Yes, there is some evidence that immoral behavior, as exemplified though political corruption, is bad for economic development. The body of literature is large and to the extent that I am familiar with it, some corruption can "grease the wheels" and hasten development. In general, massive amounts of corruption is seen as a serious hinderence to development.

https://scholar.google.com/scholar?q=corruption+and+economic+development&hl=en&as_sdt=0&as_vis=1&oi=scholart&sa=X&ei=8YxkVbyICYqzoQT5moKIBA&ved=0CBoQgQMwAA

Also, I think that the utilitarian ideal suggests that all else being equal, more wealth and consumption is better. This implies that economic development carries a moral weight. That is to say, an ethical society should develop economically, to raise the standard of living and reduce suffering.


How to derive the measurement equation for the state-space representation of a DSGE model?

Hi: I can't speak for DSGE models specifically but, in more standard "rational expectations" econometrics, the measurement equation usually comes from some assumed linear relation between the dependent variable and the expectation of some other variable. For example, one might have

$y_t = \beta x^{*}_{t} + \epsilon_{t}$

where  $x^{*}_{t}$ is the expectation of $x$ at time t so $E(x_t|t-1)$. Note that $x_t$  is often not observable so some formulation for how the expectation is generated must be assumed.  So, for example, if one assumes the adaptive expectations hypothesis for the expectation, then this implies the following relation for $x^{*}_t$:

$x^{*}_t = x^{*}_{t-1} + \gamma(x_{t-1} - x^{*}_{t-1})$

Putting the two relations together, one ends up with an exponential smoothing relation for $y_t$ which would then be the measurement equation:

$y_t = \beta \times \gamma \sum_{j=0}^{\infty} (1-\gamma)^{j} x_{t-j-1} + \epsilon_t$

Almost all of above is taken from Harvey's "Econometric Analysis of Time Series" which I highly recommend not for DSGE models but for RE type econometrics. I assume that something similar is done in DSGE models but I'm not absolutely sure because I have no experience with them. Still, this may help some.


Increase in per capita GDP always means GDP is growing?

Your reasoning seems correct.

if we consider that Gp=G/(population) and we know that Gp is increasing then we have the following possibilities:

if (population) is static or growing then we can conclude that G must necessarily be growing
if (population) is decreasing then G can be either static or actually decreasing slower than (population). In this case, the statement on the question fails


Constant Returns in a Production Function $\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$ ($R$ = Resource)

The purpose of the paper under consideration is to examine/show the "investment rule" that leads to "intergenerational equity", which with constant population translates into constant consumption.

The investment rule under examination is (last line of p. 973) "invest all net returns from exhaustible resources in reproducible capital" (and consume the rest).

Earlier, eq. $(1)$ of the paper (the law of capital accumulation) tells us that gross returns to exhaustible resources equal $f_yy$: this implies that we assume that the return per unit of exhaustible resource equals its marginal product.

But this in turn implies pricing and the existence of markets. So there must be a market for capital also. If the capital market is also characterized by marginal pricing, then , if we assume that the production function has decreasing returns to scale in capital and exhaustible resource ($\alpha + \beta <1$), then

$$f_kk + f_yy < x$$

and some part of output would have been left unaccounted for.

So the author assumes constant returns to scale in these two so that he can also assume competitive markets and marginal pricing, and per capita output exhausted in the rewards to these inputs.

This of course begs the  question: what happens to the labor market?
Well, we can get away with murder making the following assumption: There is no leisure-labor choice, labor is offered inelastically, and moreover, there is no market for labor, it is subsumed to the other factors of production, i.e. it is offered together with them and it is not paid separately: think capital owners that also work in their business without paying themselves a wage.

This of course means that the formulation with unitary labor and an irrelevant exponent $\gamma$ is sloppy and problematic, it should be absent (it would not affect the paper), and it rightfully led to the OP's question.


I interpret the question as follows:
Suppose we have a finite normal form game (with finitely many players and a finite set of actions for each outcomes). For simplicity, assume a strict ordering of payoffs for each player. That is, the payoff matrix of each player does not contain any identical elements.

We furthermore need to be clear what is meant by a positive monotone transformation of the payoffs. The first option is to transform all payoffs by the same increasing function. The second option is to apply positive monotone transformations to the payoffs of each player separately. I interpret the question as meaning the latter, but the results should also go through for the former.

We want to characterize all Nash equilibria such that in any game resulting from an arbitrary monotone transformation of the payoffs the same strategies are also a Nash equilibrium. Call any Nash equilibrium which fulfills this property "robust to monotone transformations".

With your conjecture, you have pretty much answered the question. The set of Nash equilibria robust to monotone transformations is the set of pure strategy Nash equilibria. Here is a quick sketch of why this is so:

Suppose player i plays a mixed strategy. In this case, the player must be indifferent between all pure strategies in the support of the mixed strategy. This indifference is given by an equation which is a linear combination of payoffs. Since none of these payoffs are equal, we can change any payoff by a small amount to break the indifference. It follows that mixed strategy equilibria are not robust to monotone transformations.

We still need to show that all pure strategy equilibria are robust to monotone transformations. A pure strategy equilibrium is given if all players play their best responses to all other players' strategies. This involves maximizing over a finite set of actions. Given the other players' actions, the actions of player i are in one-to-one correspondence with a set of payoffs. Since the monotone transformation of the payoffs does not change their relative ordering, the set of best responses does not change. Since this holds for every player, the pure strategy Nash equilibrium is robust to monotone transformations.


How do we prevent big companies conspiracy?

The kinds of practices you describe are frequently controlled through competition/antitrust policy. For example:

In the US conspiratorial agreements (such as price-fixing cartels, bidding cartels, or other agreements that restrict competition) or conspiring to abuse a position of dominance in a manner that harms competition is illegal under the terms of the Sherman Act 1890. Likewise, these kinds of practices are prohibited within the EU by Article 101 of the Treaty on the Functioning of the European Union.
In the US, the Clayton Act 1914 provides for the prevention of mergers that are deemed anti-competitive. A merger between "big" companies is likely to be anticompetitive because it will result in a significant increase in market concentration. More formally, the DOJ writes, "Transactions that increase the HHI by more than 200 points in highly concentrated markets are presumed likely to enhance market power under the Horizontal Merger Guidelines issued by the Department of Justice and the Federal Trade Commission." Article 102 of the Treaty on the Functioning of the European Union serves the purpose of preventing anti-competitive mergers in the EU.

The laws mentioned above are regularly used to prevent conspiracies or mergers that harm competition, and firms are often subjected to significant penalties for violations of these laws. For example, here is a list of cases in which the European Commission has intervened to prevent cartels. Here is a similar list for merger cases.

The relevant EU laws are mirrored in national law within EU member states so that member state competition authorities can also intervene to prevent conspiratorial behaviour.

The issue of patents is a slightly more tricky issue. A patent is a state-granted right to behave anti-competitively. When a patent is awarded, the state is basically saying "we are going to let you behave like a monopoly over this technology for the next 20 years". This is harmful for all the usual reasons that monopoly is bad. But, there is an important trade-off. Firms invest a lot of resources into research and development. For example, developing a new drug is estimated to cost as much as $5,000,000,000. Companies would not be willing to make this investment if they thought that, as soon as the drug were invented, others would enter the market, copy the idea, and sell a cheaper competing generic alternative. If we want firms to innovate then there has to be some reward to innovation. Thus, the compromise we have is that firms are allowed to be a monopoly for a limited period of time in return for making their inventions public by writing them into a patent filing.


For a simple counterexample let
$$
A = B = \left( \begin{array}{cc}
1 & 0 \\
1 & 0 \end{array} \right).
$$
In this game any strategy pair will constitute a Nash-equilibrium. Let
$$
a^t = \left(\frac{2}{3}, \ \frac{1}{3} \right), b^t = \left(\frac{1}{3}, \ \frac{2}{3} \right).
$$
Then the expected payoffs are larger in the symmetric equilibrium associated with $a$.

If you wish for $B$ to be strictly larger than $A$ you can easily modify this example, simply multiply all of $B$'s elements with a number larger than 1 but smaller than 2.

A counterexample where neither $A$ nor $B$ are singular:
$$
A = \left( \begin{array}{cc}
4 & 0 \\
6 & -4 \end{array} \right), \
B = \left( \begin{array}{cc}
4 & 1 \\
6 & 0 \end{array} \right)
$$
$$
a^t = \left(\frac{2}{3}, \ \frac{1}{3} \right), b^t = \left(\frac{1}{3}, \ \frac{2}{3} \right).
$$
$$
A \cdot a = \left( \begin{array}{c}
\frac{8}{3}  \\
\frac{8}{3}  \end{array} \right) >
\left( \begin{array}{c}
2  \\
2  \end{array} \right) = B \cdot b
$$


