoneymustache.com/2012/05/29/how-much-do-i-need-for-retirement/. It has been featured on the Wall Street Journal so it seems quite credible.


Are no arbitrage models and equilibrium models equivalent? This YouTube video from WHU (starting from 3:50) claims that no-arbitrage models (such as Black-Scholes and HJM) are equivalent to equilibrium models (such as CAPM or C-CAPM).
He uses the Euler equation and the stochastic discount factor (SDF) as arguments which apparently link the two types of models.
I understand that an equilibrium requires the absence of arbitrage (to clear the market) but I don't see why any market without arbitrage is automatically in equilibrium.
Put differently, I thought that, for example, the Black-Scholes model can be derived by merely assuming that the market is free of arbitrage. I understand that the BS model can also be derived from the CAPM but that these additional assumptions ($\mu$-$\sigma$ based agents $\Leftrightarrow$ quadratic utility function $\Leftrightarrow$ linear SDF) are not necessary and the poor empirical performance of the CAPM does not directly impact the validity of the Black-Scholes model (which is flawed for other reasons).
Of course, Black and Scholes assume that (log-)returns are normally distributed and this may be in line with a quadratic utility function but we can derive similar option pricing formulae using completely different distributions with more parameters (CEV and such).

...no-arbitrage models (such as Black-Scholes and HJM) are equivalent to
equilibrium models (such as CAPM or C-CAPM).

Short Answer Yes, for models where asset prices are assumed to be Ito semimartingales (where the martingale part is a Brownian integral), although a more general argument is needed than that suggested by the special cases typically encountered in finance.
Clearly, no-arbitrage is a necessary condition for general equilibrium.
To claim equivalence is then to say it is also sufficient, i.e. given any price process $P_t$ and a density process $D_t$ such that the discounted price $e^{-rt} P_t$
is a martingale after change of measure by $D_t$, one needs to find
a (say, representative) investor $u$ and a equilibrium consumption process $c_t$ such that
$$
e^{-\beta t} u'(c_t) = \lambda e^{-rt} D_t, \quad\quad (*)
$$
for some $\lambda > 0$.
In other words, one needs a "SDF/marginal utility representation" for equivalent martingale measure densities.
The equation $(*)$ is the usual relationship
$$
\mbox{marginal utility} \; \propto \mbox{price}.
$$
If you write down a heuristic Lagrangian, $\lambda$ is the Lagrange multiplier in the FOC.
In general, FOC's of this type is only necessary for optimality of $c_t$.
If $(*)$ is sufficient for optimality of $c_t$, you can take the dividend to be $c_t$ in $(*)$ and $P_t$ becomes the equilibrium price faced by the representative investor $u$.
With certain assumptions on $u$---such as concavity and the Inada condition,
Karatzas, Lehoczky, Shreve (1987) showed that this can be done when $P_t$ is an Ito semimartingale and market is complete. (See also Cox and Huang (1989).) The rigorous argument makes use of convex duality and is referred to as the martingale duality method in mathematical finance.
The Ito semimartingale case certainly covers many---perhaps most--- asset pricing models in finance.
In fact, asset prices are usually assumed to follow a very special Ito semimartingale---geometric Brownian motion, where the risk neutral density $D_t$ is itself an exponential martingale.
Then $(*)$ takes the special form
$$
\frac{dM}{M} = - r dt + \frac{dD}{D},
$$
where $M$ is the SDF.
One can then take the representative investor to be CRRA and, since powers of exponential martingales are still exponential martingales, back out an SDF $M$ without referring to a more general argument.
For example, as you already point out,
the Black-Scholes formula for pricing European call option in complete markets can be recovered from the Lucas asset pricing model, where the equilibrium cum-dividend return process follows
$$
\frac{d P + D dt}{P} =  (\mu + \delta) dt + \sigma dW
$$
with $\delta$ being the endogenous dividend-price ratio $\frac{D}{P}$.
The underlying for the call being priced is the Lucas tree.
I don't know whether the martingale duality method has been extended to general semimartingales.
When market is incomplete, after some browsing it appears that only the case of terminal utility has been investigated and it is shown that certain constraints must be placed on $u$.


What was "Pareto's proof of the immeasurability of utility"? Wong (1978, 2002, Foundations of Paul Samuelson's Revealed Preference Theory), repeatedly refers to "Pareto’s proof of the immeasurability of utility". What was this proof?

