Edit: Roel Beetsma replied! He says he got the same derivation we basically got. So the solutions seem to be mistaken and we win! Also I'm removing the strikeout below.

Note that
$$+ e\frac{\lambda - \theta}{[-1-\lambda]}$$

actually simplifies to

$$ - e\frac{\lambda - \theta}{[1 + \lambda]} $$

rather than with $\lambda + \theta$ in the numerator like you had.

So the solutions has

$$- e \frac{1 + \theta}{1 + \lambda}$$

instead of what we have above. The difference between the two is:

$$-e\frac{-1 - \lambda}{1 + \lambda} = e$$


Seemingly simple consumer theory problem

It follows indeed from the first three conditions, though I did not find a simple proof. Here is a messy one:
Observe first that, $v(tx)>p\cdot tx$ with $t>0$ is equivalent to $v(tx)/t>px$. By replacing $p$ by some multiple, we can see that the right-side can take any value without changing the left value. The function $t\mapsto v(tx)$ is strictly concave, strictly increasing, and has value $0$ at $0$. The problem is then equivalent to showing that its right derivative is infinite for all $x\gg 0$. That the right derivative exists follows from the concavity of $v$.
As it turns out, it is enough to find a single $x^*\gg0$ such that $t\mapsto v(tx^*)$ has an infinite right derivative at $0$.
To see this, let $x\gg 0$. There is some $\delta>0$ such that $x\gg \delta x^*$. Take some $K>0$. For $t>0$ small enough, we have $v\big(\delta t~x^*\big)/(\delta t)>K/\delta $ or $v(\delta t x^*)/t>K.$
Since $v$ is strictly increasing on the interior,
$v(tx)/t>v(t\delta x^*)/t>K$ for $t>0$ small enough. Therefore, the function $t\mapsto v(tx)$ must have infinite right derivative at $0$.
It remains to find the damn $x^*$. Let $\Delta$ be the $K-1$-dimensional unit simplex. For each $N$, there must exist some $x\in\Delta$ such that the function $t\mapsto v(tx)$ has a right derivative of at least $N$ at $0$. Otherwise, for $p=(N,N,\ldots,N)$ we would have $A(p)-\{0\}=\emptyset.$ Also, $A(p)-\{0\}\subseteq\mathbb{R}^K_{++}$ implies that such an $x$ must be positive in each coordinate. So let $x_n\in\Delta$ be such that $t\mapsto v(tx_n)$ has a right derivative of at least $n$ at $0$. Now let
$$x^*=\sum_{i=1}^\infty\frac{1}{2^i}x_{2^i}.$$
That the relevant series converges follows from any norm being bounded on the compact set $\Delta$ and absolute convergence implying convergence. Clearly, $x^*\gg 0$ since $x_{2^i}\gg 0$ for all $i$. It remains to show that $t\mapsto v(tx^*)$ has an infinite right derivative at $0$. Since
$$t^{-1}~v\bigg(t\sum_{i=1}^\infty\frac{1}{2^i}x_{2^i}\bigg)>t^{-1}~v\bigg(t\sum_{i=1}^n\frac{1}{2^i}x_{2^i}\bigg)$$
for all $n$ by $v$ being strictly increasing on the interior, it suffices to show that for each $K>0$, there exists some $n$ such that
$$t^{-1}~v\bigg(t\sum_{i=1}^n\frac{1}{2^i}x_{2^i}\bigg)>K$$ for $t>0$ small enough. Let $S_n=\big(\sum_{i=1}^n 1/2^i\big)^{-1}$ and note that
$$t^{-1}v\bigg(tS_n\sum_{i=1}^n\frac{1}{2^i}x_{2^i}\bigg)=t^{-1}v\bigg(\sum_{i=1}^n S_n\frac{1}{2^i}t x_{2^i}\bigg)\geq\sum_{i=1}^n S_n\frac{1}{2^i}t^{-1}v(tx_{2^i})$$
by the concavity of $v$. Since $\lim_{t\downarrow 0}t^{-1}v(t x_{2^i})\geq 2^i$, this sum is almost $S_n n$ for $t>0$ small enough. Now $S_n$ converges to $1$, so $S_n n$ is larger than $K$ for $n$ large enough and $t>0$ small enough.


income is total profit plus wage income, and the two are fixed in proportions

I.e. $y_t = w_t + \pi_t$ and  $w_t = \alpha y_t$ and  $\pi_t = (1-\alpha)y_t$, where $\alpha$ and $(1-\alpha)$ are the fixed proportions.

For $y_{t+1}$ we have $y_{t+1}=(1+g_y)y_t$ and $w_{t+1} = \alpha y_{t+1} = \alpha (1+g_y)y_t = (1+g_y)\alpha y_t = (1+g_y) w_t$ and $\pi_{t+1} = (1-\alpha) y_{t+1} = (1-\alpha)(1+g_y)y_t = (1+g_y)(1-\alpha) y_t = (1+g_y) \pi_t.$

I.e. $g_y = g_w = g_{\pi}$


Is it possible to calculate how long it will take to reach Break Even Point if the information provided is only an ROI (Return On Investment) of 10%?

Lets say your investment is $x$. What does ROI mean? The money you get per period out of your investment (= revenue).

What does break even point mean? The point (here in time) where your total costs (here it is only the investment) are equal to your total revenue.

Therefore, the time $t$ at which the total costs are equal to the total revenue is the break even point, with $r$ being the ROI.

$$ costs = revenue \\
x = r x t \\
t=\frac1r$$

You can see, it does not depend on $x$, therefore it is possible.


Your article probably resulted from an Oxfam campaign

If you believe table 2 of a January 2015 Oxfam report then about $\$250$ trillion. The campaign spliced together some information from Forbes on the richest $x$ individuals and a series of Credit Suisse reports - see the bottom of page 94

The analysis has faced criticism: it concentrates on individual assets and liabilities, e.g. a US doctor whose student debt exceeds personal financial assets counts as poorer than a subsistence farmer in Africa, and uses a very large amount of estimation, so other reasonable calculations could be very different


International shipping services count as exports?

It's complicated, and it is supposed to depend on where the ships are generally operating from, rather than the country of the firm that owns the ships or the flag under which they are flown. Under the 2008 System of National Accounts (the international standard for calculating GDP), overseas shipping is generally to be allocated on the basis of the ports out of which the fleet is serviced. See 4.13, on page 62. The document to which it refers is the Sixth Edition of the IMF's Balance of Payments and International Investment Position Manual. You'll want to look specifically at Chapter 4, 4.131-4.136.

So if we assume that the ship owner is actually operating out of Greece, in your example he would be considered to be exporting the service for which he is paid 10 million in revenue.


Interest Rate and Foreign / Domestic Investment

Great question! I know it gets little confusing.

Let's take an example of US and India. I am in the US. India's interest rate (return on investment) is 10%. The US has 8%. Then, I will invest in India -- more ROI. Hence, high-interest rate means more foreign investment.

However, my brother in India wanted to build a building there. Since the interest rate is high, taking loans is more expensive, and hence he will decide not to invest -- not to make that building. So, because of high-interest rate, domestic investment decreases.


How to prove that a utility function U(x,y)=min(x,2y) is quasiconcave?

A function $f:D\rightarrow \mathbb{R}$ is said to be quasiconcave if the following set is a convex set for every value of $a\in\mathbb{R}$:
$P_a = \{x\in D: f(x) \geq a\}$
To show that $f(x,y) =\min(x, 2y)$ is quasiconcave, we just need to show that $P_a = \{(x,y)\in \mathbb{R}^2: \min(x, 2y) \geq a\}$ is a convex set. For that we consider arbitrary $(x', y')$ and $(x'', y'')$ from the set $P_a$ and arbitrary $\lambda\in [0,1]$ and show that
$\lambda (x', y')+(1-\lambda)(x'', y'')$ is in $P_a$. Observe that $x'\geq \min(x', 2y')\geq a$ and $x''\geq \min(x'', 2y'')\geq a$, so $\lambda x'+(1-\lambda)x''\geq a$. Likewise, $\lambda 2y'+(1-\lambda)2y''\geq a$. Therefore, it follows that $\min(\lambda x'+(1-\lambda)x'',2(\lambda y'+(1-\lambda)y'')) \geq a$ and consequently, $\lambda (x',y')+(1-\lambda)(x'', y'')$ is in $P_a$.


Proof of DCF model by mathematical induction

I suspect your induction hypothesis should be an expression for $P_t$ not $P_{t+j}$ , and should be discounting $P_{t+j}$ not $s$ times but $j$ times i.e.

$$P_{t} = \left(\sum\limits_{s=1}^{j} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j}}{(1+k)^j}$$

for $j \ge 1$, which reduces to your initial statement $P_t = \frac{D_{t+1}}{(1+k)} + \frac{P_{t+1}}{(1+k)}$ when $j=1$

Now consider your initial statement replacing $t$ by $t+j$ to give  $P_{t+j}= \frac{D_{t+j+1}}{(1+k)} + \frac{P_{t+j+1}}{(1+k)}$ and substitute this into your induction hypothesis to give

$$P_{t} = \left(\sum\limits_{s=1}^{j+1} \frac{D_{t+s}}{(1+k)^s}\right) + \frac{P_{t+j+1}}{(1+k)^{j+1}}$$

which proves that the corrected induction hypothesis follows from your original statement


What exactly is the argument for Free Trade?

The theory behind free trade is denominated comparative advantage, and as mike stated, the advantage is that a country, as a whole, benefits from specializing in the fields at which it is best, as it will get more than it would if it would produce everything alone.

In practice, a big advantage is the ability to operate for a bigger market, reaping economies of scale. This is particularly advantageous for small countries.

The downside is that, although free trade will make the country as a whole richer, it wont make the society necessarily equally richer, so it wont guarantee that an economy will develop. This connects with the concept of exploitative trade and fair trade: by entering international trade, developing countries might leave their economies vulnerable to foreign companies or governments, sometimes in critical sectors such as food and energy.


Is GARP trivially satisfied with only 1 good?

I assume by "GARP" you mean "Generalized Axiom of Revealed Preference". One can take GARP to be satisfied with one good, but it's a rather vacuous type of "satisfied"; with one good the concept "preference" don't really have any meaning. You can't have any preferences if there's only one option.