A Lorenz curve is a P-P plot, so the shape of the curve depends on the underlying probability distribution. Depending on what probability distribution you're using for your model (uniform, normal, exponential, etc.), you'll end up with a different functional form for the Lorenz curve.
Given an underlying cumulative distribution function $ F(x) $ with inverse (if it exists) $ x(F) $, the Lorenz curve is given by
$$ L\big(F(x)\big) = \frac{\int_0^F{x(G)}dG}{\int_0^1{x(G)}dG} $$
In practice, when you're looking at Lorenz curves and Gini coefficients, it's common to use a Pareto distribution as the underlying probability distribution.
Here's a derivation for the functional form of the Lorenz curve for the Pareto distribution as an example of the process. Starting from the CDF of the Pareto distribution:
$$ F(x) = 1 - \bigg(\frac{x_m}{x}\bigg)^\alpha , $$
we can invert it to
$$ x(F) = x_m (1-F)^{-\frac{1}{\alpha}} . $$
Then, inserting the inverse of the CDF into the definition of the Lorenz curve above, we have
$$ L(F) = \frac{\int_0^F{x_m (1-G)^{-\frac{1}{\alpha}}}dG}{\int_0^1{x_m (1-G)^{-\frac{1}{\alpha}}}dG} $$
$$ =
\frac{\bigg[-\frac{x_m}{1-\frac{1}{\alpha}}\big(1-G\big)^{1-\frac{1}{\alpha}}\bigg]^F_0}{\bigg[-\frac{x_m}{1-\frac{1}{\alpha}}\big(1-G\big)^{1-\frac{1}{\alpha}}\bigg]^1_0} $$
$$ =
\frac{\frac{x_m}{1-\frac{1}{\alpha}}\bigg[1-\big(1-F\big)^{1-\frac{1}{\alpha}}\bigg]}{\frac{x_m}{1-\frac{1}{\alpha}}} . $$
So, for the Pareto distribution with parameters $x_m$ and $\alpha$, the corresponding Lorenz curve has the functional form
$$ L(F) = 1 - (1 - F)^{1-\frac{1}{\alpha}} . $$
Now, if you want to find the functional form of the Lorenz curve given data on some measure, like wealth or income, you would first have to fit the distribution of your data to the Pareto distribution first and estimate its two parameters, $x_m$ and $\alpha$ (see here).
Once you have your estimate of $\alpha$ ($\hat{\alpha}$), you can simply plug that into the Lorenz curve for the Pareto distribution we found above:
$$ L(F) = 1 - (1 - F)^{1-1/\hat{\alpha}} . $$


The calculation that you gave doesn't make sense. To answer this question, you should revise your definition of "present value." The definition you gave isn't specific enough to be operational:

  To me, it seems the present value cost would be the cost of what I would pay if I was accounting for the interest of the tuition.

What does present value mean? Think about it like this. How much money do you need today to pay the future costs. Well you need $\$15,000$ to to pay for the first year and $\$ X$ to pay for the second year. Why did I write $X$? Because we want to know how much money we need today to pay for the second year and that amount is less than $\$15,000$. This is because we can invest $X$ amount of dollars at $8\%$ interest so that we have $\$15,000$ in a year. So, we want to
solve
$$
X * 1.08 = 15000.
$$
Thus, the proper present value calculation is
$$
28,889 \approx 15,000 + \frac{15,000}{1.08}.
$$


Why do longer maturity treasuries normally have higher yields than shorter ones?

Existence of secondary market only reduces liquidity risk. It does not affect the inherent risk of purchasing long term debt because situation can change. For example, long term bonds have higher interest rate risk because there is higher chance that interest rate will change in long term than in the short term (and raise in the interest rate will lead to lower bond price).
People at a secondary market will not buy bonds from you at any price you will fancy. Any prospective buyer will take into account that long term debt is more risky even if there is a secondary market. Advantage of secondary market is that it makes the bonds more liquid so it primarily reduces liquidity risk but virtually all other sources of risk remain unaffected, and generally (from a perspective of a buyer) the risk of buying debt increases with the time to maturity of the debt and so the buyer has to be compensated for that (plus there is the time value of money - money now is always better than money tomorrow). For example, the person that buys the bond from you will have to either factor in the price that either they will have to hold the bond to maturity (with all risks that carriers) or find another buyer who will then have to factor in the same problem. The bond will have to end up in someone's hands at the end of the day so all disadvantages of holding that long term debt will be priced at secondary market as if the bond would be held to maturity from the date the transaction is made (save for liquidity related issues).
Consequently, in the end regardless of whether secondary market exists it generally does not change riskiness of purchasing bond save for the liquidity risk. Since longer term debt is riskier for buyer (e.g. interest rate risk etc.) they need to be compensated more for purchasing it than in the case of short term debt that carries less risk.


Why is the fed forbidden from buying gov't debt directly from the treasury?

There is no direct economic rationale for this. This is done because there is a worry that if Fed will directly deal with government it will undermine its independence.
When it comes to the independence itself most economists would argue that independent central banks perform their duties better than ones that are not independent (the same way how for example independent courts usually rule more fairly than ones that are not). This is also how Fed itself justifies its independence as you can see at their page here. Also for gauging the consensus among economists on central bank independence you can see this IGM poll amongst the top US economists (the poll is about political appointments - but that is all part of the whole Fed independence issue).


Effect of a permanent increase in government expenditure in an open economy?

The intuition for this result is pretty straightforward, and I think one can think about it in terms of saddlepoint stability in a phase diagram, although you don't need any serious technical apparatus - it's all conceptual.

Krugman and Obstfeld posit a model in which government expenditure does not affect the "full employment" level of output $Y^f$ (to which the economy must tend in the long run - note that proving this requires you to look at a phase diagram and rule out the explosive paths that fail to converge to $Y^f$). The money supply $M^s$ is also assumed to be constant. Together, these assumptions imply that the steady state price level $P$ doesn't change.

Now Krugman and Obstfeld, critically, also assume that you're already at the steady state price level $P$ when the permanent increase in government expenditure occurs. And even once the increase hits and the phase diagram changes somewhat, this $P$ is still the steady-state price level (i.e. the price at the unique fixed point of the implied system of differential equations).

In their world, $P$ is the only state variable. (Prices are sticky, but quantities and exchange rates are not; they're jump variables.) Since you're already at the fixed point's value of $P$, the other variables will immediately jump to their fixed point levels and stay there. (This is always true for models where the fixed point is a saddle and equilibrium is unique.) And since the fixed point for output is unaffected by the expenditure shock, this means we stay at the same output even in the short run.

Conceptually, the point here is that we only get nontrivial dynamics from a permanent shock if that shock changes the fixed point values of state variables. This happens, for instance, in the neoclassical growth model following a permanent TFP shock, because an increase in TFP raises the steady state level of capital above the current level of capital. But it doesn't happen here, since under Krugman & Obstfeld's particular set of assumptions there is only one state variable and it's unaffected by permanent changes in government expenditure.


Alpha interpretation in Solow growth model

I don't quite understand what you mean by "share that goes into capital", but the common interpretation is that $\alpha$ is the share of income/output spent on capital.

You can show that the following way:
Since the factors will be compensated according to their marginal products, under the assumption of competitive markets, we have (for capital):

$$
\frac{\partial Y}{\partial K}= \alpha K^{\alpha-1}L^{1-\alpha}=\frac{r}{p},
$$
where $\frac{r}{p}$ is the real rent.

So the amount of income spent on capital is equal to price times quantity, which in this case is,
$$
\frac{\partial Y}{\partial K}\frac{K}{Y}= \alpha K^{\alpha-1}L^{1-\alpha}\frac{K}{K^{\alpha}L^{1-\alpha}} \\
\frac{\partial Y}{\partial K}\frac{K}{Y}= \alpha
$$

Click here for more detail.


Does Bad Money Drive Good Money Out Of Circulation

Gresham's law has to do with commodity money. When several types of commodity money are available (e.g., both gold and silver coins), the one with the lowest quality will drive the rest out of circulation. The underlying idea is that everyone will try to hoard the "good money" while using the "bad money" for payments.

You have a full description of this issue in wikipedia.


Is there an economic reason why employees don't share salary information with each other?

dwjohnston,

To answer your question briefly: no, because it is rather a social or psychological matter than it is an economic matter.

There is already asked a good question about this subject in The Workplace community: https://workplace.stackexchange.com/questions/15397/why-is-discussing-pay-such-a-taboo-how-can-i-get-people-to-talk-about-concrete/15431#15431

Feel free to read the (also good) answers. The answer section also contains some handy tips and tricks that you and your co-workers could try.

I hope this will be useful for you.


Econometrics - marginal effect of x on y at average values

"They knew" because they have written down the models. Ignoring other variables, the log-log specification is

$$\ln y = \beta\ln x \implies e^{\ln y} = e^{\ln x ^{\beta}}$$

$$\implies y = x^{\beta} \implies \frac {\partial y}{\partial x} = \beta \frac {x^{\beta}}{x} = \beta \frac {y}{x}$$

I guess the OP can work out the other model now.


What does a central bank guarantee against its currency?

Most central banks do not provide guarantees on their currency to the public.  In fact it can be argued that historically they were never fully guaranteed as most central banks issued more receipts for gold (or other assets) than they had, so the receipts were never fully redeemable in the aggregate.

The American Fed was stressed during the great depression and had to go off the gold standard for the public in 1933.  The international gold standard was retained and dollars were redeemable for gold for a while.  This too was "overbooked" and when the rightly suspicious French started to mass redeem dollars for gold, the Americans had to get off the international gold standard as well in 1971.

Today the dollar is not redeemable for anything.  You can exchange paper dollars for coins or electronic dollars, but that is just changing the form of the same thing (the monetary base).

Because you can go to jail if you don't pay your taxes in dollars, it can be said Americans are forced to purchase dollars and thus provide extrinsic value to the dollars though public tender laws (granted still different from intrinsic value but still important).

Pegged currencies are a different matter and are the closest modern equivalent to perhaps what you are asking.  With a peg, a country must acquire a certain amount of an external currency to maintain a peg.  It could be thought of indirectly that the country maintaining the peg is guaranteeing a redemption of their local currency to an external one.  The idea is to somewhat stabilize the local banking systems which need a lot of artificial stability to survive.  Granted this doesn't work well as pegs often do run out of "reserves" and suffer "runs" just like a gold standard.