I think the "immeasurability of utility" refers to what we know today as the non-uniqueness of utility representation. The word "utility" also has a slightly different meaning there. As Pareto notes in the mathematical appendix to his 1909 book, Manual of Political Economy (limited preview on Google Books), the "utility" measure is

  the measure of pleasure provided by the combination [of quantities of consumption goods], or [...] the measure of the value in use, the utility, the rareté (Walras), the ophelimity of this consumption.

He goes on to prove that there is a lack of a one-to-one correspondence between quantities of consumption and the indices of indifference curves (which are the candidate for measuring utility). Therefore he concludes that indifference curves are merely the index of utility (i.e. pleasure), not the measure of it.


Examples of behavioural breakdowns in game theory? I'm trying to compile multiple scenarios of the breakdown of game theory. Specifically, I'm looking for scenarios where game theory predicts certain behaviours, but in real-world scenarios or experiments, people tend to behave quite differently.

Here's an example of the kind of thing I'm looking for, based on the ultimatum game:

Two players, A and B, are offered a certain amount of money, for example, 100€, to share between them.
Player A first decides the split (e.g. 50-50, 60-40, 90-10, or whatever player A decides).
Player B chooses to either to accept Player A's split, in which case each player receives the amount dictated by Player A; or Player B refuses Player A's split, in which case both players receive 0€.
The game is played only once; there is no repitition.

Classic game theory predicts that as long as Player A offers any split in which Player B receives more than 0€, Player B would accept the offer. For example, if Player A offers 99-1, then Player B would accept the offer since 1€ is better than nothing. However, from what I understand, experiments have shown that for many offers below 50-50, Player B refuses the offer. The explanation is that when Player B perceives that Player A is being unfair, Player B often prefers that both players receive nothing rather than undergoing what they perceive to be unfair treatment. Apparently, the refusal depends on how much Player B would eventually receive. (For example, Player B might refuse a 90-10 split if 100€ is at stake, but might grudgingly accept it if 1000€ is at stake.)

Could anyone please offer any examples where game theory predictions are known to differ from actual human behaviour?

Strictly speaking, in the example I gave, the problem isn't with the "theory" part of game theory; the problem is with correctly specifying the utilities. The players' real utility is a function of both the money they would receive and their perception of fairness. Because the game classically only specifies their utility in terms of money, the prediction fails because the perception of fairness is also important, yet neglected. It is neglected because it is much harder to quantify and to accurately specify on the same scale as money. In fact, I strongly suspect that all cases that I'm asking for would involve a similar element: the game theoretic prediction fails because players have some very important behavioural aspects of their utilities which are not specified in the game model because they are hard to quantify.

I cross-posted this question with Cognitive Science StackExchange because I originally posted it on the Mathematics StackExchange but didn't receive much helpful answers there.

Here is an article that reports about several laboratory experiments showing large inconsistencies between theoretical predictions and observed behavior for some payoff structures:

Goeree, J.K. and Holt, C.A., 2001. Ten little treasures of game theory and ten intuitive contradictions. American Economic Review, 91, 1402-1422.
https://doi.org/10.1257/aer.91.5.1402

There are also papers studying variants of the ultimatum game in order to quantify players’ preferences for fairness. One approach to do so is to compare (i) games in which two human players interact with each other and (ii) games in which a human player interacts with a computer (so there is no scope for social preferences):

Hoppe, E.I. and Schmitz, P.W., 2013. Contracting under incomplete information and social preferences: An experimental study. Review of Economic Studies, 80, 1516-1544.
https://doi.org/10.1093/restud/rdt010


Policy relevance of the optimal taxation given uncertainty about social welfare specification One common criticism in the optimal taxation literature is the specification of the social welfare function. The optimal taxation literature (I have the Mirrlees framework in mind) relies on first specifying a social welfare function. However, because we never observe the social welfare function (SWF) and because we're not even sure if a reasonable specification of social welfare can be represented by a utilitarian-like SWF, what kind of policy prescriptions can we get out of these types of models? How much of a problem is this? Are there potential ways in which this problem can be made less offensive?

