Can capital still be paid its marginal product in the absence of a homogeneous capital stock? Thomas Piketty's best-selling book on inequality, "Capital in the Twenty-First Century" has attracted a lot of criticism on the right for its data analysis.  Less well-known, however, is the criticism it's gotten from economists on the Left.  Piketty uses an aggregate production function, where capital is represented by a single number K.  This is a fairly standard thing to do in mainstream economics, but back in the 1960's there was a major debate in economics called the Cambridge Capital Controversy, revolving around the fact that capital is heterogenous and thus it may not be legitimate to represent it by a single number.  So some economists argued that many standard conclusions in economists, like the notion that capital is paid its marginal product, are flawed.

Paul Krugman disagrees with this critique of mainstream economics, so in this blog post he defended Piketty from the charge that his conclusions depend on assuming a homogenous capital stock:

  The thing to bear in mind, however, is that you really don’t need to
  reject standard economics either to explain high inequality or to
  consider it a bad thing.

  There are a few economists on the left who seem to believe that:

  You need to believe in the existence of a perfectly well-defined aggregate measure of capital to believe in the marginal productivity
  theory of income distribution;
  If you believe in, or even use, marginal productivity theory, you are conceding that capitalists deserve their income.

  Neither of these things are true. Nothing about marginal productivity
  theory depends on the exact truth of a simple aggregate production
  function with capital defined by a single number. And saying that
  capital gets its marginal product in no way says that the people who
  own that capital deserve what they get.

My question is about the statement in bold, where Krugman says that you can still get the conclusions of marginal productivity theory without assuming that capital can be described by a single number.

So my question is, what is the basis of Krugman's assertion?  Can someone point me to a proof of the statement "capital is paid its marginal product" which doesn't depend on homogenous capital?

Nothing about marginal productivity theory depends on the exact truth of a simple aggregate production function with capital defined by a single number

Different sorts of capital used as separate production technologies prevent clean aggregation to a representative form of capital but does not prevent capital from being paid its marginal product.

To make things simple, consider an economy with two kinds of capital $k_1$ and $k_2$ and no labor. There is only a single final good ($y$) but are two separate production technologies $y_1$ and $y_2$ where $y_1 =  k_{1}^{\alpha_1}$ and $y_2 =  k_{2}^{\alpha_2}$. Then aggregate production of $y$ is:
$$y =  k_{1}^{\alpha_1} + k_{2}^{\alpha_2}$$
If $0 < \alpha_1 < \alpha_2 < 1$ then for positive rental rates on capital $r_1$ and $r_2$ it will be efficient to choose a positive amount of both types of capital such that for each type $i$:
$$ \alpha_i \cdot k_{i}^{\alpha_i - 1} = r_i\Rightarrow k_{i}^{*} = (r_i/\alpha_i)^{1/(\alpha_i - 1)} $$

There is no single $k$ that represents capital in this economy. Sure you could define $k=k_1 + k_2$, but that wouldn't mean exactly the same thing. On the margin the two sorts of capital don't have the same product and so aggregation doesn't make sense here. But in this setting, it is likely that the rental rate on capital would be equated ($r_1 = r_2$) because why would you buy one sort of capital when the other sort paid more?


Use of Slutsky equation I know that the Slutsky equation is defined as:

$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1} + x_1^o \frac{\partial x_1^m}{\partial m}$

My problem is right now is making use of this information given (I am aware of how to take partial derivatives) but cannot seem to understand how to apply it to problem sets.

Here's an example (I'm more concerned about the steps on how to get to the answer not just the answer);

A consumer has preferences given by $U(x_1,x_2)= x_1^2x_2$

(a) Derive the demand curves for $x_1, x_2$ when prices and income are given by $p_1, p_2$ and $m$

$x_1^*=2m/3p_1$ and $x_2^*=m/3p_2$  -I think I understood how to do that

(b) Illustrate the equilibrium on a diagram when $p_1$ = $p_2$ $ =$ 1 and
$I$ = $12

the way I did this was by graphing and simply finding the equllibrium point graphically based on the Demands for goods $1$ and $2$ on the budget line

(c) Calculate the exact income and substitution effects for $x_1$ when $p_1$
rises to $3.

-The only way I'm currently able to do this is without Calculus, as described in this video which doesn't seem to sit well with me being that the Slutsky Equation is defined very clearly with use of calculus. I just don't know how to apply it.

(d) Explain your exact results using the appropriate Slutsky equation.

same problem here.

Note: I'm no simply looking for someone to "do my homework" I'm primary interest is in knowing how to apply the Slutsky equation when facing similar problems.

Utility function $u(x_1, x_2) = x_1^2x_2$.

  Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$.

Here are the demand functions for $x_1$ and $x_2$:
$$x_1(p_1, p_2, m) = \frac{2m}{3p_1}$$
$$x_2(p_1, p_2, m) = \frac{m}{3p_2}$$

  Q. Illustrate the equilibrium in a diagram when $p_1=1$, $p_2=1$ and $m=12$.

  Q. Suppose $p_1$ rises to 3. Calculate the substitution effect and income effect.

If $p_1$ rises to 3, the new equilibrium choice is $\left(\frac{8}{3}, 4\right)$. To find the Substitution effect and Income effect using Slutsky approach, we will find the equilibrium at new set of prices when the consumer has just enough money to buy the old equilibrium bundle i.e. we will find the demand at prices $(3,1)$ when income is $m' = 3(8) + 1(4) = 28$. Substituting this data in the demand functions, we get the equilibrium choice as: $\left(\frac{56}{9}, \frac{28}{3}\right)$. Here is how the situation looks in graph:

Substitution effect = $\displaystyle\frac{56}{9} - 8 = -\frac{16}{9}$

Income effect = $\displaystyle\frac{8}{3} - \frac{56}{9} = - \frac{32}{9}$

  Q. Explain your exact results using the appropriate Slutsky equation.

Slutsky equation:
Change in Demand =  Change in Demand due to substitution effect + Change in Demand due to income effect

$$\Delta x_1 = \Delta^s x_1 + \Delta^i x_1 = -\frac{16}{9} - \frac{32}{9}
 = -\frac{16}{3}$$


Can i check monotonicity with the marginal utility? For example I have a Cobb-Douglas utility function $U(x,y)$ and I want to check the monotonicity property.
Can I use the marginal utility functions to see that they are always positive to conclude that $U(x,y)$ has the monotonicity property?

Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.


What are the empirical techniques to show causation? A simple linear regression only shows correlation between two variables. To establish causation, two commonly taught methods are IV regression and natural experiments. What are the other methods people use to establish causation?

Natural experiments are usually a setting for causal inference rather than a causal inference tool per se. You often need to employ something like difference-in-difference or instrumental variables anyway even when you have a natural experiment.

 Here a list of statistical causal inference approaches (Approach: Lay description)

Instrumental Variables: Randomly assigned variable X influences Z only through Y
Difference in Differences:  If two groups have a common trend and only one group is treated then the change in the difference between the groups is the treatment effect
Regression Discontinuity: If a hard threshold determined treatment, look at difference right around that threshold
Propensity score matching: Create a control group by matching untreated observations that were likely to be treated (but not in fact treated) with treated observations with a similar probability of treatment.
Manhalobis distance matching:   Create a control group by matching untreated observations that look similar to treated. Another notable distance measure is Coarsened Exact Matching.
Synthetic control: When you have only one treated observation, create a composite of untreated observations that individually are imperfect controls but collectively act as a good control.
Synthetic Cohorts - Treat repeated observations of groups like a panel of individuals and employ panel techniques.
Selection bias modeling like Heckman correction: Assume a parametric form for selection bias and remove it, so the corrected regression results can be interpreted causally.
Sample weighting more broadly - fix bias resulting from endogenous participation and un-modeled heterogeneity by weighting sample units to look more like the true population of interest.


Ruling out boundary solutions in Utility Maximization Solving the basic Utility Maximization Problem, i.e.

\begin{align}
max_{x\geq 0} u(x) \\
s.t. \,\,\, p^Tx\leq w
\end{align}

we get the Kuhn-Tucker first order condition

\begin{gather}
\frac{\partial u(x)}{\partial x_l}\leq \lambda p_l
\end{gather}

which holds with equality when $x_l > 0$, i.e. when solutions are interior.

Is it enough to say that since preferences are assumed to be monotonous, we can rule out the possibility of boundary solutions and thus we are safe in setting the FOC with the equal sign?

No, it isn't.

A simple counterexample is
\begin{align*}
\max_x \hskip 10pt & x_1 + x_2 \\
\\
\text{s.t.} \hskip 10pt & x_1 + 2x_2 = 10.
\end{align*}
The utility function is monotonic (strictly monotonic even), but the solution is a corner solution at $(x_1,x_2) = (10,0)$.

The above answers the question, but it is worthwhile to note that defining a sufficient condition (that is not too strong) seems to be difficult.

Not even convexity is sufficient. (The utility function above is convex, and could be made strictly convex with an extremely slight bend.)

Seems like you would have to compare the level curve gradients at all the possible corner solutions to the gradient of the budget set, but that does not really help you in higher dimensions.


Does money neutrality mean that nominal GDP (output) changes following changes in money supply? If money neutrality stipulates that only nominal variables will be affected following changes in money supply, then does this mean that following such money supply changes, real GDP remains unchanged while nGDP changes, as changes in money supply causes changes in price levels, and nGDP is by definition real GDP (which is unchanged) * price level deflator?

I looked at “Monetary Economics: Theory and Policy” by Bennett T. McCallum. On page 95, he describes what he refers to as “the Classical Model”, where the money supply is set outside the model (exogenous). In the Classical Model, if the money supply is changed, nominal values in equilibrium change, but ratios like wage/prices and real values do not. (The implication being that nominal GDP changes proportionally to the money supply, but real GDP is fixed.) He describes this as money neutrality, and it matches the description in the question.

I do not know how money neutrality can be defined for models where the money supply is determined within the model (endogenous).


competitive equilibrium, Walrasian equilibrium, Walrasian auction There are several sellers holding some indivisible goods, and several potential buyers with different valuations for these goods.  I need to calculate the Walrasian equilibrium in this scenario, but first I need to understand what exactly is a Walrasian equilibrium. So I am looking for a short, formal, operational definition of this term.