Your link contains chapter 2 of the following book:

Potential Game Theory - Applications in Radio Resource Allocation

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

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


Let the strategy of player 1 be represented by $(x1_1,xDD_1,xDC_1,xCD_1,xCC_1)$ where $x1$ is the first round action of player 1, $xDD_1$ is the action taken at the information set where both players have defected in the first round, $xDC_1$ is the action taken at the information set where player 1 has defected and player 2 has cooperated in round 1, etc.. Note that something like $(x1_1,x2_1)$ (with $x2_1$ being the action taken in round 2) is never a full specification of the strategy of player 1, since we need to specify behavior at each information set separately. Define the strategies of player 2 similarly. However, a perfect Bayesian equilibrium must also specify the beliefs of the player, $\mu_1,\mu_2$. This is an important part of the specification of an equilibrium. As we will see below, the question is geared towards understanding that a different equilibrium does not require the strategies to differ. A difference in beliefs is sufficient to count as a different equilibrium.

The perfect equilibrium is given by: $((D,D,D,D,D),\mu_1)$ for player 1 and $((D,D,D,D,D),\mu_2)$ for player 2, where $\mu_1$ and $\mu_2$ are consistent beliefs at all information sets.

As has been noted in the comments, since "defect" is a dominated strategy irrespective of beliefs, even in a weak perfect Bayesian equilibrium the strategy profiles must be $(D,D,D,D,D)$ for both players. However, the following is now also a weak perfect Bayesian Nash equilibrium: $((D,D,D,D,D),\mu_1')$ and $((D,D,D,D,D),\mu_2')$ with $\mu_1'$, $\mu_2'$ consistent on the equilibrium path.

Thus, the question is not wrong, it simply shows that two weak perfect Bayesian Nash equilibria can have identical strategies as long as they differ in beliefs off the equilibrium path.


Empirical explanation on divergence between productivity and wages since the 70s

This is sort of my field, tough I focus more on education. One explanation that seems to fit the data is that the american labor market now is incorporating benefits and variable pay, so wages should decouple from productivity if total compensation doesn't - and voi lá, that's whats happening.

  One reason, perhaps, is that the character of the productivity acceleration
  changed circa 2000. Prior to that date, studies have suggested
  that the more important effect was an increasing ratio of capital
  to labor (capital deepening) as businesses substituted relatively
  less expensive information technology and communication equipment
  for labor. Since 2000, some studies suggest that the more
  important factor has been a re-engineering of business practices,
  which has increased the “skill bias” in the labor market, that is,
  the premium paid for higher levels of technical, professional, and
  managerial education and experience. For employers, variable pay
  solves, in part, the problem of monitoring the performance of such
  workers: Unlike traditional factory and retail service workers,
  whose hours at work are relatively easily monitored, many skilled
  professionals work at varied locations and times of day. Finally,
  increasing reliance on variable pay also perhaps is a type of risksharing
  arrangement between businesses and workers. If the recent
  productivity trend slows, reducing or curtailing variable pay may
  be less offensive to professional workers than reductions in base
  salaries.


If GDP is measured (as it usually is) for a period such as a year, then number of hours worked ($h$) should be measured for the same period.  Thus $h$ is dimensionless and the dimension of the right hand side is also value / time.


GDP Income Approach by industry: Public Administration calculation?

Income approach will sum all gross profits, rents, interest, wages and indirect taxes minus subsidies across the economy. If you want to separate GDP by industry then for public administration you just sum all of the above for its category.
Public administration includes among other things:

Services like transportation, maintaining law and order, road construction, police, jails, tax exemptions, medicare, social security,

Some of the above won't generate profit but some can (for example public transportation). The same holds for interest, rents, wages and so on. For GDP in public administration you just sum gross incomes in this category.


Solow model: Partial derivative of y with respect to n

You need to use the implicit function theorem. The long-run relationship implies

$$
0 = F(n,k) = s f(k) - (n+g+\delta)k,
$$

and this implicitly defines function $k(n)$. The derivative is then

$$
\frac{d k(n)}{d n} = -\frac{\partial F / \partial n}{\partial F / \partial k} = \frac{k}{s f'(k) - (n+g+\delta)},
$$

which can be evaluated once one has solved for particular long-run (n,k) combination. To finally obtain derivative of output $y = f(k)$ wrt. $n$, simply use chain rule:

$$
\frac{d y}{d n} = f'(k(n)) \frac{d k(n)}{d n}.
$$


Optimal population allocation layout over the Earth

The closest I can find to what I believe you’re looking for would be the biocapacity map of the Global Footprint Network. They don't seem to publish granular spatial data, i.e. where in a country its biocapacity is located.
Note, however, that actual economic activity does not correspond to local biocapacity because high density stock energy (fossil fuel) can be transported to population centres. (Source)


Longitudinal microdata on house migration patterns

The US Panel Survey of Income Dynamics (PSID) has this  data,  at least if pre-tax income will do.

  We use the 1999-2009 Panel Survey of Income Dynamics to estimate household move probabilities as a function of, among other things, current housing equity. The lock-in effect supposes that mobility decreases with equity, particularly as equity becomes negative. We find that while owners do move less than renters, the move probability increases as homeowners become underwater. The propensity to move out of state in particular increases dramatically for sand state homeowners who have negative equity. There is no lock in effect from negative equity.

Mobility and Mortgages: Evidence from the PSID (Colson and Greece (2013))

The NLSY may also have this data.


Price and cross price Elasticity of demand

$P_XX+P_YY=M$

Differentiating w.r.t. $P_X$,

$\frac{\partial X }{\partial P_X}P_X+X+\frac{\partial Y}{\partial P_X}P_Y=0$

Divide the whole term by $X$, considering $X\neq 0$

$\varepsilon _X+1+\frac{\partial Y}{\partial P_X}\frac{P_Y}{X}=0$

$\varepsilon _X=1$(Given)

$2+\frac{\partial Y}{\partial P_X}\frac{P_X}{Y}(\frac{P_Y}{X}\frac{Y}{P_X})=0$

Here, $$P_X,P_Y,X,Y> 0$$(ASSUMPTION)

Therefore, Cross- Price elasticity is $-2$ (some positive no.) and not $0.
So, this statement is not true.


Chicago Fed National Activity Index: underlying data download

Friend.   Yes there is.   You can use the St. Louis Federal Reserve Economic Data download website (FRED)     https://fred.stlouisfed.org/series/CFNAI

Not sure about seasonal adjustment.  For seasonal adjustment you may need a statistical computing software to do that and you would have to choice your parameters.


What is the best example social norm effect economy?

Well, it sounds like the paper is drawing a conclusion between behavioral responses towards energy consumption when they're shown public averages.

There's a problem with that paper -- unless it makes specific reference to this:
- How does he/she derive causation between variation in electricity and 'social norms'? I don't see social normality being the primary influence.

Separately there are psychological effects at play like:

• Bystander Effect, Social Loafing, Diffusion, and Moral disengagement

That very much could be at play, confounding the entire study. What if I chose to increase energy expenditures 'just' because I see that the averages aren't 'that bad.'

"Everyone else is saving well, so I can go heavy and not hurt the environment as much..."

This incentive and what I understand as the one described in the paper where people conform to a standard norm are different.

But if you're doing some homework and need some 'answer': there are many, cigarettes are very heavily advocated against right now. Someone would choose to change their habit of smoking because 'it's currently less acceptable.


How can economies grow if every account is perfectly balanced against others?

The Accounting mistake is in step 1, obviously, since you performed a single-entry and not a double-entry:
The Central Bank books are not imbalanced: schematically speaking the CB credits the account "money creation" (or "Our future") and debits its own "Cash Available": this is the essence of "the printing privilege": the CB has the right to create money out of thin air.
Then, it credits its Cash Available account and debits the customer/borrower "Commercial Bank" -because Commercial Bank don't just get money from the Central Bank, they borrow it (or the Central Bank instead of lending money, it buys shareholder stock of the commercial bank or any other so-called "open market operations").

So all books balance. As regards growth in such a balanced-book situation, the key is in step 5, "People buy good or service from company":

What tells you that the firm has the capacity to serve the demand it faces? If demand is now increased because new money have flown into the economy (and initially people treat money as a store of value/purchasing power, not just as a medium of exchange in a static situation), the firm may have to increase its productive capacity to satisfy total demand.

If it cannot do that, the firm will see an opportunity to raise prices, hence new money will just cause inflation. But if the firm can increase its productive capacity, namely, if there is people unemployed and capital laying idle, then the firm will "call to arms" these currently unemployed factors of production, and it will put them to work and produce, thus increasing output and so creating growth.

This effect can certainly be written using the balanced-book Accounting Language, but I will leave that as an exercise.


Fair and efficient allocation of "family goods"

Right now I'm not sure about the equivalence of the relabeling, and therefore the usefulness of this anwer -- see comments below.

This is the beginning of an answer and an attempt to demonstrate how strong the necessary assumptions would have to be to guarantee existence.

Let's transform the problem into one that's equivalent but a bit easier to work with. Instead of indexing over families, let's instead index over the agents (members of families). The key to this relabeling is understanding that families can be written as constraints: If agents $i$ and $j$ belong to the same family, then $x_i=x_j$ and $y_i = y_j$.

Now we're back in the standard environment with individual agents (not families) but with these familial constraints. Recall the proof of Varian's theorem, which you link in the question. It uses the existence of a competitive equilibrium from equal incomes. In this context, we would need the existence of a competitive equilibrium from equal incomes in which the familial constraints were also met. This is going to be very difficult to do. For instance, consider $i$ and $j$ are in a family, and
$$
u_i=x_i + \varepsilon y_i \:\: \text{ and } \:\: u_j = \varepsilon x_j + y_j
$$
where $\varepsilon>0$ is tiny. These preferences are monotonic and convex. Basically, one family member cares about $x$ and the other cares about $y$. If each of the two agents is purchasing $x$ and $y$ to maximize his or her utility, you would not expect $x_i^* = x_j^*$ or $y_i^* = y_j^*$ in the competitive equilibrium (see addendum at end).

This is why you certainly need some assumption on preference similarities within families (at least to use a version of Varian's proof). My sense is that if you give me any arbitrarily small difference in preferences between family members, I can construct an example around it where there exists no CEEI in which they choose the same allocation. And then, at the very least, you can't use Varian's proof.