Here is an answer based on the following interpretation of the SWF : there is no "true" SWF, but SWFs are observable in principle, they simply represent the preferences of the policy decision maker.

Under this interpretation, the policy recommendation are relevant despite depending on the specification of the SWF, precisely because different decision makers have different preferences. So there is no problem with studying different configuration of the SWF and getting different policy recommendations. In fact this is rather desirable as it makes the theory relevant to a larger number of decision makers.

Now, whether you are aiming at policy recommendations for the average decision maker, or whether you are giving policy recommendation to a particular decision maker,  figuring out an exact social preferences might be complicated. Data may be insufficient to pin down particular preferences without much uncertainty. And if you ask the decision makers to state their preferences, they might be uncertain about their own preferences.

As is to be expected, the more uncertainty about the decision maker's preferences, the less precise the policy recommendations will be. But there is still a whole lot you can say even under relatively large uncertainty. As a matter of fact, most of the literature on optimal taxation has focused on providing general policy recommendations which hold for a wide class of SWF.

A very good example is Roëll, A. A., 1985. A note on the marginal tax rate in a finite economy.
Journal of Public Economics 28, 267–272's very weak redistribution assumption (VWR), which was informally presented earlier in Guesnerie, R., Seade, J., 1982. Nonlinear pricing in a finite economy. Journal
of Public Economics 17, 157–179. The VWR is a property of the solution of the taxation problem which formalizes the idea idea that the decision maker is egalitarian in at least a very mild way. To give a taste of the weakness of the assumption, notice for instance that

  If the utility function U is strictly concave, leisure is a normal good, single-crossing is satisfied, the social welfare function $W$ is differentiable, strictly concave, increasing, and symmetric, and at a solution of the taxation problem, the lower level of income is positive, then VWR is satisfied (Brett and Weymark, forthcoming)

It turns out that the VWR is sufficient to derive many important policy implications. For instance, under the VWR (and single crossing), marginal tax rates must be non-negative at a solution of the taxation problem. Also under the same assumptions, the marginal tax rate of the highest type should be zero, and any type different from the highest type whose consumption and income are strictly positive must face a strictly positive marginal tax rate. Provided the non-normative assumptions of the model hold (e.g. single crossing, but also the fact that decision maker's preferences are representable by a SWF, more on this below), these are policy recommendation you can safely make to any policy decision maker with some form of egalitarian concerns, even if you do not know the exact degree to which they are egalitarian.

Another question you are raising is whether policy decision makers' preference can be represented by a SWF at all. To say the least, I am not a big fan of the cardinal utility framework of SWF. This is mostly because hidden in the cardinality assumption is the idea that one can somehow "objectively" compare utilities. But it is also partly because some decision maker's preferences might be not representable through SWF. However, there are alternatives to both these problems, with applications to optimal taxation. See for instance the last chapters of A theory of fairness and social welfare, by Fleurbaey and Maniquet.

The approach described in A theory of fairness and social welfare, by Fleurbaey and Maniquet is not only ordinal, it is also purely axiomatic. Decision makers may have a hard time figuring out which SWF represent their preferences. With an axiomatic approach, they simply have to decide on the axioms they like the best to "discover" their preferences. This is often much simpler than trying to figuring and spelling out your full preference relation.


How to solve a linear forward-looking equation $x_t = \beta E_t[x_{t+1}] + k$ where $\lim_{t \to \infty} x_t = 0$ and $0<\beta < 1$? How does one solve a linear forward-looking equation $x_t = \beta E_t[x_{t+1}] + k$ where $\lim_{t \to \infty} x_t = 0$ and constant $k,\beta \in \mathbb{R}$, $0<\beta <1$?

(I suspect that $E_t[x_{t+1}] = E(x_{t+1}\mid I_t)$, where $I_t$ is the Information Set, i.e. $E_t[x_{t+1}]$ is translated as "the expected value of random variable $x_{t+1}$ conditional on Information available at time $t$").

@FooBar answer implemented the "standard solution algorithm", arrived at a constant solution, and remarked that such a solution violates the limiting condition.
Strictly speaking, a constant solution would satisfy the limiting solution also, if it was equal to zero (which in our case implies $k=0$): If indeed we had as solution

$$x_t = \frac k{1-\beta}$$
then

$$\lim_{t \to \infty} x_t = \lim_{t \to \infty}\frac k{1-\beta} = \frac k{1-\beta} \neq 0$$

except if $k=0$. But this is trivial (or at least, uninteresting).

