Bellman equation for this dynamic programming problem For the following problem
  \begin{equation}\max_{(\tilde{c}_t,\tilde{a}_{t+1+s})}\sum_{s=0}^{\infty}\beta ^su(\tilde{c}_{t+s})\end{equation}

  s.t. the following restrictions

  $\begin{equation}
\begin{split}
\tilde{c}_t&=(1-\delta )Y_t+a_t-\frac{\tilde{a}_{t+1}}{R_t}\\
\tilde{c}_{t+1+s}&=(1-\delta )Y_{t+1+s}^{e}+\tilde{a}_{t+1+s}-\frac{\tilde{a}_{t+2+s}}{R_{t+1+s}}, \forall s \geq 0
\end{split}
\end{equation}$

  where:

  $\tilde{c}_t$: Consumption at time $t$

  $\tilde{a}_t$: Financial wealth at time $t$

  $Y_t$: Income at time $t$

  $R_t$: Nominal interest rate at time $t$

  $Y_t^e$: Expected income at time $t$

  Consider that $u(\tilde{c}_{t+s})$ is defined by the isoelastic utility function: $u(\tilde{c}_{t+s})=\frac{\tilde{c}_{t+s}^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}}$

  Find the optimal policy function for $\tilde{c}_t$.

I don't know how to write the Bellman equation because I have two restrictions. What would be the optimal procedure to solve this problem?

The "second" constraint appears redundant and it confuses matters. Re-arrange the first one to obtain

$$\tilde{a}_{t+1} = \big[\tilde a_t+(1-\delta )Y_t-\tilde{c}_t\big]R_t$$

This tells us that wealth in the beginning of next period is fully determined by current-period decisions and known states, without any uncertainty whatsoever: we start with the given wealth at the beginning of the period, our income and the interest rate becomes known, we decide and "set aside" the full amount of consumption (for the whole period, at the beginning of the period), and the rest becomes interest-bearing asset.

Why the second constraint, which looks into the future, confuses matters? Because in order to apply Dynamic Programming/Bellman equation, it has to be the case that our optimization problem can be formulated as a recursive one, meaning that the whole many-periods or even infinite-horizon problem can be broken down to a two-period problem. And we have this condition in the above constraint, we do not need the second one.

If $V(\tilde a_t)$ is the value function, the Bellman equation is then

$$V(\tilde a_t) = \max_{\tilde c_t}\big [u(\tilde c_t) + \beta V(\tilde a_{t+1})\big]$$

and we must find the policy function $h(\tilde a_t)=\tilde c_t$  that satisfies

$$V(\tilde a_t) = u[h(\tilde a_t)] + \beta V\left[\big(\tilde a_t+(1-\delta )Y_t-h(\tilde a_t)\big)\cdot R_t\right]$$


Can I treat pooled cross-section as panel data and do regression analysis? I work with a large survey data that is pooled cross-section, i.e, people interviewed are randomly chosen in each time period. This is not panel data. But, I aggregate the individual cells according to age and gender groups and then run the regressions. Now that I work with age-gender groups, is this considered panel data?

Yes, you can consider this as panel data, but the key is to understanding why, as this affects how you explain and interpret your panel regression. In treating each cohort as the same unit over time, you are assuming that each cohort has a constant, unobserved fixed effect. That is, people in a particular age category and gender tend to have an unobserved effect on the dependent variable that is different to other cohorts.

Please note:

Age increases over time, so, for example, the group aged 30–34 in year 1995 should be treated as the same group as the 35–39 year-olds in 2000. I suggest your time variable increases in steps of five years, as you indicated that your age categories are five-year bins.


Static game with complete but imperfect information I am confused on the concept of static game with complete but imperfect information and its consequences on the equilibrium definition.

Suppose we have 2 players. Each player $i$ chooses action $Y_i\in \{A,B\}$. Actions are chosen simultaneously by the two players. Simultaneity=static in my mind. Is it the case actually?

Each player $i$ gets a payoff $\Pi^i(Y_1, Y_2)$ at the end of the game.

We assume complete information, i.e., players know the values
$$
\begin{cases}
\Pi^1(A,B),\Pi^1(A,A),\Pi^1(B,B),\Pi^1(B,A)\\
\Pi^2(A,B),\Pi^2(A,A),\Pi^2(B,B),\Pi^2(B,A)\\
\end{cases}
$$

An equilibrium concept that we can use in this context is Pure Strategy Nash Equilibrium: $(Y_1, Y_2)$ is a PSNE if
$$
\begin{cases}
\Pi^1(Y_1, Y_2)\geq \Pi^1(\tilde{Y}_1, Y_2)\\
\Pi^2(Y_1, Y_2)\geq \Pi^1(Y_1, \tilde{Y}_2)\\
\end{cases}
$$

Question: What does it mean assuming the players have imperfect information in this case? Would that be in contrast with any of my claims above? Would that change the equilibrium notion? Is Pure Strategy Nash Equilibrium still applicable? This question is related but it does not clarify exactly my point.

Complete information means every player has common knowledge of the structure of the game, in particular, everyone's preference over outcomes and the realization of the relevant state variables.

Perfect information means every player has common knowledge of the entire history of play up to the point where they are about to make a decision.

Static games are considered to be of complete information because the players know each other's payoffs in all possible outcomes (and there's no uncertainty about any state variable here), as you rightly noticed.

