If a firm cannot cover costs and make profit then it has no profit motive to develop and sell drug XYZ. So the profit motive is a "cost plus" model. The price must cover costs plus profit whether there is a single buyer or multiple buyers of the output goods.
If drug XYZ makes a net profit in a domestic market, and if there is a marginal profit in other global markets, then prices can be set lower in some parts of the global market and there is still an incentive to provide drug XYZ for marginal profits there.
Bottom line is price floors must be set by the market and/or government purchases if long term investments are to be recovered from future cash flows. This looks like cost plus pricing of output goods. Otherwise a firm that cannot recover costs from cash flows goes bankrupt and the assets transfer to another firm with less long term debt and then the cost can come down.


Why do people buy negative interest rate bonds?

Some pictures and text are from Schroders, Marketwatch and other websites, but I don't remember all them.
Of note, Germany first sold negative-yielding bonds in Aug 2019, €2bn worth of 30-year bonds that offer no interest payments at all.
1. Better than holding cash
Cash is obviously most liquid, but some central banks like Japan impose negative interest rates on cash deposits, which trickles to the rate banks charge institutional investors.
Then highly liquid but only slightly negative yielding government bond looks better! Because the UK government unlikely will default, bond buyers are just paying a small sum to the government to guard your money, like in a vault!
2. Wagering that there are other “bagholders.”
Investors who buy negative-yielding bonds are betting on the value of the securities to keep rising. Although investors buying bonds with subzero interest rates are paying for the privilege to hold a bond, they can profit way above their cost if the security’s price rises.
For example, if you expect a central bank to buy more assets,  bond buyers can rely on the central bank to hoover up their negative-yielding securities.
In July 2020, an auction for €4 billion euros of 10-year German government bonds TMUBMUSD10Y, 0.644% sold at $-0.26%$, but at a premium price of 102.6 cents to the euro. The benchmark bund is now trading at a price of a 106.9 cents to the euro,. So  investors who bought this at auction would've gained around 4% from the price increase alone.
3. Bonds as deflation hedge
When prices fall, most asset classes don't perform well in deflation. An exception are fixed interest government bonds! Because they fixed  their coupon and principal, they retain their value and will positive real (inflation-adjusted) return, if inflation $<$ their yield. In other words, a negative yielding bond can  positive real return if there is deflation. So you can hedge against deflation with negative-yielding bonds.
4. Currency hedging can transform negative yields into positive yields.
Negative yields don’t mean negative income for some bond buyers. Unlike European and Japanese investors, US investors are often paid to hedge against fluctuations of foreign currencies because U.S. interest rates are much higher than in other developed markets like Europe and Japan.  Currency hedging can annualized return 3% for U.S. investors, like  American fund managers, buying negative-yielding  euro-denominated debt like European government bonds, according to Jens Vanbrabant, senior portfolio manager at Wells Fargo Asset Management.
In Sep 2009, the 10-year German Bund yield at -0.6% looked unattractive to US investors.But currency hedging is priced at the difference between US and German short term interest rates, which can be positive even if a central bank cuts interest rates. Since US short term interest rates are much higher than Germany's, US investors are paid money to hedge euro exposure! With currency hedging, US investors can earn 2.2% when investing in German 10 year bonds (see Figure 2 by Schroders), higher than the 1.6% available on US 10 year Treasuries.

5. Diversification benefit
What other bolt holes can weather volatile global equity markets, Brexit, US-China trade war?
Traders like volatility to a degree, but they try to diversify the risk so their portfolios aren’t fully exposed to the market's unpredictability. So buying notoriously safe bonds is a risk-averse strategy. You'll lose some money for that safety, but that loss can constitute ‘over-performance’ if other asset classes tumble. When risky assets sell off, negative yields on government bonds may be overshadowed by their proven ability to rally during recessions. Even at negative yields, bonds still are important. Figure 3 by Schroders  proves the 3-year rolling correlation between German Bunds and the German DAX stock market index. The two strong negative correlated, if you overlook the period between 2014 and 2017 when the European Central Bank’s (ECB) quantitative easing raised higher bond prices (lower bond yields) and higher stock prices.

Bonds  perform well when equities don't, vice-versa. In times of distress, you must reduce risk like this! If you are running for safety,  capital leave stocks to government bonds. Even if negative-yielding bonds lower your return compared to normal conditions, they can still reduce portfolio risk.
6. Roll down the yield curve later.
You're assuming the bondholder is holding it long term, but many don't.
In periods of uncertainty, demand for bonds increases, which makes their price increase. So you can profit from buying and selling government debt. If you think this  volatility will last, buy bonds now in the hopes that more investors flock to them later. You don't even need fellow investors to buy them! The central banks may be the ones to buy bonds to try to stimulate the economy.
Bond buyers can take advantage of the yield curve’s slope, which still can be steep even for negative-yielding bond markets in Germany and Japan. For example, a trader can buy a negative-yielding 3-year bond and sell it after a year. Since debt prices inversely correlate with yields, all else being equal, the value of the 3-year bond should be higher than, for example, a 2-year bond.
So long as shorter-term bonds' interest rates are more negative than their longer-dated counterparts,  the long-term bond's price should rise closer to maturity. But they can profit from this “rolling down the yield curve” as a short-term strategy, and must sell the bond well before maturity, because the bond will trade only at par when it expires.
For example, the Swiss 10-year bond was yielding -0.1% in Jan 2009. But at 22 August 2009, it had returned 5.6%! Negative yields can still positive return! Of course, this works both ways. Any rise in yields can make you lose money.
7. Some institutions are legally required to buy bonds! Liability matching with negative yields
The other reasons are about speculation or managing risk. But institutions can be legally mandated to buy bonds with negative yields.
Certain institutions must follow rules dictated by the banks, pensions funds or insurance companies who provide the bulk of the capital. Those rules mandate the managers to invest just in certain things like investment-grade bonds. Regulators force some clients to buy certain assets, like banks can only buy liquid assets.
Liability-relative investors, such as insurance companies and pension funds, don't care the absolute return or yields from bonds. They often buy bonds to “match”  their liabilities. The present value of these liabilities is calculated in many ways, but factors in government bond yields. For example, in Sep 2009 in the euro area, insurance industry used negative rates up to the 11 year maturity point (Source: EOPIA. See data from 31 July 2019). These liability-driven market participants can buy negative-yielding German Bunds to match a future liability . Their values will move in tandem with each other.  If they don't buy negative matching assets, they'll be exposed to significant risks if interest rates fell further.


No, they are not market makers.

A market maker is someone who (i) quotes two prices (one 'low' and one 'high'), (ii) will buy from any seller at the low quoted price (even if there is no corresponding buyer right now), (iii) will sell to any buyer at the high quoted price (even if there is no corresponding seller) right now.

This injects liquidity into the market when there is an imbalance in the number of sellers and buyers.

NASDQ and NYSE are not market makers, they are marketplaces in which market makers operate.


What factors affect the temporal discount rate individuals apply in decision making?

I think one can answer this question by looking at what affects saving rates.

Age

According to the life-cycle model of household savings*, young and old people save less than middle-aged people. Young people who are starting a new family have many expenses and a lower income due to lack of work experience. Thus, young people save less and value money more in the present (a high reservation intertemporal discount rate). Middle-aged people have a higher income and save towards their retirement (a low reservation intertemporal discount rate). Old-aged people retire and live off their savings, so they have a high reservation intertemporal discount rate.

Dependants

If you support more dependants, you will be shorter of money and will have a high reservation intertemporal discount rate.

Investment Opportunities

Better quality investments and financial institutions will improve the interest rate at which money can be saved, which increases the intertemporal discount rate at the intersection of the budget constraint line and the indifference curve of the individual.

Income

One would think that saving increases with income, but high-income countries actually save less than middle-income countries, but moreso than low-income countries (World Bank, data on Gross Domestic Savings (% of GDP)). So there is no obvious relationship with the level of income, but if the anticipated future level of income goes up, then people will save less (and consume more in the present) giving a higher intertemporal discount rate.

*Alberto Aldo and Franco Modigliani, "The Life-Cycle Hypothesis of Saving: Aggregate Implications and Tests," American Economic Review 53, no. 1, (March 1963), 55-84.


Using a Logit model to predict unknown outcome

If I understood you correctly, $\Lambda(x_i'\hat\beta)$, where $\Lambda(x) = e^x / (1+e^x)$. You can use predict in both R and Stata. Try in Stata:

clear all
*** Generate data
set obs 10
set seed 1
gen x = rnormal()
gen y = 1+x+rnormal() > 0
replace y = . in 7/10
list
*** Estimate
logit y x
*** Predict
predict phat, p
list

For clarification, I presumed that your outcome variable is in $\{0,1 \}$ (not in $[0,1]$). You observe $y_i$ for some $i$ but not for all, you fitted a logit regression, and then you want to predict the probability for every $i$. Please let me know if I misinterpreted your question.


What's the status on uncertainty shocks?

Macroeconomic Uncertainty and Expected Stock Returns is a recent paper (Bali, Brown, and Tang (2015)) finds an economically important effect on uncertainty on returns.

  This paper introduces a broad index of macroeconomic uncertainty based
  on ex-ante measures of cross-sectional dispersion in economic
  forecasts by the Survey of Professional Forecasters. We estimate
  individual stock exposure to a newly proposed measure of economic
  uncertainty index and find that the resulting uncertainty beta
  predicts a significant proportion of the cross-sectional dispersion in
  stock returns. After controlling for a large set of stock
  characteristics and risk factors, we find the predicted negative
  relation between uncertainty beta and future stock returns remains
  economically and statistically significant. The significantly negative
  uncertainty premium is robust to alternative measures of uncertainty
  index and distinct from the negative market volatility risk premium
  identified by earlier studies.