So I ask, if $k\neq 0$, what would it take to obtain a solution? The law of evolution must hold also at the limit so

$$\lim_{t \to \infty} x_t = \lim_{t \to \infty}\Big[\beta E_t[x_{t+1}] + k\Big]=0$$

$$\Rightarrow \lim_{t \to \infty}E_t[x_{t+1}] = -\frac k{\beta} \tag{1}$$

In words, as the random variable tends to zero, its expected value conditional on information available in the previous period, must equal a non-zero constant, while the variable itself tends to zero. Consider some cases, to see whether and when such a thing can happen:

A) If $x_t$ was totally unobservable, (not observed directly, and not amenable to indirect estimation though other variables)? In such a case, the information set would not contain any information on $x_t$, meaning that the conditional expected value would equal the unconditional one. So we would have,

$$\lim_{t \to \infty}E_t[x_{t+1}] = \lim_{t \to \infty}E[x_{t+1}] \tag{2}$$

This is economics, meaning that the properties of the mathematical entities should take into account what do they represent in the real world. So it would be meaningless not to assume that $x_{t+1}$ is not bounded (more over, this would mean that the $x$ random variable goes to infinity in finite time and then "comes back" to tend to zero in infinite time -and for those who may think "hyperinflation", hyperinflation is a very very small number compared to infinity). If it is bounded then the Dominated/Bounded Convergence theorem holds and we have that

$$\lim_{t \to \infty}E\left(x_{t+1}\right) =E\left(\lim_{t \to \infty}x_{t+1}\right) =0 \neq -\frac k{\beta} \tag{3}$$

So this is not a case in which we can obtain the required relation $(1)$.

B) The limiting condition is part of the information sets for finite $t$. Then immediately

$$E\left(\lim_{t \to \infty}x_{t+1}\mid I_t\right) =0$$

In words, at any point in time we know that the variable will tend eventually to zero. Can this be compatible with relation $(1)$? We could invoke again the boundedness of $x_t$, and a generalization of the Dominated Convergence Theorem, since here the (conditional probability) measure is also time-varying, and again obtain a zero expectation at the limit. But intuitively also, if we know that eventually the variable will tend to zero, at some point in time this must be reflected in $E_t$, most certainly at the limit. But then again, the required relation $(1)$ won't hold.

C) We do observe perfectly or imperfectly the value of the variable, but "we don't give proper notice": our Expectations are not Rational. In that case, the symbol $E_t$ no longer represents the conditional expectations operator, but it is just a generic symbol for the "anticipation at time $t$" (in Adaptive Learning literature this is usually symbolized by $E^*_t$) and it acquires the value it needs to acquire for the limiting condition to hold.  In other words, given the law of evolution and the limiting condition, we conclude that expectations must not obey the Rational Expectations Hypothesis, not even at the limit.

In such a case, a solution is obvious: define
$$E^*_t(x_{t+1}) = -\frac {(1-c\beta^t)k}{\beta}$$

for some constant $c$, This expectations formation rule satisfies $(1)$, and  transforms the law of motion into

$$x_t = \beta \Big[-\frac {(1-c\beta^t)k}{\beta}\Big] + k = ck\beta^t$$

which in turn satisfies the limiting condition.

So we conclude that: the specific equation, with the specific limiting condition, has a solution if expectations formation does not obey the Rational Expectations Hypothesis, but instead follows the above rule, or some other to the same limiting effect.


Empirical estimates of CRRA and CARA utility I am working on macroeconomic model and I need to calibrate it.
I am looking primarily for a statistically-founded estimate for the
coefficient of relative risk aversion in the CRRA utility
function based on macroeconomic US data (but also for the coefficient of absolute risk aversion for the case of a CARA utility function). Cannot seem to find it anywhere.
Can anybody help?

There are many estimates in the literature. For example, Havranek (2013) does a meta-analysis of avalible results and argues for a value of intertemporal elasticity (inverse of sigma in your notation) around 0.3-0.4. But it might also depend on what your goal is - the single parameter in CRRA utility controls both risk aversion and intertemporal smoothing motive, so a calibration for asset-pricing model might need to differ from let's say a deterministic growth model.