The reason why static games are usually considered to be of imperfect information is due to the extensive form representation of such games (see figure below). Since static games emphasize strategic simultaneity (making a decision without knowledge of what the other player has chosen) as opposed to temporal simultaneity (making a decision at the same time as the other player), the extensive form of such games usually feature one player moving first while the second player chooses without observing the first player's choice. This lack of information about the first mover therefore leads to imperfect information, i.e. player 2 does not have knowledge of part of the history of the game.

This is why perfect information is sometimes characterized as an extensive form game in which every information set is a singleton.


What are the differences between "Crawling peg" and "Crawl-like arrangement"? The IMF AREAER 2019 classifies 3 countries as "crawling" and 18 as "crawl-like". It's not clear to me though how these classifications were made and why they couldn't be switched around. For example, why is Botswana "crawling" while Burundi is "crawl-like"?
"Crawling peg":

Classification as a crawling peg involves the confirmation of the country authorities’ de jure exchange rate arrangement. The currency is adjusted in small amounts at a fixed rate or in response to changes in selected quantitative indicators, such as past inflation differentials vis-à-vis major trading partners or differentials between the inflation target and expected inflation in major trading partners. The rate of crawl can be set to generate inflation-adjusted changes in the exchange rate (backward looking) or set at a predetermined fixed rate and/or below the projected inflation differentials (forward looking). The rules and parameters of the arrangement are public or notified to the IMF.

"Crawl-like arrangement":

For classification as a crawl-like arrangement, the exchange rate must remain within a narrow margin of 2% relative to a statistically identified trend for six months or more (with the exception of a specified number of outliers) and the exchange rate arrangement cannot be considered as floating. Normally, a minimum rate of change greater than allowed under a stabilized (peg-like) arrangement is required. However, an arrangement will be considered crawl-like with an annualized rate of change of at least 1%, provided that the exchange rate appreciates or depreciates in a sufficiently monotonic and continuous manner.

The answer is already in your question. As the IMF paper says:

Classification as a crawling peg involves the confirmation of the country authorities’ de jure exchange rate arrangement.

So the difference between crawling peg and crawling-like is that in the first case country officially states that it is doing crawling peg, while with the crawling-like arrangement the country does not officially follow crawling peg, but behaves in a way as if it would be following it.


Example of Law of One Price holds but No Arbitrage Fails I have been told that no arbitrage is a stronger assumption than than the law of one price.

In particular, Law of One Price is equivalent to the existence of a stochastic discount factor, whereas no arbitrage is equivalent to the existence of a strictly positive stochastic discount factor.

Can someone give me an example where LOOP holds, but No Arbitrage fails?

Thanks.

Examples where this happens are always extreme and contrived. I can think of two kinds of examples. The first is where you have an asset that for some reason has a price of zero or negative but a positive payoff in some states (perhaps no other asset pays off in that state). The second is where you have two assets with positive prices that have negative payoffs in all states (or vice versa).

You can come up with other varieties, but they're always a little weird. To show you how to do this, I'll be a little more precise. To answer this question, we need to be a little more clear about what arbitrage means. A lot of books give definitions for two kinds of arbitrage (e.g. see the asset pricing book by Darrell Duffie). Let me set up some notation first, though.

Notation

Let $p_i$ be the price of security $i$ with $p = [p1,...,p_N]'$. Let $D_{ij}$ be the payoff of security $i$ in state $j$ with $D$ being an $N \times S$ matrix, where $N$ is number of assets and $S$ is the number of states. Let $\theta_i$ be the number of securities of type $i$ that are owned, with $\theta = [\theta_1, ..., \theta_N]'$.

Some definitions

"The law of of one price": For any portfolio $\theta$ that satisfies $D' \theta = \vec 0$, it must be that $q' \theta = 0$.
"Arbitrage of the First Type": There exists a portfolio $\theta$ such that $p' \theta \leq 0$ and $D' \theta > \vec 0$. This means that all elements of the vector $D' \theta$ are strictly positive.
"Arbitrage of the Second Type": There exists a portfolio $\theta$ such that $p'\theta < 0$ and $D'\theta \geq \vec 0$. This means that all elements of the vector $D' \theta$ are nonnegative.

Examples

In this first example, consider the extreme case of the existence of only one asset. Suppose that $p = 0$ and $D=1$ (one state, but we can have more). Then the law of one price holds and arbitrage of the first type exists. This is pretty contrived, though.
Consider the example of $p = [1 \quad 2]'$ and
$$
D =
\begin{bmatrix}
-2 & -1 \\
-4 & - 2
\end{bmatrix}.
$$
This satisfies the law of one price. However, when $\theta = [-1 \quad 0]'$ or $\theta = [0 \quad -1]'$, we have $p'\theta < 0$ and $D' \theta \geq 0$ and $\theta \neq 0$.


Difficult quantity theory of money question Suppose that you live on an island with 100 units of currency. The cultural beliefs of the islanders discourage an excess focus on commerce, which has created two important rules of commerce. First, you are legally permitted to do 10 transactions per year. Second, the maximum permitted value of each transaction is $10, expressed in USD.

With these assumptions, can you solve for the maximum legal FX rate of the islander's currency expressed in US dollars? I feel like you should be able to do this with some kind of simple, recursive calculation, but for some reason I just can't figure it out right now. GDP is obviously $100 USD per islander and I feel like you can know velocity by knowing the number of transactions per year, but I am not sure how.