Two questions:

Do you agree that my reformulation of the problem is formally equivalent to yous?
Can you think of any assumption weaker than assuming preference homogeneity within the family that I can try to invalidate with a counter-example?

Addendum: Remember that in a competitive equilibrium, each agent's marginal rate of substitution (MRS) equals the price ratio. Here, my agents have constant and different MRS's, so there can exist no competitive equilibrium with a price ratio that equals both of their MRS's. If each agent has an MRS that varies, then perhaps they could happen to be equal at the equilibrium price ratio. So maybe you could get away with some notion of local homogeneity of familial preferences. But you need to have them be locally homogenous at the competitive equilibrium, which is exactly what you're trying to prove exists, so it would be a bit circular.

Important note: As mentioned previously, I'm assuming that the only way to prove existence is how Varian did it, via CEEI. There may be other proof techniques that skirt these issues, but I suspect not.

Beyond CEEI: As the OP points out in the comments, proving existence of PEEFs through CEEI as Varian does is somewhat restrictive. I do not have a lot to say about proving existence of PEEFs directly, but the following is readily apparent: For any allocation satisfying your Pareto efficiency condition (ignore envy-freeness for the moment), for any $i,j$ such that $x_i, x_j, y_i, y_j > 0$,
$$MRS_i = MRS_j$$
If this weren't true, there would be a Pareto improvement. Competitive equilibrium essentially equates the MRS's through the price ratio, but you still need to equate these MRS's just to find a Pareto efficient allocation. I think the familial constraints will make this very difficult -- it's not hard to come up with an environment and familial constraints such that there exists no Pareto efficient equilibrium satisfying those constraints. In any case, this could be another partial step towards an answer: Forget about envy-freeness. First try to come up with an assumption on preferences (and maybe on familial constraints) that guarantees the existence of a Pareto efficient allocation that satisfies familial constraints. Then worry about envy.


Here;
$M^d(Y,i)$ is the demand for money.
$M_0$ is the ‘autonomous’ demand for money, i.e. demand for money that would not depend on income or interest rate.
$M_1$ is the effect that real income/output has on demand for money. It will be positive because ceteris paribus when there is more goods and services around people demand more money for carrying out transactions.
$M_2$ is the effect nominal interest rate has on money demand. It is actually negative not positive, and the reason why it is negative is that if interest rate is higher people demand less money both because it is more expensive to get (borrow) money and because with higher interest rate people prefer to save more and postpone their purchases to the future so they need less money.


Why do all apartment complexes in Albuquerque have pools?

In most places, it's probably more efficient to invest in more houses than to put pools for the same value in the houses you own. In some place, having a pool might even decrease the renting value of your house: the maintenance cost compensate the rent increase. In Albuquerque, it is possible that it's more relevant to invest in pools than in houses, hence all landowners invest in pools, hence, no more houses to rent without a pool.

With $P$ the cost of a pool, $H$ the cost of an house, $r_H$ the revenue you get for renting a naked house and $r_P$ the revenue you get for renting a house with a pool and $c$ the upkeep of a pool, every house owner is going to build a pool before renting or selling if $$(r_p - c)/P > r_H/H$$

If this isn't true (and it isn't in most places), some people we choose to build a pool for other reasons but not all houses will have pools.


Different terminology for discrete vs. continuous "marginal" concepts?

I found an answer that satisfies me in Preston McAffee's Introduction to Economic Analysis (2006):

  Marginal is just economist’s jargon for “the derivative of.” For
  example, marginal cost is the derivative of cost; marginal value is
  the derivative of value. Because introductory economics is usually
  taught to students who have not yet studied calculus or can’t be
  trusted to remember even the most basic elements of it, economists
  tend to avoid using derivatives and instead talk about the value of
  the next unit purchased, or the cost of the next unit, and describe
  that as the marginal value or cost.

So it looks like economists don't have a terminological distinction, because in their own work they tend to treat all goods as near enough to continuous anyway (I suppose because it's easier and because markets are often large). The only reason they talk about e.g. the price of "one more unit" of a good, in introductory classes and textbooks, is that they don't expect the students to be comfortable with calculus yet.


Why does a lack of pipelines cause Canadian crude oil to trade at a discount compared to US crude oil?

If you take the oil price as a given, meaning not set by Canada but by the market, then the costs of more expensive transport will eat the profit margins. For example, if the price of the oil is 100 USD per barrel, 20 USD would be costs for Canada, while for US producers the cost is, for instance, only 10$.

Keep in mind that a project investment analysis should take in account the cost of the pipeline divided by the number of years that it will be used to deliver that oil, adjusted for potential changes in oil demand, and including costs of insurance and removal at the end on the life of the pipeline. When all is taken in account, it might be that the oil will always be sold at discount when coming from Canada, and building the pipeline is not such a good investment after all - but only a full analysis would tell.


What is the government's ideal rate of growth for property prices?

I can't speak for the rate at which you should set your prices.

However, borrowing some knowledge for my class of monetary economics, I would say pricesetting has a lot to do with expectations.

Just like in the exchange market, when people expect prices to rise a certain self-fulfilling prophecy occurs which raises prices. Why is that? When people expect prices to rise it gives them time to adjust themselves to the rise in prices.

For instance: Let's suppose that the dollar is at 1,5dollars/€
For some reason people expect the exchange rate is going to rise. when I say rise I mean an exchange rate of  for instance 1$/€ . Eventhough when the number becomes smaller because you need to spend more dollars to get the same quantity of euros.

People expect the exchange rate to rise. Well some people are scared of the price rise so they anticipate that by buying the euros at the current low price. This in turn raises the demand for euros and therefore raises its price. This is the self-fulfilling prophecy mechanism I was talking about.

I realise I haven't answered your questions completely. But I hope it helped you to know that even if you do you at which rate you should set the prices, it is still important to factor in the expectations of the population, however difficult that can be.


This may be rather late, but it seems from an article at the time that this was due to supply disruptions in Canada, Nigeria, and Venezuela. Canada was experiencing wildfires which shut down oil production, Nigeria suffered from attacks in the Niger Delta (the main source of Nigerian oil), and Venezuela was experiencing severe political crisis. With a supply decrease, oil prices were bid up.


Why do we need the specific utility assumption?

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

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


At what rate can a multi-billionaire liquidate his assets without significantly affecting the stock price?

One place to start would be to look at the OnBalanceVolume technical indicator.

This is by no means reliable, but it’s a place to start.  It tells you how much of the volume hit the bid vs how much hit the ask.

On a very good day, MSFT on balance volume is 2 million shares out of ~34 million shares traded.

So presumably you could sell some relatively small percentage of the on balance volume, say 5 or 10%, without affecting the price too much.

If you sold 100,000 shares a day at $184 a share, that works out to $18million per day.

If you did that for 200 trading days per year that would be $3.6B per year.

Also note that large blocks are often traded in dark pools which may or may not be recorded in the volume.  More than likely, he sold his shares in dark pools, in large blocks,  to institutions, at the end of day closing price, over a period of months.  If this were the cases, On Balance Volume tells you nothing.  Still, it’s a place to start.


Do heterogeneity in consumption baskets lead to very different inflation rates for different consumers?

Your intuition is entirely correct. Inflation is calculated based on a "standard basket", whereas real baskets vary among different consumers. As such, the incidence of changes in prices affect consumers differently.

A good treatment of the issue is this piece by the IFS, UK. Basically, distributional effects of inflation would not exists if:

households buy the same
price of all items change equally

As you can imagine, both are not true. The report calculates then the rate of inflation faced by different households, taking Sept of 2008 as example, leading to the following result:

This is, inflation can be very different across households. Check the document for more details and insights.

Finally, an example of a theoretical treatment (aka model) is here. The authors model a monetary economy where the cost of credit is a non-linear function of income, such that the rich have a lower cost per unit of credit. This means that rich buy their consumption with a greater proportion of credit than cash, compared with the poor. Then, as inflation is basically a tax on cash, this means the same inflation affects the rich and poor differently. In fact, inflation is regressive. This is a well-known fact in political discussions (e.g. here).


Comparing rates of return for different assets using $V=Pe^{rT}$

First of all stop using discrete interest in your intermediate steps to keep your work simple. (As a final step you can convert continuous to discrete if you want to.)
The reason you get different results when you change the start and end dates is because the longer the time period the more the result (growth rate of the totals) will approach the biggest growth rate. By implication the dates do not matter; only the length of the period, so 1901 to 1911 gives you the same result as 2001 to 2011. I didn't test this but it seems like it would be true.
You said you want the overall growth rate.
You can take the average (the mean).
You can take the trimmed mean, which is the average after discarding the highest and lowest rate because may be they are anomalies. Since you don't have a lot of data, you might not want to do the trimmed mean.
You can take the price weighted mean of all data or the trimmed data. I would use initial prices, to be specific. Since you might want to buy the whole street (big spender) the price weighted mean might be more relevant to you. If you're only interested in buying one randomly selected house, the mean or trimmed mean is better.
You can take the median if you think higher and lower rates are less useful for your purpose but you would probably want more than 4 houses in your set if you take a median. If you're only interested in buying one house and you have a median-sized budget then the median rate is more interesting for you.


Why is a higher inflation rate bad for international competitiveness?

Competitiveness simply means how expensive goods of a country are when compared. To correctly compare we would need to transform everything into a(ny) single currency. To do this correctly we need two elements: a) the actual price of the good and b) its exchange rate. If a country has higher inflation, then nominal prices increase, the goods are more expensive, less desirable and therefore due to (a) competitiveness decreases.

However over time purchasing power partiy (called PPP) holds. If it holds (typically does in the long run) then due to higher inflation in a country, the currency will likely lose value (depreciate) and competitiveness will increase due to (b), in which case it will eventually cancel the aforementioned effect (a) in which case your confusion is completely justified, there should be no effect.