If a utility function is homothethic,
$-\frac {\frac {\partial u}{\partial x} (tx)} {\frac {\partial u} {\partial y} (ty)} = -\frac {\frac {\partial u}{\partial x} (x)} {\frac {\partial u} {\partial y} (y)} $

In other words, the MRS must be homogenous of degree zero.

Hence, for $MRS(tx,ty)=2(tx+ty)=2t(x,y)=t MRS(x,y)$ is homogenous of degree 1

While $MRS(tx,ty)=\frac{(tx)^{2}+(ty)^{2}}{(tx)(ty)}=\frac{t^{2}(x+y)}{t^{2}(xy)}=\frac{x^{2}+y^{2}}{xy}=MRS(x,y)$ is homogenous of degree 0.


How to model the change in fossil fuel price?

You are not going to be able to forecast oil and gas prices with any useful accuracy at all. Lots of people try, and as far as we know, every one of them has failed. If anyone has ever succeeded, then they've become extremely rich through it, and they're not telling anyone the secret. If you did succeed, then your energy economics model would be completely irrelevant to you, as you'd be too busy trading the oil market, and eyeing up which tropical island you should buy next.

The best you can hope for is consistency in your model. And then sensitivity-test your model based on a broad range of assumptions.
For example, my usual 95% credible range for the price of a barrel of oil in 2030 is USD(2016) 5-500.

So, what do you look for, for consistency? Demand and supply. Have short-run and long-run curves for both. If you're forecasting more than five years ahead, you must have some kind of uptake model for substitutes too, as well as short-run and long-run supply curves for them. Depending on how complex your model is, you might want to model storage too, particularly if modelling in less than 5-year time steps.


General government debt vs central government debt?

Both concepts refer to different levels of government. For the U.S., for example, central government debt refers to the debt by the federal government, while general government debt refers to overall debt, including states and municipalities. Depending on how fiscal federalism is designed, different levels might be affected differently by the business cycle.

The OECD offers the same indicators, with some information on how they are defined. I copy from there:

  Data are within the framework of the System of National Accounts (SNA), a set of internationally agreed concepts, definitions, classifications and rules for national accounting. Using SNA terminology, general government consists of central, state and local governments, and social security funds. State government is only applicable to the nine OECD countries that are federal states: Australia, Austria, Belgium, Canada, Germany, Mexico, Spain (considered a quasi-federal country), Switzerland and the United States. Debt is a commonly used concept, defined as a specific subset of liabilities identified according to the types of financial instruments included or excluded. Generally, it is defined as all liabilities that require payment or payments of interest or principal by the debtor to the creditor at a date or dates in the future. All debt instruments are liabilities, but some liabilities such as shares, equity and financial derivatives are not debt.
  Debt is thus obtained as the sum of the following liability categories, whenever available/applicable in the financial balance sheet of the general government sector: currency and deposits; debt securities; loans; and other liabilities (i.e. insurance, pension and standardised guarantee schemes, other accounts payable, as well as, in some cases special drawing rights - SDRs). According to the SNA, most debt instruments are valued at market prices, when appropriate (although some countries might not apply this valuation, in particular for debt securities).
  The treatment of government liabilities in respect of their employee pension plans is diverse across countries, making international comparability difficult. In particular, according to the former 1993 SNA, only the funded component of the government employee pension plans was reflected in its liabilities. However, the 2008 SNA recognises the importance of the liabilities of employers’ pension schemes, regardless of whether they are funded or unfunded. For pensions provided by government to their employees, countries have some flexibility in the recording of the unfunded liabilities in the core tables; this has also been followed by the ESA 2010, its European equivalent (although a new supplementary table will be added showing liabilities and associated flows of all pension schemes, whether funded or unfunded). Some OECD countries, such as Australia, Canada, Iceland, Sweden and the United States (including others whose data source is the IMF Economic Outlook) record employment-related pension liabilities, funded or unfunded, in government debt data. For those countries (except non-OECD ones), an adjusted government debt ratio is calculated by excluding from the debt these unfunded pension liabilities. Government debt used here is recorded on a gross basis, not adjusted by the value of government-held assets.
  The SNA definition of debt differs from the definition applied under the Maastricht Treaty, which is used to assess EU fiscal positions.
  Gross domestic product (GDP) is the standard measure of the value of goods and services produced by a country during a period.

The reason for countries reporting only one indicator could be that the respective statistical office simply only report one number. For small countries, general government debt might be by and large equated with central government debt if there is no relevant fiscal capacity below the central government.

The explanation for Cyprus might be that the central government had to bail out banks recently, leaving the non-negligible budgets of municipalities unaffected, but this is a bit speculative.


A very famous study in this direction is Card and Krueger (1994).
They look an increase in the minimum wage in New Jersey in 1992. While New Jersey raised the minimum wage from USD 4.25/h to USD 5.05/h, the minimum wages remained at $4.25 in adjacent Pennsylvania.

You should have a look at the subsequent research.


Why does the EU need a clearing centre to circumvent US sanctions against Iran?

Because the US has threatened to sanction European companies that trade with Iran.  Most European companies that are interested in trading in Iran also trade with the US, so would be vulnerable to various kinds of sanctions.


Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good?

That's a standard case of a Consumer Price subsidy. Its effect will be that of market expansion: higher quantity at higher equilibrium price, irrespective of whether it will be lump-sum (a fixed monetary amount) or a fixed proportion of the price.

A) LUMP SUM SUBSIDY
Consider a monopolist. The monopoly faces a downward-sloping demand curve, say

$$q_d = a -b(p-S) = (a+bS) - bp$$

where $S$ is the lump-sum subsidy. Here the demand schedule shifts upwards by $bS$ maintaining the same slope. Profits are (assume linear costs for simplicity)

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [(a+bS) - bp]$$

and the first order condition  for a maximum gives

$$(a+bS) - bp -b(p-c) = 0 \implies p^* = \frac {a+ bS +bc}{2b} = p^*|_{S=0} + \frac {1}{2}S$$

Profit-maximizing price is increasing in the lump-sum subsidy. Now substitute this into the demand equation

$$q_d^* = (a+bS) - b\frac {a+ bS +bc}{2b} = \frac {a-bc}{2} + \frac{b}{2}S = q_d^*|_{s=0} + \frac {b}{2}S$$

Quantity demanded also increased. So the lump-sum subsidy leads to market expansion.

B) PROPORTIONAL SUBSIDY
Here we have
$$q_d = a -b(1-s)p,\;\;\; 0\leq s<1$$
The demand schedule changes slope (becomes more "flat"), maintaining the same intercept.

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [a -b(1-s)p]$$

and the first order condition  for a maximum gives

$$a -b(1-s)p -b(1-s)(p-c) = 0 \implies p^* = \frac {a+ b(1-s)c}{2b(1-s)} = \frac {a}{2b(1-s)} + (c/2)$$

Equilibrium price is (non-linearly) increasing in the proportional subsidy. Substitute this into the demand schedule to get

$$q_d^* = a -b(1-s)\frac {a+ b(1-s)c}{2b(1-s)} = a -\frac {a+ b(1-s)c}{2} = \frac {a-bc}{2} + \frac {bc}{2}s$$

Here too equilibrium quantity is increasing in the subsidy.

For a sector like Health services, where increased market access is of concern (in some countries) due to societal values (and/or cold calculations as regards the material benefits to society in having more healthy members), and where "quantity demanded" is not based so much on desires but rather on not-negotiable needs (illnesses, accidents, etc), the rationale for such subsidies is that the quantity increased will mainly come from new "consumers", previously excluded from the market due to the pre-subsidy equilibrium price in combination with their income level.


How can a crypto currency crash affect liquidity?

As an initial note, I have not seen any plausible claims that a crypto crash would cause liquidity problems for the financial system. As a reult, I would need to guess to the mechanism.

I do not think “liquidity” is what anyone should worry about right now. For a firm we need to distinguish between illiquid and insolvent.

Illiquid: the firm does not have enough short-term assets (“cash”) to meet short-term liabilities.
Insovent: the net worth (equity) of the firm drops below zero.

A bank loan is an illiquid asset. As a result, loan losses do not directly cause illiquidity; they cause insolvency. Of course, people will not normally lend short-term to an insolvent bank, so it will be illiquid later. (If the concern is about money creation, impaired deposit growth would only be a consequence of insolvency.)

Can a crypto crash cause a general wave of insolvencies? That is difficult to answer, but we can look at the possibilities.

The credit risk at the time of writing appears to from individual investors borrowing to buy cryptocurrencies. The only business borrowing that I am aware of is by the crypto exchanges - they borrow to support customer borrowering. Such loans are covered by two parties - the customer, and the exchange. We cannot double-count these debts as a result.

As for customers, banks either lend against collateral, or cash flow.

No sensible bank would lend against most crypto-currency collateral: most of the cryptocurrencies are designed so that authorities cannot seize them. Unless there is an exchange-traded fund (which can be used for margin debt), such crypto-based lending will be negligible.

Banks could lend against other collateral, such as a second mortgage. However, such loans are backed by both the home as well as household cash flow. People take out second mortgages to go on vacation or to fund small businesses (which mainly fail); there is always a safety margin for the banks in such lending.

Unsecured lending is lending against income. The recent wave of banks blocking cryptocurrency purchases via credit cards is a sign that some are getting uncomfortable with such risks. However, losses would only materialise if the involved investors are cash-flow impaired (lose their job in most cases). Although such cases will happen, any losses need to be looked at in the context of other sources of loss, such as those caused by medical emergencies in the United States.

In order for these losses to pose a systemic risk, it would probably require large job losses, which by itself poses a risk.


Interpreting the ratio of two variances in a Bayesian decision problem

Excuse the word-play, but the interpretation of $n'$ is a... posterior one. Meaning, the important thing is not how $n'$ is defined (ratio of variances, although this will prove consistent with the interpretation), but how it functions in the posterior mean and variance.

What does it do? For the posterior variance, it is easiest: firstly, it appears as an addition to the expression's denominator, an addition comparable to the actual sample size. This permits the authors to talk about "equivalent number of sample observations". Secondly, its effect is to lower the variance, which is a metric of dispersion but also of uncertainty. This effect now permits the authors to talk about "the amount of information implicit in the prior distribution": more information in the prior - lower uncertainty in the prior - lower uncertainty in the posterior - and indeed, $\sigma^2_0 \downarrow \implies n' \uparrow \implies \text{Posterior Variance} \downarrow$.