The velocity should just be the amount of times the average unit of currency exchanges hands per year. That is $10$ if we take the highest possible number.

Money supply is $100$ units of island currency ($IC$), so that the left side of

$$MV=PY $$

turns into $100*10$ Nominal GDP,
so $YP$, is \$$100$ per Islander, or
$$\$100/XR*n $$ in $IC$,
where $XR$ is the exchange rate of $USD/IC$ and $n$ is the amount of people.

Then we get
$XR=1/10*n$.

However, I'm a bit lost for what $n$ would be in this case. I'm guessing you got this far yourself?


Can heterogeneous prices be consistent with competitive equilibrium? I do not want to put many constraints on the structure of the problem, except by defining a competitive equilibrium as such where firms are price takers in the factor and good(s) market, and, perhaps (?), that firms make no profits.

For example, is it possible to have something like "competitivistic" competition, where consumers have love for variety and firms produce differentiated goods, yet, instead of monopolistic competition, firms do not charge a markup, because there is free entry, so that firms end up producing at zero profit?

Or for example, a model of consumers geographically immobile, and where differentiated transportation costs across regions mean firms need to charge a markup to pay for that cost, and where, again, free entry deter firms from exploiting consumers?

As a starting point, you might take a look at Stigler's model of price dispersion. Quoting from a resource I found online,

  George Stigler’s 1961 “The Economics of Information” begins with:

  "One
  should hardly have to tell academicians that information is a valuable
  resource: knowledge is power. And yet it occupies a slum dwelling in
  the town of economics."

  Stigler documents the existence of price
  dispersion empirically, and proceeds to sketch various theories that
  might be applicable to price dispersion. Stigler is careful to
  distinguish price dispersion, which results from imperfect information
  on the part of consumers, from price discrimination, which involves
  distinct prices for different types of consumer preferences

  There are various reasons that price dispersion might arise. Clearly
  some consumers must be willing to pay higher prices than others, that
  is, distinct consumers have distinct reservation prices. (Note that a
  reserve price – the minimum bid in an auction – is a very different
  concept.) Differences in reservation prices can arise because of
  differences in knowledge – some consumers have access to price lists,
  others don’t – or search costs. If some consumers have lower search
  costs, then these consumers will search for lower prices.

You might also check out the Wikipedia articles on Price Dispersion and Search Theory.


Data on bilateral real exchange rate Where can I find bilateral real exchange rates? For example per USD?

I can only find real effective exchange rates, but it is an index, not a bilateral rate.

Check out this link from the United States Department of Agriculture.

This data set contains annual and monthly data for exchange rates. It includes both bilateral nominal and real exchange rates for 79 countries, plus the European Union, as well as real trade-weighted exchange rate indexes for many commodities and aggregations. All series are updated quarterly. Data series start at the beginning of 1970 and run to the last available data point.

As pointed out by @luchonacho, real exchange rates here are derived by multiplying the nominal exchange rate by the ratio of the U.S. to local currency Consumer Price Index


Under the gold standard, how did the redemption of gold work in practice? For example, in the US:

  In 1834, the United States fixed the price of gold at $20.67 per ounce, where it remained until 1933. (EconLib)

Does this mean that anyone with US\$20.67 could walk up to any bank in the US and demand an ounce of gold? Conversely, could anyone with an ounce of gold demand US\$20.67 in paper money from any bank?

Or were the rules of redemption more restrictive than that?

It depended on the year and if a banking suspension was in effect in the town.  In 1834 it was absolutely true that you could walk up to a bank and demand one ounce of gold or in some time periods silver for \$20.67.  Of course, you would have to accumulate \$20.67 which was a lot of money.  The Treasury would demand upon receipt of a banknote any and all gold represented by those notes at all times during this period.

You should double check the amount because I thought it was pegged at $20 over that time period.

  Temin, Peter.  The Jacksonian economy  Norton New York  1969

EDIT

The converse was not true unless it was a minted coin.  The period was known as the free banking era.  I hadn't noticed the converse question.  You should read Temin's book.  There are also other books on the free banking era.  Paper money was not printed by the United States until Lincoln authorized the creation of demand notes under an 1861 Act.  See

  Statutes at Large.  37th Congress, 1st Session. Chapter XLVI.


Something Terrible is Happening - But When Is it Likely to have Happened? We live in continuous time $t$ and something terrible is happening at a poisson rate of $r(t)$.

How can I compute the length $T$ such that with a probability of $P$ (for example, 0.99), at least one terrible thing has happened?

I know that for a constant rate $r(t) = \bar r$, the expected value is $1/\bar r$. How can I go on?

I assume that $r(t)$ is continuous.

The idea is that, this Poisson process with time-varying parameter $r(t)$, as the limit of Bernoulli trial with time-varying probability of success, is memoryless:

For a generic finite partition $\mathscr{P}$ of $[0, T]$ as $\{[0,t_1),[t_1,t_2),[t_2,t_3) \ldots,[t_{n-1},T]\}$, let the $n$th cell following a Poisson distribution with a fixed parameter $r(t_n)$, so
 $$P_{\mathscr{P}} = 1 - \underbrace{e^{\sum_{k=1}^n-r(t_k)(t_k-t_{k-1})}}_{\text{Probability of no accidents}}$$. So  $P = 1 - e^{-\int_0^Tr(t)dt}$. Notice that the Riemann integral $\int_0^Tr(t)dt$ is the limit of a net indexed by the set of all finite partitions of $[0,T]$ with vanishing maximal length of cell with respect to number of cells. $f(x)=1-e^{-x}$ is continuous, so this limit preserves.


nash equilibrium public good - is this correct? Two players, 1 and 2, simultaneously choose their consumption of a public good. Given the consumption choices, g1 and g2, player 1 derives a marginal benefit of MB1 = 10 - (g1 + g2), while player 2's marginal benefit is given by MB2 = 8 - (g1 + g2). The unit price of the public good is equal to 4.