Has it truly been established that the restaurant industry operates on thin margins? For as long as I can remember I've constantly seen the truism that restaurants operate on "razor sharp margins" repeated endlessly by the media, the restaurant industry, and the average joe. This seems to be something that people take for granted, yet there are and have been so many restaurants, many extraordinarily profitable and consistently so, and so little data ever presented to back up this truism when it is bandied about.

I've never seen any restauranteurs being open with their margins or disclosing how much they are losing or profiting from the business and the reasons for said profit/loss in a validatable & analytical way. Yet I always see this truism used to justify everything from paying employes $2 an hour and offloading their wages to the consumer, to exorbitant meal prices, and frankly its usage in making complaining about literally anything in the restaurant business seem legitimate. But when pressed, nobody ever backs it up with actual data. Does this data even exist or has this been an extremely effective guerrilla marketing ploy?

Why should I accept that the industry operates on thin margins? Why are they famously so much thinner than any other industry (or more loudly thin), and why is this so widely accepted as true? Has this truly been established from a data-based economic standpoint and if so does that data remain relevant year after year? If the answer is "yes" it has indeed been established, I ask for the how, when, where, and why--if only established once and not continually--it is relevant.

(pre-coronavirus economics only)

TLDR: Interesting question, but difficult to answer. The restaurant industry is made up of so many firms and many are small and self-owned. This makes collecting data difficult. The data I did collect shows restaurants operate with margins that are tight, but not as tight as commonly believed.
A potential hypothesis of the reason behind this is restaurants are often small enterprises run by a family. Those businesses may operate on healthy margins in good times but struggle excessively in bad times. Spending at restaurants is highly variable and falls dramatically during recessions. This triggers a wave of failures across smaller restaurants who do operate on tight margins and are unable to secure financing thus causing them to fail. This occurrence might be a large contributing factor to the commonly held notion that restaurants operate on extremely thin margins.

Forbes wrote a piece on restaurant margins in early 2018. The data they complied showed margins to be 6.5% for the most recent year. That is low but certainly not razor thin. However, 2018 was deep into an economic expansion that restaurants were certainly benefiting from. The numbers from earlier years is much tighter. In 2014 it was 2.3%, 2.3% again in 2015, 2.8% in 2016, and 3.3% in 2017. Those numbers paint a different picture which is much gloomier for restaurants.

The dataset posted in the comments does not contain information on profit margins but it does contain information on the proxy measurements return on equity and return on assets. The dataset was published February 26, 2020 and contains a selection of 861 restaurants from across Europe.
For the United Kingdom the average return on equity for a restaurant is 18% and the average return on assets is 4.47%. Spain has an average RoE of 19.3% and RoA of 7.3%. France has an average RoE of 4.45% and RoA of 1.9%. Besides France, these are robust numbers.

The Australian Taxation Office collects data on many industries including the restaurant industry. Average total expense for a small restaurant is 85% (65,000-500,000) and for a large restaurant (2,000,000+) 92%.

There is an interesting comparison of margins at McDonalds and high-end restaurants over on Quora.


Can the Machina Paradox be solved by expanding the choice set? In another question, the Machina paradox is mentioned as a possible counterexample to the expected utility model:

  Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory.

  A person prefers a trip to Paris to watching a television program about Paris to nothing.

  Gamble 1: Win a trip to Paris 99% of the time, the television program 1% of the time.

  Gamble 2: Win a trip to Paris 99% of the time, nothing 1% of the time.

  It's reasonable to suppose that given the preferences over items, the second gamble might be preferred to the first. Someone who lost the trip to Paris might be so disappointed that they wouldn't be able to stand watching a program about how great it is.

However, it seems to me that this can be resolved by expanding the decision space to account for possibly state-dependent utility. For example, consider a model with two time periods,
$t=0$ and $t=1$. The first represents before the resolution of the uncertainty surrounding winning the trip to Paris. The second time period is after the resolution of the gamble. Now, model this potential outcomes as follows:
$$
\begin{align}
A &= \{P, \emptyset\} \\
B &= \{P^C, T\} \\
C &= \{P^C, N\},
\end{align}
$$
where $A$ corresponds to the outcome where you win the trip to Paris (and then it doesn't matter what you do after that), $B$ is the outcome where you don't win the trip and you watch TV afterwards, and $C$ is the case where you don't win and you do nothing afterwards. Then, although you might like Paris over TV over nothing in one time period (...?), when considered together over time (because of some sort of complementarities) you prefer $A$ over $B$ over $C$.