Dynamic co-movements of stock market returns, implied volatility and policy uncertainty (Antonakakisa, Chatziantonioua, and Filisb (2013)) finds that policy uncertainty, as measured by Baker, Bloom, and Davis (2013), has an unconditional correlation with the Vix of about 0.43. But using a DCC_Garch they find episodes when this correlation was negative, which may be surprising.


Where to find more detailed data for International Comparison Program 2011?

These data are not publicly available. According to Section VI of this PDF, the more detailed the data, the more restricted they are.

For the less restricted data, there is ostensibly some painful application process that researchers can go through in order to get access. But in practice, unless you're a big shot or someone who has connections with the World Bank, they will be reluctant to share the data with you. (That was my personal experience — I went through the whole long application process, there was some back and forth for about a month, and eventually they decided not to share the data with me.)

And for the most heavily restricted data:

  Access to Sub-National Data and Individual Price Observations with supporting metadata is restricted in general. However, users may address applications of access to these data directly to the countries in question.


Do contemporary Marxist economists still deny the utility/value theory which underpins classical economics?

I am not familiar with contemporary Marxist economists, but the labour theory of value as formulated in Das Kapital does not deny that the usefulness (use value, in Marx' terms) of an item is subjective. Moreover, the fact that an item is useful to someone (however this usefulness is determined by each individual subject) is a precondition for it to be exchanged on the market.

For items that are exchanged on the market (commodities) the exchange value is the ratio between the quantity of one good that is exchanged for a unit of another good. Money is a special commodity
chosen by a society to measure the exchange values of all other commodities (price).
Items that are not exchanged (e.g. air, or something that is produced for personal use but not sold) have no exchange value / price, however useful they may be.

The prices of commodities that are sold only occasionally (e.g. a painting) may vary considerably and are rather unpredictable.
On the other hand, commodities that are produced, exchanged and consumed on a regular basis (and this is the topic of Marx' analysis) tend to have more stable prices.

According to the labour theory of value (LTV), this equilibrium price is related to the
average amount of labour necessary in a given society to produce each unit of the commodity (value).
The reason for this is that, if producers sell a commodity at a low price that does not reflect all the labour that went into making it, they will make no profit or even go bankrupt; on the other hand, if they try to sell at a very high price that is totally unrelated to the necessary labour, there will probably be competitors selling the same commodity at a lower price (unless there is a monopoly).
This situation is different from the case of a painting, because the buyer of a painting cannot find another seller selling the exact same painting at a lower price: LTV considers equilibrium prices of reproducible commodities under a given average productivity.

Once you have a certain number of people who decide to buy a certain product (whatever personal / subjective reasons they may have) and an average productivity for that product, you can use LTV to explain certain aspects of prices.
E.g. LTV would explain why a toothbrush costs less than a Boeing 747 even though the demand for toothbrushes is much higher than that for Boeing 747s by saying that the production of a toothbrush requires (much) less labour.

So, as far as I understand, even in Marx' original formulation there is no contradiction between the LTV and a subjective theory of value (use value) in that the LTV does not deny that the judgment about the usefulness of a given product is subjective.


Can I add a variable that varies only with time in Least Squares Regression model with a time-fixed effects term?

Your model has $\beta_3 * t$, which is a linear time trend, not time dummies. If that's correct, you are controlling for only a linear trend. Because oil prices do not have a perfect linear trend, you can include them.
But I am not sure you really want the linear trend specification instead of time dummies (say, $\beta_{3t}$). For a model with common time effects (year dummies), time series variables are not allowed due to perfect collinearity. Thus, if the time series variables are the key variables, you have a trouble. But if you only want to control for them, you can just omit them because any common time effects will be eliminated by the time dummies.
Country-pair fixed effects are similar. You cannot use time-invariant variables if those fixed effects are included.
One possible solution is to take the correlated random effects approach, but there is the issue of RE $\ne$ FE. You will have to defend your RE model. That's a different issue.


Why can high variance help you in Heckman (1998) but not Aigner and Cain (1977)?

Say you have two distributions, $G$ and $H$ with equal means supported on (say) $[0,1]$ and $H$ is second-order stochastically dominated by $G$. Then, for any concave increasing $u$, we'll have $$\int udG \geq \int udH$$ This would correspond to Aigner and Cain, where the risk averse firm would prefer to hire an agent from distribution G.

However, now suppose that $G$ and $H$ are distributions of talent (with the same stochastic ordering i.e. $H \succ_{C} G$). One equivalent notion to this idea is that distribution $G$ can be obtained by ``fusing" together part of the mass of $H$, i.e. by collapsing part of the mass to its respective barycenter. Given this, it is easy to see that there is always a cutoff $\eta \in [0,1]$ such that $H(\eta) \geq G(\eta)$.

This corresponds to Heckman's result--if the job is sufficiently demanding, then it will be (at least weakly better) to be part of the high variance population. Moreover, note that this by no means depends on the expected quality of the agent. There are simply more of the highest quality people in the group with distribution $H$ than in the group with distribution $G$. They are not just more likely to exceed the cutoff, there are more of them that exceed the cutoff.

Another way to think about the hiring problem is as follows: say a firm wants to hire one person but receives applications from $10$ applicants, $5$ from group $A$ and $5$ from group $B$. A candidate's quality is unknown and it it ranges from $[0,1]$ continuously. Suppose also for the sake of simplicity that the firm may observe the candidate's group (and so knows what distribution he/she is drawn from): the distribution of quality for group $A$ is given by $G$ and the distribution of quality for group $B$ is given by $H$. Interviewing a candidate fully reveals a candidate's type, but the firm has the budget to interview only $4$ candidates. Whom should they interview?

The answer is simple: all four candidates interviewed should be from group $B$ (which has distribution $H$). Why? Because we can think about a candidate's quality as a random variable $X_{i}$ and thus the firm wants to maximize $\mathbb{E}[Z]$ where $$Z:=\max\big\{X_{1},X_{2},X_{3},X_{4}\big\}$$ Since $H \succ_{C} G$, $$\mathbb{E}_{H^{4}}[Z] \geq \mathbb{E}_{G^{4}}[Z]$$

Note also that $H \succ_{C} G$ $\Rightarrow$ $Var(H) \geq Var(G)$ but not $\Leftarrow$. In this sort of thing, I think it is stochastic dominance and not variance that is the apposite metric.


Is ongoing economic growth possible without additional non-renewable inputs?

I don't think anyone knows how to "make" an economy grow, but there's a simple channel through which economic growth can occur under fixed non-renewable inputs: innovation.

Economic growth does not require increasing use of inputs; if there's profit to be gained through more efficient use of fixed quantities of inputs, or by increasing output of renewable resources, people will have an incentive to figure out how to do so.

Regarding whether fixing a minimum price of a nonrenewable resource is a good idea: possibly, but likely no. Typically, the most efficient ways to reduce the use of a particular input are to either charge the marginal social cost of the use of that input as a tax (allowing quantities to vary freely, which you disallow in your scenario), which incidentally has the effect of setting a minimum price but which requires that you know the marginal social cost of the use of that resource, or to do cap-and-trade, where you fix the quantity available and let market prices determine the most efficient allocation. Setting a minimum price in a cap-and-trade scenario can result in inefficient underuse of the resource.

If you're interested in the theory of the price of nonrenewable resources, you might want to start with Hotelling's Rule.


Can there be sensible choice behavior that violates the Weak Axiom of Revealed Preference?

I think that your example relies on a misunderstanding of the meaning of the choice function. $C(\{x,y,z\})$ does not contain the elements that the decision-maker would choose simultaneously if he could. He is only allowed to select one element at a time (otherwise, why not consuming all three items ?). Therefore $C(\{x,y,z\}=\{y,z\}$ does not mean that he selects both shoes. It means that, if you ask him to choose one element inside $\{x,y,z\}$, he might either pick $y$ or pick $z$, being indifferent between the two. In that case, your story would not make much sense, since the value of a single shoe would be the same in both choice sets.

Regarding your question, there are some well-known violations of WARP that rely on behavioral biases, in particular the attraction effect and the compromise effect. These expressions describe situations where the comparison between two goods is affected by the nature of a third (unselected) good in the choice set. You can easily find some references (experimental evidence, models) on these phenomena.


This is a macroeconomic tautology, and like many others, the words often do not have their commonly expected meaning.  In particular saving and investment are not directly about money but about stuff measured in money terms

Suppose you had an economy based purely on grains of wheat.  The real activities are:

Plant wheat grains, grow them and harvest them, getting more grains as a result
Consume wheat as food
Store unconsumed wheat for future planting or consumption

In this case, macroeconomic saving is the amount that production exceeds consumption while macroeconomic investment is either increasing stored stocks or the planting of grains for future production.  (If you were wondering, grains which rot after having been purchase by consumers still count as consumption, while grains which rot while in producers' storage count as negative investment.)  So in this case, both saving and investment  measure the same thing, seen from different perspectives.

Clearly you could measure this particular economy purely in grains of wheat, but if you want to do so in money terms then you would need to know the price of wheat, where say a million grains are worth $\$5$.  This would then allow you to incorporate the production and consumption of other stuff into your calculations, such as barley where say a million grains are worth $\$3$ and yields are different, or beer produced from barley and bread from wheat, or anything else.  But saving representing unconsumed production, and investment representing the other non-consumption uses, would remain equal, almost by definition


Extension on the suspension of national debt ceiling

Debt is between US govt and The Fed.

If the Fed reset the debt back to 0 and bail out the govt, nothing changes. The next debt epoch starts.

Viscious if my understanding is correct. No other govt in the world has this freewheeling !!


Did real incomes drop significantly since the 1950s?