Wikipedia just links to a page about General equilibrium theory, which is very long and verbal, but I couldn't find there a formal definition of a Walrasian equilibrium.

This page also hints that a Walrasian equilibrium is similar to Competitive equilibrium, but I didn't understand whether they are identical or different, and if they are different - what is the relation between them?

There is also a page about a Walrasian auction, but again, I am not sure what is the relation between this and a Walrasian equilibrium?

The definition in the about.com dictionary looks promising: "An allocation vector pair (x,p), where x are the quantities held of each good by each agent, and p is a vector of prices for each good, is a Walrasian equilibrium if (a) it is feasible, and (b) each agent is choosing optimally, given that agent's budget. In a Walrasian equilibrium, if an agent prefers another combination of goods, the agent can't afford it."   But, they do not define what they mean by "feasible".

I would very much appreciate an orderly explanation of the relation between all these different terms.

The definition from about.com is what you are looking for. Feasible means that total consumption of each good does not exceed total endowment (initial holding) of that good in the economy. Terms Walrasian and Competitive equilibrium can be used interchangeably here and Walrasian auction is entirely different concept.


First Price Auction (Expected Payoff) I'm trying to solve followed first-price auction problem.

Bidder's pdf is
$$
f(v_i)= \begin{cases}
\dfrac{1}{8}v_i, & \text{if} & 0\leq v_i\leq4\\
0, & \text{if} &  \text{otherwise}\\
\end{cases}
$$

Bidders only know their own values. One gets 0 if lose the auction and ($v_i-b_i$) if win the auction game.

The question asks the symmetric bidders' bid under Bayesian Nash equilibrium and expected payoff of bidders and seller.

So, one bidder will maximize as follows:

$$\max_{b_1}\,\,(v_1-b_1)\Pr(b_1>b_2)$$

If we take the derivative with respect to $b_1$ we get

$$b(v_1)=\dfrac{\int v_1f(v_1)\,dv_1}{F(v_1)}$$

We know from question $f(v_1)=\dfrac{1}{8}v_i$ and we can find $F(v_1)=\dfrac{1}{16}v_i^2$. If we replace them, we get

$$b(v_1)=\dfrac{\int v_1\dfrac{1}{8}v_1\,dv_1}{\dfrac{1}{16}v_1^2} = \dfrac{2}{3}v_1$$

So this is the bidding of bidder 1. However, I couldn't find the expected payoff bidder 1 and bidder 2. My question is how can I use PDF or CDF to find the expected payoffs of the bidders.

Let me answer here putting together all the hints in the comments.

You've figured out that the equilibrium bid for a type $v$ bidder is $b(v) = \frac{2}{3}v$.

The bids are strictly increasing in $v$, so bidder $i$ wins whenever $b(v_i) \geq b(v_j)$ or $v_i \geq v_j$.

The expected payoff for a type $v$ bidder is thus
$$
(v -b(v))P(b_1 \geq b_2)
$$
or
$$
(v - \frac{2}{3}v)P(v \geq v_2) = \frac{1}{3}P(v \geq v_2)
$$
So what is the probability that $v \geq v_2$?

By definition of CDF $P(x \leq v) =  F(v) = \int_0^v f(x) dx$

So $P(v_2 \leq v) = F(v)$.

Thus expected utility for a type $v$ bidder is:
$$
u(v) = \frac{v}{3} F(v)
$$


What if price were a function of buyer's income? If everyone's income were public information, would purely profit-seeking sellers benefit from charging different prices to people based on their different incomes? My intuition says yes because a buyer's demand for some good at a given price depends on that buyer's situation in life, including income. It seems to me that prices could be better optimized if customer information on an individual level were taken into consideration.

But I'm not sure about this. I lack the mathematical know-how to think about this thoroughly. It seems plausible that the optimal price would be the same for all customers due to competition causing prices to converge ...or something along those lines. The Wikipedia page on price discrimination brings up the issue of discount buyers reselling goods for a profit, but I think this could be avoided by limiting the quantity of a discounted good a customer can buy, and it's not an issue at all if the good can't be resold. That page also refrences coupons as an example of price discrimination because only people with a low enough income can benefit from spending time on clipping coupons.

So I think there are already some successful forms of mild price discrimination, but it doesn't seem all that common, possibly because there isn't a way for customers to prove his/her income. Could price discrimination based on income work at a larger scale if people could prove their incomes, and could this help reduce inequality?

This is just perfect price discrimination. Happens sometimes with niche drugs (where drug makers can know the ability of people to pay). In this scenario, consumer surplus is completely captured by suppliers.


Economic term for relation between drug and companion diagnositic I'm looking for economic terminology which must exist, although I don't know what it is.  (I want to look up articles on a particular situation involving two real-world companies.)

Here is the situation:  Company A is developing a diagnostic test which will warn of a particular type of deadly antibiotic resistance.  Company B is a pharmaceutical company developing a drug to treat that kind of resistance.

Is there an economic terms that describe concepts relevant to this situation?   The important features are that

1)  Product B will never be bought without product A.
2)  Customers will only buy A with the intention of possibly buying B.
3)  Product B will certainly sell at a very high price if product A works perfectly, because A will identify those patients for which B is life-saving.  A deserves some of this return, but the many customers buying A each will have only a low-probability of being one of the few to benefit.

Similarly, one could imagine a related case that there are exactly two bicycle wheel manufacturers, and by law A can only sell front wheels and B can only sell back wheels.  There will be some sort of odd price hostility between A and B, because the two products are each useless without the other.  The natural thing to happen is for the producers A and B to merge, although wrangling about the terms of a merger may make this impossible.

In my real-world case the cognizant shareholders of A should oppose any buyout by B at anywhere near current market prices, because A's technology will extend to many many other applications.

Product A and B are perfect complements. As the Wikipedia entry states:

  A perfect complement is a good that has to be consumed with another good. [...] One example is a left shoe and a right; shoes are naturally sold in pairs, and the ratio between sales of left and right shoes will never shift noticeably from 1:1, even if, for example, someone is missing a leg and buys just one shoe.

The example of the shoe matches the one of the bicycle you give.

Additionally, the two goods do not need to have a symmetric degree of complementarity. Again, quoting Wikipedia:

  In the case of video games, a specific video game (the complement good) has to be consumed with a video game console (the base good). It does not work the other way: a video game console does not have to be consumed with that game.

So, for example the base good can be the diagnostic and the complement is the drug.


How did reducing the employment gender gap impact the US economy? Premise
In the United States, women have been replacing men in the workforce, or rather were from sometime prior to 1976 until the mid-90's. (The general upward trend is from the recent recession, sorry, I'm recycling a graph from another presentation).

Presumably, this occurred when a more qualified woman got a job over a male counterpart as women entered the workforce.
Given that, one would expect the economy to grow substantially as overall productivity rose.
Yet US GDP growth did not appear to outpace the world over that period, since not all the world liberalized their gender roles simultaneously.

Question
Why is that?
Is there a flaw in the reasoning, or were the effects simply not visible at the scale I've looked because the data was too volatile or confounded by external factors.
I'd be interested in any studies that dealt with this topic and offered potential causal mechanisms for breaking the presumed link between (again, the presumed) increased efficiency, and output growth.
Thanks!
Sources:

LNS10000001 - Civilian noninstitutional population, male
LNS11000001 - Civilian labor force, male
LNS10000002 - Civilian noninstitutional population, female
LNS11000002 - Civilian labor force, female
Google

I'm eyeballing that the unemployment rate of man rose by 3% while the rate for woman fall by 13. Clearly not all of those newly working woman replaced man.

You choose a rather long period with incredible changes to the US economy; The one I think responsible for the effect you saw being the shift from manufacturing to service.

The Situation in 1977 can be seen on Page 10 (4 in the original numbering) of the 1977 Census.
It shows 44% of all employees working in manufacturing and 39% in service (Retail + "Selected services industries".

Contrast that to the 1997 Economics Census (Page 20 or 14) where "Retail Trade" alone employed more people than manufacturing - and Services almost double compared to manufacturing.

That shift favored woman (and probably was only possible due to more woman choosing not be stay-at-home moms) which explains the diminishing employment gap.

(The methodology of those reports has changed considerably so the exact percentage rates can certainly not be compared -- but the shift is so big that the conclusion stands unchallenged)


European Government Bonds correlations Why do European Government Bonds 10Yr (or at least some of them) have such a high level of correlation?

My guess is because they are denominated is Euros and their yields respond to the European Central Bank monetary policy decisions.


Do banks make profits by selling bonds to the central bank? I am a bit puzzled by the quote below, from here (emphasis mine):

  Modern monetary theorists have helped clarify the obvious point that private banks are not essential to the design of the system. It would be possible for the Federal Reserve to bypass the banks which profit by selling bonds to the Fed and, instead, invest the proceeds, for instance, in a public “Infrastructure Bank” [...]

So, central banks pursue open market operations by selling/buying bonds from banks or other financial institutions involved in these operations. Maybe the author is referring to the "buying side" of these operations, whereby the central banks buy short-term debt from banks?

Does this mean that banks only sell these bonds if they expect a profit? Does that also mean that banks only buy these bonds at a profit? How does this profit materialise? Through a discount/premium over the price at which these bonds are bought/sold? Or is the author simply referring to the fact that these bonds have positive nominal interest rates? (but these bonds are usually from the government, so it is not the central bank who is paying the interest)

Does this mean that banks are in every single transaction profiting from such operations?

(sorry, a lot of questions but they are very much related).

No, banks don't directly profit from selling bonds to the central bank. These bonds are sold at face value, there is no discount or premium, this is not part of the system.

My best guess of what the author means is that by selling bonds to the central bank, commercial banks get cash, which they loan out and make a profit on the loans. So commercial banks profit from selling bonds to the central bank in an indirect way. Attributing banks' profits from lending to the central bank is a bit of stretch though and not a very appropriate statement in my view, so I fear I might have misunderstood the author.