My question is this. Is this a reasonable way to resolve this paradox? What are ways in which people have tried to resolve this?

No, I would not say that this resolves the Machina paradox, because it is exactly the same as the Machina paradox: the paradox does indeed require from you to look at the three possible outcomes. The M-C/W/G book discuss only the $B$ and $C$ outcomes because it is there where the paradox focuses on whether a violation of the axiom of independence may happen.

But most importantly, Machina did not argue that all people will have preference ordering $A>C>B$. He argued that it is reasonable, for evident psychological reasons, to expect that some people may... So some others will have the ordering $A>B>C$, which does conform to expected utility framework.

The first will say "I cannot watch a movie about Paris after losing the trip - I will smash the TV!" The second will say "Well, tough luck. At least I will see it on screen and keep dreaming about it". Both seem like behaviors that could be anticipated by "usual" human beings.

The point of the paradox is not to show that Expected Utility (EU) is invalid for all people -only that it may be violated in reasonable situations, i.e. situations that may characterize a lot of people and may happen often.

What paradoxes like this examine and contemplate, is the degree to which EU represents adequately the "majority" of people in some sense, and so whether it is valid/useful/not-misleading as a core theoretical assumption in economic models, or not.
And this is a matter of degree, a quantitative matter. This is true for almost all assumptions in theoretical models in social sciences.


How does fed cutting federal fund rate affect student loan If fed cuts rate, it means it's cutting federal fund rate right? then why do loans like credit card and stuend loan? those aren't like Mortgage rate which the fed cuts its rate by buying MBS

The Fed funds rate is the rate at which commercial banks can borrow reserves on the overnight market (see this explanation at Investopedia). As such it affects all other interest rates banks charge since when they can borrow more cheaply they can also lend money cheaply to  consumers.
This affects among others also student loans (when they are provided by commercial banks - some places have government provided and subsidized $0\%$ interest student loans). However, of course if you already had a loan in a past it will only be affected if you took it at variable interest rate or if you are taking out a new loan, the fixed interest rate loans won't be changed retroactively.


A valid argument against free trade? Recently I read Bertrand Russell's Political Ideals (1917). Although he expresses strong left-wing ideals there, he supports free trade and criticizes protectionism.

It seems that not any left-wing theorist agrees with Russell on this. As an example, I quote from A Darwinian Left (1999) by Peter Singer:

  What capitalists failed to accomplish by a century of repressive measures against trade union leaders, the World Trade Organization... is doing for them. When barriers to imports are removed, nationally based trade unions are undermined. Now when workers in high-wage countries demand better conditions, the bosses can threaten to close the factory and import the goods from China, or some other country where wages are low and trade unionists will not cause trouble.

Now, I know that there may be different opinions on this and that's the nature of a complex subject like economics. However, I can't resist the desire to know what serious students and scholars of economics think of this argument by Singer.

I can get to more technical papers, but it's not a big secret what you have quoted. E.g. The Economist quotes MIT professor John Van Reenen:

  But just because the size of the pie expands [due to free trade], it doesn’t mean that everyone is better off. There are going to be some losers whose slice of the pie is so much smaller that they would have been better off with less trade. However, because the overall size of the pie has got bigger, the government can compensate the losers which can still make everyone better off. [...] But remember: thanks to free trade, you can afford that [better welfare programs], because the overall size of the pie is bigger.

And Harvard prof. Dani Rodrik writes:

  there is one Stolper–Samuelson-like result that is
  extremely general, and which can be stated as
  follows. Under competitive conditions, as long as
  the importable good(s) continue to be produced at
  home – that is, ruling out complete specialization –
  there is always at least one factor of production that
  is rendered worse off by the liberalization of trade.
  In other words, trade generically produces losers.
  The proof of this result is simple enough to be
  stated quickly here. [the proof is still like half a page]

  More recent work in trade theory has emphasized
  heterogeneity among firms and workers. These
  models have additional margins for redistribution,
  between firms and workers that otherwise look
  quite similar. Grossman, Helpman, & Kircher (2017), for example, enrich the Stolper–Samuelson
  framework by considering heterogeneity within
  broad worker categories. ‘‘Managers’’ and ‘‘workers’’
  must combine in teams, and their productivity
  depends on the quality of the match. Trade liberalization induces re-matching and generates distributional effects within occupations and industries,
  in addition to the standard effects across broad
  factors of production and industries.
  Hence in all these models, redistribution is the
  flip side of the gains from trade. No pain, no gain.