The question asks to find a Nash Equilibrium of the game, then calculate the efficient level of public good provision and compare it to the NE.

I can understand the efficient provision of public good which will be nothing but the sum of marginal benefits set equal to the marginal cost. In this case, it turns out to be 7. What will be the Nash equilibrium?

An update with my attempt:

Player 1's best response function: g1 = 6 - g2
Player 2's best response function: g2 = 4 - g1

Added these two to get the total level of public good - it comes to 5.

Then I equated the two response function to get: g1 = g2 - 2

Substituted it in g1 + g2 = 5 equation with getting g2 = 3.5 and g1 = 1.5

I don't know if this is correct.

For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.

So your steps to solving this game should be:

Determine player 1's best response function ($g_1$ as a function of $g_2$)
Determine player 2's best response function ($g_2$ as a function of $g_1$)
Now you have two equations with two unknowns. Solve this system of equations algebraically to get the Nash equilibrium $g_1^*$ and $g_2^*$.

So how can you get the best response functions? Take player 1 for example. Player 1 wants to set $MB_1 = MC_1$. You have $MB_1$, and $MC_1 = 4$. So set them equal to each other and solve for $g_1$ to get player 1's best response function $g_1^*$ as a function of $g_2$. Then similarly for player 2 to get $g_2^*$ as a function of $g_1$.

That's not quite all, though. In this game, the players (presumably) can't contribute negative amounts. So the real best response functions will be of the form

$$
g_1^*=
\begin{cases}
some.function.of.g_2 \text{ if } some.condition.of.g_2\\
0 \text{ otherwise}
\end{cases}
$$

So, first, try to solve your two best response functions together algebraically, assuming that both $some.condition.of.g_2$ and $some.condition.of.g_1$ hold so neither is 0. If you don't get a solution, or if that solution makes one of them negative or violates one of the $some.condition$s, then instead try setting $g_1 = 0$, find $g_2^*$, and see if that gives you a Nash equilibrium (i.e. if $g_1^*(g_2^*) = 0$). Then try setting $g_2 = 0$, find $g_1^*$, and see if that gives you a Nash.


Possible regression models for a dependent variable that is a proportion I am writing an applied econometrics paper. The dependent variable is a percentage of people who answered in the affirmative to a survey question asked in 60 different countries.

One possible model is a beta regression model for proportional dependent variables.

What are some other methods that are suitable for proportional dependent variable regression analysis?

Let $y$ be the proportion in $[0,1]$ instead of the percentage. I think the issues here are possible nonlinearity and censoring. You can try including quadratic terms on the right hand side. At the same time, you can try the following models.

I. Linear models: If no $y$ is exactly 0 or 1, linear models will be OK. The following four options come to my mind.

$y=\beta_0 + \beta_1 x + u$ if no $y$ is close to 0 or 1.
A log model $\ln y = \beta_0 + \beta_1 x + u$, which is helpful when some $y$ values are close to zero but all are far from one. Note that interpretation changes.
Transform the dependent variable to $-\ln (1-y)$. This is helpful if $y$ are all far from zero but some close to one. But this looks a bit unnatural to me, and I would consider the next logistic model instead.
The logistic model $\ln \frac{y}{1-y} = \beta_0 + \beta_1 x +u$. This usually helps if there are many $y\simeq 0$ and $y\simeq 1$. Interpretation is done in terms of logits.

II. Tobit models: If some $y$ are exactly 0 or 1, you can try Tobit models (help tobit in Stata). Remember that normality is assumed for the error term before censoring. Also, using Tobit models means that "I think $y$ could be bigger than 1 (smaller than 0) if not censored."


How does the economy of a city work? I'm interested in learning how the economy of a town is analyzed. For instance, some occupations serve only people in the town, for instance the town grocer, baker, restaurant, dry cleaner, etc. Their income must come from other people in the town. But other people's jobs will have them sell externally, like someone with an online store.

I've read things discussing how if Amazon's HQ2 brings 50,000 high paying jobs to a city, that also will cause there to be, say 100,000 new secondary jobs (restaurants, dry cleaners, etc that cater to the additional 50k people)

What is this area of study called and what papers or books are good to learn more about it?

Answering your question involves the various sub-fields of Regional Economics, such as Spatial Competition theory (Hotelling, 1929), Economic Geography, Urban
Economics (Alonso, 1964), New Economic Geography (Krugman, 1991), preferably in an integrated way.

Indeed, each of these subfields relate to considerations that complement each other. As stated by Thisse (2010), integrating space in economic analyses relates to (i) the use of a computable general equibrium (CGE) framework to endogenize the formation of incomes and relative prices, (ii) the realism of the modeled geographical space, the necessity of being multidimensional, asymmetric and empirically grounded and (iii) the number of encompassed spatial scales. Otherwise, it is difficult to figure out how spatial forces shape the so-called Space-economy (Isard, 1949) and, a fortiori, what policy or what institutional change can influence them if they are judged to go in an undesired direction.