However since PPP does not always hold and especially not immediately, higher inflation reduces competitiveness.


Because what Trading Economics states to be M0 is not really M0 but the monetary base (MB). Compare the data in their graph with the data from the Fed (e.g. here). These are some of the values from the monetary base (variable BOGMBASE):

2016-09-01  3735888
2016-10-01  3572132
2016-11-01  3629770
2016-12-01  3531565
2017-01-01  3595455
2017-02-01  3746401
2017-03-01  3856260
2017-04-01  3821654
2017-05-01  3774390

They correspond roughly to the ones in the TE graph.

Whilst M1 is indeed larger than M1, the monetary base does not need to be. This is the entry in the Wikipedia article for monetary supply in the US:

M0: The total of all physical currency including coinage. M0 = Federal Reserve Notes + US Notes + Coins. It is not relevant whether the currency is held inside or outside of the private banking system as reserves.
MB: The total of all physical currency plus Federal Reserve Deposits (special deposits that only banks can have at the Fed). MB = Coins + US Notes + Federal Reserve Notes + Federal Reserve Deposits
M1: The total amount of M0 (cash/coin) outside of the private banking system plus the amount of demand deposits, travelers checks and other checkable deposits

A similar entry from the Fed website indicates that M1 does not include the whole of MB. The relevant quote is:

  The monetary base is defined as the sum of currency in circulation and reserve balances (deposits held by banks and other depository institutions in their accounts at the Federal Reserve). M1 is defined as the sum of currency held by the public and transaction deposits at depository institutions (which are financial institutions that obtain their funds mainly through deposits from the public, such as commercial banks, savings and loan associations, savings banks, and credit unions).

It follows that the data from Trading Economics is misleading. It might be worth pointing this out to them.


Absolute Value of Total Factor Productivity in an Aggregate Cobb-Douglas Production Function

Applying a regression model

$$\ln Y = \ln A + a\ln K + b\ln L$$

on a standard macroeconomic time-series data set from a country would immediately provide an estimate for $A$, essentially an average over the sample.

The problem is that experience has shown us that we don't have $A$ but $A_t$, i.e. it is time-varying. As such, estimates of $A$ as though it was a constant seem rather meaningless or even misleading.

More-over, in a time-varying coefficient framework, what is important is the change, rather than the absolute value, since it is essentially a mark-up index on the production function, and not something that can be meaningfully measured in absolute units. In this case, we take $A_0=1$ (the first period in the sample) and we track the percentage change over the sample.


Production choices in Hayek's theory of information-bearing prices

Having mulled over this for a while, I think the answer is actually quite simple. While luxury goods might have some of the highest margins and highest prices per unit, they don't actually make the producers the most money because of the low quantities in which they are sold. So I think my summary might have been ill formulated. Instead of “produce what is most expensive”, a better summary would be “produce what is most profitable”.

Clearly luxury goods are not what is most profitable. Currently this would rather be high-tech services such as data analysis and reliable forecasts.


Establishments are firms at the facility level. Think about it like branches: If a firm has three different branches, each of these branches counts as an establishment. If a firm has no branches, the firm itself is the (sole) establishment.


General Equilibrium with Linear Production

If the firm takes output price and wage as given, then its supply is infinity if the price is larger than the wage, and it is zero if the price is equal, or lower, than the wage. If the price is larger than the wage, the firm by increasing output can make arbitrarily large profits, and thereby, in particular, cover its fixed cost. If the price is equal or below the wage, then there is no way of making profits, and therefore, the unique best option is to set output equal to zero.


Do countries have to run on deficits? What is their possible economic future?

One of the main reasons for the general tendency to run deficits is political economy. Governments are elected for a limited period and try to stay in power by benefiting their electorate. An alternative explanation is that high debts reduce the scope for the subsequent government. So there are good reasons to impose some restrictions to the deficit into the constitution.

I find the comparison to big companies misplaced. Governments fulfil different purposes than companies. Many of their 'products' are public goods (public infrastructure, education, defense, social security) which cannot be provided by private actors due to external effects. A strictly profit-oriented view of public services therefore falls short of what governments are supposed to do (which does not say there is probably lots scope for efficency-enhancing in most administrations). As a sidenote, running huge deficits is not uncommon to big companies, especially in digital services. As long as creditors are convinced these debts will be paid, companies can be run on losses for quite a while.


How can I show whether the following function is Increasing, Decreasing, or Constant returns to scale?

You have $$F(x_1,x_2)=x_2\sin\bigg(\frac{x_1}{x_2}-\pi\bigg).$$
If you multiply all factors by $t$ you get
$$F(tx_1,tx_2)=t x_2\sin\bigg(\frac{tx_1}{tx_2}-\pi\bigg).$$
Simplify the fraction and you are almost done.


Will inflation rate be the same if we use different base year?

Not at all.
Assuming You wanted to calculate the inflation rate from between 2003 to 2004, you only need to know the CPI for the starting and ending dates.  So the CPI index in 2003 is 1.84 and the CPI index is 1.889 in 2004.

The formula is: (end -start)/start

so we have (1.889-1.84)/1.84 = giving us 0.02663 or a percentage change of 2.663% -This is the inflation rate-.

If we moved the base year to 2003 and assigned the CPI of 1, we can use this calculate the CPI in 2004.

(X-1)/1 = 0.02663
X-1 = 0.02663
X = 1.02663 * 100 is a CPI for 2004 of 102.663.

Made this table for you to get an idea of how it is calculated based on your example.


Are the figures quoted in this liblabcon's blog post accurate? (topic: UK banks)

Source:- Bankers cost this country £456.3bn in fraud

  In the UK, where the government bailed out Royal Bank of Scotland Group Plc (RBS) and Lloyds Banking Group, the total outstanding support explicitly pledged to Britain’s banks stood at 456.3 billion pounds ($730 billion) at the end of March, or 31 percent of GDP, the National Audit Office said in a July report. The amount was down from a peak of 1.16 trillion pounds.

Source:- Politicians had to bail the bankers out with £1.2tn, leaving the next generation with 3 times more tuition fees to pay.

  The Treasury has pledged to spend £1.2 trillion on the bail-out since the crisis began. But the real outlay has been much smaller. By March it was committed to spending £456.33bn: £123.93bn in loan or share purchases, which required an actual cash injection from the government to the banks, and £332.4bn in guarantees and liabilities. It costs taxpayers up to £5bn a year just to service the loan that the crisis incurred.

The idea seems to be more to create noise rather than putting facts into perspective. The 2 lines in isolation are damning, but when you read the whole thing it seems to make a bit more sense. Seems more at fear mongering rather than putting the facts into perspective.


I think you use the inverse (1-MPC) of the MPC to derive the marginal propensity to save (MPS) and create the investment function.  So, I= 60 + .2*1000 = 260, or an additional 200 from the autonomous level.


How much more can the US Federal Reserve spend on COVID-19?

There’s no real dollar cap on what the Federal government can do, rather the limit is on what is available to buy, and how much it is willing to spend. (There are various institutional limits on spending, but those limits can be raised. This is what allows the size of government to grow in line with increasing nominal GDP.)

Examples

It can only buy as much medical equipment as is available for sale. (As noted in a comment, it can use emergency powers to increase the supply, but purchases are still limited by what is in inventory and what can be produced in a given period of time.)
If it lends to a corporation (a big part of announced packages), there are limits to how much they want to lend to them, based on the size of the firm, etc.

One can point to limits that are created by the structure of the bond market. This limit is fuzzy, and there is no well-defined “cap.” One can point to the experience of World War II government finance as an example where the government modified how borrowing operations worked to eliminate restrictions created by the bond market.


Whatever happened to Henry George's ideas?

If you're referring to a land-value tax, I'll just quote The Economist:

Other economists like it too. Adam Smith said “nothing could be more reasonable”; Milton Friedman termed it “the least bad tax”. Winston Churchill said scornfully that a landlord “contributes nothing to the process from which his own enrichment is derived”. The Organisation for Economic Cooperation and Development, a Paris-based thinktank for industrialised countries, supports the idea. So too did a recent working paper by the International Monetary Fund and the Mirrlees Review of British taxation by the Institute of Fiscal Studies. The mayor of New York City, Bill de Blasio, hopes that taxing vacant lots by value will help deal with urban blight in the Bronx and elsewhere.
[...]
The idea has wide-ranging support among modern economists, and more than 30 countries have some form of land-value taxation, usually in harness with other property taxes, on buildings.

While the idea of a land-value tax certainly has its drawbacks, it's far from being discredited.


Monetary expansion leads to decrease in inflation?

In a nutshell, I think you're right.

The crux of your professor's argument is that, according to the classical dichotomy, the real interest rate will remain unchanged. However, that is only true in the long run, and we shouldn't be looking at the long run for this question. We should be looking at the short run. In the long run, the nominal interest rate as well as the inflation rate will remain unchanged, so the real interest rate will remain unchanged.

In the short run, the nominal interest rate will decrease, which increases output (according to the Investment-Saving relation) and causes inflation (according to the Aggregate Supply relation).

In the medium run, with output above its natural level, wages will rise due to a shortage of labour and thus prices will increase further. This happens until output is back at its natural level. This increase in price reduces the real stock of money as well, so that the nominal interest rate returns back to its original level according to the Investment-Saving relation (a leftwards shift of money supply in the money market).

In the long run, with prices stable at a higher level, there are no further increases in prices, so inflation becomes 0. (This is a static model, whereas in the real world the economy is dynamic and constantly experiences price increases.) Thus, real interest increases back to its original level.