What the article seems to promote is already a reality in communist China, where state-owned banks lend at favourable rates. There can be significant problems with this approach of cutting out private banks, as the Chinese bad loan problem shows (although private banks can also accumulate bad loans, but maybe less so due to competition), but this is a different discussion.


How encouragement design works practically? I'm trying to understand the encouragement design but I have some doubts. For example, if I want to introduce a voucher for teachers that can be used for professional development(the treatment) and I want to estimate its impact on student performance(dependent variable) obviously I have some problems, since not all the individuals in the treatment group will actually use the voucher. How can I introduce the encouragement design in this case? for example is it right if I send (randomly in the treatment group) some invitations to some professional developments program and I use their participation to these invitations as instrumental variable? And what's the difference with intention to treat?

The fact that you can only encourage but not force the teachers to participate puts you squarely in the world of Local Average Treatment Effect (LATE) and Intention to Treat analysis.

Essentially if you use the voucher as a instrumental variable, you will get an estimate of the effect of the voucher on "the sort of person who you can encourage to take a class by giving them a voucher".

Read the Instrumental Variables section in Mostly Harmless Economics, to get a good easy to understand explanation of exactly the situation you are interested in.


In panel data application, when using Fama and MacBeth regression is preferable over the fixed or random effect model? thought When discussing panel data, many econometric books, usually, focus just on fixed or random effect model as means of estimating regression for panel data. Despite this tendency, I have seen many papers use Fama and MacBeth regression for this purpose, an approach I previously thought its application is constrained to asset pricing models like CAPM.
Now my question is: in panel data application, when using Fama and MacBeth regression is preferable over the fixed or random effect model? Is there a statistical test shedding light on this issue? Another question is: in Fama and MacBeth regression can individuals under study be different in different time periods? It would be very kind of you, if you guide me through a link or book dealing whit this subject.

I cannot precisely answer your questions because I do not know which exactly regressions you want to perform as @jmbejara says and which papers are you referring to that use Fama-MacBeth regression. Are they on financial or other literatures? I haven't seen Fama-MacBeth on other literatures (I do not follow any other literature to be exact), so please post the papers. Obviously, if in other literatures one can justify the analogy of the risk-premia concept, Fama-MacBeth can be applied there, too.

I answer your three main questions in reverse.

3. It would be very kind of you, if you guide me through a link or book dealing whit this subject?

If you are interested in financial panel data, there are plenty of resources to consult. Some are:

The amazing book of Bali, Engle, and Murray (2016). "Empirical Asset Pricing: The Cross Section of Stock Returns", because it explains the Fama-MacBeth procedure (among others) step-by-step and I find it the most important of all the resources (ToC available here).
For more information on the standard errors, Petersen (2008, published in the RFS) with his article "Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches" (it also seems available online here) compares different approaches and highlights their intricacies. This paper is crucial for any financial panel data research paper!
Goyal (2012) in "Empirical cross-sectional asset pricing: a survey" also explains the Fama-MacBeth approach with different applications.
Chordia, Goyal and Shanken (2015) in "Cross-Sectional Asset Pricing with Individual Stocks: Betas versus Characteristics" because it has an extensive discussion for the Errors-in-Variables problem of the Fama-MacBeth procedure.
Jagannathan, Skoulakis and Wang (2010) (also a draft) contains more "mathematical/econometric" formalization.

2.  In Fama and MacBeth regression can individuals under study be different in different time periods?

Yes, and this is one of the reasons that Fama-MacBeth is appealing, since you can run it on unbalanced panel datasets. In a balanced panel, every individual has data for the same/every time period, while in an unbalanced this is not a requirement.

1. In panel data application, when using Fama and MacBeth regression is preferable over the fixed or random effect model? Is there a statistical test shedding light on this issue?

Usually, in finance, a fixed effect concerns a firm effect (dummy for firms), while Fama-MacBeth is designed to account for a time effect (Petersen (2008)).
Random effects are rarely used in the financial literature (Petersen (2008)). As he says: "When the residuals of a panel regression are correlated, not only OLS standard error estimates are biased but also the coefficient estimates are inefficient (the estimates do not exploit all of the information in the data). Researchers can improve efficiency by estimating a random effects model using a GLS approach."

To sum up:

I find Fama-MacBeth appealing for accounting for time-effects (it's easy to calculate time-varying betas, for example)
it has easy intuition for the financial literature, and
it can be applied to unbalanced panels.

People use the Hausman test to decide between fixed/random effects models, but I find the intuition and justification of the choice of the proper model more appropriate. I am unaware for a specific test which includes Fama-MacBeth but I am far from an expert in the field.


Reduced Form of an econometric model, identification problem and test Looking for some help to understand the following problem and how to use the reduced form in econometrics

Consider a model for the health of an individual:
$$health = b_0 + (b_1)age + (b_2)weight + (b_3)height + (b_4)male + (b_5)work +
(b_6)exercise + u$$

assume that all variables in the equation with the exception of exercise are uncorrelated with u.

A)Write down the reduced form for exercise, and state the conditions under which the parameters of the equation are identified.

B)How can the identification assumption in part c be tested?

Is it correct to assume:

$$exercise = b_0 + (b_1)age + (b_2)weight + (b_3)height + (b_4)male + (b_5)work +
  u$$ as the reduced form?

and is the condition for identification of parameters simply

$E(exercise|u)=0$

and how can I test it? But moreover what is it good for?

The question doesn't make much sense to me as stated. If the problem says that exercise is endogenous (correlated with error term), you can't assume the opposite in the solution. Plus, one usually speaks about reduced vs. structural form in the context of IV estimation. If exercise is endogenous, you need an instrument for it (variable that predicts exercise, but doesn't affect health otherwise) to obtain causal effects. For example, if some people in your sample randomly won gym membership coupons, that could be a valid instrument.

Identification assumptions would then be

coupon really does predict exercise
coupon is orthogonal to $u$

What is called structural form would be two equations, one your original model, the other regression of exercise on coupon and other explanatory variables from the original model (the first stage). Reduced form would be when you substitute first stage into the main equation, so you regress health on age, weight, ... , work and coupon (but not exercise, as that has been substituted out). Reduced form is sometimes used for explaining properties of IV estimation, but AFAIK it's not much used in practice.


Calculating growth rate of capital when not in steady state Given a Cobb-Douglas production function, the annual population growth rate, savings rate, alpha, annual depreciation rate, and annual technological progress rate, how would one calculate the growth rate of capital?

We're assumed to not be in steady state so we would need a growth rate in order to calculate how long it takes to reach the true steady state.

Perhaps I misunderstood something but it seems to me you can calculate the growth rate by using all the parameters to calculate $K_{t+1}$. Then get the growth rate by taking $\frac{K_{t+1}}{K_t}$.

You will never reach the true steady state, as outlined in my answer here.


On the interpretation of the coefficients in a log-log model as an average In the linear regression model $E(y|X)=x'\beta$, and so we're capable of interpreting a $\beta_i$ as $\Delta E(y|\text{all fixed except } x_i)=\beta_i \Delta x_i$. I'm thinking in discrete changes.
Hence, we're able to interpret the estimator of $\beta_i$ as an estimation of what happens on average.

I was wondering if this interpretation of the coefficients' estimators is also preserved when we have a log-log model... $ E(\%\Delta y|\text{all fixed except } x_i)\approx \beta_i \%\Delta x_i$

Any help would be appreciated.

The short answer is no, this is not so when the dependent variable is in log, unless you define the $\%\Delta$ as (100 times) log-difference. The reason is that $E(\log y) \ne \log E(y)$.

We don't need a multiple regression model to show this so consider $\log(y)=\beta_0 + \beta_1 x + u$. It is the dependent variable in log that matters, so let us not take log to $x$ for notational brevity.

It is true that $\%\Delta y \approx 100 \beta_1 \Delta x$ if $\Delta u = 0$ (ceteris paribus) when $\beta_1 \Delta x \approx 0$. But we cannot interpret it in terms of $E(y|x)$. The reason follows.

As you can see, we have $y = \exp(\beta_0 + \beta_1 x + u)$. When $x$ increases from $x_1$ to $x_1+\Delta x$, $y$ increases from $y_1 = \exp(\beta_0 + \beta_1 x_1 + u_1)$ to $y_2 = \exp \{ \beta_0 + \beta_1 (x_1+\Delta x)+u_2 \}$. Importantly, $u$ can also change because $u$ is not held fixed, which is why I wrote $u_1$ and $u_2$.

Now, what is the change rate of $y$? We have
$$
\frac{y_2}{y_1} -1 = \exp(\beta \Delta x) \exp(\Delta u) -1 \approx (1+\beta \Delta x) \cdot \exp(\Delta u) -1,
$$
where $\Delta u = u_2-u_1$. Here, the left-hand side is the change rate of $y$. You can verify that the change rate of $y$ is approximately $\beta \Delta x$ if $u_1 = u_2$, i.e., if $u$ is held fixed so that $\exp(\Delta u) = 1$. But the problem is that $u$ is not held fixed so $\exp(\Delta u)$ can be very different from 1. When $x$ and $u$ are independent, the average growth rate is $\exp(\beta \Delta x) \cdot E(e^{\Delta u}) -1$.

How different would this be from $\beta \Delta x$? If $u\sim N(0,\sigma^2)$ and $u_1$ and $u_2$ are mutually independent, then $u_2-u_1 \sim N(0,2 \sigma^2)$, and thus $E(e^{u_2-u_1}) = \exp(\sigma^2)$. For example, if $\beta_1 \Delta x_1 = 0.01$ and $u\sim N(0,1)$, then the average growth rate of $y$ is $\exp(0.01) \times e - 1 \approx 1.75$, that is, approximately 175%, which is hugely different from 1%. Interesting, isn't it?

Simulations: Use R to do the following simulation.

set.seed(1)
n <- 1e6
x1 <- rnorm(n)
x2 <- x1+1
y1 <- exp(1+0.01*x1 + rnorm(n))
y2 <- exp(1+0.01*x2 + rnorm(n))
100*mean(y2/y1-1)
# 175.2406
mean(log(y2)-log(y1))
# 0.01004419

When $\beta_1 \Delta x = 0.01 \times 1 = 0.01$, the average growth rate is not 1% but 175%. But $\Delta E(\log y)$ is approximately 0.01.