For the posterior mean we have (setting $N \equiv  n' + n$)

$$\text {Posterior Mean} = \frac{n'\mu_0 +\sum_{i=1}^nx_i}{n'+n} =
\frac{n'}{N}\mu_0 + \frac{n}{N}\bar x$$

i.e. a convex combination of the prior mean and the sample mean - so the magnitude $n'$ weighs the prior information against the sample information, as though you had two separate samples from the same population of sizes $n'$ and $n$, and considered their pooled average. Again, this can be interpreted as "information implicit in the prior distribution expressed in terms of virtual/fictitious sample size".


Expansionary fiscal policy in a low mobility economy

Where do you see that this expansionary policy is itself causing interest rates to rise?

If capital flow restrictions are damping inflationary pressures then the increased stock of money minus the decrease in buying power might cause an increase in imports. This increase in imports increases demand for foreign currency and decreases demand for domestic currency. This method of exchange subverts the capital flow restriction and the depreciation follows.


Derive utility function with both substitutes and complements

The general way to do this would be using a nested CES Function.  CES Wikipedia

For your example you could define the utility of sandwiches (S) and condiments (C) to be

$$ U(D,H,M,Y) = (a_1 S^{\frac{s-1}{s}} + a_2 C^{\frac{s-1}{s}})^{\frac{s}{s-1}} $$
then you can define $S$ and $C$ "nests" as
$$ S = (b_1 D^{\frac{\rho-1}{\rho}} + b_2 H^{\frac{\rho-1}{\rho}})^{\frac{\rho}{\rho-1}} $$
$$ C = (c_1 M^{\frac{\eta-1}{\eta}} + c_2 Y^{\frac{\eta-1}{\eta}})^{\frac{\eta}{\eta-1}} $$

$s$ determines whether S and C are complements ($s \rightarrow 0$) or substitutes ($s \rightarrow \infty$), or neither ($s=1$).

The same holds for $\rho$ and $\eta$ in the "nests".  $a$, $b$, and $c$ establish the relative importance of each item within a nest.

Extending this to $N$ you can have an arbitrary number of nests and an arbitrary number of elements within each nest.  This way may look complicated, but it has a lot of flexibility and gives you nice derivatives when solving maximization problems.


One quick way to do this is through inverting a distribution c.d.f.. For example, the distribution Beta(alpha,beta) has density concentrated at small and large values when alpha and beta are less than one. Thus, you can first generate equally spaced grid, and then use the inverse function of Beta c.d.f to map the grid to unequally spaced grid. You can change the percentiles you want (60-70%, e.g.) by setting the appropriate alpha and beta.


Is the Convex Combination of Two Pareto Optimal Allocations Also Pareto Optimal?

To complement densep's answer, here is a schematic Edgeworth box illustration of what can go wrong. The points on the dashed line are convex combinations of the Pareto optimal points $(x_1,x_2)$ and $(\hat{x}_1,\hat{x}_2)$, but the marked point is not Pareto optimal.


How to prove dynamic inefficiency in the Solow model?

I give the intuitive explanation in words first, then below is the proof.

Define a Pareto improvement to be where consumption is no lower in every period and higher in at least one period. We're said to be dynamically efficient if there's no possible Pareto improvement and dynamically efficient if there's a possible Pareto improvement.

Assume we start off already in steady state.

Proposition 1. If the savings rate is below the golden-rule level (or equivalently, the steady-state capital stock is below the golden-rule level), then we're dynamically efficient.

Proof. If we switch to a lower savings rate, the new steady-state level of consumption will be lower, and so this cannot be a Pareto improvement.

Conversely, if we switch to a higher savings rate, then consumption in the initial period of the switch will be lower, so that again this cannot be a Pareto improvement.∎

Proposition 2. If the savings rate is above the golden-rule level (or equivalently, the steady-state capital stock is above the golden-rule level), then we're dynamically inefficient.

Proof. We claim that switching to the golden-rule savings rate is a Pareto improvement.

First, note that by switching, the new steady-state level of consumption will be higher.

Next, note that in the initial period (after we've switched), consumption is higher than before, because we've lowered our savings rate.

In fact, in the initial period, consumption will be higher than the new steady state. Consumption will then fall monotonically towards the new steady state level (which is higher than old steady state).

Altogether then, by switching to this lower savings rate, consumption in each period is higher than the old steady-state level (or in other words, than it would've been had we not made the switch). This is thus a Pareto improvement.∎


The short answer is No.

Every single year, except 2009, for the past 55 years of continuously recorded economic history, the world has been getting richer. The -2.1% global recession in 2009 was made up in 2010 with 4.1% growth. I was just working with the World Bank's World Development Indicators, which track global GDP growth, and I double checked.

We measure prosperity and growth by the quantities of goods produced. Money is irrelevant. Every year we, humans, produce more and more goods and services and that makes us richer. Not every country grows steadily though. Some countries are stagnant, like Greece, and some grow remarkably rapidly over very long periods of time, like China. Moreover, there is increasing inequality withing many countries (measured by GINI coefficient). So it is very natural for many to feel like everything around them either stands still or degrades. But on average, the world keeps getting richer.

If you are in Switzerland, you may have felt the shock from recent currency appreciation. Sharp currency appreciations hammer exporters and industries reliant on tourism, which make up a large portion of Swiss economy.


[...] are there any ways in which Macroeconomics help us to understand the behavior of individual units?

Putting aside the totological cases in which a utility would directly integrate macroeconomic signals such as, e.g. key-rates, AFAIK, no there are not.

As you mention, Macroeconomics is about aggregate level considerations and/or identities That being said, you will find theoretical frameworks, such as  General Equilibrium Theory that imply/represent microfounded behaviours with the intention of adding them up instead of directly dealing with accounting-like macro aggregates.


Macroeconomic model of the open economy - NCO and loanable funds

This 11 page reference describes capital flight on page 10:
https://www.le.ac.uk/economics/teach/ec1001/Lecture%20Notes%20given%20to%20Students/Lecture%20Notes%2013%20Chapter%2032.pdf
The conventional logic for capital flight and demand for loanable funds:

Since net capital outflow is also part of the demand for loanable funds, (during capital flight) the demand for loanable funds rises.

If I understand correctly, according to conventional economic logic, the act of saving means units in the economy spend less income on consumption and this saving increases the supply of loanable funds. This is the primary assumption which is challenged by the so-called heterodox economic logic. The heterodox logic argues that government deficit spending policy and market finance deals provide new funds from "thin air" for investment and consumption activity without anyone having to save funds from prior income sources. In other words spending decisions with finance and government budget policies drive the mix of investment and consumption. If spending for investment and consumption go down then so does future income and saving as the outcome or result of those prior financial planning decisions.
Supposedly the equilibrium demand for loanable funds expressed as the sum of spending for investment and the net capital outflow.
The way I think about this is top down as follows:

National Wealth (W) = Non-financial Assets (K) - Foreign Financial Claims (FFC)

Domestic investment means the increase of Non-financial assets K regardless of the finance source. Capital flight means foreigners are trying to reduce their financial claims held against the domestic non-financial assets. The foreign financial claims are financial assets which are issued as liabilities of governments, banks, or other units in the economy during the processes of international trade and investment. In the modern world international trade is only partly coupled to international investment flow so capital flight can disrupt trade and other features of the domestic economy.
The domestic loanable funds market would be used to finance investment projects which increase K over time as investments become new wealth. But the net domestic financial wealth is zero because for every financial asset there is a matching liability in another economic unit. Therefore the valuation of non-financial assets K is the wealth of the nation less the foreign financial claims against this wealth.
I don't think all the conventional economic logic necessarily holds if one examines how domestic and international trade occurs in the accounting context with government deficit policies. I do not agree that government deficits reduce supply of loanable funds for example. The United States government does not have a net worth comparable to any other unit in the history of the world as far as I am concerned because its role is not to save for retirement nor is it to profit from commercial activities. The United States holds vast tracks of land with mineral wealth which are not given a value under national accounting customs. So the SNA Integrated Macroeconomic Accounts (SNA-IMA) show the United States government with large and growing negative net worth since the 1970s and yet there has been no significant capital flight from US dollars. The factor which stabilizes global trade is US government budget deficit increases to prevent a flight away from US dollars when the dollar based financial markets threaten a systemic crisis.


Finding an intercept on percentage based data and using dummy variables

As Jamzy noted, run an OLS regression on grades against whatever variables you have.

$$\text{grades} = \beta_0 + \beta_1 x_1 + \cdots + \beta_i \ \text{race} + \cdots$$

$$\text{race} =\begin{array}{cc}
  \Bigg\{ &
    \begin{array}{cc}
      0 & mixed \\
      1 & Chinese \\
    \end{array}
\end{array}$$

or vice versa with the 0 and 1. $\beta_0$ will be your intercept that you are looking for. If it ends up negative, try taking the log of both sides and see if a linear regression still has a good fit for you there.


Issues with an application of definite internal and marginal utilities

The mistake you are making appears to be treating $\theta_i$ as a variable while in reality it is a specific value of the derivative.

So
$$\int_0^{q_i} V'(q)dq = \int_0^{q_i} \theta_i dq \;\;\; i=l,h$$

is wrong, since $V'(q)$ equals $\theta_l$ evaluated at $q_l$ only (and equals $\theta_h$ evaluated at $q_h$ only).


When the government prints more money, then how does this extra money end up in the hands of the people, they still are being paid the same amount of salary annually by their employers.

This is really two questions:

How does the money get into the economy? The US government increases the money supply by having the Fed buy things with dollars that did not previously exist.  Typically they buy financial assets, pushing their price up and putting new money into the hands of the previous owners of those securities (often the US treasury, which means it is injected into the economy in the form of government spending).

How do individuals get this money if their incomes are fixed?  For a given individual, their income may be fixed in the short run.  However, across the economy there are many people whose incomes fluctuate with sales or who are up for raises that depend on sales or profit.  Or who get a higher wage when they change jobs.  As the money supply increases, consumers have more money in hand and spend more money, raising wages for those whose wages can go up using these mechanisms. Your intuition is right about people with truly fixed wages, though: in the short run, they do not participate in the money supply increase.  This is why people on a fixed income (like elderly people living on a pension) hate inflation.