Most of the US real earnings data which  go only as far as mid 60s. According to the statista data presented in this article by world economic forum the evolution of real hourly earnings in the US for production and non-supervisory workers looked like this:

If we extrapolate to 50s then the real earnings are now higher overall. However, this being said the real wages first peaked in the 70s at $\\\$23.24$ then fallen to about $\\\$20$ and recently in 2019 they finally again reached the previous peak from the 70s. Hence, we cannot really say they dropped significantly but rather they tended to stagnate.
There is no single agreed upon explanation for this. Some authors argue this is due to technological change and decline in demand for low skill workers (see Fernandez 2001), other arguments include declining union membership (see David,  Katz, and Kearney. 2006),  or international competition (see here). Others argue that CPI overstates inflation and that this can be also due to increase in non-monetary benefits such as health insurance (see here).  A good explainer is also provided by this brookings article.


Does the concept of Capitalism increase the net wealth?

Here is a graph of labor productivity per hour for the United States since WWII.

This is fairly typical of capitalist economies. Even Karl Marx and revolutionary socialists recognize that increasing labor productivity is a general characteristic of capitalism. So yes, the pie keeps getting bigger and bigger as a result of this in combination with population growth.
The question of inequality is a somewhat separate one. If the whole pie is growing big enough, the absolute size of my slice can grown even if my relative share as a percentage of the whole is getting smaller. In other words, the total wealth of society keeps increasing, and so in absolute terms, even the poorest see some benefit. In relative terms, that gets more complicated.


How to efficiently study and learn Economics?

I am a ph.D Student in Economics, and I am sharing my experience.

It is not necessary to be able to prove everything. Each time you will have to tackle a specific issue you will have to go back to handbooks and articles that have been made.

I think what is important to memorize is the global story of each chapter of these kinds of handbooks (as Mas Colell for instance). The most useful things to memorize is probably the tool you will constantly use in Microeconomics (such as optimization, linear algebra, topology), Econometrics, Games Theory and so on.
Maybe craft your own handbook based on what you learned from different lectures, books and write as many tips as you can when you are inside the subject. Then if you have to go back to your notes you will be way faster.

If you have this basis, what matters is to be able to take some perspective with your subject, and to be the most relevant is the hardest thing.
Studying economics permits to shed lights on (not all) human behaviors, never forget that behind your proofs and lemmas !


Assuming that $p_i \neq p_j$ you just apply the formula;
$$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\sigma}} \right)\frac{p_i}{\frac{p_i^{-\sigma} I}{P^{1-\sigma}}} =\sigma$$.
The way how you gave the problem neither $P$ or $I$ contain $p_i$ so you treat them as constants during differentiation.


One book that may help you out is Macroeconomics and the Phillips Curve Myth by James Forder.  His narrative is similar to the one you refer to, but the terminology is a little different.  He argues that pre-1968, the economic literature had a more nuanced understanding of inflation that incorporated factors like profit and union power to a greater extent.  I'm not sure the ideas would be completely new to you, and their authors probably wouldn't have described them as "political theories", but Forder argues that there is a great deal of academic work, including empirical work, that quite suddenly fell out of historical accounts of the literature.  If nothing else, I think his reference list could get you started.


Allocative efficiency -- correlation between MB and p in populace with wealth inequality

My textbook asserts that a competitive market would produce an efficient allocation of resources. Resources would be moved from less profitable products to more profitable ones, and this would necessarily increase the monetary value to consumers of the total products sold.

The textbook is correct (provided the assumptions behind "competitive market" and other implicit ones hold).

First, assume the distribution of wealth is given. Then, based on their budget constraint, consumers will have a certain demand for goods. Firms will compete for such demand, offering the lowest price that earn them an economic profit. In competitive markets, such profits are zero. Resources are allocated such that consumers are maximising their utility at the given prices. The allocation is Pareto Optimal.

This result is summarised in the First Welfare Theorem.

Second, through some form of lump-sum taxation, we could redistribute wealth in order to change that allocation. A competitive market will make the equilibrium Pareto Optimal too. This is the Second welfare theorem (see link above).

So, your intuition that the wealth/income distribution affects prices and the final allocation is correct. But the key to allocative efficiency is in the mechanism (market), not in what is being allocated.

I hope I am answering your questions.


I agree with the comments that suggest that the question is almost indecipherable. However, it should be easy, given the cdf, to derive the bounds. Why? Well we know for a cdf $F$, we must have $F(\underline{S}) = 0$ and $F(\overline{S})=1$. We can use the latter to work backwards from the upper bound i.e.

We guess that $F(s)$ is of the form

$$F(s) = k\bigg(\frac{S-v_l}{v_h-S}\bigg)$$

We must have $F(\overline{S}) = 1$ and using your solution, we have

$$\begin{split}F(\overline{S}) = F\bigg(\frac{7v_h+3v_l}{10}\bigg) = k\bigg(\frac{\big(\frac{7v_h+3v_l}{10}\big)-v_l}{v_h-\big(\frac{7v_h+3v_l}{10}\big)}\bigg) &= 1\\
k\bigg(\frac{7v_h+3v_l}{10}\bigg)-kv_l &= v_h-\bigg(\frac{7v_h+3v_l}{10}\bigg)\\
\frac{7kv_h - 7kv_l}{10} &= \frac{3v_h-3v_l}{10}\\
7kv_h - 7kv_l &= 3v_h-3v_l\\
k &= \frac{3}{7}\end{split}$$

Thus, I suspect that your mixed strategy for the high type is given by the cdf $$F(s) = \frac{3}{7}\bigg(\frac{S-v_l}{v_h-S}\bigg)$$


Empirical justification for a constant-returns-to-scale production function in the Solow model

Constant returns to scale are mostly mathematical convenience. Empirically they can occur but there is no guarantee that they will. Although you should note that Solow model can be tweaked to accommodate both increasing and decreasing returns to scale. See Neto, Claeyssen, & Júnior (2019) paper as an example of people doing that, so this assumption is mainly there to make the model easier for students.
For example, Basu & Fernald (1997) found in their study that on average US firms in their sample had slightly decreasing returns to scale (although they are quite close to be being constant). However, this is by no means something that has to hold it is just empirical observation. Empirically you can have constant, increasing or decreasing returns to scale  and it depends on the time and place.
If you want to read more studies than just the ones I listed above you can simply use google scholar using the keyword returns to scale and then country and period you are looking for (if you would want to know it on more micro level you can add industry).  The literature on this subject is too wide to provide any exhaustive overview.


Is unemployment benefit part of national income measurement?

It is included. It is a transfer payment, so the trick is to ensure that the income flow is only counted one time in the national accounting.

Statistics Canada, National Gross Domestic Product by Income and by Expenditure Accounts link to description

  The Income and Expenditure Accounts record the distribution and use of
  income by the six main institutional sectors: households, non-profit
  institutions serving households, general governments, non-financial
  corporations, financial corporations and non-residents. These accounts
  articulate revenues to the sector (including current transfers from
  other sectors, such as employment insurance received) and current
  expenditures of the sector (including transfers to other sectors, such
  as income taxes paid to government).

("Employment insurance" is the Canadian equivalent of unemployment insurance.)


What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy?

If in equilibrium, a player "chooses a mixed strategy" that plays $H$ and $T$ with positive probability, $H$, and $T$ must be both optimal choices. It is a standard result that for a (subjective or objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected utilities being linear in probabilities. So the problem you mentioned can never occur.
So expected utility maximizers do not ever have a strict incentive to randomize, which raises the question of how one can interpret Nash equilibria in mixed strategies. One popular interpretation is that a player's mixed strategy is really representing a shared probabilistic belief other players have over the choices of that player. One can then define Nash equilibrium as a consistency condition on beliefs, though it is not clear why this condition should obtain in practice. Oner idea, Harsanyi's purification idea is that with a certain probability the player has private information that makes $H$ or $T$ uniquely optimal, but for the other players it will look like choosing $H$
and $T$ randomly. Especially in zero-sum games, there is also the idea that players deliberately use randomization devices such as coins so that their plans cannot be found out. Lastly, there is the "mass action" interpretation that can already be found in Nash's thesis. According to this interpretation, you are randomly matched with players from a huge population that play a fixed pure strategy, and the mixing probabilities represent the population fractions playing each pure strategy.


Derivation on p.99 of Salanie, The Economics of Taxation (2nd edition)

Why are you doing $\frac{\partial^2 V}{\partial w'^2}$ ?

Even if it is said that $w^{'}=w$ at the optimum, it should be taken different when you differentiate it for first order conditions. So, you differentiate it according to $w^{'}$ and $w$.


What would happen if fractional banking was phased out?

Fractional banking acts as a multiplier on the money available for loan takers.

Actually, this is subsequently to having taken loans that the money multiplier increases from 1. Fractional banking potentially increases the "money supply", it does not do so performatively. Incidentally this is even why, western economists are currently discussing the possibility of fighting the next downturn with helicopter money.

If I have borrowed, say, a bike, and that I do not plan to use it during the next week, why preclude someone else from borrowing it from me? And the risk that I change my mind and need my bike during the next week is not unmanageable. Especially that the new borrower may have good opportunities to increase to collective wealth.

  What would happen to the economy if fractional banking was gradually abolished?

Resources would be suboptimally employed and we would probably enter a long period of recession. Especially that the world currently "needs" inflation and that credit is the main conventional inflation-engine. That being said, at some point, the entire economic system would likely synchronize with this new credit pace, finally leading the world into a new economic regime, slower, with less volatility and with longer business cycles.


Large consumer goods companies not using advertising, and what their reasons are

I'd argue that the best example is Jiffy Mix, which spends literally no money on advertising, even in the form of the implicit advertising fees paid to direct marketing affiliates. They just put their product on a shelf and people buy it.

We'd expect to see this no-advertising strategy more often in industries where advertising has a significant impact on the cost of the product, and where consumers are very price-sensitive. As Jiffy is very nearly a commodity good, this holds.

In Zara's case, it's "fast-fashion", a segment of the fashion industry where the goal is to provide popular styles of clothing (so product differentiation, a major goal of advertising, does not apply), at low cost (they're targeting price-sensitive consumers), so this strategy is in some sense not a surprise.