Note 1: The above discussion is for the log-level model. Little is changed for the log-log model.

Note 2: If you define the growth rate as the log-difference, the expected growth rate equals the expected log-difference = $\beta \Delta x$.


Confusion regarding remedying negative externalities of consumption through taxation This diagram in my textbook illustrates two solutions to a negative externality of consumption - here, the consumption of cigarettes. It states that MSC is shifted upwards by the specific tax. Since the deadweight/welfare loss is defined as the area demarcated by MSC, MSB and the quantity demanded, this would seem to imply that the deadweight loss is not reduced at all (as per the diagram). This seems rather strange.

Here's an alternate diagram I think makes more sense.

Since an increase in taxes will effect a change in the behaviour of individuals, I would argue that it is the MPC (and not MSC) curve that is affected by taxes (hand-drawn Diagram 1). Diagram 2 shows the same picture, but using an ad valorem tax. This will cause a reduction in the deadweight loss. One could argue that a tax will also cause an increase in MSC (effects on employment of tobacco farmers etc.), but, by my reasoning, for the policy to be effective ΔMPC>ΔMPC (otherwise the welfare loss will not be reduced). See Diagrams 3 and (w/ ad valorem) 4 for this scenario.

Does this make sense?

Your reasoning is basically correct. In fact, when I teach externalities to my students, I draw something very much like your diagram 1.

The textbook is confusing because the MSC and MSB curves are supposed to show the social benefits and costs of this particular activity. Taxes and subsidies can't possibly change these costs/benefits because they don't change the fundamental real effects of the activity (one unit of smoke is equally disgusting, regardless of whether it is taxed or not). The tax just means that in addition to this cost, some money is changing hands. But that's not really a social cost because the money will end up belonging to someone.

I don't think it's too helpful to start thinking about the idea "that a tax will also cause an increase in MSC (effects on employment of tobacco farmers etc.)"

Those workers sell their labour in a labour market with a price mechanism. Unless we think there is some market failure in that labour market, the market achieves allocative efficiency. We don't usually talk about externalities when people loose out from price increase or quantity falls in a well-functioning market because those things are efficient and there isn't a very strong case for intervention.


Dimensional analysis for the qdf/quantile function corresponding to the pdf/CDF for the size distribution of income I have previously posted a very similar question on Stackoverflow, but based on responses there I have decided that the real nub of my question is economic. I will give a longish introduction, mainly to distinguish what I know (introduction) from what I am asking (last 3 paragraphs, skip to it if you like).

Suppose I have a pdf f(x) or a CDF F(x) that represents the size distribution of income. Income is a flow of money (dollars or some unit of currency), per some unit of time, to particular individuals. I therefor think the proper unit of measurement is dollars per person-year.

In order for the probability to come out in dimensionless units, f(x) must be measured in person-years per dollar. Integrating this over x also yields a dimensionless share for the CDF because the units of an integral are the product of the units of the integrand y = f(x), here measured in person-years/dollar, and of the differential dx, here measured in dollars/person-year.

Integrating xf(x) therefore puts the units on the mean income in $/person-year; this seems reasonable.

From this I have drawn conclusions on the units and interpretation of the quantile density function q(y), quantile function Q(y), and the integral of yq(y)dy (i.e. the the inverse function analogue of the partial moment function of order 1) which if correct would be very helpful to me, but in which I have limited confidence. I’d be grateful if someone could confirm or deny these conclusions, perhaps in the context of some discussion/explanation of interpretation of dimensional analysis in the the context of distribution functions.

I believe the units of the y axis and of q(y) are necessarily the same as those in the previous pdf/CDF analysis, i.e. the y values are still measured in person-years per dollar and x = q(y) is still measured in $/person-year. This implies that integrals of q(y) over some interval in general, and Q(y) (which is equal to the integral of q(y) from 0 to y) in particular, are both measured in unitless shares. However (uncertain interpretation #1) F(x) is in unitless shares of total income, while Q(y) is in unitless shares of the population for a year.

Finally, (uncertain interpretation #2) this would have the integral of yq(y)dy in units of person-years per dollar. But I am having trouble interpreting this. The corresponding x quantity is the mean (or for a bounded x, the partial mean) of income per person-year. So this would be the mean number of person-years--for what? for a single dollar? for the mean income?

Consider the truncated version of this function, where the integral of yq(y)dy from zero to q* is divided by F(q*). The corresponding truncated function of x yields the mean for those with incomes of x* or less, and $/person-year seems like reasonable units for a mean income. But I want the the y version to come out in number of people, or number of person-years, corresponding to the share of person-years over the same interval, and the "per dollar" in the denominator, which I need to make the CDF(x) and Q(y) unitless, does not have an interpretation which is obvious to me.

Your random variable is $X$ = "Yearly Income per capita measured in dollars". So $X$ is a real-valued function. Its domain is usually left unspecified, while its range is measured in "dollars per person-year", as the OP writes it.

The density and the distribution (pdf/cdf) functions of $X$ ,say $f_X,F_X$ have as their domain the range of $X$. So the domain of $f_X,F_X$ is a set that carries the unit of measurement "dollars per person-year".

Being strict in dimensional analysis, indeed we could say that the density must then be measured in "person-years per dollar" (irrespective of whether this is meaningful or not).

So the expected value that is the integral $\int_0^{\infty} x f_X(x) dx$ keeps having "dollars" as a measurement unit (I just assumed $X$ is non-negative), applying the rule the OP states (the one unit is the reciprocal of the other, and we are left with the third which is again "dollars per person-year").

Moreover the integral that is the cdf $F_x(x) = \int_0^x f_X(s) ds$ has a range that comes out "dimensionless", since probability is a relative measure.

(Digression: Note that by definition, the distribution function of a random variable measures the probability of the random variable being smaller or equal than a threshold, $F_X(x): = Prob(X \leq x)$. Additional assumptions are required to "tranlsate" this probability as representing also "income shares", meaning I guess "percentage of population with per capita income below or equal to $x$")

Turning to the Quantile function, it is defined as the inverse of the distribution function (generalized inverse to cover all cases, but let's keep it simple). Being the inverse, it uses as its domain the range of $F_X(x)$, so the domain of $Q(p)$ carries "probability" as unit of measurement, while its range is the domain of $F_X$. And the domain of $F_x$ is the range of $X$: the Quantile function has the same range as $X$ and so it is measured in "dollars per person-year" too (i.e. its range caries the "dollars per person-year" unit of measurement).

it follows that the derivative of $Q(p)$", $q(p) = dQ(p)/dp$ has the same units of measurement as $Q(p)$, while the integral $\int_0^1 pq(p)dp$ is measured in "dollars per person-year" also.


Would PPP adjustments between countries be irrelevant if all countries had the same relative prices? I'm trying to understand the relationship, if any, between the fact that countries have different relative prices (e.g. different 'food price-to-general consumer prices' ratios) impacts the need for purchasing power parity adjustments between countries.

That's correct, the PPP adjustments are made to account for the fact that living in some places is more expensive than in others. So if you are comparing two places with equal relative prices, their PPP index should be the same, and so it is irrelevant if you use the adjustment or not.

Note that saying that prices are the same in different countries can be quite tricky, since typically goods that can be found in some countries are not even sold in other countries. But in principle, the PPP is trying to adjust for the fact that the same 100 American dollars have a larger purchasing power in some countries than in others.


Expectations in Gali's classical monetary model In Gali chapter 2 we have the following constraint to the classical monetary model

$P_t C_t + Q_t B_t \leq B_{t-1} + W_t N_t-T_t$.

Then it seems that this is treated as an equality. Therefore my question is: are we assumig that $\partial U/\partial B_t >0$? So households would achieve the optimun at the equality.

Another question is, in general if I have the problem

$Max E_0 \sum_{t=0}^{\infty}\beta^t U(C_t,N_t)$

Should I solve the OP

$Max \sum_{t=0}^{\infty}\beta^t U(C_t,N_t)$

And then take expectation to the relations I obtain by optimizing?. For example if we add the constraint

$P_t C_t + Q_t B_t \leq B_{t-1} + W_t N_t-T_t$ to the OPs, in the problem without expectation I get $Q_t=\beta \frac{\partial U/\partial C_{t+1}}{\partial U/\partial C_{t}} \frac{P_t}{P_{t+1}}$ and taking $E_t$ I obtain Euler's equation. Is the procedure of solving the problem without expectation and then take expectation correct?

EDIT:
If it is incorrect to forget the expectation and optimize, then I am confused about this: How to relate real rate of return on capital to bond interest rate: Lagrangian
. As there the procedure to solve the problem seems to forget the expectation.

Second,
if I do $\frac{\partial }{\partial C_t}E[U]=E\left[\frac{\partial U}{\partial C_t}\right]$. I obtain $Q_t=\beta\frac{ E_0\left[\partial U/\partial C_{t+1} \right]}{E_0\left[\partial U/\partial C_{t} \right]}\frac{P_t}{P_{t+1}}$ because I can not get rid off $E_0$. Nevertheless this would be equivalent to $Q_t=\beta E_0 \left[\frac{\partial U/\partial C_{t+1} }{\partial U/\partial C_{t} }\right]\frac{P_t}{P_{t+1}}$ if $E_0 \left[ \frac{\partial U/\partial C_{t+1} }{\partial U/\partial C_{t} }\right]=\frac{ E_0\left[\partial U/\partial C_{t+1} \right]}{E_0\left[\partial U/\partial C_{t} \right]}$. Is this because $C_t$ is independent of $C_{t+1}$?.

Third, I realized that if I solve $Max \sum_{t=0}^{\infty}\beta^t E_tU(C_t,N_t)$ s.t the constraint as before and proceed with $\frac{\partial }{\partial C_t}E[U]=E\left[\frac{\partial U}{\partial C_t}\right]$ I obtain the correct Euler's equation. Is this the correct problem to optimize? but what about Gali's propose?