Overall inflation shifts wealth from lenders to debtors and from those with fixed incomes to those with adjustable incomes.

  why does the price of the goods increase?

The price of a good is the equilibrium between the people buying and the people selling.  A change in the money supply does not change how many goods there are to sell but it does change how many dollars the people buying them have.  If people have more money, they will be willing to pay more, so the price goes up.


Why do companies sit on cash while they have debt?

Debt is cheap. Flexibility is valuable.

They hold debt + cash up to the point where the value of flexibility is still greater than the net cost of servicing the debt minus any interest earnt on the cash.

It saves them the transaction costs of re-raising debt when they need it, had they paid it down early.

It's cashflow that typically kills businesses, rather than profitability. So holding extra cash, makes long-term business sense: it's the ultimate safety net.


Romer (1986): Increasing Returns to Long Run Growth
Lucas (1988): On the Mechanics of Economic Development
Romer (1990): Endogenous Technological Change
Jones (1995):  Time Series Tests of Endogenous Growth Models

These are all classic papers in this vein of endogenous growth and questions of cross-country convergence/divergence.


Econometric model for industrial production

Depending on the goal of your analysis, you may want to take a different approach; but, often, more regressors do not necessarily improve a model. Rather than throwing in as many potentially relate variables into your regression, you may want to start with a more theoretical approach to try an determine the underlying supply side and demand side factors that drive industrial output.

It seems to me that the regressors you have included can be broken down in this way:

Supply Side: producer price index and interest rates
Demand Side: population, inflation, interest rates, unemployment, and seasonal adjustment
Other: auto and truck sales and PMI

While I am sure that the "Other" regressors are correlated with your dependent variable, do you really think they are determinants?

That said, if you were to include a lagged term for auto sales you could make the argument that the more cars/trucks bought in the last period would influence the demand for gas and electric in the current period. Additionally, you may have an argument for the PMI, but my point is that you should try and think of the underlying variables driving the supply and demand for industrial production.

Given the above comments, I would say you could add these potential variables:

Supply Side: Government investment in infrastructure, gross fixed capital formation (for the relevant industries), oil prices
Demand Side: Per capita income, manufacturing exports, oil prices

I am sure there have also been many studies like this in the past, and the previous literature is always a good place too look for additional ideas in the modeling process.


The period of the business cycle in which the real GDP is increasing is called the?

The trough is not a period, it is simply the lowest point. The opposite of this is a peak, i.e. the highest point. The answer you are looking for is "Expansion" or "Boom". The period in which GDP is falling is called "Recession" or "Contraction".

Also note that the business cycle is typically not defined using real GDP, but rather nominal GDP.


What keeps minimum wage rates in balance?

How is the minimum wage determined?

Yes and no. It is a political choice, but with economic reasoning. In the end, that's the case for every economic rule, maybe with the exception of monetary rules, as we try to preserve independence for central bankers (but it isn't completely true for the US FEDs).

When setting the new minimum wage, you compare minimum wages with other states, and with your own (earlier) state, to get a feeling what's possible and what is not. In the end, the minimum wage is a trade-off between raising earnings for some parts of the population, and decreasing employment chance for others.

How does the minimum wage affect employment?

The degree of trade-off (if you will, the transition rate between the two) is unclear. In the data, we tend to see no employment effect of minimum wages, but an increase in earnings (Dube et al (2010): (Ungated version). In fact, we see a large number of estimates around 0:

Many economists suspect (based on a strong prior that there should be an employment effect) that this is due to poor data (Matt Rognlie's comment), or long-term effects of minimum-wages (adjustment costs in technology) that we are not capable of measuring/estimating precisely (Sorkin (2015) (IDEAS)).

Why doesn't the minimum wage affect employment?

The counter-effect to some part is the political process. People are not stupid, they will accept raises in the minimum wage to some degree, but not if you go bananas. Also, there are natural counter-forces in the economy:

minimum wages are nominal
minimum wages are lower bounds

To that end, if you increase the minimum wage, eventually inflation will wash off the real impact. Also, raises in the minimum wage tend to become substitutes for (non-forced) increases in the wage level.

Have a look at the following graph, which depicts the relative impact of minimum wages. The author plots the maximum of minimum wages at a time period, deflated by average hourly wage in the private sector among production and non-supervisory employees (no managers). It really looks as if any spike (raise in the minimum wages) gets deflated quite quickly, doesn't it?


1st of all, regading your remark

  ... , it is solely interest rate differential brought about by monetary
  policy that cause exchange rates to fluctuate...

That's just because it these models the internet case is isolated (like the joke of economists needed to change a light bulb)

As to your question, its always a combination between interests/ exchange rates and trade:

Per your example:
When China lowers rates, effectively increasing money supply, the Yuan decreases and make exports more profitable
Once exports occur, dollars come back to China, and converted to Yuans, making the Yuan increase back.
To keep exports more profitable to China, a constant increase of money, lower interests rates, are needed. This, in turn, can create local price surges. Essentially the local economy subsidizes exports

What China does though, is not exactly that,  even if similar
China holds its foreign exchange rates low, and its interest higher. That sounds a bit contradicting, but its possible, if China's government absorbs dollars.
So on one hand, Chinese exports are cheap, and on the other hand, the Yuan is a strong local currency. Furthermore, due to yhe e, export accelerated growth, wages and wealth has increased...


What interesting insights can be gleaned from specific data on imports and exports.

Studies in international trade are already using trade data at a particularly disaggregated level, at the transaction level.

For instance, this very new paper on Global Firms points out that research in international trade has changed dramatically over the last twenty years, as attention has shifted from countries and industries towards the firms actually engaged in international trade. They outline a framework that allows firms to be of positive measure and to decide simultaneously on the set of production locations, export markets, input sources, products to export, and inputs to import. They use this framework to interpret features of U.S. firm and trade transactions data and highlight interdependencies across these margins of firm international participation.

However, their data is from the U.S. Customs Bureau data and quite confidential. Can you give the source of "ton of open data available on on how much of a particular good is imported/exported"?


The person taking out the 1 million dollar mortgage will take the funds and use them towards buying a 1 million dollar home. If the home is sold by Joe who keeps his money under the pillow then in this scenario money supply does not increase at all. If, on the other hand, the party selling the home will deposit the 1 million dollars in his/her bank account with bank A, assuming a 10% reserve ratio, bank A will keep 100,000 dollars as reserves and lend out the remaining 900,000. Bank A lends these 900,000 to Bob, who will take them and deposit them with Bank B. As long as Bank B will again keep 10%, 90,000 in this case, and lend out the remainder - 810,000 to some other customer and if this chain continues in this fashion with no one keeping money under the pillow, then the overall money supply will increase by 1,000,000 + 900,000 + 810,000 + ... = 1,000,000/0.1 = 10,000,000. The banks will end up increasing their reserves by 100,000 + 90,000 + 81,000 + ... = 100,000/0.1= 1,000,000.


Nash Equilibrium of modified Keynes' beauty contest

Assume the players have to choose integers, otherwise a best response may not exist. Let the payoff of winning be $\alpha\cdot[\frac23\text{ of the average}]$, $\alpha>0$. Consider the two player (A and B) case, and let's verify whether choosing above $0$ is optimal.

Suppose A chooses $x>0$. Then B can guarantee a win by choosing $x-1$, since $\frac23(x-\frac12)$ is closer to $x-1$ than to $x$. However, if B chooses $0$, A choosing $x=1$ is a best response (A gets zero anyway). So we will have a NE where not both players choose $0$, and the one who chooses $0$ gets positive payoff.

This equilibrium can be generalized to the $n$-player case. Let $n-1$ players choose $0$ and the remaining one choose $1$. In this equilibrium, those who choose $0$ would share the positive payoff (each gets $\frac{\alpha}{n-1}\cdot\frac23\cdot\frac1{n}$). They are best responding because choosing any $x>0$ would imply zero payoff. The one who chooses $1$ is also best responding since he gets zero anyway. (Of course this is not the only equilibrium. There is also one where everyone chooses $0$, which is obvious.)

Admittedly, the above equilibrium relies crucially on the assumption that players are only allowed to choose integers. But this is due to the fact that payoff is a function of the choices (compared to a fixed amount in the original version of the problem). Suppose players are allowed to choose any real number in $[0,100]$. Then if A chooses $x>0$, B would best respond by choosing $x-\epsilon$, where $\epsilon=\min\{y:y>0\}$ (B would win of course, but she also wants to maximize her earning by making her choice as close to A's as possible).
However, such an $\epsilon$ does not exist. Therefore, the only equilibrium is for everyone to choose $0$.

The intuition is this: however large the winning is, you don't get it if you don't win. But winning requires that you choose small. So the incentive to win overwhelms the incentive to win big.


Use of mathematics and imprecise definition of terms

Edesess is attacking what is really just a straw man of economics. I'm not sure he really understands the field. To start, economics is not math. We're not claiming that it is. It's more of an "applied" science. Economists have never claimed that these definitions are precise in the way that mathematics is. These definitions are modeling constructs---they're for applied work. They're use is temporary in a way. The point is to try to convey an idea in a more precise way than just in words---but everyone knows that they're not a precise as we would like and not as precise as they ultimately should be. They're meant to be debated and later refined. But, as all applied scientists know, you've got to start somewhere and sometimes ideas are best conveyed through simpler---if less detailed mean.

Coming up with better definitions is a huge part of the economic science. Consider these examples. When the Cowles Foundation was founded in 1932 its motto was "Theory and Measurement" (the motto was first adopted in 1952). Measurement is not an easy thing to do. As another example, a lot of Larry Kotlikoff's work has dealt with how a lot of fiscal measures are not economically well-defined concepts.

  Einstein taught us that neither time, nor distance are well-defined physical concepts. Instead, their measurement is relative to our frame of reference – how fast we were traveling in the universe and in what direction. Our physical frame of reference can be viewed as our language or labeling convention. ... Kotlikoff, along with Harvard's Jerry Green, offered a general proof of the proposition that deficits and a number of other conventional fiscal measures are, economically speaking, content-free, concluding that the deficit is simply an arbitrary figment of language in all economic models involving rational agents.