Income and substitution effect for perfect substitutes

An indifference curve for perfect substitutes is a straight line. In fact it is the line defined by $y=const-x$, for a utility level of $const\in\Bbb R$. We maximize the utility when our budget line is tangent to the IC line. But they are both straight lines, so there are a few cases (considering a situation with only 2 goods):

the prices are not equal ($P_X< P_Y$ for example, then you would always choose good $X$)
$P_X> P_Y$ for example, then you would always choose good $Y$
the prices of the goods are equal $\implies$ our budget line is parallel to the IC line and the utility maximization point is anywhere on the budget line.

For cases 1 and 2:

Let's consider the case where good $Y$ is cheaper and becomes relatively more expensive after a price change.

We can use Hicks' method to determine the SE and the IC by reducing artificially the consumer's income so that he ends up consuming a basket that is on the original IC curve before the price change (I think Slutksy method is impossible to implement, because we would have to reduce the budget so that the original basket is chosen but with the new prices it is impossible, unless the good is an inferior good I guess, but can they are perfect substitutes so they are both inferior, which I'm not sure if it's possible because an inferior good must take up most of your budget)

We go from equilibrium $E$ to $E'$ and the equilibrium $E''$ is the one due to a reduced budget by the method of Hicks.

For the third case, in theory the original equilibrium can be anywhere on the budget line. In practice it is probably in a fixed position and you can use the same method to derive the IE and the SE.


Given how the problem is set up, I think they want you to take A as given when you choose price.  If that is the case, then you can find the comparative static $\partial P / \partial A$

We can think of p in the Foc as being a function of all the other parameters: $A\alpha-2A\beta P(A,\alpha,\beta,c)-A\beta c=0$

Now differentiate this wrt A: $\alpha-2\beta P(A,\alpha,\beta,c) -2A\beta (\partial P / \partial A)-\beta c=0$

Now you can solve for $\partial P / \partial A$


Sources for macro economic forecast data

GDP: If I remember correctly, Congressional budget office offers 10-12 yr forward potential gdp every quarter, from which you can derive predicted potential growth.

PopGrowth: U.S. Census has this data

Inflation: Fed Res/Philidelphia provides inflation forecast

Account deficit: All the sources for 5 or more years forecast in int. trade are either proprietary or highly unreliable. Best of luck in your search.


It is normal that the slope of the LM is related to the elasticity of money demand. If in an economy, the elasticity of money demand is so high, for little changes in the interest rate will make fluctuate the demand so much.

If you think about the extreme case, you will understand much easily. Suppose that the elasticity of money is near to infinity, (which makes the slope of your LM near to something horizontal on (interest rate, money demand) coordinate plane. (so in this case, your response to change of an interest rate is so huge.)

The symmetrical extreme case is that this curve becomes vertical, which makes that your money demand never changes whatever the change of interest rate.


I'll offer a little guidance to get you moving on the problem your own.

You've set up the Lagrangean incorrectly.

The maximization problem you have looks like:

$$\max \ U(x_1, x_2) \\
\text{s.t.} \ p_1 x_1 + p_2 x_2 \leq y $$

The Lagrangean should read:

$$\mathscr{L} = U(x_1, x_2) - \lambda (p_1 x_1 + p_2 x_2 - y)$$

The Lagrangean you offered had utility as a function of $X$ and $y$, but you can see your utility function does not have $y$ as an argument in it. So this is just a notation error to keep track of. As for the part with the Lagrangean multiplier, you can imagine that if the inequality constraint is binding, then $p_1 x_1 + p_2 x_2 - y = 0$, so in that case, maximizing the Lagrangean is just maximizing utility minus $0$, but it's a special $0$ that helps to incorporate the budget constraint into the problem. (As for the non-binding case, you can think about what is happening to the optimization problem yourself.)

Your first order conditions are correct though. So that tells me you've mostly made some typing errors. To answer your question about whether the Lagrangean is okay here, the Lagrangean will work as long as the optimum is an interior solution. So if you suspect either $x_1, x_2$ will be equal to zero, then you may have to break out Kuhn-Tucker's theorem. So think about under what conditions Lagrange's theorem holds.


Impact evaluation of a policy when different individuals adopt the policy at different times

The description is bit vague  on how many organizations we are talking but you could still do differences-in-differences (DiD) for all treatment-control-pairs that adopted it at the same time  separately and then running meta-analysis on the results.
As far as I understand based on Angrist & Pischke (2009) Mostly Harmless Econometrics ch.5, DiD solves any possible self-selection bias as long as the identifying assumption of common trend is satisfied. That is in DiD it it is required for both treatment and control to follow common trend pre-treatment. If the changes in outcome variable ($y$) pre-treatment are virtually same between treatment and control, then even if the levels of $y$ are different the estimates won't be biased. This of course will have to be tested in each case.
The above will however only be feasible if you do not have to run too many of the individual DiD otherwise it would be unpractical.
An alternative could be the event study design (see MacKinlay, 1997) that is used quite often across finance literature. It allows for events to occur at different times and it does not matter that firms/organizations experiencing the event are heterogenous and possibly self-select for the event (this approach is often used to evaluate M&A effect on stock returns for example), but on the other side this approach is very data intensive and depending what precisely is $y$ it might not be appropriate, or also if there is event clustering.
In order to implement regression discontinuity design (RDD) there must be some cut-off point that determines whether treatment is assigned to subject or not. For example, RDD would be appropriate if there is a policy that requires any power plant producing more than let's say 1.21 gigawatts is required to adopt some $CO_2$ curbing technology and below that threshold it is not required, and you want to examine the effect of such policy on some outcome variable like air quality. There is also so called fuzzy RDD which does not require sharp cut-off in some deterministic variable as long as probability of adopting the treatment is significantly different  (see ibid. (2009) Mostly Harmless Econometrics ch.6). The feasibility of this will again depend on data availability and other details, but if the organization themselves have full control over probability of treatment it will not be appropriate.
Lastly, as you could have probably already guessed from the discussion above the answer the choice of methodology is always context dependent. In order for anyone to give you concrete answer you would have to share virtually every single detail of the study which is not feasible, so take the above as some pointers with references (since you tagged the question as a reference request) rather than telling you exactly what to do.


Difference between long run coefficient and non stochastic steady state coefficient ARDL model

yes, the term that you showed for the ALDR non-stochastic steady state:
$$\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$$
is long-run multiplier or sometimes also called long run equilibrium coefficient (see Verbeek 2008 Guide to Modern Econometrics 4th ed.  pp 340). As  far as I can understand there is not much difference between the two concepts in the context of ARDL model and they are used interchangeably.
The term non-stochastic steady state has wider application especially in macroeconomics (e.g. in growth theory), but often steady state results are just called long-run equilibrium because in most models steady state is attained only in long-run equilibrium.


Is the dictator game studied by game theory or by decision theory?

I would say that the dictator game is of interest to a decision theorist more so than to a game theorist, although the dictator game can be formally modeled in a game theoretic framework.

Decision theory studies how behavior, especially those that deviate from the rational ideal assumed by standard economic theory, can be modeled, as well as the implications of these behavioral models.

Game theory, in contrast, focuses more on the interactive aspect of decisions. The fact that the dictator game can be formalized as a game has more to do with the broad applicability of game theory than the dictator game's intrinsic game theoretic value.

Based on the purpose of your research, you should be looking up references in the behavioral/experimental literature.


I think the best way to answer you is to build a simple model. That's usually what economists do to answer such questions. I hope it will be useful.

Assume there is only one good in the economy, the 'consumption good', that people are all of the same generation or the same age and that one lives only for two periods (youth, and old age).

Then the answer to your question depends on the durability of this consumption good. How long can it remain useful to consumption after being produced ? Or, to think as an economist, is it expensive to keep the good useful ?

If it can for no cost remain useful for more than one period, then it can be rational for all workers to save simultaneously half of the monetary wage they receive as a compensation for their labor in period 1. Then at the end of period 1 firms will sell only half of the output produced by the workers. They will store the rest of it (we have assumed that this storage does not cost much). Then this stock of the consumption good will be available in period 2 and will be sold to the workers, who will buy it with their monetary savings. So the story is exactly the same as it would have been without money, if workers would have stored directly the commodities they produced.

If the consumption good is not durable, or is expensive to store, then this scheme is not possible. In reality, of course, many commodities of first necessity are not durable (goods but also services), and many durable goods are expensive to store. But that's not so much of a problem because we live in a world where generations do overlap, so the young produce for the old. When there is a demographic disequilibrium inside an aging country, then it should be able to trade with a 'younger' country, provided it has accumulated enough 'savings', i.e. valuable durable assets.

Edit. So far I only answered the question : is it possible for every one to live without working at the same time ? But the number of active workers can decrease with time if savings are used to invest in capital goods which make labor more productive.


Does the Marginal Propensity to Import = negative Marginal Propensity to Tax?

Hint. Is it always the case that MPC + MPS = 1?  Or does this hold only in a certain type of model with assumptions that simplify from reality?


How can energy consumption be an indicator of per capita availability of capital?

In economics, capital refers to things that are used to produce other things.

So, capital includes tangible things like a delivery truck, an office building, or a factory. These are things which often require energy to run.

Hence, if a developing country has low capital per head, then it probably also has few delivery trucks, office buildings, and factories. And hence, it probably also has lower energy consumption.


Determining cost from production function, wage, and rental rate

What's tricky is that the problem has a kink. Below a certain production value no capital is used. The demand functions for $k$ and $l$ are as follows:

If $Q \leq 180$
$$L(Q) = (Q/20)^2$$
$$K(Q) = 0$$

If $Q > 180$
$$L(Q) = 81$$
$$K(Q) = (Q - 180) / 2$$

Which you can get from thinking about what happens to the marginal cost of producing with labor just below and above $L=81$ (goes from below 4.5 to above 4.5) and comparing it with the the cost of producing a marginal unit with capital (which is always 4.5).