As for your first question, we are not just assuming $\partial U/\partial B > 0$. We simply assume monotonicity of preferences, which for the model here is a plausible condition. In simple terms this means that "more is better". Hence, the consumer will always choose to spend all of his budget for current or future consumption, given by $C$ and $B$ respectively. Hence, the constraint holds with equality. In these types of models we pretty much always assume equality of the budget constraint.

As for your second question, solving without expectations and then taking the expectation is strictly speaking incorrect. It will give you the correct result here though. What you can do here is to take the derivative in the expectation. For these models that usually works in the non-stochastic setting as here. If you have for example stochastic income with log-normal distributions you would have to be more careful. What you can do then, depending on the case, is solve the expectation before taking the derivative or use the Leibniz Rule.

Hence for your problem you don't forget the expectation in the first place, which corresponds to $\partial U / \partial C_{t+1}$. Instead for your model you can take $\partial E_0[U] / \partial C_{t+1} = E_0[\partial U / \partial C_{t+1}]$. As you can see, it will yield the same result as simply taking the expectation later on $\partial U / \partial C_{t+1}$. However, that is the wrong way and you'll probably lose points in an exam if you write it down that way.


Logic Fallacy in a single government purchase creates infinite GDP? Suppose the government increase its spending by $100 million. And suppose we are talking about intermediate level macroeconomic class. Let the marginal propensity to spend be 0.5.

Now the total GDP should increase by 100/0.5=200 because the increased spending increases people's income and thus further increase the spending, so on and so forth. But wait, what if the marginal propensity to spend is 0! Then we have an infinite increase in GDP! OR if this sounds just wrong, let us consider where the unspent money get saved in banks. And banks will lend the money out which induce firm to invest more. So the investment will increase people's income, and hence the same effect as before. So in the end, we see that the increase in government spending will increase income by the same amount, and then increase spending again by the same amount (the saving also get invested, so it increases the GDP), and etc.

This is clearly not the case in the world we live in. What is the catch?

It seems that you misunderstood the concept of marginal propensity to consume. The marginal propensity is only valid for an infinitesimal increase in income. As soon as you move away from the original starting point where the marginal propensity was 1, there may be a different marginal propensity. The multiplier however assumes that the marginal propensity to consume remains the same at all income levels (since it is essentially an infinitely repeated application of the marginal propensity to consume).

The above problem with such analyses is why economists switched to models with so-called microfoundations in which the firm and consumer behavior is explicitly modeled via profit/utility maximization. In a model with fixed resources it would quickly become clear that a marginal propensity to consume of 1 for all income levels would violate market clearing conditions.


Can we have incomplete markets with a continuum of securities? Imagine there is a continuum of firms in the economy. Each draws its productivity from the same stochastic process. The stochastic process has unbounded support.
The only securities in the economy are the firm's shares.

The claim is that markets are incomplete because the cardinality of the states of the world is $\aleph_2$ and the cardinality of the securities is $\aleph_1$.

However one could also argue that since there is a continuum of firms, the strong law of large numbers applies and one knows exactly the proportion of firms that will get each shock, hence there is no risk (and markets would be complete)

Which of the two statements is correct?

Thanks a lot for the reference. I think the result does hold. Here is what I found:
The validity of such a law of large numbers what subject to some debate in the 1980s.
See Judd (1985), Feldman and Gilles (1985) and Uhlig (1996) for representative papers.

Luckily Feldman and Gilles show that for a probability space $(Y; B(Y ); \Pi)$ there exists a continuum of random variables $y(i;.) : \Omega \rightarrow Y$ such that for all $i$ the random variable is distributed according to $\Pi$ and that for all $\omega \in \Omega$ and all $D \in b(Y)$
$$m(i \in I : \{y(i; \omega) \in D ) = \Pi(D)$$
that is, the population income distribution is nonstochastic and given by $\Pi$ The continuum of random variables cannot be pairwise independent.


How to calculate capital investment? So i have this question:

After spending 3 million on research, Better Mousetraps has developed a new trap. The project requires an initial investment in plant and equipment of 6 million. This investment will be depreciated straight line over 5 years to a value of zero, but when the project comes to an end in 5 years the equipment can , in fact, be sold for 500,000. The firm believes that working capital at each date must be maintained at 10% of the next years forecasted sales. Production costs are estimated at 1.50 per trap and the traps will be sold for 4 each. Sales forecasted are given in the following table. The firm pays tax at 35% and the required rate of return on the project is 12%. What is the NPV?

The image attached is the answer to the problem under the form of a table.

The only thing i do not understand in this calculation is why , under year 5 in the table, in the capital investment row, it says that the capital investment is 325. Where is this number coming from ? Thanks!

Presumably the $\$325K$ represents the sale of the equipment.  It is positive while the purchase of the equipment was shown as $-\$6,000K$

The question says $\$500K$, but that there has been depreciation down to zero (also shown in the table five times as $\$1200K$)

The depreciation has led to lower taxes being paid, so tax at a rate of $35\%$ is due on the sale of the equipment, and $\$500K \times (1-0.35)=\$325K$ is the net benefit to the company from the sale


Growth theory / Initial GDP: positive coefficient sign I estimated growth regressions for several EU countries. In each of them, the sign of "initial GDP"'s coefficient is statistically significant but positive, which contradicts growth theory as well as most of other published papers on this topic. I tried to change the variables, the period, etc. but it remains positive.

I can't find any explanation for this result in the literature, which would imply that there is no growth convergence among European countries.

Did anybody already face this problem? How did you solve this? I can't find the source of this problem. Personally, I replaced "initial GDP" by "lagged (one period) GDP" but it would be better if I could keep "initial GDP" among the explanatory variables.

It's really difficult to give a satisfactory answer without having the data at hand. I would do two things:

Check carefully the data and computation of growth rates. By experience, it happens that some "particular" result is the consequence of data or computation issues. You may for example observe any issue by plotting your growth rates and initial GPDs with the name of the countries.
Use your data but exactly the same period, same countries and same methodology than the literature and check if you still find a contradictory result. Then, you can mix: use the same source of data, same countries, same period but your methodology, etc...


How do government purchases crowd out the private sector? I am curious about how government purchases crowd out the private sector (if it does). If we look at a graph of the composition of US GDP, it seems that government purchases and investment are somewhat negatively correlated. It's obvious during WWII that government purchases skyrocketed while investment shrank significantly.

I am wondering about the intuition behind why this happens. If I had to guess, it's because at a given snapshot in time, the capital and labor in the economy is limited, and if the government is spending more money to use capital goods and labor, then there are less capital goods to go around for the investment. Is that line of thinking correct?

Just to add some more besides the answers above:
Short answer: by influencing interest rate.
Explanation:
In the post Keynesian literature what has come to be appreciated a lot is that government purchases, backed by increased government debt, increases interest rate. Why? Well, that's in itself an interesting explanation and perhaps should be a separate question (interesting, because contrary to what people think that "more govt borrowing means more govt bonds in the market, means lower bond prices and hence higher interest rates" is not at all a satisfactory explanation).
Private investments are assumed to be a function of interest rate. So some of the investment projects (particularly, those at the margin) that were earlier profitable may not be anymore due to increase in interest rate.
This is what is essentially, crowding out of private investment.
This does not require the assumption that Savings have to be fixed. Instead, in Keynesian framework, what happens is that $Y, S, C$ all goes up (because crowding out is not necessarily full, depending on the interest elasticity of money demand - slope of LM curve). It's just the share of $G$ increases and that of $I$ decreases.
Coming to your assessment that historically, at least during some periods, $I$ and $G$ have been found to be positively correlated. Indeed, investment is not just a function of interest rate. It is also a function of sentiments of future of the economy. When govt's start building infrastructure or take measures to improve supply constraints it boosts the confidence in the economy and encourages investment.
That doesn't make concept of crowding out incorrect. Because the concept is applicable when, ceteris paribus, $G$ increases.
Lastly, your own idea that "the capital and labor in the economy is limited, and if the government is spending more money to use capital goods and labor, then there are less capital goods to go around for the investment." is the somewhat closer to the classical view. This assumes that economy is at its full potential and there are no rigidities in the labour market. Without going into details, this view over the year has been seriously challenged. However, it should still be considered as an alternative explanation.


Can measures of "efficiency" be objective? So I've noticed the reliance on the term "efficiency" on some cases of economic practice. But I've also recognized problems, e.g. problems of abstraction or problems of pseudo-objectivity, related to making claims about "economic efficiency".

I wonder whether "efficiency" can ever be an objective quality? Or whether it's in practice a very complicated concept.

E.g. while slaves could be efficient in terms of output, using slaves would negate many other aspects of humans and thus is not necessarily a "complete" view on "efficiency". Also, while "exploiting" laborers for gains might offer efficiency, it again might neglect that laborers have more aspects than "mere cold production". And that those other aspects (e.g. life quality) might have correlation between how they perform in work.

Also, since communities or societies cannot necessarily be shown to have "objective goals", then concepts such as "economic growth" may seem absurd. Because objectively, there's nothing to fundamentally motivate it.

Efficiency can be a very "objective" concept because it is conditional on whatever "non-economic" environment production will take place. Consider the "production possibilities frontier" of a society, that says "given the resources we have, we can produce $100$ of X only, or $70$ of Y only, or some combination of the two, in a year".  As long as we are on the production possibility frontier, we are efficient.
Right. How do we measure how much we can produce? It looks like a totally objective "engineering" exercise (however complicated). "We have available $1000$ labor hours and with them we can produce $100$ units of $X$, it is simple as that".

Hmmm... and why do we have $1000$ labor hours available, and not, say, $2000$? How did we determine that? We determined it by considering perhaps some "average" number of hours worked per person times the working population of the society. Ok, and how did this "average" number of labor hours per working person cοme about? In practice it is strongly affected by the institutional environment in the specific economy. But his institutional environment is taken as given in determining whether we are efficient or not.

So this concept of efficiency is conditional on technology and socio-economic institutions (market structure is included here).

But can the socio-economic institutions be assessed as to whether they cause inefficiency themselves? They certainly can, but we just moved one step higher than the previous concept of efficiency. And here things get cloudy, because the socio-economic institutions are never free of ideology, values and the like.