Also, take another example of current interest. Lars Hansen's recent work (winner of the 2013 Economics "Nobel" prize) has focused on the difficulty and ongoing failure to define certain economic concepts, including "bubbles" and systemic risk. See his essay "Challenges in Identifying and Measuring Systemic Risk". I'm a fan of the dictum he relays, attributed to Lord Kelvin,

  I often say that when you can measure something that you are speaking about,
  express it in numbers, you know something about it; but when you cannot
  measure it, when you cannot express it in numbers, your knowledge is of the
  meagre and unsatisfactory kind: it may be the beginning of knowledge, but you
  have scarcely, in your thoughts advanced to the stage of science, whatever the
  matter might be.

He notes that "an abbreviated version appears on the Social
Science Research building at the University of Chicago." So, yeah, economists (as social scientists) definitely take this seriously.

So, the point is that economists are well aware of the problems in these "definitions." They are a part of ongoing research in the field; sometimes they're ignored if people don't think they're first-order to the problem; etc...


If strict preferences map onto $\Bbb{N}$ what do weak preferences map onto?

First of all, I don't know if there is a convention as I never use weak preferences myself. But I can give you some suggestion that should work:

Do you have a finite number of goods, lets say $n$? Then $\mathbb{N}$ is clearly enough:
Just map the best good to 1, the second best to $n$, third best to $n^2$,... .
If you are indifferent between 2 goods (lets say they would be ranked third overall) assign them to $n^2$ and $n^2+1$. As @MettaWorldPeace already pointed out: it is kind of "wasteful" to use $\mathbb{N}$, so can we use less? And clearly we can never need more than $(n-1)^n+1$ numbers, so a mapping on $\{1,2, ..., (n-1)^n+1 \}$ would be sufficient (you can use even less numbers, but it is annoying to write down, so I just leave it at that).

What if we have a (countably) infinite number of goods? Here we can use the same trick, using $\mathbb{R}$ (or, if you like to, use $\mathbb{Q}$) just assign the best good $1$, the second best $2$, ... . If there are indifferences (i.e., you have three second best goods, the first one is assigned $2$, the second one $2.1$, the third $2.11$(or whatever you like)). Again, as before it is kind of wasteful, what else can you use? You could use a mapping on $\mathbb{N}\times \mathbb{N}$.

The same idea clearly works for every other number of goods, but I guess you can figure that out yourself.

Remark: This is not more than a different way to write preferences. You can not use this, e.g. as a utility function. However, all of these examples have the property that you can compare the ranking by simple mathematical operations (i.e., comparing $\lfloor \frac{x}{n}\rfloor$ for the first example).


Finding restrictions on parameters for a demand function

Demand is positive, so $A>0$.
If $p_1$ goes to $\infty$, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\alpha < 0$.
If $p_2$ goes to 0, $x_1$ cannot go to $\infty$, since $p_1x_1$ is bounded by $M$. Thus $\beta\ge 0$.
If $M$ goes to 0, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\gamma > 0$.
If both prices and income are scaled up by the same factor, then demand doesn't change, so $\alpha+\beta+\gamma=0$.


Is there an accepted metric for "the value of the dollar"?

It sounds like you want the Trade-Weighted Dollar Index, which is available from 1973 on for major currencies. It's the value of the dollar against a basket of other currencies, weighted by their share of US trade.


Many or few variables when testing for Granger causality?

The implication is, the model should be adequately specified, without omitting relevant variables. Once the model is adequately specified, Granger causality will be easier to establish if the model has fewer variables than if it has more. (I am not taking a stance regarding how reasonable the quoted statements are but rather just rephrasing/interpreting them.)
This characterizes the systems you may be modelling (few or many variables) and the modelling choices (avoid omitting variables). In applications of time series econometrics (e.g. in macroeconomics or finance), usually you cannot change the underlying system (make it contain fewer variables). Then you do not have much choice over the number of variables in your model -- unless you are willing to violate the suggestion to avoid omitting relevant ones and risk working with an inadequate model and consequently obtaining invalid results.


If protectionist policies hinder economic growth, why do we need border taxes at all?

From pure trade theory perspective there are cases where tariffs can improve welfare of a country provided it is big economy (economy that can affect international prices). And there can be also other arguments made in favor of some protectionism such as infant industry argument (protecting new industry until it is competitive), national security (you don't want to sell arms to your enemy - it would be foolish if US would let its firm sell intercontinental ballistic missiles to Iran), and tariffs can be useful for countries with week central governments which might have easier time to collect taxes on borders (see Krugman et at International Trade: Theory and Policy for overview of these arguments and others).
However, the above being said most economists believe that there should be free trade. Krugman (Nobel Prize winner for his work area of international economics) put this best when he written in his 1987 paper "Is Free Trade Passe?":

If there were an Economist's Creed, it would surely contain the affirmations "I understand the Principle of Comparative Advantage" and "I advocate Free Trade."

The reason why this is so is that even if tariffs can be sometimes improve welfare of a country, these results are contingent on other countries not retaliating (so you don't have trade war), or would be very hard to implement politically (often infant industry argument is abused by firms that do not need it but have political clout) and so on.
Hence generally economists prefer free trade. This consensus still holds, the paper above is from 1987 but more recent poll amongst the top policy economists show overwhelming consensus for free trade. This is why organizations such as WTO are set up with explicit goal of reducing tariffs and other trade barriers across world.

This being said, most economists would still argue that there are exceptional situations. For example, as mentioned above you don't want to sell arms (or crucial component of military equipment) directly to your enemy (although this can be still misused - Trump administration used this argument when it implemented  Huawei chip ban and this was criticized by some economists). WTO allows for increases in tariffs for countries that break trade treaties or raise their own barriers unliterary (as sort of deterrent).  Recently, some good arguments have been made in favor of environmental tariffs which would be applied to countries that do not have their own environmental policy.
In a specific case of Russia and EU there can be arguments for tariffs on the above mentioned environmental grounds or as a response to aggressive international policy that Russia is pursuing (landgrabs in Ukraine, threatening sovereignty of some Baltic EU members). However, this being said in principle if there would be free trade area between EU and Russia (with exception of dirty goods or some common environmental policy) it would be to the net benefit of both parties in long-run (if we ignore the political issues).


Understanding Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion

From $U(W+11)−U(W)\le U(W)−U(W−10)$ we get that $\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}$, which is what the sentence before the bolded italic part says.

Now by concavity of $U(.)$ we know that $MU(W-10)\ge\frac{U(W)-U(W-10)}{10}$, and also that $MU(W+11)\le\frac{U(W+11)-U(W)}{11}$.

Therefore $MU(W+11)\le\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}\le\frac{10}{11}MU(W-10)$.

So $MU(W+11)\le\frac{10}{11}MU(W-10)$, wich is just the bolded italic statement.


Ireland wholesale electricity prices are higher during the weekends

From looking at the data I have access to, I'd say that things stop early on Friday afternoon in Ireland. So the Friday evening peak is lower than evening peaks on other weekdays: it's the only day of the week where the evening peak is about the same price as the morning peak.

As you haven't provided the metadata of the data you're using, nor links to find that out, I've looked at the SEMO data I know about.

The Single Electricity Market Organisation (SEMO) for Ireland gives prices by trading period (half-hours, starting at midnight), for every day of the year. So I took a 3-month period at random, 1 Feb - 1 May 2015, and looked at the average price for each half-hourly period, for each day of the week. Note that this is a simple arithmetic mean: there's no weighting by volume, no special handling of public holidays, no weather adjustments.

Friday, Saturday, Sunday have heavier lines, because they're the ones I'm emphasising. The four weekdays Monday-Thursday are very similar to each other. Saturday and Sunday each have later and lower morning peaks than Mon-Fri.

But Friday's evening peak is the lowest evening peak of the week. Much lower than any other day. Whereas in-day prices normally peak around 18.00-18.30, due to most industry still being powered, most commerce still being powered, and most homes being active, that's not the case on Friday: peak prices are a lot lower.

Note that there is no single price of any one MWh of electricity. Electricity gets traded many times over, starting maybe a year in advance, a month, a week, a day, an hour in advance, and then within the half-hour trading period itself there will be varying sell and buy prices. So there are huge dangers in trying to draw inferences from two different sets of price data, particularly when you don't have the metatadata that tells you what the data actually represents. The SMP price data I've used here is the System Marginal Price. There's a good description of pricing and scheduling from SEMO here (pdf), including on page 9 the calculation of the SMP.

Electricity prices are complicated. Unit commitment is complicated.


Books/Papers on Macroprudential Regulation

It's not really introductory in the sense that it focusses on what Macro-Prodential means (reducing risk at the cost of efficiency), but rather applied: Martin and Philippon (2014) analyze the great recession in Europe and simulate to what end more macruprudential policies would have ameliorated the situation.

  We provide a first comprehensive account of the dynamics of Eurozone
  countries from the creation of the Euro to the Great recession. We
  model each country as an open economy within a monetary union and
  analyze the dynamics of private leverage, fiscal policy and spreads.
  Our parsimonious model can replicate the time-series for nominal GDP,
  employment, and net exports of Eurozone countries between 2000 and
  2012. We then ask how periphery countries would have fared with: (i) more conservative fiscal policies; (ii) macro-prudential tools to
  control private leverage; (iii) a central bank acting earlier to limit
  sovereign spreads; and (iv) the possibility to recoup the
  competitiveness they lost in the boom.


In Austrian theory, does constant, predicted deflation distort credit markets?

As with inflation, so long as deflation is expected, there is no distortion. The expected deflation is simply built into interest rates.

For example, if I expect deflation to be 1% per year and I am looking for a 5% real return on an investment, then I'll look for investments with a 4% nominal return.


rotating and exchanging x for y's in regression

This is a somewhat "dated" subject in introductory econometrics, I suspect because, in econometrics the models come from theories and arguments that try to a priori establish causality and not just association.

Anyway, the issue is analyzed in Maddala's Econometrics textbook.

In the 2001 3d ed. the issue for simple regression is presented in ch. 3. inside section 3.4, under the title "Reverse Regression".

Writing

$$\hat y = \hat a + \hat b x,$$
and

$$\hat x = \hat a' + \hat b'y$$

one can quickly find that

$$\hat b \hat b' = r^2_{xy}$$

i.e. the product of the two is equal to the squared correlation between $x$ and $y$. If this correlation coefficient is close to unity then the two regression lines will be close.

One can also establish a relation between the two constant terms easily.