How can a sovereign debt ratio be high, but debt-servicing ratio be low?

This is because the cost of servicing debt depends critically on the interest rate on that debt. When interest payments are low the cost of servicing debt will be lower as well.

A general debt servicing ratio can be expressed following this Fed study as:

$$  DSR = \frac{i_t}{1- (1+i_t)^{-s}} \frac{D_t}{Y_t}$$

While the authors from Fed apply this debt service ratio for households -  the formula is generally valid and can be easily repurposed and applied to government or any other organization. It is not necessary the best way of measuring servicing ratio for government (as explained below), but its a good way to understand the intuition.

You can think of the $D_t/Y_t$ part as debt to GDP ratio. The $i$ is the interest on debt, and $s$ is the average remaining maturity. To see how even with high debt ratios the debt servicing cost ratios can be low lets consider few examples. Lets suppose for the sake of argument the $s$ is always $1$ to simplify calculations. Moreover, lets assume that country has debt-to-GDP ratio of $200\% $ (thats quite high - according to the World Bank data it would be higher than the most recent debt-to-GDP of any country they report - although the WB data are not very recent). Now lets consider 4 scenarios, scenario 1 where interest rate is $5\%$, scenario 2 where interest is $1\%$, scenario 3 where $i$ is $0\%$ and 5 where it is $-0.5\%$.

In scenario 1 the $DSR$ will be: $\frac{0.05}{1- (1+0.05)^{-1}} 2 = 2.1$

in scenario 2 the $DSR$ will be  $\frac{0.01}{1- (1+0.01)^{-1}} 2 = 2.01$

In scenario 3 the $DSR$ will be  $\frac{0}{1-(1)^{-1}} 2  $ which is undefined as it would require you to divide but zero, but the limit of this function from both sides approaches 2 which is the debt to GDP ratio - implying there is no extra cost of servicing the debt beside paying down the principal

In scenario 4 the $DSR$ will be  $\frac{-0.005}{1- (1-0.005)^{-1}} 2= 1.99 $ implying that the servicing debt is now even cheaper than just paying down the principal - you get paid for borrowing.

So as you can see that low interest rates make debt servicing progressively cheaper and the rate is non linear - moreover, the above examples all include average debt to maturity 1 for simplicity but actually allowing for it to be higher the effect would be even stronger due to compounding.

Hence while the debt-to-GDP ratio can be very high, borrowing at low (or even negative) interest rates results in progressively lower debt servicing ratios.

As mentioned above the first formula is not necessary the most appropriate for measuring the servicing costs of government. A common debt servicing ratios applied to sovereigns reported by institutions such as World Bank is servicing cost over export, defined as:

$$X_t = S_{t}/E_{t} $$

And this ratio is more relevant when it comes to the government since as for example Frank and Cline point out:

  the rationale for the use of the debt service ratio [authors reference the above service costs to export ratio] as
  an indicator of a country’s debt servicing capacity is that an increase in
  the debt service ratio indicates increased vulnerability to foreign exchange crises. Any shortfall in foreign exchange earnings or capital imports which is not covered by exchange reserves must be met by reducing imports: since aebt service is a fixed obligation, the higher the
  debt service ratio, the greater is the relative burden on import reduction
  for a given shortfall in foreign exchange.

This being said, I think that the intuition behind the quote can be better understood from that repurposed household debt service cost ratios and caries to this. Moreover, as correctly pointed by @BrianRomanchuck in his +1 answer there is no standard definition for countries with no external debt which is important thing to keep in mind.

However, all above being said from the context of the article I dont think the author had in mind any accounting debt servicing ratio. Based on the context of the article "debt-servicing ratios were low" it seems to me that the author just wanted to say the relative costs of servicing debts are low thanks to the low interest rate which logically also implies that any servicing cost ratio (which will presumably have servicing cost in numerator) will be low. The article is an interview, I think the author just could not think of better way to say it on the spot, I would not read too much into it.


Reading list for physics undergrad who'd like to cover the economics related to the hedge fund/ private equity field

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.


Are American poverty rates increasing? What about global poverty rates

US

A snapshot of Poverty for the US is here. The key is Figure 5, which shows alternative measures of poverty, using different data and methods. (see text for details). Basically, they show that there is plenty of variation from alternative criteria, ranging from 0% to 5%. This is likely to be an explanation of why sometimes evidence seem to be contradictory.

For the evolution of poverty, see this study. Unfortunately, it stops at 2010, but it is very comprehensive. Figures start in page 53. Basically, poverty has fallen recently, specially when measured in terms of consumption. This could be an indication that before the Great Recession, poorer individuals were living on loans/debt. There is plenty of evidence that consumption inequality is lower than income inequality, because individuals have a desire to "keep up with the Joneses" (the so-called Veblen Effect).

World

Here is the evolution of World Poverty since 1820:

Here is in relative terms (as % of world population):

Here is the current (2014) snapshot of poverty around the world:

An amazing analysis of World Poverty, in simple slides, is here.

A careful analysis of existing evidence is found here.


Monopolies are just a mathematical misunderstanding

$PQ(P)=TR$, Total Revenue.

$\frac{∂Q}{∂P}P+Q$ is the derivative of $PQ(P)$ with respect to $P$.

$MR$, Marginal Revenue, is the derivative of $TR$ with respect to $Q$.

So in general $\frac{∂Q}{∂P}P+Q \neq MR$


Does aggregate demand increase investments?

Once again, the fixed thing is interest rates. There are numerous ways in which a government can control an interest rates: one of which is to control the money supply. I feel that in your case, you are assuming that the amount of money in the economy and the speed of money flow is fixed. However, we can control the interest rates by controlling the money supply. When we want to make an interest rates at a fixed term, we want to pump more money into the economy and thus increasing the supply of money and keeping the interest rates as low as it used to be. Thus it is possible for that to happen. Please note that I have also assume that the interest rates dixed and the economy is not running at the intermediate or full capacity


Is it worth while to get a master's in math (or applied math) before pursuing phd in economic theory?

The following answer is based mostly on gut-feeling and I urge you to read some of the papers found in the last section to base your decision on something quantifiable.

Coursework (-)
Having an additional degree in math will be completely irrelevant in the coursework, where your current courses will be sufficient preparation. Indeed, a game theory/advanced econometrics summer course may be more useful as preparation.

Thesis Advisor (+)
The degree will make it easier for you to find an advisor who is willing to invest time in working with you. If you move from a philosophy/math/economics undergrad degree into a top-school economics Ph.D. you will most likely compete for the attention of supervisors with students from Europe and Asia who already have Master's degrees in some field. Some research from the last section of my answer suggests that in most departments a lot of resources are spent on the very best students.

Research (+++)
The math degree will be helpful in your research. From the fields you listed as your interests a Master's degree in math does not seem like a bad idea. All fields mentioned are very math-heavy and often use math that is not taught at undergraduate level. However, if you later decide to work in other fields such as macroeconomics, your math Master's will not help you much.

Job Market (+)
Your decision will also affect your job market prospects. The fields you have outlined have a fairly small job market compared to let's say applied econometrics or macroeconomics. If you do not stand out from the crowd it may be difficult to find a position as assistant professor. Thus, having a math master's may help you write an outstanding job market paper. However, you will be older when you enter the market. For very few employers, this may be a secondary criterion.

Tenure (+)
When you obtain an assistant professor position after your Ph.D., you will need to write a number of strong papers within several years. If you at that point need to acquire additional math skills this will slow down your research in this crucial time.

Lifetime income (--)
You will most likely lose at least one year of income from doing the degree. If your degree is two years or 1.5 years, you may at most be able to make up for one year via a shorter Ph.D. by being more productive. It is not unlikely that you will lose 2 years of income. However, doing the Master's may allow you to enter better Ph.D. programs, write better papers, and achieve a better job placement, earning you more money. But this effect is hard to estimate.

Readings
Be sure to check this website to make the right strategic decisions. There is a lot of research on what makes Ph.D. students in economics successful.


Showing existence of a Nash equilibrium in pure strategy

No. Consider the matching pennies game
$$ \begin{array}{c |c}
& H & T  & \\ \hline
H & (1,-1) & (-1,1)\\
T & (-1,1) & (1,-1)\\
\end{array} $$
which we know no PSNE exists.
Define the function
$v_i(\cdot,\cdot)$ by
$$
v_i(H,0) = 5 \\
v_i(x,y) = 0 \quad \forall (x,y) \not = (H,0)
$$
At $\delta = 0$, we have the game
$$ \begin{array}{c |c}
& H & T  & \\ \hline
H & (5,4) & (-4,1)\\
T & (-1,5) & (1,-1)\\
\end{array} $$
which has a PSNE of $(H,H)$.
For any other value of $\delta$, we're back in the original game so no PSNE exists.
Redefine $H = 0, T = 1$ and you have your monotone decreasing requirement.


Rosen's unique equilibrium conditions: Multi dimensional strategies?

These results can indeed be extended to a more general setting. In this case, define the $\nu$-th player's strategy vector $x^\nu \in R^{n_\nu}$ and so $x=(x^1,\dots,x^N)^T$ of a size $n_1+\dots+n_N$ for a $N$ player game.

In this context, $\nabla_\nu \varphi_\nu(x)$ is a vector of size $n_\nu$.

The pseudo-gradient (denoted $g(x,r)$ in the paper) is still the vector concatenated by all the gradients of $\varphi$ i.e. $g(x,r)=(\nabla_1 \varphi_1(x),\dots,\nabla_N \varphi_N(x))^T$. In this general case, the positivity constraint in the definition of diagonally strictly concave means that each component of a vector is positive.

Now, the reasoning of Rosen can be followed in the exact same way with the notations above.