So to say "the structure causes inefficiency" has the possible rebound "given your values which is not my values". In theory we could even quantitatively assess that -but we would have to perform real-world experiments, something like "let's introduce slavery, and see whether in a long term perspective we produce more or not". Ok. Are you willing to play slave in this scenario (that will last many-many years), so that the world will benefit from the objective knowledge we will gain?


Is lowered probability of spending equals savings? This a really basic one and logic says: Yes given enough iterations.
But I am looking for validation

Situation: There is a 7% probability of expenses of 1000 occurring. I have a tool that then applied to the process has a 95% probability to reduce the chance of expenses occurring by 58-78% (mean at 60%).
How do I calculate savings?

Not going into Risk-Adverse case as this is sensitive to how we define the Government's utility function

$\mathbb{E}[xpnse] = 0.05*(0.07*1000)+0.95*(0.6*(0.07*1000))$

Follows from Law of Large Numbers


How to adjust salaries of more than two countries to a common year given that the data was collected in different years? I want to compare salaries of more than two countries. The data I am using is based on the Occupational Wages around the World (OWW) database. Let's say I want to look at the salaries of the nurse. The salaries are specified in US Dollars and not all of them are collected in the same year, so we have sort of an incomplete data. As an example:

Year: 2008, Country: Azerbaijan, Annual Salary (USD): 1678.15
Year: 2007, Country: Singapore, Annual Salary (USD):  19523.57 (no 2008 data)
Year: 2006, Country: China, Annual Salary (USD): 3338.208 (no 2008 or 2007 data)

According to the author of the database (Freeman and Oostendorp), to compare living standards, we have to deflate wages using Purchasing Power Parity (PPP) instead of exchange rates for the US dollars (It seems to me that their data is in local currency, rather than in USD as the example above). However, it is not clear to me how we adjust these salaries to a common year, say 2009.

Is it correct to do the following to my data above? Or is there a better way to compare salaries over countries and time to a common year? I use World Bank Indicator (http://data.worldbank.org/indicator) to compute things.

First, we multiply the USD salary with their corresponding LCU/USD exchange rate [Indicator: "Official exchange rate (LCU per US$, period average)"]
Then, we divide by the PPP-correction [Indicator: "PPP conversion factor, GDP (LCU per international $)"]
To adjust across time, then we multiply the result by ratio of GDP per capita, PPP between the common year (2009) and the year the data is collected (2008) [Indicator: "GDP per capita, PPP (current international $)"]

I would greatly appreciate any help.

Your three stage procedure is not correct.

In effect, since the data is already in USD, to apply PPP you need to revert it back into LCU (local currency units). That is your step 1. Have in mind this might introduce errors if the one you use is not the same the authors used.

Then, since wages are not for the same year, ideally you do need a proxy for wage growth of each occupation, for each country and year, so that you can compute wages in LCU for the same year. You are attempting to do this in step 3 using GDP growth, which is not ideal (and the order is wrong too). I would rather use an inflation index. This is not ideal either, but the World Bank provides an homogenised one. Alternatively, you could get wage growth for occupational groups (or aggregate) from national statistical offices, but this is a time-consuming exercise.

After putting wages into the same year, you proceed with step 2, where you convert LCU to USD PPP, using a common year.


How can foreign protectionism affect local FDI I am citing an article here:
In a speech at the G-20 in Istanbul, TİM’s Mehmet Büyükekşi said, “Protectionism is increasing in every market around the world and it is affecting not only Turkish exports, but also Foreign Direct Investment [FDI] in Turkey.

I really can't see a link between protectionism in another country and FDI in Turkey. What is the link here? Also, he later says that devaluation of euro against the dollar affected total FDI value. That makes a little more sense to me, because it can be that amount in euros prepared for some investment in Turkey suddenly became less valuable. I hope that is correct, but I don't get the protectionism part?

A protectionist policy in other countries could take the form of providing more incentives for companies to invest domestically instead of abroad. For example, U.S. companies face penalties if they invest abroad compared to investing in the U.S. or they have more benefits from investing in the U.S. now compared to before. Anyhow, companies outside of Turkey could have more inventives to invest in their own countries, which in turn means they have less incentives to invest abroad, which in turn means they have less incentives to invest in Turkey, which in turn means they invest less in Turkey.


Do banks take money to repay deposits from required reserves? Say I deposit 200 dollars in the bank and 20 dollars is set aside for the required reserves. Other customers also deposit money and more money goes into the required reserves.

If I go and get the 200 dollars back, can the bank take money from the required reserves, or can that not be touched unless a bank run occurs?

When you get your \$200 back, the bank's required reserves go down by \$20 as well, so the bank can take that amount back from its reserves.

Operationally, the bank wouldn't go to the central bank and get this, as they probably would have \$200 in cash laying around already.


Has the "dismal theorem" practical implications? The so-called "dismal theorem" asserts that we do not appropriately account for catastrophic scenarios which have very small probability of occurrence. It has been studied in details by Martin's Weitzman, notably in his article "Additive Damages, Fat-Tailed Climate Dynamics, and Uncertain Discounting".

The articles of Martin Weitzman rely on a large dose of mathematics, and my skills do not allow me to get everything, nor of course to question the reasoning and deductions of Weitzman. His conclusion is broadly that "The take-away message here is that reasonable attempts to constrict bad-tail fatness can leave us with uncomfortably big numbers" (p19)

I would like to know if there are actually any practical implications from this dismal theorem. In particular concerning climate change. I thought so until @Dole pointed out that the dismal theorem could also be used to justify trillions of dollars in investment in a anti-asteroid defense system. I would appreciate any insight on the conditions to apply this theorem, if it can ever be used. Any relevant literature would also help me.

My conclusion based on reading his paper is that the utility function of an individual or society can't be of the CRRA form presented in the paper. That would indeed lead to scenarios where you could not get out of the bed in the morning, as minimizing the tiniest probability of an enormous risk would warrant infinite sum of money.

I will attempt to explain the mathematics of the paper. First the utility as a function of consumption is of the form:

$$ U(c) = -c^{1-a}, \space a>1$$

This is also called a constant relative risk aversion utility function. Constant relative risk aversion with regards to consumption means that a person prefers a bundle of 1 utils over uncertain bundles with expected value of 1 utils. The same applies to 2 utils to the same degree proportionally. Note that the utility is $-\infty$ when consumption reaches 0.

Now, if you want to calculate the exact utility that a person receives you simply multiply the probability of each bundle with the utility it grants:

$$P_1*U(C_1)+P_2*U(C_2)... = \sum_{n=1}^{\ b}P_n U(C_n)$$

Where P denotes a probability of a given bundle and U(c) is the utility.

Finally, consider a case where there is a possibility of a bundle that has consumption value of 0, with probability greater than zero:

$$ P_x U(c_x) = -\infty $$

It does not matter what the probabilities and utilities of other bundles are as you can't recover from minus infinite utility.

The same principle applies to continuous probability distributions, so just replace the sum sign with an integral and consider case where it doesn't reach 0 probability when c=0.

You may be interested in readin Nordhaus's response:
http://aida.wss.yale.edu/~nordhaus/homepage/documents/weitz_011609.pdf (By the way, he uses the exact same case scenario as I did).


Is the study of the volunteer sector inappropriate for a labour economist? In the basics of labour economics the driving factor for increase in labour supply (for the workers in a given market) is increase in wages.

The volunteer sector consists of general and skilled labour that provide their services for free, meaning there is no wage incentive.

what I would like to suggest is that volunteer labour has to be exclusively viewed through the scope of consumer theory as income is not a driving factor for labour supply and volunteer labour must be viewed as another consumption good or service which faces utility maximising constraints.

I would agree that volunteered labor is a utility enhancing activity. Seen as a good, its price could be the opportunity cost, i.e. the wages foregone (for not working for pay). But then this is not really different from the concept of "leisure", since "leisure" in economics does not mean "fun and games", but rather "time spent not working for pay".

An interesting extension could be the potential value of volunteered labor as professional experience towards future employment for pay (in line with Arrow's "Learning by doing" efficiency concept). Then the issue becomes intertemporal, and also, income considerations arise: I want to volunteer, it generates utility directly for me, but also, I'd rather volunteer and build up professional experience that may lead to higher wages tomorrow, than try to work now and have lower pay due to smaller experience.


Emergence of notion "primitive business" in the era of Industry 4.0 and globalization It has always been the conventional wisdom that businesses should be fit to survive in the market competition, that only the best businesses can survive beyond horizon of several years, that survival of some enterprise certifies both its style of conducting business and its general meaning for the society.

But Industry 4.0 and globalization has created 2 interesting trends:

Automation, robotics and artificial intelligence are providing opportunities to automate jobs, but some investment is required. So - for the first time in the human history the humans compete against machines and sometimes human labour lower the standards of remuneration to survive in this competition. So, in that way the primitive businesses emerge - those enterprises that rely on the cheap human labour and that simply do not make investment. So, if someone has no capital and if someone has enough cruelty to create mean jobs, the he or she can build up primitive business and survive the market competition relying on the cruelty of primitive, low wage jobs. The general society suffer - the society provides cheap labour, the society does no receive taxes from low wage job, the society looses could-be scientists, technologists, ethical businessman, artists and so on, so on.
Globalization and "race to bottom", international and interregional competition relax the standards on labour and renumeration, sometimes countries or regions asks for any job the enterprises can provide and this is again the opportunity for the unethical people to create enterprises with precarious jobs.

So - both those trends do very much good to the society, but those trends also create opportunities for the unhealthy, unethical companies without investment and use of technologies to emerge, so - in short - primitive businesses can exists, survive and even thrive in this new era.

My questions are:

Is there economic research about primitive business or maybe they are called differently in the economic research? There is research about financially-constrained households and how their option space is bounded and how they disturb the usual assumptions about the benefits of free market, free will and utility maximization. Maybe there is literature about primitive businesses as well?
Is there normative economic research that considers the constraining the power of primitive businesses - both - by the increased regulation (e.g. common minimal standards) and by increased opportunities for the workforce to escape the trap in falling the braces of primitive businesses.

p.s. I am aware about the usual thinking that any jobs are good in the era when automation automates away the jobs and that people should work whatever that means. But there is escape from the low standard jobs either by Universal Basic Income or by Universal Dividend (better yet) and people will not become vegetables in these settings, because, the mentally healthy people, according to Maslov et el, prefer self-realisation (science, arts, travelling, volunteering, etc., etc.) and not laziness (and there should be treatment for mentally unfit people).