Are Preference Relations Robust to Aggregation

I presume you mean to take the total (or average) willingness to pay for each alternative to be the social willingness to pay. $WTP_{soc}(a)=WTP_{1}(a)+WTP_{2}(a)+WTP_{3}(a)$ Then $WTP_{soc}$ is a representation of a social preference relation $a\succsim b \Leftrightarrow WTP_{soc}(a)\geq WTP_{soc}(b)$. This preference is transitive and complete. Individual and aggregate preferences are consistent in the sense that they satisfy the Pareto criterion and the independence axiom of Arrow's impossibility theorem. The way we have tricked the impossibility theorem is that we have taken an ordinal preference relation and turned it into a cardinal concept (WTP).

The question is whether the mean willingness to pay is a meaningful concept in any way. For example, a poor person's willingness to pay for a good may be relatively low, but the marginal utility very high. For a discussion of interpersonal comparisons of utility, see the following article:

Harsanyi, John C. Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Springer Netherlands, 1976.


In microeconomics : is this the contradiction in the atomicity of firms ?

A price-taking firm takes prices as given, but that does not mean that the firm cannot influence prices; it just means that the firm ignores its own impact on prices.

Now the question is how sensible it is to assume that firms take prices as given. The usual view is that it is a reasonable assumption when the impact of a firm on prices is small enough that profit-maximizing behavior for given prices does not differ much from profit-maximization under actual prices. This is usually the case if there are many firms that are small relative to the market. There are various ways to make this precise. One can also work with models with a continuum of firms in which a single firm hs literally no impact on prices.

As a side remark, Steve Keen's book reveals a strange mixture of incompetence and dishonesty. Keen is ignorant of basic calculus. Keen, sometimes with coauthors, has come up with a nonsensical theory of the firm. Here is a note by Paulin Anglin pointing out several of the many problems. One of these problems is acknowledged here. Later, Keen (and Standish) wrote a survey of Keen's history of criticising the theory of the firm in which they did not acknowledge Anglin's criticism. I recommend reading books from people with higher scholarly standards than Keen.


Why does having a single market price maximize quantity traded?

The competitive equilibrium price does not necessarily maximize the amount of goods produced. (The answer to your actual question ends here.)

What the competitive equilibrium price does is reach an efficient level of production. Any situation where the production level differs from the equilibrium level is Pareto-inferior to the equilibrium. This means that starting from the competitive equilibrium, you can transfer money from some of the market actors to other market actors in such a way that everyone will be better off than in the situation with non-equilibrium production levels.

In the example you gave buyer 1 (reservation price 10) and seller 1 (reservation price 9) have a collective surplus of \$1, and buyer 2 (reservation price 3) and seller 2 (reservation price 2) have a collective surplus of \$1, meaning both pairs 'feel' they gained \$1 by trading. But in the competitive equilibrium, buyer 1 and seller 2 have a collective surplus of \$ 8. If they give buyer 2 and seller 1 one dollars each, everyone would still be more happy than in the other situation. So the second sale is in some way inefficient.

A single price is not necessary to achieve Pareto-efficiency, for example first degree price discrimination yields the same results.


Why would a government bail out private banks?

Because the banks are not loaning their own money but that of depositors. One may also argue that the depositors should know the risks and if their bank makes bad investments it is partly their fault so they cannot complain should they lose their deposits. But a functioning banking system is essential to the current economic system. The bankruptcy of a big bank would cause panic and banks runs, which because of fractional reserve banking would probably result in the insolvency of other banks, banks that perhaps made good investments but ones that cannot immediately be liquidated. After the closing of the banks considerably less savings would be deposited in financial institutions. This will hinder consumption because credit/debit card traffic would cease and getting investment funds would also be considerably more difficult.

So the governments have considered the damage the bankruptcy of a bank would cause to their economy and decided it is better to bail them out. ("Too big to fail.") Of course since banks have a very large revenue stream they can also afford lobbying for intervention as well, but this does not mean that the intervention is not beneficial to the public. Furthermore, if you lose your deposits you may very well (wrongly) blame the government, so a bail-out is also politically prudent.

If you are interested in what could be done I recommend reading about the Glass-Steagall act.

If you are not satisfied with my answer there are any number of conspiracy theories that offer unconvincing alternative explanations.


What are the arguments against a zero corporate tax?

It is possible to structure business entities in most if not all countries as pass through structures (e.g., limited liability corporations, sole proprietorships, or general partnerships).  Such entities are not taxed at the corporate level.  Tax liability instead flows through to the principals who own not shares but partnership interests.

Your question should therefore be restated: what advantage is there to reducing the corporate tax rate to allow shareholders to pay less tax than if profits flowed through to their personal income tax returns?

The answer to that question is data-dependent.  In some economic environments, it is useful to encourage corporate investment through corporate tax cuts.  We are not in such an environment today, however.  There is now a very wide and deep literature on how corporations have been saving more than they have been investing since the 2007-09 financial crisis.  Corporations have been hoarding cash or distributing capital to stockholders in the form of dividends and share buybacks more than they have been investing in future productivity and profitability growth.  Corporate tax cuts would therefore likely increase wealth inequality because they would flow through to already wealthy shareholders.

The only argument for reducing the corporate tax rate today has to do with international competitiveness.  The top tax rate for U.S. corporations is 35%.  In Ireland it is 18%.  If you ran a giant multinational, you might want to "invert" and re-domicile to Ireland or any other lower-taxing country.  The U.S. tax code has so many loopholes that through clever use of intercompany debt, net operating loss carryforwards, and operations overseas, it is possible bring the effective tax rate for a U.S. domiciled multinational closer to 10-15% over longer time horizons.  We should obviously attempt to set corporate tax policy to limit inversions, but many companies already pay less than they would if they were headquartered overseas.


When building a social accounting matrix, do you input domestic intermediate demand or total intermediate demand?

It depends on the structure of the particular SAM, although in the case of the diagram you've provided, intermediate inputs should include any imports (as should final demand and exports - in other words, you should be using the Combined Use table to populate this SAM, not the Domestic Use table).

You can work this out by looking at the commodity balance identity.  Rows of the SAM must sum to the same total as the corresponding column of the SAM.  For commodities, the SAM structure requires:

Domestic sales + Tariffs + Imports = Intermediate inputs + Final Demand + Exports

i.e. total supply of commodities = total use of commodities (note that this is also the row balancing condition for the Supply-Use tables).  Since imports are included on the left hand side, they need to be included on the right hand side as well.


Does Modern Monetary Theory (MMT) provide a useful insight into how to manage the economy?

That depends on your definition of MMT, because it is not generally agreed on what it even is. You will find some arguing it is just a macro/monetary theory (such as the Wikipedia page) but then I seen MMT proponents on this site arguing it is a whole new paradigm that encompasses all macroeconomics and microeconomics. In addition, the MMT is not a theory in a way that is commonly understood in science, that is as far as I know there is no rigorous agreed upon MMT model that is testable. This makes it difficult to address it, because it is hard to criticize something that has no agreed upon framework as one can always simply respond to any criticism that it is not about the right interpretation of MMT.
However,  this being said, the main tenets of MMT, as laid out by advocates such as Stephanie Kelton seem to include assertions that:

debt and deficit for countries issuing their own currency does not matter.
government can fund virtually arbitrary amount of real spending through monetary financing.

This is rejected by conventional economists. In fact, an IGM poll among more than 40 top mostly Ivy league US policy economists shows that literally none of them agrees with the MMT propositions

Modern Monetary Theory

You can also have a look at critiques of MMT by other top economists such as Krugman, Rogoff, Summers or Mankiw (2019).
Consequently, at least when it comes to the above mentioned tenets, conventional economists reject the modern monetary theory and do not consider it useful. This is not to say that some policy recommendations might not be valid. For example, as far as I know many MMTers advocate for more government spending and higher deficits - that is not necessary bad idea especially in recession, and it is recommended policy by conventional macro models (e.g. see Mankiw Macroeconomics 8ed or Blanchard et al Macroeconomics: a European Perspective) but most conventional economists would still worry about deficits getting too out of hand and would caution against too much monetary financing (outside perhaps liquidity traps). Or most conventional economists and conventional macro models would argue it is not possible to simultaneously control inflation through taxes while simultaneously stimulating economy with monetary financed spending, in most macro models to avoid inflation you would have to raise taxes by amount exactly offsetting the effect of government spending (see Romer Advanced Macroeconomics), and to the extend government uses distortionary taxes (e.g. non-lump-sum taxes) it would leave economy worse off.
However, as stated at the beginning it is not clear what MMT is. For example, as pointed in the comments your description of MMT is quite different from what many MMT proponents asserts. For example, you state:

Government of any country with sovereign control of its own internationally accepted currency can print money to create as much inflation as it wants at will (depending on who receives that money).

But as far as I know most MMTers reject the idea that inflation is caused by expansion of money supply and rather argue inflation is consequence of different factors (such as institutional factors, or lack of competition that allows firms to hike prices and so forth). For example, according to the recently published 'MMT textbook' by the  most public proponents of MMT, Mitchell, Wray, and Watts:

“Conflict
theory situates the problem of inflation as being intrinsic to the power relations between workers
and capital (class conflict), which are mediated by government within a capitalist system.”

Regarding your other question:

It appears to me that this must be true to some extent, but my concern is that the feedback mechanism where the extra money generates inflation might not be smooth and free flowing for various reasons and a vast excess of money might accumulate in an unstable way in certain areas of the economy. After a period of time some trigger event might cause a sudden uncontrollable cascade flood of that money that could destroy public confidence and the currency itself. Is this an unfounded fear?

This is difficult to address because it is not clear what you are trying to say here. For example, it is not clear how money could "accumulate in certain areas of the economy" (whatever that is even supposed to mean - people from selected sectors hoarding money under matrasses?)  or what is meant by 'flood of money' - once money circulate they are already in the economy and affect the economy.
However, this being said there are many prominent economists who argue that following policies advocated by MMTers such as Kelton or   Mitchell, Wray, and Watts  it would eventually after some time lead to excessive inflation (see again the articles by the critics in previous part of the answer). There are also empirical examples of countries which attempted to excessively monetize their debt and fund spending via monetary expansion where this led to such high inflation that in the end it led to currency substitution (e.g. see Kamin & Ericsson 2003) about the Argentinian case), but I think it is more likely most developed countries would simply reverse course and change policy before going that far.