Why the split at $Q = 180 / L = 81 $?
 $$\frac{MP_{L}}{MP_{K}}=w/r \leftrightarrow  \frac{10L^{-1/2}}{2} = 5/9 \rightarrow L = 81$$
Note that $ Q(k=0,L=81) = 180$


Question Regarding Equilibrium Price & Surplus

A supply curve is typically derived from a cost function, and you're leaving out some important costs such as storage and depreciation, which would help relieve some of the seeming counter-intuitiveness around a supplier choosing to "sell more at a loss" compared to selling less.


Has an increase in housing supply in popular cities including Amsterdam led to an increase in house prices?

Let's just say that economists don't really agree with this thesis, and most city-level empirical evidence points in the opposite direction. E.g.

  We ultimately conclude, from both theory and empirical evidence, that adding new homes moderates price increases and therefore makes housing more affordable to low- and moderate-income families.

  [...]

  A few studies use panel data and find that the imposition of more stringent land-use controls
  leads to lower supply and higher prices. Jackson (2016) uses longitudinal data from California’s
  cities to assess the effect a city’s adoption of additional land-use regulations has on the number of
  new construction permits issued, and finds that each additional land-use regulation adopted
  reduced multifamily and single-family permits by an average of more than 6% and 3%, respectively,
  and that regulations reducing allowable density had even larger effects. Zabel and Dalton
  (2011) use longitudinal data from localities in Massachusetts and find that increases in minimum lot
  sizes are followed by significant increases in prices. Looking at longitudinal data on municipalities
  in the Boston (Massachusetts) metropolitan area, Glaeser and Ward (2009) find that the adoption of
  stricter local regulations leads to higher house prices, but the coefficient falls in magnitude and
  loses significance once they control for population demographics. They point out that this is
  expected, if homes in other jurisdictions are seen as perfect substitutes. Thus, whereas supply
  restrictions may increase prices in a market as a whole, they may not increase them disproportionately
  in the particular locality where they are imposed due to spillover effects across jurisdictions.

  Several other researchers use instrumental variables to try to more clearly assess the causal
  effects regulatory restrictions have on housing supply and prices. Ihlanfeldt (2007) uses such an
  approach to study regulation in localities in Florida and finds that predicted regulations significantly
  increase the price of single-family homes. Saks (2008) uses instrumental variables and shows
  that increases in labor demand lead to less residential construction and larger increases in housing
  prices in metropolitan areas with more restrictive housing supply. Hilber and Vermeulen (2016)
  show that changes in demand lead to increases in local house prices rather than increases in
  supply in municipalities in England with greater regulatory restrictions, measured by the refusal
  rate of proposed residential projects and the number of project approvals delayed more than
  13 weeks.

  In sum, the preponderance of the evidence shows that restricting supply increases housing
  prices and that adding supply would help to make housing more affordable. [...] although it is surely true that land is constrained, especially in certain
  markets (Saiz, 2010), land can be used more intensively to allow for more housing. The limits on the
  land with which housing is bundled make housing different from many goods, but the difference is
  one of degree: the supply of housing can and does increase even in constrained markets, and
  prices should generally fall in response (see the review by Dipasquale, 1999; Mayer & Somerville,
  2000).

  [...] most housing filters down, or loses
  value as it ages, representing new supply in submarkets at lower price points. [...] Indeed, recent research shows that filtering
  was the primary source for additions to the affordable rental stock between 2003 and 2013,
  whereas new construction was the largest contributor for the higher priced rentals, and tenure
  conversion was the largest source for moderately priced rentals (Joint Center for Housing, 2015, fig.
  14). Further, Weicher, Eggers, and Moumen (2016) report that 23.4% of the rental units that were
  affordable to very low-income renters in the United States in 2013 had filtered down from higher
  rent categories in 1985. Another 21.8% were conversions from formerly owner-occupied homes or
  seasonal rentals.9 Most of the higher priced rental units that filtered down to become affordable in
  2013 were moderate-rent units in 1985, but 15% of those that filtered down were high-rent units in
  1985. Note that filtering occurs over a shorter time frame too; among affordable units in 2013,
  19% had been higher rent units as recently as 2005.

  Recent research analyzing the incomes of successive occupants of homes also suggests substantial
  downward filtering, particularly of the rental stock due to tenure conversion; as the owner-occupied
  stock ages, a portion converts to rental (Rosenthal, 2014). Rosenthal also finds, however,
  that filtering rates are considerably lower in areas with high house price inflation, although
  downward filtering still occurs.

  In short, new construction is crucial for keeping housing affordable, even in markets where
  much of the new construction is itself high-end housing that most people can’t afford. A lack of
  supply to meet demand at the high end affects prices across submarkets and makes housing less
  affordable to residents in lower-cost submarkets.

Finally getting to the discussion of the precise thesis you ask about:

  Some skeptics argue that even if additional supply could help make housing more affordable in the
  short run, it won’t in the long run because the additional supply will induce more demand, especially
  among buyers or renters wealthier than the existing residents in the neighborhood (Redmond, 2015).
  The claim is analogous to the argument that building more highways will not reduce congestion
  because the lower cost of travel will simply cause more people to drive or to take that particular route
  (Gorham, 2009). In this case, the argument is that by making the jurisdiction more affordable, adding
  housing supply will attract new demand—both from current residents who would otherwise leave,
  and from people living elsewhere who will now choose to move to the jurisdiction. Further, the argument goes, lower rents and prices may also induce latent demand—people who are living with
  roommates or family members may choose to form their own households (Ellen & O’Flaherty, 2007) or
  people may choose to invest in pied-à-terres in a city. That additional demand will drive prices back up
  until supply can again respond, causing housing to be more affordable, at best, only cyclically,
  according to the argument, and increasing the density of the jurisdiction, with the attendant costs
  of congestion.

  Although building additional highways does appear to induce more demand (Duranton &
  Turner, 2011), in the case of housing, additional demand is unlikely to completely offset the new
  supply. Such an offset requires demand curves to be perfectly elastic—or, in other words, it
  assumes that neighborhoods and jurisdictions are perfect substitutes and that there are no
  constraints on the ability and willingness of households to move. That is unrealistic. Moving
  homes is not like driving a few extra miles (Lewyn, 2016), and costs associated with moving may be
  high. Any additional demand induced by new housing is limited by personal and economic
  constraints on the ability and willingness of households to move, restrictions on immigration, and
  uncertainty and other factors that might inhibit renters and buyers from renting or buying in the
  market in which housing supply increases. Indeed, mobility rates have fallen sharply over the past
  several decades, and although the reasons for the decline are being debated, the decline reveals
  significant constraints on the ability and willingness to move.

  Thus, in the long run, whereas some additional households may be drawn from outside (or from
  within the city) to buy or rent homes as supply increases, it is highly unlikely that prices will end up
  at the same level that they would have reached absent any new supply. Finally, as noted above, the
  empirical evidence shows that allowing more supply leads to lower housing prices; if adding
  supply induced sufficient additional demand to offset the increased supply, the studies would
  not find an association between supply and prices.

It can happen indeed in some very limited circumstances, and with localized [e.g. neighborhood] effect, which is probably on what this faulty generalization is based on:

  Many renters in neighborhoods where market-rate housing is proposed express concern that the
  construction of new housing will actually make their affordability problems worse by raising rents or
  house prices, fueling gentrification, and potentially displacing existing residents (Atta-Mensah, 2017;
  Savitch-Lew, 2017). Hankinson (2017) theorizes that renters’ opposition to local additions to supply is
  driven by such worries; he argues that it is plausible that the construction of an attractive new building
  will increase prices locally (by improving the physical landscape, bringing new amenities to the
  neighborhood, and signaling that the neighborhood is improving), even as it reduces them citywide.

  Testing this proposition empirically is quite challenging, given that developers will naturally be
  attracted to areas where prices and rents are rising. There is evidence that improvements to
  blighted housing can, in some circumstances, increase surrounding property values, even when
  the new or improved housing is subsidized, low-income housing (Diamond & McQuade, 2016;
  Schwartz, Ellen, Voicu, & Schill, 2006). The new housing studied, however, typically replaced
  vacant, abandoned buildings and littered vacant lots, in essence removing a disamenity. [...]

  There is little empirical evidence about the net effect new market-rate housing has on the prices or
  rents of nearby homes, and what exists may not be causal. One recent study examines the effect of
  market-rate single-family homes newly constructed on infill sites, and finds that newly constructed
  single-family homes can have positive impacts on the sales price of other single-family homes nearby,
  but the effect varies with context (Zahirovich-Herbert & Gibler, 2014). A study of multifamily high-rise
  infill developments in Singapore found positive price effects on nearby houses (Ooi & Le, 2013), as did
  a study of single multistory apartment buildings constructed in Helsinki (Kurvinen & Vihola, 2016).
  These studies all consider property values and not rents, and none is able to prove a causal relationship
  given that market-rate developments aim to target neighborhoods where they expect property
  values to improve. Unfortunately, we found no study examining impacts on rents, although one study
  by the California Legislative Analyst Office concluded that additional market-rate construction is linked
  to lower displacement rates (Taylor, 2016). Examining low-income neighborhoods in the Bay Area
  between 2000 and 2013, these researchers found that the production of market-rate housing was
  associated with a lower probability that low-income residents in the neighborhood would experience
  displacement. Although a singular study, the findings suggest that for neighborhoods in high-demand
  cities, blocking market-rate construction may place greater pressures on the existing stock.

  In short, although it is clear that the construction of new homes will moderate price and rent
  increases citywide, neither theory nor empirical evidence provides clear guidance about when
  localized spillover effects might occur and when they might actually cause an increase in the prices
  and rents of immediately surrounding homes.