There surely is research about the complement(s) of the (low tech) "primitive businesses", i.e. the high-tech and/or innovative businesses.
The US CES for instance publishes a BDS-HT (Business
Dynamics Statistics - High Tech) report. You can compare these with the core BDS data and get an idea how the high-tech sector differs. The US CES method of high-tech business classification is based on the number of STEM employees.

Ultimately, High Tech industries account for about 4% of firms and 6% of employment.

So there are a lot "primitive businesses" by this standard, even in the US.
A different method is used by CES for finding/reporting "innovative firms", BDS-IF. This is based on patents (BDS-PF) and trademarks.
I'm guessing the German Industrie 4.0 that motivated your question is probably similar to BDS-HT, but I don't know their exact methodology.


Effect of positive inflation shock on the exchange rate How does a positive inflation shock affect the exchange rate? Does the exchange rate appreciate or depreciate? I am looking for an intuitive explanation.

Before intuition. let's determine the outcome. By the Uncovered Interest Rate Parity

$$ i_t  = i^*_t - [s^e_{t+1 | t} - s_t] \tag{1}$$

where $i_t$ is domestic nominal interest rate, $i^*_t$ is "foreign" nominal interest rate, and $[s^e_{t+1} - s_t]$ is expected appreciation of domestic currency in percentage terms ($s$ is the natural logarithm of "foreign currency per unit of domestic currency" exchange rate).

The "Fisher equation" for the relation between real and nominal interest rate in expected inflation is

$$i_t = r_t + \pi^e_{t+1|t} \tag{2}$$

where  $r_t$ is domestic real interest rate, and $\pi^e_{t+1}$ is expected domestic inflation. Putting $(1)$ and $(2)$ side-by side we have

$$r_t + \pi^e_{t+1|t} = i^*_t - [s^e_{t+1|t} - s_t]$$

Assuming that there are no real effects by an inflation shock, and that the international economy couldn't care less (so also $i^*_t$ remains unaffected), using the change operator we have

$$\Delta \pi^e_{t+1|t} = \Delta (s_t-s^e_{t+1|t}) \tag{3}$$

where now the right-hand side represents "change in expected depreciation" of the domestic currency (note that the right hand side may be algebraically negative in which case it is "negative depreciation" = appreciation).

Now, it is reasonable to argue that expectations for tomorrow will take into account what happens today. Moreover, usually current inflation shocks either affect in the same direction expectations about future inflation, or not at all.

So an inflation shock happens in period $t$ which increases the left-hand side of $(3)$. Then so must also increase its right-hand: a domestic positive inflation shock increases expected depreciation of the domestic currency. In other words it creates expectation of currency devaluation. The currency may still be expected to appreciate, but less than before the inflation shock.

Mathematically this can come about in different ways but realistically one should expect that what will happen is that $s^e_{t+1|t}$ will go down, i.e. that the domestic currency will be expected to devalue.

The main intuition behind this comes if one realizes that, for the people abroad, domestic currency is just another product, that has a price. If due to inflation this "product" is worth less (because now it can buy less of utility-enhancing or productive goods), this should be reflected in its price, which is the domestic exchange rate $s$ (price of domestic currency measured in units of foreign currency), which should get lower.

This is the rationale also behind the closely linked "purchasing power parity". Of course, this is a highly stylized analysis and all sorts of real-world imperfections and rigidities will affect what will actually happen. But the tendency is the one described above.


How to find daily/weekly data on the state of the economy I am interested in economic modelling of the correct state of the economy.  Figures such as unemployment and gdp come out with a considerable delay.  What are good publicly available datasets that are available daily/weekly that are known to be proxies for unemployment and gdp growth?
Ideally these data would be available for a number of countries. I am only interested in datasets I can download in full.

That frequency is very hard to come by for GDP and employment, especially in many countries.
The only thing I know that allows that level of measurement is prices: check https://www.pricestats.com/ for daily measures of inflation, purchasing power parity and real exchange rate. It's available for many countries, both in developed and emerging economies.


Difference between objective and subjective distributions in asset pricing models? How does one differentiate between "objective" and "subjective" probability distributions in asset pricing models?

In asset pricing, economists often make a distinction between the "subjective" and the "objective" probabilities of future states of the world. What do they mean by this?

Probability judgments are based on an investors' private information sets, i.e. subjective probabilities. In contrast, objective probabilities are the underlying real probabilities that the world will end up in some state. It is necessary to make the distinction because traders do not have perfect information. See Markowitz (1991) for further explanation.

But objective probabilities remain a remarkable concept. If the market reveals objective probabilities, then these are in fact known, and traders can update their subjective probabilities accordingly. But if all traders agree on the probability distributions, they would have very little reason to trade. Hence one implication of portfolio theory is that there is too much trading going on. See Odean (1999) for empirical work on this, or Grossman & Stiglitz (1980) for a theoretical critique.


Why does Slutsky compensation "overcompensate" the consumer? Suppose I have a Marshallian demand function $x_M(p_x^0,p_y,m^0)$. As I understand it, Slutsky compensation is defined as

$$T_S = \Delta p_x \cdot x_M(p_x^0,p_y,m^0)$$

Can someone explain why this compensation overcompensates the consumer?

Here's a figure to explain:

Starting from the old price line, where the optimal consumption bundle is point $A$, we increase the price of $y$ to get the new price line.

The Slutsky compensation says that we have to give the consumer enough extra income so that he can afford to old bundle ($A$) at the new price. Thus, we shift the new budget constraint out to the dashed line.

The reason this "overcompensates" is that, taking the dashed line as the budget constraint, the consumer could afford bundle $B$, which gives him higher utility than he started with.


Reading list for physics undergrad who'd like to cover the economics related to the hedge fund/ private equity field I have had  relatively little exposure to economics( nothing more than EC101) and would greatly appreciate help in answering the following questions:

Which sections of economics are more or less very relevant related to the hedge fund/ private equity fields?
Which books would one recommend that would help one get a solid grasp(almost to the level of an economics undergraduate) of the sections in 1. ?

Thanks

Macroeconomics is the part of interest to fixed income hedge funds, and a secondary concern for other parts of finance. It is also the area with the greatest theoretical splits. I am in the post-Keynesian economics camp. Within post-Keynesian economics, I would list two books as a good starting point.

(I would guess that “mainstream” economists are in the majority for answering questions here. I would let them list what they view as the best introductions to macroeconomics; I have obvious theoretical reservations about the mainstream texts that I have read.)

The text Monetary Economics by Wynne Godley and Marc Lavoie is an introduction to stock-flow consistent modelling. It is aimed at undergraduates, and gives an overview of post-Keynesian economics from a mathematical perspective. The models themselves are simple, but a lot of key information is embedded in the text itself. It explains the post-Keynesian critiques of the theoretical mainstream. My background was in applied mathematics, and the treatment in Monetary Economics made the most sense to me; the mathematical conventions used by economists elsewhere are somewhat non-obvious to outsiders.

The next book is Stabilizing a Unstable Economy by Hyman Minsky. It’s more of a popular book than a textbook (largely non-mathematical), but it provides a readable introduction to Hyman Minsky’s views on the relationship between finance and the economy. The 2008 Financial Crisis was labelled a “Minsky Moment,” reflecting Minsky’s popularity among financial market commentators. Stabilizing an Unstable Economy may not be Minsky’s best book (collections of his articles might be better), but it offers the best overview.


Laffer curve in one period economy Let us consider a perfectly competitive one-period production economy consisting of one firm, N>0 household(s) and one government. Each household is endowed with x unit(s) of physical capital and with h unit(s) of time that can be allocated to work: e and/or turn into leisure time: l at a rate of one for one. The preferences of the representative household are represented by the following utility function:

commodity is produced by the single perfectly competitive firm using the following Cobb-Douglas production function:

$$Y=AK^{\alpha}H^bL^{1-\alpha-b}$$
where Y denotes the aggregate level of market production, K represents the aggregate level of private physical capital, H represents the aggregate level of public physical capital, L stands for the aggregate labour input defined as the total amount of time allocated to work, and ,b ∈ (0,1), A>0 are parameters. The firm rents from households both labour at the real wage rate w and physical capital at the real rental rate . Each household earns a real dividend income that corresponds to a fraction 1/N
of the firm’s profit P , pays a physical capital tax t $\in$(0,1) proportional to his real
physical capital income. Let us assume that a fraction  ∈ (0,1) of the total tax revenue T is invested into the public physical capital:  =  and a fraction 1 −  ∈ (0,1) is consumed by the government:  = (1 − ).

My question

  Would production possible without the government: t= 0? Do households have incentives to rent the physical capital to the firm if t= 1?

I wrote the governemnt’s revenue = $T= t(NρΧ)$

And then I think to use laffer curve to answer this question.

For example

when t=0, T=0, so H=0 so Y=0. There is no production.

When, t=1, T=NρΧ so H>0 and Y>0

And hh’s phisical capital revenue=(1-t )ρX. So when t=1, there is no revenue for households. So they don’t want to rent their physical capitals.
In order to find optimal $t^*$

I need to maximize $T= t(NρΧ)$ w.r.t $t$. But I could not get any reasonable value.

——

An edit:

Should I consider X depends on t here. That is X=X(t)

Then    $$ \max tNρΧ(t)$$ subject to $t$.

By FOC. NpX(t)+tNρΧ’(t)=0

There exist $t^*$ such that $NpX(t*)+t^*NρΧ’(t*)=0$ Holds.

Is this way correct? Or

How can I solve this question?

Disclaimer: I still have to educate myself on utility theory (same as I disclosed here).

  Would production possible without the government: t= 0?

It cannot be determined with the information given. The problem's definition of $T$ is unclear or incomplete, as the only component of $T$ mentioned in the problem is the physical capital tax rate $t$.

Assuming that the only source of government revenue is the physical capital tax, then no, production would not be possible if $t=0$ because --as you point out--, the latter implies that $T$, $H$, and therefore $Y$ are zero.