Do companies have non compete agreements between each other?

This is known as dividing markets or market allocation, and it is against the law in the US, the EU, and I imagine in most countries with antitrust laws.


Production function involving profit maximisation

Max

Then you have: Max

Which gives you:

So L = 100, and you have a loss of 15000


It appears to suggest applying the way shareholders vote in a company to the political setting of democracy: each citizen will have as many votes as his/her monetized wealth. So the rich will have officially "more say" in what gets decided or who gets elected in public office.

Certainly, the general consensus is that economic power does play a role in affecting political outcomes anyway. This approach would make that an official, institutional rule.


Do the last one, the first one is just the same thing but you will not be using the in-built adf function. The second one does it better and you have a choice of including lagged differences to control for possible autocorrelation. If your data is monthly, give it a lag order of 12.


An index of the The Economist's Schools Brief series

The Economist has some articles online from 1999 onwards: this is a partial selection of some which seemed interesting, though not quite what you are looking for
1999

https://www.economist.com/unknown/1999/10/21/finance-trick-or-treat
https://www.economist.com/unknown/1999/11/04/moneyed-men-in-institutions
https://www.economist.com/unknown/1999/11/11/stocks-in-trade
https://www.economist.com/unknown/1999/11/18/monopoly-power-over-money
https://www.economist.com/unknown/1999/12/09/the-future-of-finance

2013

https://www.economist.com/schools-brief/2013/09/05/explaining-the-schools-briefs
https://www.economist.com/schools-brief/2013/09/07/crash-course
https://www.economist.com/schools-brief/2013/09/14/lending-weight
https://www.economist.com/schools-brief/2013/09/21/controlling-interest
https://www.economist.com/schools-brief/2013/09/28/sovereign-doubts
https://www.economist.com/schools-brief/2013/10/03/calling-to-accounts

2016

https://www.economist.com/leaders/2016/07/21/breakthroughs-and-brickbats
https://www.economist.com/news/2016/08/04/which-economists-have-had-the-greatest-impact
https://www.economist.com/economics-brief/2016/07/23/secrets-and-agents
https://www.economist.com/economics-brief/2016/07/30/minskys-moment
https://www.economist.com/economics-brief/2017/08/03/gary-beckers-concept-of-human-capital
https://www.economist.com/economics-brief/2016/08/06/an-inconvenient-iota-of-truth
https://www.economist.com/economics-brief/2016/08/11/where-does-the-buck-stop
https://www.economist.com/economics-brief/2016/08/20/prison-breakthrough
https://www.economist.com/economics-brief/2016/08/27/two-out-of-three-aint-bad
https://www.economist.com/sites/default/files/econbriefs.pdf

2018

https://www.economist.com/finance-and-economics/2018/04/12/economists-understand-little-about-the-causes-of-growth
https://www.economist.com/finance-and-economics/2018/04/19/economists-still-lack-a-proper-understanding-of-business-cycles
https://www.economist.com/finance-and-economics/2018/04/26/many-results-in-microeconomics-are-shaky
https://www.economist.com/finance-and-economics/2018/05/03/economists-focus-too-little-on-what-people-really-care-about
https://www.economist.com/finance-and-economics/2018/05/10/barriers-to-entry

This clearly misses the older articles.  Google turned up

http://marshallinside.usc.edu/joines/549/articles_pdf/ec901103.txt.pdf (Paradigm Lost, November 3, 1990
http://people.tamu.edu/~aglass/econ452/Economist2a.pdf (The Miracle of Trade, 27 January 1996)
http://walkerd.people.cofc.edu/Readings/Markets/schoolsbrief.pdf (Feb 17 1996 - State and Market)

Added: The Economist relaunched the series in 2020, which (including some of those listed above) can be found at

https://www.economist.com/schools-brief/


you can do a belief elicitation.
See more on this here:

Rauhut, H., 2013. Beliefs about lying and spreading of dishonesty: Undetected lies and their constructive and destructive social dynamics in dice experiments. PloS one, 8(11), p.e77878.

and here:

Gächter, S. and Renner, E., 2010. The effects of (incentivized) belief elicitation in public goods experiments. Experimental Economics, 13(3), pp.364-377.

If you don't want to invite them into the lab, you can use Amazon mTurk and oTree.


For time series analysis a classical text is the time series analysis from Hamilton - although nowadays the text is slightly outdated it has an excellent and detailed overview of models that are still in use or serve as basis for more advance models.

The time series and panel data econometrics from Pesaran is also excellent - it’s also more contemporary but also deals with models not used in monetary economics.

If you have no background in econometrics the introduction to modern econometrics from Wooldridge is really good. Or also Verbeeks guide to modern macroeconomics has good treatment of time series models and in terms of difficulty it is somewhere between Wooldridge and Hamilton or Pesaran.

For a more practical handbook I would recommend international macroeconomics and finance from Nelson, and if you care mainly about forecasting and policy evaluation the kind usually done in central banks or ministries of finance economic forecasting and policy from Carnot, Koen and Tissot is quite good practical text


Comparing baskets of goods in this exercise

So we are told that bundle A (3,7) is optimal at prices p=(4,4) with wealth w=40

Consider bundle B (5,3) at prices (4,1.5) with wealth 24.5

By Weak Axiom of Revealed Preference (WARP), we know that if both bundles are affordable under both sets of prices and wealth, and we chose A when B was available, then A is revealed preferred to B.

Both A and B are affordable under those price wealth pairs, so B cannot be optimal.

What you did graphically is equivalent to the math.


Are there fundamental reasons why (exponential) economic growth is highly desirable?

Because according to utility theory: more is better- or at least not worse (nonsatiation- or free disposal).

Your question seems to actually be: why is growth good?

Firstly, It's axiomatic. Utility theory very formally assumes the law of nonsatiation/free disposal.  Things that result in the death of the cosmos are bad not because we are dead, but because we cannot consume and improve our happiness afterwards. Evidince: We observe that consumption improves happiness.

Second, in cases of limited resources, we have situations where optional growth paths are negative-But this is only because it allows for increased overall consumption.


A panel survey is a kind of longitudinal study.

In effect the survey sample (known as the panel) is asked repeated questions over time.  This allows the measurement of change over time for the units of analysis.

For example you could measure unemployment by taking a random sample of the working age population and asking them whether they are working or looking for work.  Later you could take a different random sample, and get a new number at the later date. This would be a series of cross-sectional studies and could give you an estimate of the net change.

With a panel survey, you could ask the same people at different times, giving you an estimate of gross employment flows and other more interesting details of change over time.  The difficulty is that some members of the panel might leave, and if this departure was related to the question of interest then it could bias the results. In addition, even if the panel was typical at the start, it might cease to be in future simply because all its members had become older.


How does hyperinflation affect investment?

Hyperinflation has great negative impact on the entire economy, as it makes it very difficult for households and businesses to plan long term projects and protect their wealth and earning power, reducing the real growth of the economy.

Inflation increases the nominal yield of investment securities and erodes the value of a currency. You can get more information on inflation effects on investment from this PIMCO primer.


How do I derive the aggregate demand function given two utilities functions?

If you have $J$ consumers therefore $J$ demands for a good $X$. Denoting the individual demand of each consumer with $x_j^*$ as you have it, if $X$ is the aggregate demand, it is just the sum of every individual demand:
$X=\sum_{j=1}^{J}x_j^*$
Then for your case it's: $x_1^*+x_2^*=\frac{(I_1+I_2)}{2P_X}$, and the same with $Y$.


Determining the Gini Coefficient in a 2 person economy

Gini Coefficient is the ratio of the area that lies between the 45 degree line (line of equality) and the Lorenz curve, and the area below the line of equality. Given the data in the problem Lorenz curve connects points $(0, 0)$, $(50, 20)$ and $(100, 100)$ in the graph. To find the Gini coefficient, we find the area between the line of equality and the Lorenz curve and divide it by the area of the triangular region that lies below the line of equality.

Gini coefficient $\displaystyle = 1 - \frac{\color{blue}{0.5\times 50 \times 20} + \color{red}{(100 - 50) \times 20} + \color{gray}{0.5\times (100-50) \times (100-20)}}{0.5\times 100 \times 100} = 0.3$

Here is the graph:


Does the Fed/SEC have tools to address shock-propagation from correlation of fund flows?

I’d start by briefly noting that “the Fed” didn’t say anything; that’s a research paper (by some smart people), but it’s disclaimed and doesn’t necessarily reflect the views of anyone other than the authors.
Having said that, I would then rephrase your question as follows: “what happens if everyone decides to cash in their retirement funds at once?” The answer to that would be: they would all get a very poor return, but thankfully that’s unlikely.
Very little of the question as posed has anything to do with active vs. passive strategies. “Passive” players aren’t different types of players, they’re the same types engaging in different strategies because they have different beliefs. If I invest my 401(k) in a Vanguard index fund vs. an actively managed fund with the same targets, I’m still the same person with the same risk tolerance, and I’m still subject to the same shocks from the real economy.
It’s always extremely important to keep in mind the underlying motivations and stylized behaviors of different types of players, and thankfully, retail investors do tend to avoid panic selling their holdings at market lows. This is simply because they tend to have long time horizons (most passive investors are simply saving for the future and not drawing on their assets for current income, in which case they’d hopefully have shifted to fixed income). This also largely takes fire sale dynamics off the table. Fire sales happen generally when leveraged players are forced to mark-to-market. Most retail index investors are completely unlevered, so draw-downs mostly represent buying opportunities. So because the underlying type hasn’t changed, the fact of being invested in passive strategies doesn’t really change much.
If that were not true, you’re right, the investors would suffer huge losses, but even then, it wouldn’t really be a financial crisis, and there wouldn’t be any good logic for official sector intervention. I say that it wouldn’t constitute a financial crisis because losses on equity are just someone’s investment losses (and likely soon someone else’s gains). From a financial stability standpoint, equity does not matter, debt matters. People often conflate the two, because when debt is distressed, so is equity, but that is confusing a symptom (distressed equity) with the disease (distressed debt). Keep in mind that the post-2000 equity market losses rivaled the 2006-2009 mortgage market losses (which were debt) in magnitude, but only in the latter did we see a financial crisis. In contrast, large-scale inabilities for companies to access debt markets reflect a broad tightening of credit conditions— a liquidity crisis, which is what Bagehot described and what is generally an object of concern for central banks. So while the official sector will intervene when debt markets break down, it is not within the official sector’s purview to attempt to allocate gains or losses.
Having said all that, in the spirit of trying to fully answer your questions, there are ex ante tools that exist to basically discourage opportunistic redemptions from funds. They do not really apply differently to passively vs actively managed funds, though; they instead exist to ensure that one’s redemptions don’t negatively impact other fund holders. There are a whole litany of such tools, including gates, fees, swing pricing, minimum balance at risk, and others.