I would reformulate your question as

  What to do to model a city, a priori itself within a system of cities in interaction.

Incorporating CGE theory

The modeling architecture to develop in order to spatialize a GE is implicitly specified by Thisse (2010), who, instead of soft-linking compact descriptions of the economy and spatial formalisms, insist on the necessity of their full entanglement. This intellectual movement is the same as the movement toward hybrid models approach to couple GE models and engineering-based energy models (Hourcade et al., 2006). In the case of spatial economics, this hybridization passes through the use of New Economic Geography (NEG), which intrinsically implies GE theory. Indeed, GE concerns are what makes Economic Geography New (Krugman, 1998).

Involving geographical realism

At the scale (implicitly) involved in Urban Economics (UE), i.e. the urban scale, you can read Anas and Xu (1999), Nitzsche
and Tscharaktschiew (2013), Choi et al. (2015), Anas and Liu
(2007). If we collapse the internal structure of a metropolitan area into a single point, we conventionally jump to the field and the scale of analysis of New Economic Geography (NEG). Because at its early stages the analytic tractability of the spatial processes governing the interactions of areas was a key objective, one NEG’s notable feature is its lack of realism with
respect to the geographical distribution of areas, e.g. distributed in a mono-dimensional world as in Krugman (1993)’s Racetrack economy or in a part of it as in Fujita et al. (1999b)’s Line economy, as well as in a bi-dimensional world as in Puga (1999)’s side-by-side Equidistant Economy (EE). Since then, adding more real spaces components to NEG has been thought of as the main step for future development (Behrens and Thisse, 2007; Fujita and Krugman, 2003). You may also want to read Behrens et al. (2007), Bosker et al. (2007), Stelder (2005), Brakman et al. (2006), Sheng et al. (2016), Mercenier et al. (2016).

Involving multiple spatial scales

Both UE- or NEG-related models rely on exogenous signals coming from upper geographical scales, be them national, regional or global. But it is harder to find modeling approaches in which these links are explicitly involved. Attempts to bridge this gap can be found in Waisman et al. (2013), Allio (2015, 2016) and Faucheux (2018).

Waisman, Allio and Faucheux integrate (i), (ii) and (iii) in their modeling exercices.


What would happen if a country bans the trading of its currency against foreign currency? The iranian rial has been losing its value rapidly recently. The dollar went from 40,000 rials to 60,000 rials in less than 6 month, most of the loss of value happening in the last few weeks. Iran responded by banning the trade of its currency until further notice.

I wanted to ask the economists here their opinion on the implication of this ban? Also example of precedents of such bans by other nation states would be appreciated.

news source:
https://en.radiofarda.com/a/iran-ban-sale-foreign-currency-exchange-bureaus/29168476.html

This is quite a common response to an exchange rate crisis (which normally also suggests broader economic problems).  The consequences depend on how diligently the government applies the ban, the relationship between government and businesses that trade internationally and how far the official rate strays from the true (black market) rate.

First, it is almost always possible to buy and sell currency on the black market, and perhaps legally in neighbouring countries.  The black market rate is likely to follow the downward trajectory of the currency prior to the ban.  Exchange controls create risk for currency traders, so buying and selling on the black market tends to be expensive (i.e. the rates for buying and selling move further apart).

Second, lots of businesses urgently need foreign currency to import inputs and goods for retail sale.  They will try and find a strategy for obtaining that currency.  They may be able to buy it on the black market.  If the government is able to sell some local currency at the official (overvalued) rate, it may also sell foreign currency at the official rate to firms it sees as a priority (e.g. oil importers or friends of the regime).  For instance, tourists entering East Germany were forced to buy a certain quantity of local currency per day of their visit at the official rate, giving the government a source of forex.  If these strategies are not possible, companies may try to create symmetry in their import/export businesses, so that they can use the foreign exchange they earn themselves to cover their imports.  Government may or may not intervene to prevent this.  Importing firms without a viable strategy are likely to fail.

All in all it makes international trade more difficult and expensive, so prices of imported goods are likely to rise even more quickly than they were under the sliding exchange rate.

The most reasonable motivation for this type of exchange control is to prevent people and companies from shifting their savings into another currency, which could cause a general banking collapse.  If you see the exchange rate falling continuously, you would be tempted to convert your savings from local currency to dollars, which hold their value.  If everyone does this, the exchange rate slide accelerates, banks can't cover their liabilities, loans are called in, and banks and businesses may fold.

The Soviet bloc is a common example of the use of official exchange rates but they've been used by governments in trouble all over the world, recently in various African countries, a few in South America and less recently in Asia.


Is product opportunity frontier the same as PPF? In CFA level 1 reading 14, they have this term called Production Opportunity frontier and it is defined as

Curve describing the maximum number of units of one good a company can produce, for any given number of the other good that it chooses to manufacture.

The graph is a straight line

Is it the same thing as what is commonly known Production Possibility Frontier with the curved line?

Yes. The "curved line" clearly fits your definition as well. The PPF is curved if (but not only if) there are diminishing returns to inputs such as labor and capital. See also
https://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier


If a Chinese person wants to buy an iPhone or any international goods, which GDP per capita would be a better indicator? Which GDP per capita of a Chinese citizen would better reflect their affordability- Nominal GDP per capita or PPP GDP per capita of China? My question is would PPP would matter much in case of buying and selling international goods and services?