My point is not to say that these theoretical models 100% apply to the complex realities of international trade, but rather that economists are aware that under certain conditions some segment of the workforce can be worse-off with free trade... unless something else happens.

Nevertheless, the prevailing opinion among economists, or at least US academic economists surveyed by IGM is that the long term benefits of free trade outweigh the short-term negatives (including [un]employment in some sectors).

As the survey additionally collected fill-in comments/responses, they are an interesting reading, but the first one (alphabetically, from Acemoglu) probably summarizes the critique you inquire about:

  Economists often understate short-term employment costs, which are significant and unequally distributed, but probably less than benefits.

As an example of empirical work evaluating such effects in a concrete context, Caliendo et al. (2015)

  We
  find that China's import competition growth resulted in 0.6 percentage point reduction in the share
  of manufacturing employment, approximately 1 million jobs lost, or about 60% of the change in the
  manufacturing employment share not explained by a secular trend. Overall, China's shock increases
  U.S. welfare by 6.7% in the long-run and by 0.2% in the short-run with very heterogeneous effects
  across labor markets.

Related enough, a more recent IGM survey of a panel of European economists (some of which do work in the US though) shows that they are not blind to the effects of inequality on [liberal] democracy. The latter survey also poses questions on whether higher government spending is justified from this perspective. So compensating or at least caring for losers (not necessarily stemming from free trade alone) is also a topic that economists are not opposed to. (Whether they should talk more about it, is a different issue.)


Simple Keynesian model of income determination When determining equilibrium, we take consumption equal to C= ¢ + c' y where ¢ is the autonomous consumption or minimum consumption that would take place even in absence of income.
What could be the consequences on equilibrium income if the autonomous consumption ¢ was negative?

I cannot see any way to have negative autonomous consumption. I believe autonomous consumption's extreme is zero if there's a kind of superheroes who don't need water, electricity and food to live and they save everything they earn. But in a case like this, there's  no reason for having aggregate supply at all as nobody will buy the goods in first place.


Can an economy grow without its population growing? Assuming natural resources are not a limiting factor, can a country's economy keep growing if the country's population is not growing? What about the global population?

Yes, there can be economic growth without population growth.

As a reminder: gross domestic product (GDP) is the total value of all of the goods and services produced within an economy over a period of time (usually a year). GDP growth is simply the tendency for the GDP to increase over time.

Roughly speaking, there are two ways for GDP to increase. The first is for each worker to keep producing the same stuff, but to increase the number of workers (i.e. population growth). But the second is to keep the population the same and increase the amount produced by each worker. This increase in productivity might happen because

workers become intrinsically more productive (i.e. through education)
workers have better tools to work with (e.g. through technological progress).

If you live in a subsistence society in which people tend the fields by hand and produce 100 bushels of grain per year and then someone invents a combine harvester that allows them to produce 500 bushels per year then the output of the economy will increase even if the population stays the same.


Elasticity and logarithms Let's consider a relationship between $ y $ and $ x $, $ y = a x^b $. Taking log on both sides, we have $$ \log y = \log a + b \log x $$

Now, my textbook, Nicholson and Snyder's Basic Principles and Extensions derives the relationship between elasticity and the logarithm of the two variables thus:

$$ \eta = b = \frac{ d \log y}{d \log x} $$

Now, I understand that $ d \log y = \frac 1y dy $ and $ d \log x = \frac 1x \ dx $. So I understand why we can write $ \eta = \frac {d \log y}{d \log x} $. What I don't understand is: why does $ b $, which is the power on the variable $ x $, equal $ \eta $?

Here's a snapshot from the book:

Because $a$ is a parameter, and so
$$
\eta = \frac{ d \log y}{d \log x} = \frac{ d \log a + d \ b \log x}{d \log x} = 0 + b.
$$