Question regarding preferences in Gale and Shapley (1962)

Gale and Shapley barely make any assumptions about preferences. They don't need a functional form, simply an ordinal ranking of the options. Moreover, there are no transfers in this setting, only the discrete matching options.
Even Kelso and Crawford do not use quasi-linear utility. They assume some utility function that is increasing and continuous in salary, but not necessarily linearly increasing.
In Shapley and Shubik (1971) monetary transfers enter linearly.
However, I wouldn't call this quasi-linear utility, because in quasi-linear utility most people have the following functional form over a good $x$ and money $m$ in mind: $u(x,m) = v(x) - m$, where $v$ is a continuous function.
In their model, the $v(x)$ is simply the money value of house $x$, where the houses are district options.


It's usually considered a short term solution when you would rather do something quick than engage in structural reform. Economics consider this to be a very poor way of increasing competitiveness, since you're effectively harming the purchasing power of your own citizens. Let me explain.

Say a business in the People's Republic of China has to pay its workers in its local currency, and can get most resources locally, so also paid in local currency. On the other hand its finished product is sold abroad, in a foreign currency.  What can the government do to help this business make money? It can go and buy vast amounts of the foreign currency on the local money markets (i.e. from all the businesses that do export). To get this money, it will pay the holders of foreign currency a slightly higher rate in the local currency. The more money it wants, the higher the rate. This of course, by basic supply and demand logic, devalues the local currency. This means it is easier for the company to pay its workers and local suppliers. The workers and suppliers won't be able to buy as many foreign goods, but if they buy local products, they won't be as much affected.

Now the government has a giant pile of money (you can think of it as savings extracted from the workers through the currency manipulation) and a lot of suddenly competitive exporters. To prevent other countries from easily doing the same, it will go and prop up the foreign currency (for instance by buying government debt at low rates).

Why would a country go through the trouble of doing that? Several reasons.
1. The country is dominated by an oligarchy of company owners and their friends and relatives, who want their businesses to do well, and don't care if this makes the ordinary workers relatively worse off. The ordinary workers must have no effective political representation for this to work, of course.
2. Your long-term policy slowly (or quickly) drives foreign businesses into bankruptcy, allowing you to corner an industry and effectively reap in large profits in that sector in the future, especially if initial set up costs are significant in terms of physical capital or intellectual know-how.

So why is this a poor solution long-term? Eventually, even the dumbest foreign governments will start to notice what you're doing and begin to impose punitive tariffs, or devalue their own currency in turn. Besides, doing it for decades will make your currency worthless. Italy's pre-euro currency, the Lira, got devalued so much, that Italian people wisened up to it, and asked for yearly salary increases to match. To keep its exports competitive, the devaluation rate had to be increased, and thus inflation rose and before you knew it, you had 500,000 lira banknotes. Taking the devaluation option for decades made it difficult for the devaluing country politicians to effectively implement the kind of structural reforms needed to make the country genuinely more competitive, discouraged savings, encouraged foreign currency hoarding by the population, depressing its long-term growth potential.

Whether a giant economy like China will be more successful with this policy by driving its competitors' companies bankrupt due to the sheer scale and sophistication of its operations is an open question. They have a lot of very smart people handling this, and as long as their markets such as the US do not retaliate and as long as inflation is in check, they're good.


Why have trickle-down economics failed to trickle down to the middle-class and poor in recent years?

Trickle-down economics are widely criticized and many believe that it doesn't work. Asher Edelmanon explains brilliantly in the first 2min of this video (or in the written version here).

The basic logic is that, in practice, doing this only leads to a higher accumulation of wealth by the wealthy.


Is "technical chart analysis" for stock development something that is scientifically supported?

There are not a lot of studies on this topic.  There were studies in the 70s that would tend to disconfirm the idea of technical analysis as it was proposed at the time.
The studies generated random sequences of numbers in a time series and had technical analysts then make decisions believing that the time series were real time series.  They didn't detect the substitution.  You cannot make systematic profits on a true random series.
Unfortunately, that is a weak falsification.  There is no evidence they got the time series correct. It may be the case that equity time series are different and that signals exist that are legitimate.
With that said, there is a bigger problem with actual time series.  Block orders are "taken off the tape," which means that block orders are not reported when they happen.  The time series is "as reported" rather than "as occurred."  Block purchases or sales are reported after the fact at the weighted average price.
It is not unreasonable to believe that the bid and ask do contain information even if the price record does not.  Whether it creates a "pattern," is another matter.
I am not convinced by the studies so far in either direction.  I do believe, however, that no retail investor should ever try it.  There is a giant information asymmetry between retail investors and institutions.  I don't believe that any retail investor can overcome the informational differences present in the system.
Indeed, during the tech bubble of the 90s, the State of Georgia did a study of 100% of all Georgia daytraders by looking at their statements.  They found that 90% of Georgia day traders lost 100% of their money and only one percent made a significant profit.

Arditti, F.D., McCollough, W.A. (1978), Can Analysts Distinguish Between Real and
Randomly Generated Stock Prices? Financial Analysts Journal, 70-74.

Barber, Brad M. and Terrance Odean, 2000, “Trading Is
Hazardous to Your Wealth: The Common Stock Investment
Performance of Individual Investors,” Journal of Finance
55 (No. 2, April), 773-806.

Bartlett, M.S. (1946), On the Theoretical Specification of Sampling properties of Autocorrelated Time Series, Journal of the Royal Statistical Society, Series B, 8, 27-41.

Grandia, V. (2002), The Search for the Golden Rule: A Reality Check of Technical
Analysis in the Dutch Stock Market, master thesis, University of Amsterdam.


The total factor productivity (TFP) would be $a+\epsilon_t$ where $a$ is the average TFP and $\epsilon_t$ (where Solow residual is technically actually $\Delta \epsilon$) tells us how TFP varies across time. Let me explain:
First, the $A$ should also be function of time in time series model as technology can change (I doubt you want to impose restriction that technology has to be constant and if so then having time varying residual would not make sense) so actually the production function should look like this:
$$Y_t = A_t K_t^{\alpha} L_t^{\beta}$$
Hence log linearizing would give us:
$$y_t = a_t + \alpha k_t + \beta l_t,$$
where lower case letter denote logs $ \ln X =x$. Now when you make mistake is in specifying your OLS. The $a_t$ actually is the residual. Since we can only observe $k_t$ and $l_t$ we cannot include $a_t$ in regression and it will be the residual because it can be calculated as:
$$y_t - \alpha k_t - \beta l_t = a_t, a_t \equiv TFP$$
So actually $a_t$ is the residual $\epsilon_t$. So the specification would be:
$$y_t = \alpha k_t + \beta l_t + \epsilon_t.$$
However, the above specification is unnecessarily restrictive as it forces TFP to have 0 mean (although we  can always rescale any variable to have zero mean this could bias $\hat{\alpha}$ and $\hat{\beta}$). As a result we can add a constant term $\beta_0$ to the above regression.
$$y_t = \beta_0+ \alpha k_t + \beta l_t + \epsilon_t.$$
In this case TFP ($\ln A_t$) would be $\ln A_t = \beta_0+ \epsilon_t$ where $\beta_0$ represents the average factor productivity and $\epsilon_t$ would be the deviation from the average over time (see Van Beveren, I. (2012). Total factor productivity estimation: A practical review and sources cited therein - the source talks about panel data applications but I think the basic explanation holds up even in pure time series even if time series has its own issues that require attention). Also as mentioned at the beginning if you want to assume TFP is constant $A_t=A$ then $\epsilon_t=0, \forall t$.
Lastly, Solow residual is actually defined in growth terms so actually it is $\Delta \ln A_t = \beta_0 +\epsilon_t - (\beta_0 + \epsilon_{t-1}) = \Delta \epsilon_t$, since Solow residual is defined as productivity growth (see Barro & Sala-i-Martin Economic Growth 2nd ed. pp 434-435).
PS: if you actually are going to perform the estimations on a time series you should take into account that all series will most likely be $I(1)$ and estimate the whole model in first differences where the interpretation of constant would be average rate of growth of TFP. In the above I did not explored this issue in order to avoid unnecessarily adding more confusion.


Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]

The equation $100\sqrt{x_1} = 8x_1^{3/2} - 5$ is cubic in $\sqrt{x_1}$.  Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $5$, enabling division by $\sqrt{x_1}$ yielding:
$$100 = 8x_1$$
The first approximation is then $x_1 = 100/8 = 12.5$. Since $100\sqrt{12.5} \approx 354$ is much greater than $5$, it can be inferred that the exact solution will be very close to $12.5$.  So try successive approximation, perhaps starting with $x_1=12.6$.


How is currency depreciation related to import tariffs?

Assume that a chinese company is selling a product for 100 RMB or 16 USD.

With the 10% tariff the cost for an american importer rises to 17.6 USD.

If the the RMB depreciates 8% and the price in RMB is unchanged the importer pays 8% less in USD or 16.192 USD.

Depreciating the RMB by 8% offsets (almost all of) the effect of the 10% tariff.

This calculation does not account for the fact that chinese imports get more expensive. If the good is manufactured using imported oil the cost of that oil will not be offset by the depreciated RMB.


A question about Fixed effects estimation

Having read up on your question it seems the fixed effect is fixed. If this is indeed the case it will have zero variance and hence zero covariance with any variable.


Decomposing Changes in ROA into Changes in Operating Margin and Asset Turnover via Total Factor Productivity (TFP)

The formula for OM is:

$$ OM = 1 - \frac{1}{TFP\times PCR}$$

In the baseline year, you have that Operative Margin (OM) = 0.247. Further, by assumption, price change ratio (PCR) = 1. Then:

$$TFP = \frac{1}{1-OM} = \frac{1}{1-0.247}=1.328$$

Then, you are told that the TFP changed by 2.6%. Therefore, the new value for TFP is 1.3625 (equivalent to 1.328 $\times$ 1.026). With this new TFP, and assuming that PCR = 1, you can find the new value for OM:

$$ OM = 1 - \frac{1}{1.3625\times 1}$$
$$ OM = 1 - 0.734$$
$$ OM = 0.266$$

There you get the 26.6% result.