To me, the method to get PPP value could be crude. If you're only interested in an iPhone, here's what I would do.

Get the price of an iPhone sold in China (in CNY), divided by the nominal GDP per capita for China (in CNY). You'd get (roughly) the fraction of annual income a Chinese person would need to spend on an iPhone. That should give you an indication of whether that's "cheap" or "expensive"
To compare across countries, you'd just do the same and see if that fraction is higher (more expensive) or lower (less expensive).

Keep in mind though that, with existing income inequality, GDP per capita doesn't really tell you about the actual income an average person gets.


Books on Technological Progress and Growth I've done some google search for good, reader friendly albeit technical, books on technological progress/change and economic growth. However, I've not found what I want. If you are using or have heard about a title on the above topics, do please list them below in your post.

I'd recommend ADVANCED
MACROECONOMICS
Fourth Edition by David Romer.

Its a textbook for advanced macroeconomics, however it covers the topics in a very precise way. Id recommend reading chapters 1  in order to review the basic Solow model and then chapter 3 to see how the model changes when accounting for human capital and endogenous technological growth.

Hope this helps.


Competitive equilibrium with production Consider an economy with four goods, two individuals and two firms. Firm 1 produces good $x$, firm 2 produces good $y$.
Consumers' utilities are $u_1(x,y,z,w)=\min\{x,2y\}$ and $u_2(x,y,z,w)=\min\{2x,y\}$.
Production function of firm 1 is $x=(1+a)z$, production function of firm 2 is $y=w$, where $a$ is a small positive number.
Consumer 1(resp. 2) owns firm 1(resp. 2). Initially, consumer 1(resp. 2) has $1$ unit of good $z$ (resp. $w$).
The question asks for an equilibrium.
My approach is first stipulating a price, then try to deduce the equilibrium, but I fails to do so, since there are 4 goods here and they are not symmetric.
Any help is appreciated.

First, note that both firms have constant returns to scale. This implies that firms must make zero profit in equilibrium. If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist. So who owns which firm does not really matter. We know from the first welfare theorem that every equilibrium must be Pareto efficient. With the given utility functions, this means that both firms have to produce a positive amount. But this pins down the relative prices of $x$ and $z$, and of $y$ and $w$, respectively. So this reduces the problem to one in which you have to find two prices. You can set one of the remaining prices to be $1$, and then calculate the excess demand for one good. FInd the price that makes this excess demand zero. As a consequence of Walras' law, this price will clear all markets.


Why utility rather than expected utility in Cochrane's "Asset Pricing"? Cochrane "Asset Pricing" Chapter 1 p. 6 says

We model investors by a utility function defined over current and future values of consumption,
$$
U(c_t,c_{t+1}) = u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})]
$$
where $c_t$ denotes consumption at date $t$.

Later, this utility is maximized subject to a sort of a budget constraint

$$
\max_{\xi}\ u(c_t) + \mathbb{E}_t[\beta u(c_{t+1})]$$
where
\begin{aligned}
c_t &= e_t - p_t \xi, \\
c_{t+1} &= e_{t+1} + x_{t+1}\xi.
\end{aligned}

I am used to maximization of expected utility rather than raw utility. Moreover, the expression on the right hand side of $U(c_t,c_{t+1})$ looks just like expected utility where the expectation is conditional on the information available at time $t$:
$$
u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})] = \mathbb{E_t}[( u(c_t) + \beta u(c_{t+1})].
$$
Question: Why does Cochrane not call $U(c_t,c_{t+1})$ expected utility then?

As mentioned in the comments this comes down to stylistic choices, since as you correctly pointed out:
$$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$
However, in principle both expressions are correct. The first expression states that the $U_t(c_t, c_{t+1})$ is a composite function of present utility of consumption $u_t(c_t)$ and expected utility of future consumption $E_t(u_{t+1}(c_{t+1}))$, whereas the second expression is directly expected utility of present and future consumption. When we condition on presently known information those two are equivalent here.
The notation that Cochrane uses is not used everywhere  but there does not seem anything wrong with it either from mathematical perspective or economically.


Relationship between Real Interest Rate and Employment The country of Kingsland is considering the introduction of a
  compulsory retirement saving scheme. Under this scheme all workers are
  required to save ten per cent of their annual wages and salaries until
  they retire. Use the supply and demand model (and a diagram) for
  saving and investment to explain the likely effects of this scheme on
  national saving, investment and the real interest rate in Kingsland.
  Explain the effects on employment of the saving scheme. (You can
  assume that Kingsland is a closed economy).

Since the level of national savings has increased, I understand how the savings curve shifts to the right and thus decreases the real interest rate and increases the levels of savings and investment in the economy. I am unable to understand how I can relate this back to the level of unemployment in the economy.

Recall that central banks use interest rate to control inflation by inducing spending or saving  with decrease or increase in interest rate.

Also,note the theory proposed by the short-run Phillips curve.

Though this graph has  some controversy in terms of its empirical soundness. It is the basis of how we tie interest rate to unemployment.

abstractly you can think of it as a "causal chain" being:
$$\text{interest rate}\rightarrow\text{savings rate}\rightarrow\text{spending}\rightarrow\text{inflation}\rightarrow\text{unemployment}$$

Hope this helps.