Most prominently, it was Bob Wilson who criticized the role of common priors in game theory. Starting from the "Wilson doctrine", some work in mechanism was done in that direction. Note that this research endeavor is not hopeless: in a popular auction format, the second-price auction, it is a (weakly) dominant strategy to report your true value. That is, your belief about others' types or strategies does not matter.
Chung and Ely (ReStud 2007) have a paper on this topic:

Robert Wilson criticizes applied game theory's reliance on
common-knowledge assumptions. In reaction to Wilson's critique, the
recent literature of mechanism design has adopted the goal of finding
detail-free mechanisms in order to eliminate this reliance. In
practice this has meant restricting attention to simple mechanisms
such as dominant-strategy mechanisms. However, there has been little
theoretical foundation for this approach. In particular it is not
clear the search for an optimal mechanism that does not rely on
common-knowledge assumption would lead to simpler mechanisms rather
than more complicated ones. This paper tries to fill the void. In the
context of an expected revenue maximizing auctioneer, we investigate
some foundations for using simple, dominant-strategy auctions.

Another popular approach is by Bergemann and Morris (2005) (several papers).
However, I guess this is not so much what you have in mind, but they have to be mentioned here.

The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying mechanism design on richer type spaces.
We ask when ex post implementation is equivalent to interim (or Bayesian) implementation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is implementing a social choice function (not correspondence) and (2) in a quasilinear environment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance.
In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results offer new insights into the relationship between dominant strategy implementation and Bayesian implementation.

Usually in mechanism design, there is private information to be extracted and you need to take a stand on how to model it. That could be Bayesian or not, but the classical Nash equilibrium concept is suited for full information settings. The "prior-free" mechanism design literature pushed by computer scientists is concerned with worst-case analysis.


"If markets are competitive, the rate of return on capital equals its marginal product, $f'(k)$ minus depreciation $\delta$"?

This is not proven in Romer but it is a well known result. To derive it mathematically you need to take the following steps:

First, the capital as in Romer depreciates so the evolution of capital will be given:

$$k_t = k_{t-1} + i_t- \delta k_{t-1} \tag{1}$$

where $k_t$ is the present stock of capital, $k_{t-1}$ previous stock of capital, $i_t$  is investment (where investment is equal to savings $s$) that increases the capital and $\delta$ is depreciation.

Next to increase the stock of capital the producer/investor needs to pay price $P_t$ for output that is going to be saved and converted to capital through $i_t$. Assuming that we will hold labor constant the reward that producer gets for this sacrifice is the marginal product of capital - as when labor is held constant the output $y_{t+1}$ will increase only by the additional marginal product that the increase in capital brings $f'(k)$. Moreover, we can assume that in the next period the capitalist can sell their remaining capital $(1-\delta)$ as well at price $P_{t+1}$.

Consequently the net nominal return on the investment will be: $P_{t+1}(f'(k_t) + 1 -\delta) - P_t$ and the nominal rate of return to investment will be given by:

$$ (P_{t+1}(f'(k_t) + 1 -\delta) - P_t)/P_t \tag{2} $$

This can be simplified as:

$$ (1-\pi_t) (f'(k_t) +1-\delta) -1 \tag{3}$$

where $\pi_t$ is the inflation rate $(P_{t+1}+P_t)/P_t = \pi_t$.

Whether the producer/investor is interested into the above investment depends on how other returns such as those on bonds or on debts used to finance investment behave. Lets call those returns $R_t$.

Now if we assume perfect markets, rational investor will invest into capital when the nominal return on capital is higher than the returns on bonds/debt $R_t$, but as economy accumulates more and more capital the marginal product of capital declines as it becomes more scarce (and vice versa if the return on bonds/debt is higher) and thus in competitive markets rational agents will invest up until:
$$ (1-\pi_t) (f'(k_t) +1-\delta) -1 = R_t \tag{4}$$

Furthermore, the nominal rate on return on bonds $R$ has to also satisfy the fisher equation:

$$(1+r_t) = (1+R_t)/(1+\pi_t) \tag{5}$$

where $r_t$ is the real rate of return. The fisher equation basically says that the real rate of return must be equal the nominal rate minus inflation as the above function can be approximated as $r_t \approx R_t - \pi_t$ using the fact that $\ln(1+x)\approx x$ for $x\approx 0$ (this comes from Taylor expansion). This must hold as people in the market should care about real returns and thus they should expect to be compensated for inflation when they set nominal interest rates.

Solving the Fisher equation (4) for the $R_t$ and substituting into the equation (5) which equates the nominal returns between capital and bonds/debt gives you the desired result:

$$r_t = f'(k_t)- \delta \tag{6}$$

The last equation says the real return to capital will be equal to marginal product minus depreciation.

The intuition is simple, if markets are competitive then you will invest into the capital as long as it provides higher returns than other investment. But the more you invest into the capital the smaller its marginal product and hence nominal return will be. At some point people invest into capital so much that the marginal product minus depreciation (which has to be taken into account as it decreases the value of the capital) is just equal to the real return. Also if capital would started with lower return than bonds/debt people would just invest into those until the returns will be equalized. The assumption of perfect market is important as in presence of market imperfections the equality established by eq (4) might not necesarry hold (or to be more precise it would have additional parameters which would end up in the result as well).


How does economics cope with production processes that have large lumpy investment requirements?

Indeed, a lot of the marginality conditions cannot be used directly in the presence of discrete choice levels, or fixed costs of adjustment. There are many: number of children, working or quitting, number of cars, paying the fixed cost of adjusting capital, consuming or selling durable goods, number of hours worked, workers hired, units produced, delivery trips, etc.

Marginal conditions are an approximation to reality. Economists would argue that for a plant that hires 1000 workers, we can think of the decision of hiring the 1001 worker as best though of as adding a marginal until of labor.

There are a wealth of studies into these topics: Scarf's K convexity and the S-s inventory replenishing policies, Caballero and Engel's Fixed Costs of Adjustment of Capital, etc.

A book that tries to unify some of these ideas is "The economics of inaction".

From a dynamic programming point of view, there are two useful intuitions that might help: a) In the presence of fixed costs, the firm's policy function is discontinuous and the value function is continuous and convex, but not smooth; b) the problems with fixed costs are best split into two: do I act? If I act, where do I go?

Empirical studies use this last intuition to argue that if you want to study marginally optimal capital investment decisions, you should look at situations in which firms have decided to adjust their capital level. Presumably, after paying the fixed cost of acting, firms will choose a marginally optimal policy.


Why is saved money sitting on a bank account bad for the economy?

The Saving Glut of the Rich (68 page working paper)
https://scholar.harvard.edu/files/straub/files/mss_richsavingglut.pdf

A prominent idea in the history of economics is that an excess of savings relative to investment opportunities can present challenges for an economy (e.g., Hobson (1902), Keynes (1936), Eccles(1951)). A recent manifestation of this idea is the “global saving glut,” or the argument that a rise in savings by certain countries is a primary culprit for the decline in interest rates to low levels around the world (e.g., Bernanke (2005), Summers (2014), Rachel and Smith (2017)).

This study shows an equally important saving glut of the rich. In the United States, the rise in savings by the top 1% of the income or wealth distribution since the 1980s has been substantial, it has not boosted investment, and it has been associated with dissaving by the government and the rest of the household sector. The saving glut of the rich has important implications for the concept of national savings, the evolution of the financial sector, and the demand for U.S. produced safe assets. It also warrants consideration in explanations of extremely low interest rates and high debt levels around the world.

The rise in savings by the top 1% has not been accompanied by a rise in net domestic investment. Instead the additional savings have been absorbed in a less conventional manner through dissavings by the government and the rest of the U.S. household sector.  In particular, savings by the bottom 90% have fallen significantly since the early 1980s and the government has run larger deficits, especially after the Great Recession.

It may be surprising that Americans in the top 1% of the wealth distribution hold so much U.S. government and household debt in their portfolios, but the unveiling process clarifies how this is in fact the case.  For example, holdings of business equity by rich Americans represent a substantial claim on such debt.  The reason is that non-financial businesses have increased their holdings of money market funds and time deposits substantially since the mid 1990s, and these time deposits and money market funds are claims on debt through the financial system.  More generally, government and household debt have been financed by rich Americans through direct bond holdings, bond mutual fund holdings, business equity, time deposits, money market funds, and defined contribution pensions.

If it is "good" when savings correspond to investment in the non-financial sector, and it is relatively "bad" when savings do not correspond to investments in the non-financial sector, then the accumulation of financial assets by the top 1% households would seem to "crowd out" investment by saturating the working class with debt and depressing the increase of wages or discharge of debt that would be necessary to stimulate more aggregate demand.


Determine the homogeneity of a demand function

All (marshallian) demand functions are homogenous of degree zero in prices and income. This is a well known result.

That means that $x(\lambda p, \lambda I) = x(p,I)$, where $p$ is a vector of prices (i.e. x could be a function of any number of prices, including $P_x$ and $P_y$, which were given in your example).

So if we multiply all prices and income by the same amount, demand doesn't change. This is very intuitive. Imagine the government would just add a zero to all prices and to your paycheck. There is no reason for you to change your demand as nothing really got cheaper nor did your income increase in real terms.

To see this formally:

Proof:

$x(p,I) = argmax \; U(x)$ s.t. $px \leq I$.

$x( \lambda p, \lambda I) = argmax \; U(x)$ s.t. $ \lambda px \leq \lambda I \Leftrightarrow $

$x( \lambda p, \lambda I) =  argmax \; U(x)$ s.t. $px \leq I = x(p,I)$
$Which \; is \; the \; definition \; of \; homogeneity \; of \; degree \; zero. QED. $

The last step follows because we cancel $\lambda$ in the budget constraint on both sides.