Given the endowments
Individual 1 : $(4,5); U(4,5) = 20$ and Individual 2 : $(6,5); U(6,5)=30$

Given that there is no more than and 10 units of Good X and 10 units of Good Y in the economy. If I allocate Individual 1 one with $X$ units of Good X and $Y$ units of Good Y then Individual 2 has $(10-X)$ units of Good X ($10-Y$) units of Good Y.

Now the set of Pareto superior allocations to the given endowments, if they exist will satisfy these two simultaneous non linear inequalities.

$ XY \geq 20$ Red region

$ (10-X)(10-Y) \geq 30$ Blue region

Graphically plotting these inequalities we observe: The small common region which you see between red and blue areas is actually the infinite set of all Pareto superior allocations to the given endowment.

Now our job is to find out that allocation which is Pareto optimal, that is that convex lens shaped common region should not be present.

This point could be achieved where both curves are tangent to each other.

Which are required Pareto optimal efficient/superior allocations.


Regarding the Expenditure Function Underlying a Bliss Point

Usually the change in variables $X=x-\delta$ allows to write
$$ e(p,u) = \max_{x \geq 0}\{ p'x : U(x-\delta) \geq u \} =  \max_{X \geq -\delta} \{ p'X : U(X) \geq u \} + p'\delta $$
and so $ e(p,u) = E(p,u) + p'\delta.$


How central banks "create" negative interest rates?

I am not sure about the institutional limitations on policies, but this is easily done in principle. Banks hold deposits at the Federal Reserve banks (reserves/excess reserves). If the Fed sets the interest rate on those deposits to a negative level, that pricing would ripple out everywhere.

For example, assume the rate was -1%. Banks would buy Treasury bills until their yields were comparable to -1%, as they could replace excess reserves with bills to reduce the cost associated with the negative rates. These lower rates then affect the traded yields of other instruments traded in the bond/money markets.


Why should prices come down (in theory) in an educated society with a well-run government?

Is there any economic theory that would support that claim?

As far as I'm aware - no.
In science of economics it's considered that predictable mild inflation is better than deflation (i.e. prices going down), as deflation causes higher unemployment and risks to spiral into self-systaining cycle which can be problematic to break. Also, inflation allows employers to decrease real wages of workers over years by just not raising wages (or raising not high enough). And unlike deflation, inflation stimulates people to invest money, rather than sit on them doing nothing, knowing that they will be able to buy more with them if they will just wait.
There is possibility that the rich dad supported the gold standard, for which it's indeed normal to experience deflation when economy grows (so well-run government would really just mean one that supports gold standard). But gold standard opens its own can of worms and very controversial choice at the best. For an example, there was established link between countries recovering from the Great Depression and their abandonment of gold standards, earlier abandonment of gold standard led to faster recovery


Proof of "the SPNE of a sequential game might not necessarily be Pareto Optimal"?
I don't get it, your example is a proof of this statement. So what else do you need?

If you need another example, just take the prisoner's dilemma, and turn into a sequential game with imperfect information. Then, the NE is equal to the SPNE and you have an equilibrium without Pareto optimality


Can a currency with constraints on the amount a person can posses boost the economy?

Reduce work and hence let someone else work for him (increase employement)

But that won't necessarily work.  Say the person is Bruce Springsteen.  He's making enough from royalties to hit the max.  So he doesn't need to tour at all.  But if he doesn't tour, there isn't necessarily someone else that can tour instead of him.  It's just as likely that his fans will just stay at home and play recordings.

Even worse, remember that a Springsteen concert involves many other people that either don't make money from his recordings or not as much:

His band mates.  He's the face of the band and may well make the largest share of the proceeds.
Roadies.  They make no money from recordings and are only paid for their duties on the live show.
The venue crew.  Ticket sellers, concessions, security, cleaning, etc.  None of these get any money if he doesn't play live shows.

Beyond that, there are ways to get around such a limit.  Instead of buying things with the currency, barter your services directly for what you want to obtain.  Or have your employer pay you in benefits rather than salary.  So you and another person may have the same salary, but she or he gets driven to work by a limo paid for by the company while you drive your own car.

  Spend more money so he/she can keep up his/hers income (boost economy)

It seems unlikely that most rich people actually keep large cash balances.  They either invest or spend their money already.  So any boost from this is going to be minimal.  Another problem is that this boosts the effective money supply, which is going to cause inflation.  So the government will print less money in response, which of course removes the boost.

  Profit from social security, since increasing cash flow leads to (possibly) higher tax-income.

This seems unproven.  As I said earlier, it could just as well reduce employment.

  probably get faster a new job, since profitable companies can only spend more cash on either assets or empolyees

This doesn't follow from what you wrote, but let's assume that you do something else to make this so (e.g. limiting currency accrual of companies as well as people).  Then how do companies get investment capital?  If they can't pay dividends, then who buys stock in the company?  How does a company start and grow?  Realize that many companies have a minimum size.  FedEx is one of the most egregious.  They need to be nationwide.  They don't work as a local concern.  But other companies have a similar problem.  You can't build your own car and sell it to compete with Ford.  It would take one person years to manufacture a car and how would you pay for parts and materials?  Design?

So instead of getting a new job faster, this would mean that new jobs would be quite rare.  You can only get a new job if an existing company expands.  And it could only expand in fits and starts.  They'd have to buy land on the layaway plan.  Then put up the building in fits and starts.  Buy one machine here and another there.  Then, after months or years of preparation, they can finally start hiring employees.

Under this rule, a company can't save for a rainy day.  So if business is slow, it might go out of business entirely.  All of its employees are laid off.  The best find jobs quickly, possibly displacing other workers.  The lower quality workers stay on the job market.


Changes to lookup code/symbols on the NASDAQ?

You are tracking futures on natural gas. Therefore, the name changes because you look at contracts with different maturities: the first has a maturity in March 2016, whereas the second that you mention is due to expire next month (February 2016).

Future contracts are derivative which reflect accurately the expectations of market participants at a defined maturity. You should try to track indexes instead of futures if you want to track commodity prices without changing the tickers and without looking too much at expectations.


Are there stable improvements of the Ramsey-Cass-Koopmans model?

The economic essence of saddle-path stability, is that it requires conscious behavior and decision making from the part of economic agents, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, economic systems are unstable.

Contrast this with the basic Solow growth model: there, fixing arbitrarily the savings rate (i.e. depriving ourselves from making a decision on it period per period) makes the system properly stable. But once we want to optimize with respect to the savings rate, we find out that there is a single path that leads to long-term equilibrium.

Now, perusing the pages of a book like Barro and Sala-i-Martin's "Economic Growth", which contains many-many extensions of the basic RCK growth model, one will not find a model with intertemporally optimizing agents that is also mathematically stable.

ADDEDNDUM
Elaborating, in response to a thoughtful comment by the OP, my approach to the issue of "stability" of an economy is the following: The fact that economies tend to "correct course" is not an indication, let alone proof, that they posses some inherent stabilization mechanism independent of the purposeful action of the economic agents.
If this was the case, then indeed, we should have models that possessed the mathematical property of stability (I remind again that "saddle-path stable" although it uses the word "stable", is a mathematically unstable system).

On the other hand, economies do not plunge into chaos every day. The solution to the puzzle from my point of view is not that we "continuously fine tune the economy" -such dreams and visions have been tried and squashed (irrespective of what ideological/philosophical (dis) agreements one may have with socialism, the factual lesson to be learned from the failure of centrally planned economies is that an economy is such a complex system that no one can "plan and then fine-tune it", especially in our era).
My answer is that economies, after being hit by external shocks or internal institutional failures or technology upheavals, or.., tend to return to a dynamic path that tends to some fixed point by the aggregate consequences of decentralized purposeful action of economic agents that try not to fix the economy as a whole, but their own economic present and future. Economies survive not "because they inherently do", not because the government is "steering the boat with a certain and effective hand", but because individuals try hard every day to survive.

And the RCK models reflect just that: if one performs comparative statics on it, assuming an external shock that suddenly puts the economy out of the saddle-path, the prediction of the model as to what will happen in order for the economy to return to the (new) saddle path towards the (new) long-run equilibrium, is more or less what we observe in reality. But this happens not because the government makes it so, but because individual economic agents realize that they must do so in order not to eventually self-destruct.

I will indulge myself and I will link to a little exposition I crafted a year ago for a class of graduate students, to show them how the simplest RCK model can help them think about the economic crisis (and subsequent depression) in Greece -which was sudden enough to be described as a shock. The exposition can be found here.


Is non-monotonic local non-satiation supported by consumer theory in economics?

Suppose we're talking about some bundle of consumables, $x$. If preferences are monotone, either weakly or strongly, the general idea is that more is better. With local non-satiation the idea is that, no matter what $x$ you have, there is always some small change in $x$ that would make you better off.

Is your question, does there exist rational preferences where:

More is better does not hold.
You can always make a tiny change that leads to a preferable bundle.

Imagine a consumption set on the surface of an infinite dimensional unit sphere, where the dimensions are indexed by the counting numbers and represent different varieties of goods.  The consumer prefers goods found in higher dimensions.  So the 1 unit of good on the second dimension is preferred to the 1 unit of good on the first dimension.

Thus, the consumer is never satiated because they would always prefer one unit of a good found in a higher dimension, and there are infinitely many dimensions.  But these preferences do not satisfy monotonicity, because you must always give up a little bit of at least one good to acquire more of another.

Hence, these preferences are rational, because you can rank all pairs on the surface of the sphere, and transitivity holds trivially.


How does the Fama and French 3-factor model explain stock covariance?

It is not clear to what covariance this question refers to.

As in the CAPM model the covariance among assets is reduced to $\beta$, through the covariance with the market index.