Reading list: Housing Economics Hi Im looking for some books or articles or books on the basics of housing economics.

Where should I start?

Alex Schwartz's "Housing Policy in the United States"
https://www.amazon.com/s?k=housing+economics&i=stripbooks&ref=nb_sb_noss

https://www.researchgate.net/post/Is_there_any_classic_theory_or_book_for_housing_policy_and_housing_market

Lund Brian ' Understanding Housing Policy' 2017

Housing Economics
1st Edition
Authors: George Fallis
https://www.elsevier.com/books/housing-economics/fallis/978-0-409-82940-2

Housing and Mortgage Markets in Historical Perspective (National Bureau of Economic Research Conference Report)
by Eugene N. White, Kenneth Snowden, et al.

That's what I found.  Good Luck.  There are some forums that may have a better answer for you.


Do granting subsidies always cause allocative inefficiency? I realise that subsidies cause allocative inefficiency if the market was already allocation efficient pre-subsidy due to deadweight loss.. but is this always true?

what if the subsidy was granted due to market failure (as is usually the case), that would mean that granting the subsidy would improve allocative efficiency, right? In this situation, does it still cause a certain degree of deadweight loss?

Your intuition is correct: in cases of market failure when the price under-represents the marginal value to society as a whole (for example if positive externalities are present) then granting the subsidy would improve allocative efficiency. As long as the subsidy is not greater than the positive externality, then the only loss in that case arises from the overhead in managing the subsidy - the bureaucratic cost. This is typically less than the inefficiency resulting from the market failure, so society gets a net gain from the subsidy.

The same thing applies with taxes and negative externalities. For the original work on this, see Pigou's seminal "The Economics of Welfare" (1924)


How are real income and utility the same thing? The textbook I'm using, "Microeconomic Theory: Basic Principles and Extensions", treats utility and real income as the same thing in the chapters on compensated and uncompensated demand. I would like to understand the logic behind it. To me real income always meant "Nominal Income/CPI" but I don't think this can also be translated to utility.

Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services.
For example, let us assume there are only two goods $x_1$ an $x_2$ and consumer is given budget $p_1 x_1 + p_2 x_2 = m$, where $p_i$ are prices for good 1 and 2 respectively and $m$ is an income.  Let us assume Cobb-Douglas utility function:
$$ u(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}$$
Here after solving the consumer optimization problem we would get optimum consumption of $x_1$ and $x_2$ as (I skipped the steps but you can easily derive it by setting up an Lagrangian):
$$x_1^*= \alpha \frac{m}{p_1};x_2^* = (1 -\alpha) \frac{m}{p_2}$$
Consequently, given optimal consumption of $x_1^*$ and $x_2^*$ is:
$$u(x_1^*, x_2^*) = \left( \alpha \frac{m}{p_1} \right)^\alpha \left((1 -\alpha) \frac{m}{p_2} \right)^{1-\alpha} $$
So the utility will end up being function of nothing else than a function of income $m$ an prices $p_1$ ($\alpha$ is just parameter). A utility function expressed in terms of prices and income even has its own special name 'indirect utility' typically denoted as $v$ (e.g. $v(\textbf{p},m) =\max u(\textbf{x})$ such that $\textbf{px} = m$ see Varian Microeconomic Analysis 3rd ed pp 99.)
Consequently as shown above utility does depend on income. Utility is generally not equal to real income, and I don't have the copy of the textbook you mention and you did not gave us the page for that claim, but they might have just used a utility function where utility would be proportional to real income. For a didactic purposes even assuming $v(m,\textbf{p})=\frac{m}{\textbf{p}}$ can be reasonable depending on setting even if it is not realistic because it simplifies calculations for novices (e.g. supply and demand are also typically not linear but your textbook likely draws them as linear functions).


Distance transported used to price goods in Marxism A friend mentioned that in Marxism, the distance a good was transported was used to determine the price of said good.

I can't find anything on the Internet to support this claim, although I might be misremembering his statement.

Can anyone think to what he was referring?

Quantities of products are not increased by transportation. Nor, with
  a few exceptions, is the possible alteration of their natural
  qualities, brought about by transportation, an intentional useful
  effect; it is rather an unavoidable evil. But the use-value of things
  is materialised only in their consumption, and their consumption may
  necessitate a change of location of these things, hence may require an
  additional process of production, in the transport industry. The
  productive capital invested in this industry imparts value to the
  transported products, partly by transferring value from the means of
  transportation, partly by adding value through the labour performed in
  transport. This last-named increment of value consists, as it does in
  all capitalist production, of a replacement of wages and of
  surplus-value.

Source: Capital Volume II, Chapter 6: The Costs of Circulation,
I. Genuine Costs of Circulation, III. Costs of Transportation

My understanding of Marx's argument is the following: Transportation does not alter the quantity/quality of products significantly. But, products need to have a use-value in order to also have an exchange value. Thus, an orange for example produced in Spain has an exchange value in UK, only if it arrives in UK, so that it can be bought and consumed. So, transportation adds value to the products, and this is done (as with all products) partly from the labor value (transportation workers) and partly from the means of production (means of transportation: airplanes, ships, cars, trucks...).

Since transportation adds value to the product, and since price is a mirror of value according to Marx, then we can say that transportation tends to increase the price of the product (which is a logical assumption on average).