There's a more recent study on the core issue of the question, not covered in the above review:

  Preliminary results using a spatial difference-in-differences approach suggest that any induced demand effects are overwhelmed by the effect of increased supply. In neighborhoods where new apartment complexes were completed between 2014-2016, rents in existing units near the new apartments declined relative to neighborhoods that did not see new construction until 2018. Changes in in-migration appear to drive this result. Although the total number of migrants from high-income neighborhoods to the new construction neighborhoods increases after the new units are completed, the number of high-income arrivals to previously existing units actually decreases, as the new units absorb a substantial portion of these households. On the whole, our results suggest that—on average and in the short-run—new construction lowers rents in gentrifying neighborhoods.

See also a blog discussing it (which also does a good job at debunking the roads analogy); in summary, the gist of the latter study is in this chart:

  This chart shows that rental prices for apartments close to the new building fell relative to the prices for apartments located further away. The dashed “before” line has a negative slope, suggesting that prices declined the further you got from the site of the new building.  The solid “after” line has a positive slope (prices increase the further you get from the new building).  Overall, prices are higher (the solid line is above the dashed line), but prices actually went down next to the new building, and increased far less than in the area further away from the new building.

  These data are a strong challenge to the induced demand theory.  If a new building made an area more attractive, one would expect the largest effect in the area very near the building. But, consistent with the traditional “more supply reduces rents” view, the addition of more units in an area seems to have depressed rents (or at least rent increases) compared to buildings in the surrounding area.

In other words, even the theory that prices might increase locally in the neighborhood of a new building doesn't hold up, on average, to a difference-in-differences test. Without a difference-in-differences approach one can of course draw the wrong conclusion since overall demand in a city may be increasing faster than the overall supply.

Even more recently, another paper on NYC (not yet published in a peer-reviewed venue, but was mentioned in a recent NYT article):

  I find that for every 10% increase in the housing stock within a 500-foot buffer,
  residential rents decrease by 1%. The rent reduction is caused by the completion of new
  high-rises rather than their approval. Across neighborhoods, the impact is smaller in more
  central areas, presumably due to more elastic demand. Within neighborhoods, the impact
  is smaller for lower-rent buildings. Finally, the negative impact appears to be driven by
  supply effects rather than dis-amenity effects, like changes in neighborhood physical features,
  blocked views, or shadows. New housing units alleviate the growing demand for existing
  housing units and moderate the rapid growth of residential rents.

  Residential property sales prices also decrease when new high-rises within 500 feet
  are completed. [...] Because 99% of new high-rises are condos and rental
  buildings, they do not significantly affect sales prices of co-ops and 1-5 family homes. The negative impact is bigger for closer substitutes, further confirming the negative impact is due to supply effects.

  To address the hypothesis about amenity effects, I find new high-rises and their
  high-income tenants attract new full-service restaurants, cafes, and coffee shops. These consumption amenities likely make neighborhoods more attractive and potentially increase rents
  and sales prices (Couture and Handbury, 2017). However, the amenity effect is dominated by the supply effect, given that rents and sales prices still fall on net.

As was mentioned in a OP's comment below, living in a big city vs a small town (and vs the prairie) is an amenity effect. However, there seems to be a limit to this effect after which more housing supply lowers the (rental) price.

The amenity effect has been studied for decades but alas often in isolation, most prevalently perhaps with the the Roback model, which basically adapted Rosen's more general spatial equilibrium model to "nontraded" goods like housing. ("Nontraded" because Roback only considered effects on rents.) Using this model, one essentially gets (rental) price-amenity gradients. Roback assumed that "the cost of changing residences is zero" (i.e. costless migration). Modelling "partially open" cities, i.e. with non-zero (and non-infinite) relocation costs across its boundaries is a relatively more recent endeavour. Likewise for considering the combined effect of housing supply and amenities on rents. As one such theoretical work notes

  as housing is supplied more elastically [...] the rent effect of amenities approaches zero.


Identifying Nash equilibria in extensive form game

As far as I know, No.

While I am not sure why you would want to find non-subgame perfect Nash equilibria in an extensive form game, I am sure you would need to convert it to normal form to do it. Extensive form of a sequential game carries more information than normal form, specifically which moves do not exist within the sequence. In a normal form representation of the sequential game you have to show every possible move available to every player, even the moves that do not exist. So basically when you convert a sequential game from extensive form to normal form, it becomes another game where you then look for Nash equilibria.

If you are doing a two player game where each player gets one move, you can do it in your head just by looking at the game tree, but if the game is any more complex, you would need to do it step by step, first converting it into a normal form.


What is the difference between intensive margin and extensive margin in labor economics?

In Labor Economics, "Extensive margin" refers to "how many people work". "Intensive margin" refers to "how much a given number of people work, on average". To copy from a freely available recent study by Blundell, Bozio and Laroque 2011,

  "...we split the overall level of work activity into the number of individuals in work and the intensity of work supplied by those in
  work. This reflects the distinction between whether to work and how
  much to work at the individual level and is referred to, respectively,
  as the extensive and intensive margin of labour supply. At the
  aggregate level the former is typically measured by the number of
  individuals in paid employment and the later by the average number of
  working hours."

Evidently, it is an important distinction, especially when one wants to analyze changes in employment, measured say in total number of hours worked: Say, did they increase? Why? Because more people work, or because the same number of people work more? And if the answer is "both", then what portion is attributable to a change in the intensive margin, and what portion to a change in the extensive margin? (this is why the word "margin" is used).

Of course "intensity" as a word has a more general meaning, as for example the one mentioned in another answer here -it may be the case that people work "faster" and so in the same time interval, they produce more. Such changes are usually put under the concept of "changes in efficiency" (for example in Growth theory this is how they are called). In Labor Economics the terms have the meaning given above.


Differentiate a positive externality and the absence of a negative externality. Tax or subsidize?

A negative externality arises when the private net marginal benefit (i.e. the marginal benefit minus the marginal cost) of an activity exceeds the net social benefit. In such cases, the self-interested private decision maker will increase their participation in the activity even though it is socially inefficient for them to do so.

A positive externality arises when the private net marginal benefit (i.e. the marginal benefit minus the marginal cost) of an activity are smaller than the net social benefit. In such cases, the self-interested private decision maker will not increase their participation in the activity even though it is socially efficient for them to do so.

This is a semantic distinction to the extend that if one thinks activity $A$ has a negative externality then one can define a new activity $B$, which is simply the "act of not doing $A$" so that $B$ has a positive externality. On this basis, one can argue that every externality is positive or that every externality is negative. For example, many people think that education has a positive externality because educated people make better citizens (e.g they make more informed voting decisions that benefit others). As a matter of semantics, one could argue that this is, in fact, not a positive externality and that what is really going on is that people who do not educate themselves are exerting a negative externality on those who do by virtue of their ignorance.

Whilst there is some merit to this reasoning, I do not find it helpful. Often, when we study the effects of behaviour, we are interested in comparing those effects to some baseline or benchmark in which the behaviour is absent. When communicating economics to others, it is usually the case that some benchmarks are more intuitive that others. We could, for example, rewrite all of consumer theory in terms of "the dis-utility people experience from not having goods" and look at the "problem of non-consumption dis-utility minimisation". Doing so would be formally equivalent to the more conventional approach of consumption utility maximisation (only the language changed), but would probably be less intuitive for people trying to understand the economics.(*) At least to me, it is more intuitive to think that people actively choose to undetake some level of education and exert a positive externality on others than to think that everyone receives infinite education by default and the choice of abstaining results in a negative externality.

Besides education, another example that I think fits most intuitively into the positive externalities box is network effects. If I buy a telephone then all of my phone-owning friends are made better off because now they can use their telephone to call one more person that they couldn't reach before. It seems weird to think of the negative externality of not owning a phone.

In terms of taxes versus subsidies: to get to the socially optimal intensity we need to ensure that the private net marginal benefit is zero precisely when the social net marginal benefit is zero. In the case of a negative externality this can be done either by increasing the private marginal cost (via a tax) for the activity or by increasing the private marginal benefit of not participating in the activity via a subsidy. For example, we could either subsidize low carbon firms or tax heavy polluters. As far as aligning incentives are concerned, the two are equivalent. In most practical cases, the more important consideration is likely to be that of budget constraints and politics:

In the case of a tax: can the person you are taxing afford to pay the tax and can the tax be levied without seeming vindictive (for example, I think that taxing people without a university degree would be a no-no on these grounds).
In the case of a subsidy: can the government raise enough popular support and funding for a subsidy without leaving people with the impression the some parties are receiving unfair government hand-outs?

In most cases, thinking about these political and financial constraints makes it clear whether a subsidy should be used. Sometimes a combination of both is used. For example, in the UK the government both taxes petroleum consumption and subsidises electric car ownership.

(*) Nevertheless, economists do often find it useful to convert the utility maximisation problems into their dual expenditure minimisation problems, which is somehow similar. This technique, though, is usually reserved for more advanced students who already have a well-developed intuition for the economics.


What are some growth theories that are considered the most correct at the moment?

Oded Galor's interesting work on Unified Growth Theory subsumes the Solow model, the Malthusian model and endogenous growth theories in a very stylized way. This model is of course "long term" and ignores institutions.

It is a OLG model where the main problem from the household side is a choice between quantity versus quality of offspring (human capital intensity or number of children), increasing human capital is costly. From the factors side there is a fixed factor land, that plays the role of the Malthusian restriction when technology growth is non existent or low, human capital, and a labor augmenting technological or productivity factor that grows exogenously, this is understood as technological progress.

The model generates a first malthusian trap, that is only escaped after there is some population reduction shock (think black death) or some technological shock. Then the Solow stage, and then a endogenous growth phase. Population dynamics plays a vital role, in the malthusian stage population is stagnant and dependent on the fixed land factor, then in the second stage it grows exponentially and in the third stage households choose quality over quantity and the population decreases. Think of Middle Ages, 20th century US and Japan in this century respectively as examples of this stages.
Anyway I am a little rusty and it is not my field but this work is amazing.
Slides: Unified Growth Theory and Comparative Economic
Development