$\beta_i=\sigma_{i,M}/\sigma_M^2$

Then, if CAPM is correct, we could explain the covariance relating the $\beta$s

$\sigma_{i,j}=\beta_i*\beta_j*\sigma_M^2$

In the context of multifactor models something similar occurs, since, in order to use those models, it is necessary to estimate $\beta$s, or sensitivities, related to each proposed factors.


How does what companies produce and sell factor in GDP?

This formula of GDP is an expenditure approach, which classify the country's output by category of final consumers. Therefore, you should determine who demands your product (ie who spends money to purchase your production): If $q_C$ is consumed by households, $q_I$ by firms, $q_G$ by the Government and $q_X$ is exported, then your new GDP is:

$$ GDP' = C \times p\cdot q_C + I\times p\cdot q_I + G\times p\cdot q_G + ( X\times p\cdot q_X - M) $$


The concept of a free lunch is a philosophical/political phrase (not an axiom or law in economics). It was popularized by Nobel Prize winning economist Milton Friendman

The concept depends too largely on how you define free to make a general statement about the concept. For example, while Oxygen is readily available for your consumption from the atmosphere at a cost of zero, Friedman argued there was always an opportunity cost involved (you could be breathing ultra-pure oxygen from an oxygen tank to avoid pollutants in the atmosphere that will give you cancer) so technically there is a cost.


Should I deflate a nominal variable when forecasting it?

In macro, the standard approach to forecast variables which are intrinsically nominal like GDP seems to be to separate the forecast. This is, you want to have a model for real GDP and for inflation. The final forecast for nominal GDP is the "combination" of both. This is for instance the standard practice of the Fed (of Atlanta) and the Bank of England (pages 39 to 41).

In your case of a financial variable, you could model the real (deflated) variable, and then, as inflation is given by the economy, just take the official inflation forecast of your country from the central bank. They are the real experts on this so is a safe thing to do in my opinion. Also, the market expectations of inflation, crucial in finance, are very much informed by central bank estimates. This approach gives you a cleaner estimation of your model too. As you say, your determinant is a rate, and thus inflation is just molesting in the left hand side.


Solving differential equation in The Economics of Superstar (by Rosen)

The differential equation is of the form

$$y' + f(x)y = q(x)$$

The correct answer in our case is

$$p = -s$$

so that you know what you are targeting.
Namely,  it does not depend on $z$. You can verify that it satisfies the differential equation.

Then the author just plays around like

$$p= - s \implies p -p = s-s \implies p - \frac p z z = \frac s z z - s $$

$$           p = \frac p z z + \frac s z z - s = \frac {p+s}{z}z - s$$

$$\implies p = vz -s $$

...and I suppose this playing around with identities is helpful to what he does in the paper.


Your formula $\mu = \sum\limits_{i=0}^n x_i$ just denotes the total amount of goods, so this just means that $\mu$ units were distributed among $n$ agents. Scarcity (in this context) means that no matter how you distribute these $\mu$ units, as long as everything else is unchanged (ceteris paribus) at least one agent would like to get more, i.e. $\forall (x_i)_{i = 0}^n$ where $\mu = \sum\limits_{i=0}^n x_i$ there is an agent $i$ and a positive $\epsilon$ such that
$$
x_i \prec_i x_i + \epsilon.
$$
This is context dependent, there may be other definitions as well (e.g. ones that allow for production).


Is it possible to quantify & cost the ideological obstruction to climate change?

Yes, it is possible to cost it. The range of costs could be estimated by taking the range of estimates of the marginal social cost of carbon (in €/tCO2e), and the range of estimates of the net excess amount of greenhouse gases [GHG] released due to delay (in tCO2e), and multiplying these together.

One starting point would be to ask how long it would have taken to get to zero net GHG if there had been international agreement in 1989, at the start of the IPCC, to put the world on track to tackle climate change (a deal that was killed by the USA, USSR, UK, Japan at the time), and what the total net emissions would have been on that trajectory, versus how they'll be if we achieve a 1.5°C or 2°C trajectory now.

I suspect no one's done it because it's not obvious what the value of the exercise would be: in the absence of a time machine, it doesn't really inform any decisions now.


Sen's property $\alpha$ holds when limited attention in choice?

My math is a bit rusty, but I believe you can show that, for a well-behaved (e.g., open and connected) feasible set $T$ and a consideration set $T'$, $x \in S \subseteq T' \subseteq T$, e.g. Sen's property $\alpha$ holds.

That doesn't seem like a very interesting question/result though, as it just follows directly from the definition of "well-behaved".  Maybe I'm missing something.


Cartel size and collusive stability with non-capitalistic players lists the reasons why it's highly unlikely that your question finds an answer for profit-seeking firms (see Friedman, 1971 for a threshold on cartel stability)

  It is widely accepted from both the theoretical IO literature (e.g., Tirole, 1988) and policy reports (e.g., Ivaldi et al., 2003) that high market concentration is a facilitating factor for (tacit as well as explicit) collusion. In addition to coordination being likely more difficult in larger groups, the intuition that the incentive to collusion shrinks with too many competitors is fairly simple: as the number of firms grows larger, the individual share of cartel profits shrinks monotonically and therefore implicit collusion becomes harder to sustain (cf. Tirole, 1988, ch. 6).

So you may find "toy" answers for $n=4$ or the like but probably not for reasonably big $n$.

However, the same paper also proves that

  An increase in cartel size makes implicit collusion among labour-managed firms easier to sustain.

which is exactly what you're stating in your question.

As the authors sum up,

  In words, the critical threshold for $LM$ firms is decreasing and convex in n and tends to zero as n becomes arbitrarily large. On the other hand, the critical threshold for profit-seeking firms is increasing and concave in n and tends to one as n becomes arbitrarily large. To complete the description of the critical thresholds w.r.t. n, we may also observe that

So according to this reference, an answer to the question is yes, on the condition that the firms studied are not profit-seeking but $LM$-firms.

Note: A $LM$-firms is an enterprise that operates under the ultimate control
of those who work in it, which as for consequences, among others, that the firm aims at maximising profit per worker rather than profit.


How can 1 BTC gain value in USD if inflation is happening?

I'll try to fit this in an answer. I think you are confusing nominal and relative value. Inflation means that the 'value' of your money depreciates relative to other things. For example, in the year 2000 I could buy a bread for 1 USD. In the year 2015 I can buy a bread for 2 USD. This means that the dollar lost half of it's value relative to bread. But nominal 1 USD is still 1 USD and 1 bread is still 1 bread.

So in your example: today 1 BTC = 578.25 USD. If there is inflation of the USD (the value depreciates), then all else being equal BTC should appreciate against the USD (go up in nominal USD-value). So to work further with the bread-example. If in the year 2000 1 BTC would buy 500 USD then in 2015 1 BTC should buy 1000 USD. So the USD went down in value relative to BTC.

for more information:
https://en.wikipedia.org/wiki/Purchasing_power
http://www.investopedia.com/terms/c/currency-depreciation.asp


is "output" another way of saying "real gdp"?

In the UN System of National Accounts (SNA 2008),
$$
\begin{aligned}
\text{GDP}       &= \text{Total Value-Added}, \\
\text{Output}    &= \text{Total Production}.
\end{aligned}
$$
Since $\text{Value-Added} = \text{Production} - \text{Intermediate Consumption (IC)}$, we have $$\text{GDP} = \text{Output} - \text{Total IC}.$$
(IC refers to goods completely consumed as inputs during production.)

Notes:

As already noted in the other two answers, in practice, economists, journalists, and other writers often simply use the term output as a synonym for GDP, even though this is strictly speaking wrong (or at least inconsistent with SNA usage).

As stated in "National Accounts: A Practical Introduction" (UN, 2003):

GDP is sometimes called in economic textbooks "output" or "net output". However, output has a different meaning in national accounts.

Contrary to what the other two answers may have suggested, the difference between GDP and output has nothing to do with home production. Instead and as shown above, the only difference between the two is IC.

The term real in economics (usually) refers to whether the figures have been adjusted for inflation. This has nothing to do with the difference between GDP and output. For any year, we have

$$
\begin{aligned}
\text{Real GDP} &= \text{Real Output} - \text{Real Total IC}, \\
\text{Nominal GDP} &= \text{Nominal Output} - \text{Nominal Total IC}.
\end{aligned}
$$

The term net (contrasted with gross) in SNA refers to whether (or not) depreciation has been accounted for. So, $$\text{Net Domestic Product (NDP)} = \text{GDP} - \text{Depreciation}.$$

It would be somewhat strange to speak of net output (this would mean we've accounted for depreciation but not IC). And indeed SNA makes no use of this term.
But again, as in Note 1, net output is sometimes loosely (and wrongly) used as a synonym for GDP. But this is especially mistaken and confusing because such use of net seems to refer to whether IC has been accounted for, whereas in the SNA, net refers to whether depreciation (rather than IC) has been accounted for.


Why is a 'linear' regression model, with AR(1) error terms, a non-linear model?

To be precise, a model is said to be a linear regression model when it is linear in its parameters. For extension, a model is said to be non-linear when it is non-linear in its parameter (Wooldridge, 2010, p.397).

As such, a linear model can have non-linear variables. A standard example is the Mincer equation, where wage is a linear function of education, experience and experience squared.

In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because its parameters can still be recovered using linear methods. In fact, you do not need non-linear methods at all. Your model is over-identified as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.


That's a coincidence, because they assume nondepreciating capital. If $\delta>0$ was positive we'd have
\begin{align}
\dot K = Y - C - \delta K
\end{align}
which gives
\begin{align}
\dot K = 0  \quad \Rightarrow\quad C = Y - \delta K.
\end{align}

But the static optimization would still read
\begin{align}
&U(C(t)):=\max_{c(t-n,n)}\int^\infty_0 u(c(t-n,n))dn \\
&s.t.\quad \int^\infty_0 u(c(t-n,n))dn \leq C(t)
\end{align}
Actually they define an indirect utility function with subject to a resource constraint. In every period (thus static) the sum of individual consumption cannot exceed aggregate consumption.