But that prompts the question: Does the definition of production $Y$ even make sense if  = ? Obviously not: Production should still be possible because private physical capital $K$ is not affected by the condition $t=0$.

  Do households have incentives to rent the physical capital to the firm
  if t= 1?

Yes, they may have an indirect incentive even if $t=1$: The real wage $w$. If nobody rented physical capital, there would be no demand for labor. Consequently, no household would earn income from labor even if a household's utility function indicates a preference for work over leisure.

I gather that the problem is a textbook exercise, and/or that not all the assumptions are reflected in the inquiry. Thus, a portion of my answer possibly goes beyond the scope of that chapter or section.


Definition: Extractive Goods (Comtrades HS-Categorization) I'm currently writing a paper on China in Africa and I'd like to compare Africas Exports to China with its exports to Europe and the USA. More specifically I'd like to find out if the Exports to China consist of more natural resources than those to Africas other trading partners. However I find it hard to distinguish between "extractive goods" and non-extractive ones. So far I used Comtrades HS-categorization and the categories:

5 mineral products
9 wood and articles of wood; wood charcoal; cork and articles of cork; manufactures of straw, of esparto or of other plaiting materials; basketware and wickerwork
14 natural or cultured pearls, precious or semi-precious stones, precious metals, metals clad with precious metal and articles thereof; imitation jewellery; coin
15 base metals and articles of base metal

Do you think this wouldd be ok, or do you have any sources that state which categories to define as "extractive"?

I know this is a little late, (just joined this forum two hours ago) but if it is still relevant. You should look at the Australian Department of Foreign Affairs and Trade TRIEC system (Google "DFAT TRIEC") It is an established system and useful for high level groupings. You might try to use their groupings and hopefully be able to answer the questions from the analysis.


Difference between multinomial logit and conditional logit I'm trying to decide whether to run a multinomial logit or a conditonal logit (McFadden, 1973). I have data from a choice-based conjoint study in which each of the respondent's choices was between a pair of products with varying characteristics, including price. Each of the characteristics is a continuous variable, if that matters. I want to estimate a marginal willingness to pay for each of the characteristics. Typical explanations of how the two models differ are as follows:

  Multinomial logit models a choice as a function of the chooser's characteristics, whereas conditional logit models the choice as a function of the choices’ characteristics.

By this logic, I would lean towards a conditional logit given that I'm trying to estimate a marginal willingness to pay for each characteristic. On the other hand, the values I estimate for this depend entirely on the preferences of the respondents, so you could say that I'm really estimating something relating to the preferences of respondents, rather than anything innate about the choices' characteristics.

Does anybody have a more crisp understanding of the differences between the models and/or reflections on which would be more appropriate in this setting?

It would be much better to have at least a head of your dataset to be sure, but from the description conditional logit is the way to go (though, in practice, random coefficients logit should be used to avoid IIA - independence of irrelevant alternatives).

General form: Think about both models as special cases of some general discrete-choice model. I've got accustomed to convention that index $i$ is for individuals and $j$ is for alternatives.
$$ \pi_i(Y = j|\mathbf{z})= \frac{\exp{(\boldsymbol{\beta}^\top}\mathbf{z}_j)}{\exp{(\boldsymbol{\beta}^\top}\mathbf{z}_1) \: + \:... +\: \exp{(\boldsymbol{\beta}^\top}\mathbf{z}_J)}$$
Notice that $\mathbf{z}_j$ and $\boldsymbol{\beta}$ are vectors of length $J \times K$, where $K-1$ is the number of features [for details, see Agresti(2002:300)]. It is important to understand that we have one term for each alternative in the denominator of this softmax function.
We can derive mlogit and clogit by setting to zero different elements in these vectors. Thus we get different models.

For simple mlogit with $J = 3$ choices (alternatives) and one individual characteristic (chooser's attribute) we can find probability of choosing each alternative. In example, the probability of choosing second alternative by some individual is:
$$ P_{i2}(x_i)= \frac{\exp(\alpha_2 + \beta_2 x_i)}
{1 + \exp(\alpha_2 + \beta_2 x_i) + \exp(\alpha_3 + \beta_3 x_i)}$$
It is important to note that there is the function of one variable $x_i$ (it doesn't change between alternatives for the same individual). Another point is that coefficients are different for all alternatives. Finally, we may use intercepts $\alpha_j$ in mlogit, because they are different and can be identified (if we set $\alpha_1$ and $\beta_1$ to $0$ - that is why 1 in the denominator).

For clogit the situation is quite different, because we have different values of the same feature for each alternative. So the model with 3 alternatives and one feature takes the following form (i.e. probability that individual $i$ chooses second alternative):
$$ P_{i2}(\mathbf{x}_j)= \frac{\exp(\beta x_2)}
{\exp(\beta x_1) + \exp(\beta x_2) + \exp(\beta x_3)}$$
Here we deal with the function of three variables ($\mathbf{x}_j$). As far as these values are different it is enough to fit only one $\beta$, which is the same for all alternatives. As such, we do not need a base alternative (no unity in the denominator). Finally, we cannot identify any intercept in such a model, because any $\alpha$ will simply vanish in the softmax ratio ("thanks" to exponent property).

What things are also important for both models? The common way to fit both models is maximum likelihood, though in conditional logit this likelihood is also conditional (it simply means that we impose the condition that one and only one alternative should be chosen by each individual).

How to understand in practice, which model to apply? Well, you will not get results if you put data for one model in another model command (statistical software will complain that there is no variability in data).
If you have your data in long form, it might look something like this:
Y   c_logit_feature    m_logit_attrib
--------------------------------------
0           3                 10
1           5                 10
0           7                 10
--------------------------------------
1           3                 12
0           5                 12
0           7                 12
--------------------------------------
...
--------------------------------------
0           3                 18
1           5                 18
0           7                 18
--------------------------------------

Usual package will be able to fit clogit from such setting (using last column as grouping indicator). But mlogit usually wants different form and this fact (in my experience) confuses most of users. If we label our three alternatives as "a", "b" and "c", mlogit setting derived from the above table is going to be:
label  m_logit_ready
--------------------
"b"         10
"a"         12
...
"b"         18

Modern statistical packages (like R or Stata) are able to work with different formats but the general idea stays the same.
Hope that helps.


Inflation basics and understanding In some new country, there's 1 million credits in circulation, and Product X costs 10 credits. If I were to print, say, 500,000 credits and build a new swimming pool with this, how exactly would inflation increase? Why would the cost of Product X increase (to approximately 15 credits?)?

If prices and thus inflation are set by humans, then how would they know to hike up prices? Is it the increase in 'supply' of money, since whomever received some portion of the 500k stacked on top of what they already had and now they have more to spend?

The grey-haired "Quantity Theory of Money" is a good initial way to gain some clear understanding for inflation. This theory is just one equation,

$$PQ = VM$$
where

$P$ is the price level, $Q$ is quantity produced, $M$ is money supply, and $V$ is "velocity of money", which represents how "fast" money circulate, given the transactional technology and other institutional aspects of an economy. For example, in an economy where bank branches are interconnected by a real-time on-line system, money circulates faster compared to an economy where bank branches are interconnected only through, say, telegraph, which in turn move money faster compared to an economy where banks are not interconnected at all, and money has to move physically from one person to another.

Assume that we produce $Q=10$ pieces each at a price of $P=15$ kudos. Do we need $10 \times 15 = 150$ kudos to complete all buy-and-sell transactions? No. If velocity of money is, say $V=3$, we only need $M=50$ kudos to facilitate all transactions, because money circulates "three times around" (before the next production cycle is completed).

Now, in standard economic theory, economies are close to "full employment level of production factors" (which may allow for friction unemployment of humans, machines etc).

Now, assume we increase the money supply to $M=100$. The right-hand-side of the equation becomes $VM = 3 \times 100 = 300$. What happens to the left-hand-side? Well, since production cannot increase (because all available production factors are already employed), it has to be the Price level that will increase to $P=30 \implies PQ = 30 \times 10 = 300$.

Intuitively, buyers with more money in their hands will increase their demand (the demand schedule will shift rightwards in a diagrammatic representation), but supply won't change, because it cannot change. Suppliers will increase their prices since there is now "competition among buyers". Hence, inflation.

This is the most simple explanation about how increasing the quantity of money in a full-employment situation, will create inflation.


Infinitely repeated game with stationary and symmetric equilibrium We have two players playing a repeated game. At every period, each player decides to stay or to quit. If both decide to stay, then they both receive 1. If either decides to quit, then the quitter receives 10 for the period while the other player receives 0. The game terminates once someone decides to quit.
Utilities are discounted with the factor $d$. We want to solve for the equilibrium in which both players choose to stay with probability $p$ at every period.
I hope someone could help me check my approach or offer any different approaches.
Since the game could terminate at any period, we first need to find the expected payoff for either player. Suppose the value of the game is $v$. For either player at any period, the game has a probability $p^2$ to continue on, and a probability $1-p$ that the player quits and receives the payment.
So $v=p^2(1+dv)+(1-p)10 \implies v=\frac{p^2+(1-p)10}{1-p^2d}$.
Now, at every period, each player must be indifferent between staying and quitting.
For that the be the case, the payoff between quitting and staying muse be equal $10=p(1+dv)$.
Plugging in $v$, we have $10=p(1+d\frac{p^2+(1-p)10}{1-p^2d})$.
Solve. $p=\frac{10}{1+10d}$.
If $d\leq 0.9$, both players would always play stay. Otherwise, they would mix according to $p$.

Why don't you attack the problem straighrforwardly and investigate the profitability of a deviation?
Suppose there was a Nash equilibrium in which both players decide to stay forever. Both players obtain payoff $\frac{1}{1-d}$ on path.
This really is a Nash equilibrium if anf only if there is no profitable deviation. Consider the deviation to quit in the first period, yielding the deviation payoff $10$. If a later deviation to quitting or a smaller probability to quit was profitable then so would be this pure earliest deviation.
Hence, always staying is a Nash equilibrium when $\frac{1}{1-d}\geq 10 \iff d\geq 0.9$.