Question about gordon growth and expected stock prices

Plugging in the numbers into the Gordon's growth formula does not tell what will happen to the price of the house -it just verifies that the numbers given do satisfy the formula.

To see what will happen to the price in the future, you have to go to the definition of the return on an asset

$$r_{t+1} =  \frac {P_{t+1}+D_{t+1}-P_t}{P_t} = \frac {\Delta P_{t+1}}{P_t} + \frac {D_{t+1}}{P_t} \tag{1}$$

The right-hand side now is clearly "Asset percentage appreciation(Capital Gain) + rate of return on asset value" which is indeed the Total Return on an asset expressed in percentages.

But we have assumed (and the numbers given verify the initial condition), that Gordon's formula holds so

$$P_t = \frac {D_{t+1}}{r-g} \implies P_t(r-g) = D_{t+1} \tag{2}$$

Substitute $(2)$ into $(1)$ to get

$$r_{t+1} = \frac {\Delta P_{t+1}}{P_t} + \frac {P_t(r-g)}{P_t} = \frac {\Delta P_{t+1}}{P_t} + r-g \tag{3}$$

Now, reasonably we should identify "expected (total) return per period" $r$ with "total return next period" $r_{t+1}$.

Then  $(3)$ becomes

$$r_{t+1} = r = \frac {\Delta P_{t+1}}{P_t} + r-g \implies \frac {\Delta P_{t+1}}{P_t} = g \tag{4}$$

which in our case, means that the value of the house will appreciate $5$% per year. Intuitively, since expected Total Returns are $10$% and what we get out of the house as income ($D_{t+1}/P_t$) is $5$%, the other component of Total Returns, house price appreciation, must cover the distance between the two, which is another $5$%.

Note: Whether this increase in price refers to "asking" price or "equilibrium" price is another matter that involves a more elaborate modelling of the situation.


The basic idea of Easterlin hypothesis is that 10 units of additional income does not require 10 units of additional happiness. The idea is close to the decreasing marginal utility of consumption.

Also, commiting suicides are less in countries where there are so many conflicts. Commiting suicide is more often in countries where prosperity level is higher.


If I understand correctly the question, the trick to calculate the equilibrium (relative) price of trade in the Ricardian model of comparative advantage is to specify demand once the opportunity costs have been determined. Prices of internationally traded goods, like other prices, are determined by supply and demand.

Consider two countries, two goods (computers and textiles), and given unit labor requirements such that in equilibrium the world price of computers will be between 0.5 and 2 textiles per computers. The opportunity costs define the bounds of equilibrium relative prices of trade (0.5 and 2), while the structure of demand determines the equilibrium relative price (p).

For instance, p=1 can be an equilibrium price such as one country will specialize in computers and the other in textiles. This price will be determined by the interaction of the relative supply and the relative demand of computers and textiles. This interaction is discussed in the Chapter 3 of International Economics: Theory and Policy of Krugman, Obstfeld and Melitz. An interesting video with a concrete example can be found here.


Money in utility function - Value function

Your calculation has two typos a) you type two times the minus sign related to $f'$ and b) you write $(1+\delta)$ instead of $(1-\delta)$. If we correct for these we have

$$u_{c}(c_{t},m_{t}) + \beta V_{\omega}(\omega_{t+1})\left[\frac{f'(\omega_{t}-c_{t}-m_{t}-b_{t})}{1+n}(-1)+\frac{1-\delta}{1+n}(-1)\right]$$

Taking out the minus sign and $1/(1+n)$, and compacting $f'(\omega_{t}-c_{t}-m_{t}-b_{t})=f_k(k_t)$  we have

$$u_{c}(c_{t},m_{t}) - \frac {\beta}{1+n}V_{\omega}(\omega_{t+1})\left[f_k(k_t)+1-\delta\right]$$

which is the expression in the book.

As regards the definition of output per worker:
Given the clarification in the comments, aggregate output during period $t$ is

$$Y_t = F(K_{t-1}, N_{t})$$

while $N_t = N_{t-1}(1+n)$.

I guess constant returns to scale are assumed so per capita magnitudes are

$$y_t = \frac{Y_t}{N_t}  = F\left (\frac{K_{t-1}}{N_t}, \frac{N_{t}}{N_t}\right ) = F\left (\frac{K_{t-1}}{N_{t-1}(1+n)}, 1\right )$$

Now, consider the notational/conceptual problem here: we will tend "automatically" to write $K_{t-1}/N_{t-1} \equiv k_{t-1}$ due of the same index, but $K_{t-1}/N_{t-1}$ is economically meaningless because $K_{t-1}$ is not combined in production with $N_{t-1}$. This ratio describes "capital used in period $t$ per worker in period $t-1$".

Anyway, if we clearly declare in building the model that we define $k_{t-1}$ in this way, then we end up with

$$y_t = f\left(\frac {k_{t-1}}{1+n}\right)$$

and the budget constraint for period $t$ is correct while the one for period $t+1$ is not. How this affects the first-order condition with respect to consumption?

(Notice that with these notational coventions, "capital per worker during period $t$" is $k_{t-1}/(1+n)$ -I have argued elsewhere why it is preferable for modellers to adopt the meaning that $k_{t-1}$ represents the value at the beginning of period $t-1$ and so is used in the production of period $t-1$).


Can A Utility Function Take On Negative Values?

A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function.

Also, there are different decision theory frameworks that allow the utility function to be transformed. You alluded to something like this in your question. In the traditional framework without uncertainty, the utility function is defined up to a monotonic transformation. Under certain kinds of uncertainty, we get Von Neumann–Morgenstern utility functions that are unique up to affine transformations. You can read more about this elsewhere. For now, the consider the following definition of a utility function. It is taken from Advanced Microeconomic Theory by Jehle and Reny (3rd edition):

A preference relation $\succeq$ is defined as follows:

where the axioms references are these:

Axiom 1: Completeness. For all $x^1$ and $x^2$ in $X$, either $x^1 \succeq x^2$ or $x^2 \succeq x^1$.

Axiom 2: Transitivity. For any three elements $x^1$, $x^2$, and $x^3$ in $X$, if $x^1 \succeq x^2$ and $x^2 \succeq x^3$, then $x^1 \succeq x^3$.


What's the logic behind dividing rental price of capital and wage rate by price level?

Price level is the general level of prices in an economy. It is a variable that indicates what is the purchasing power of money (see Blanchard et al. Macroeconomics: a European Perspective). You can think of it as a variable that tells you what average prices are compared to some baseline.  Price level is often measured using the consumer price index (CPI).
As your textbook correctly stated economic agents primarily care about real profits, wages etc. The reasoning behind this is that what matters is not what is the number of zeros on your paycheck but what you can buy with it. For example, suppose that you are offered a job that pays ${\\\$}1,000,000$ salary per month. Is that salary high or low? Well if the average price level $(P)$ is so high, due to inflation, that even apple or postage stamp cost ${\\\$}1,000,000,000$ you would probably not be willing to work for that one million per month, because it would not be enough to even buy you one apple. However, if price level would be such that stamps and apples cost only ${\\\$1}$ that kind of salary would be great because you can purchase whole host of apples with it.
What really matters for decisions is how much goods and services you can buy with profit an wage you get. Consequently, in economics we often divide nominal variables by price levels in order to adjust them for any effects of inflation.


Why would mortgage rates go up if interest rates go down?

Your ceteris paribus logic is correct however that not how the real world works.
And yes this happened in Canada spring 2020, It's because risk premiums went up. We could imagine that if the bank had inflation expectations they would want a real ROI > 0 and thus those could move in opposite directions


What are the most recent devopments with applying fractals to economics?

Mandelbrot's original application to cotton prices, The Variation of Certain Speculative Prices, claimed that the price fluctuations followed a Lévy stable distribution with a stability parameter $\alpha$ below 2. ($\alpha=2$ corresponds to the normal distribution.)

To be honest, I'm not sure how this was an application of "fractals" where previous contributions were not. While the sample paths of the Lévy stable process we was investigating have fractal characteristics—for instance, a non-integer fractal dimension—so does good old Brownian motion, which has fractal dimension of 1.5. Indeed, the relationship between fractal dimension $D$ and the stability parameter $\alpha$ is $D=2-1/\alpha$, so that Lévy stable processes with $\alpha\in (1,2)$ actually have a $D$ closer to 1 than Brownian motion—in a sense, they're less fractal!

The contribution of Mandelbrot here was, I think, to illustrate the rich variety of possible stochastic processes that asset prices can follow. (Processes that happen to exhibit a variety of fractal features.) He proposed, for instance, both Lévy stable processes as in the cotton price paper and, later, fractional Brownian motion.

Since this isn't my field (and I'm relying on half-remembered undergraduate knowledge at the moment), I can't say too much about the cutting edge of research here. I do know that "tempered" Lévy stable distributions are still quite popular for modeling. (These distributions make tails enough thinner that all moments exist, whereas the usual Lévy stable distributions with parameter $\alpha$ have no moments $\alpha$ or higher. Not having a finite second moment is a big problem for financial applications, which is why this modification is necessary.) Variants of fractional Brownian motion remain common as well. This Wikipedia article, although not too well-written, discusses why such stochastic processes can be useful for financial modeling.

It's safe to say that Mandelbrot has a very, very active legacy in finance.


Who said that growth would stop because of satiation?

In an essay entitled Economic Possibilities for Our Grandchildren, Keynes famously argued that people would one day only need to work 15 hours per week (see bottom of p 4).  This was linked to an argument about satiation (see p 3, half way down), although he was careful to say that this applies to absolute needs, and that relative needs (ie needs to feel superior to others) may be insatiable.
Note that Keynes' essay is just pages 1-4 of the linked document.