Regarding distance, I am not sure if that is explicitly mentioned, but again we can deduce from the previous that: bigger distance --> more transportation needed (given constant technological advancement in a period) --> more transportation labor needed + more transportation means of production --> more added value --> tendency for bigger price.


Why is it that when a bank provides loans to people, it is considered a 'liability' on the bank's balance sheet? (Khan Academy Video-Related) I am watching the Khan Academy Banking Video Series:
(https://youtu.be/On3c86V5A_E) .
(https://www.khanacademy.org/economics-finance-domain/core-finance/money-and-banking/money-and-banking/banking-and-money/v/banking-7-giving-out-loans-without-giving-out-gold?modal=1)

If a person who is being loaned money opens a checking account at the very same bank that loaned it to him.. why is that a liability on the bank's balance sheet? Aren't liabilities only what the bank would owe to others (such as its customers who save their money in the bank and are hence loaning it to the bank?)...........
I understand that the concept of making it so that the money never has to leave the bank means that the person taking a loan from the bank just opens up a digital checking account, etc.. but Sal never made a mention that technically that checking account isn't a liability even if he's drawing/representing it on the liabilities-side of the balance sheet (the righthand side)

I would've posted a question in the khan academy website's comments, but these videos are quite old (and so are the comments) and from my previous experience my questions don't get replied on in those type of contexts on the khan academy website. Thus, I am asking on here.
.....
If this is the wrong place to post this, please let me know and direct me to where it should be posted. Thank you so much for your time, everyone!

A checking account is considered a liability because bank essentially owes the amount on the checking account to the owner of the account.

If a bank loans someone money that is considered an asset on the banks balance sheet but if that person decides to open checking account at the same bank checking account becomes liability. The loan will still be recorded as an asset on balance sheet but the new checking account is still considered a liability because the money in it belong  to the borrower.

Also purpose of accounting is to track all asset and liability and equity changes. Even if a company would borrow money to company it itself owns  - which is more or less economically just juggling money around, it needs to be recorded in accounting.


Empirical justification for a constant-returns-to-scale production function in the Solow model I am currently learning the Solow model of growth, where a constant returns to scale production function for the economy is assumed. I realise that the assumption makes further analysis much simpler, by allowing us to write output per unit productive labour simply as a function of capital per unit productive labour.
Apart from the mathematical convenience, is there any empirical reason for evoking the CRS assumption? Where can I read studies that have tested this claim?

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


Conforming mortgage loan limits by county and year US law restricts the Government-sponsored enterprises (GSE) Fannie Mae and Freddie Mac to purchase only those residential mortgages smaller (by origination balance) than a limit, the "conforming loan limit". This limit is generally speaking fixed by the number of units in the residence and the year of origination (or purchase by the GSE). For example in 2015:

However, in some counties, where the median sale price of homes is high enough the limit is set higher and these are known as high cost counties. Additionally, limits are higher for Alaska, Hawaii, Virgin Islands (since 1992) and Guam (since 2001). The "jumbo conforming loan limit" for 2015 was:

Loans above these limits are known as "jumbo loans".

I would like to classify particular mortgages where I know the state, county, the year, the number of units, and the loan amount into conforming, jumbo conforming, and jumbo loans. Is there a cross-walk using this data available? If so, where might I find it?

For some years, like 2015, this is straightforward, just download the files from the FHFA website. But that site only has years 2008 - 2014. Unfortunately, conforming loan limits have existed since 1970 when Fannie Mae was authorized to purchase residential mortgage loans. And many of the important mortgage data sets go back long before 2008 including LPS, HMDA, and Corelogic. As such, I need a much longer time series for these limits.

In case this feels like a pure data request and not economics, please note that these limits are used in numerous economics and finance papers. To name a few:

The impact of the agencies on conventional fixed-rate mortgage
yields
The Interest Rate Elasticity of Mortgage Demand: Evidence from
Bunching at the Conforming Loan Limit
The Effect of Conforming Loan Status on Mortgage Yield Spreads: A
Loan Level Analysis
Does Regulatory Capital Arbitrage, Reputation, or Asymmetric
Information Drive Securitization?

and many more.

The high-cost area conforming loan limits within the contiguous United States were created by HERA (the Housing and Economic Recovery Act of 2008) and therefore do not exist prior to 2008. For periods prior to the enactment of HERA, just use the historical nationwide conforming limits. There was also a temporary bump to the conforming limits enacted a few months before the passage of HERA, in the Economic Stimulus Act of 2008.

The high-cost areas that you identify (Alaska, Hawaii, Virgin Islands since 1992, and Guam since 2001) had, by statute, high-cost area loan limits that were capped at, and as a result effectively always, 1.5 times the conforming loan limits for the contiguous US states. Somewhat amusingly, the legal text responsible for the cap at 1.5x is actually still in the law; the authority granted to the Director of the FHFA to modify conforming loan limits was just tacked on to the end. You can find this text at 12 USC § 1717(b)(2)(C).


Risk neutrality in single-agent choice problem under uncertainity Consider the following static single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. First, let nature draw a realisation $v$ of $V$ from $P_V$.
Then, let the decision maker (DM) choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the DM gets a payoff $u(y,v)$.

An optimal strategy of the decision problem above is $P_Y\in \Delta(\mathcal{Y})$ such that, $\forall y\in \mathcal{Y}$ such that $P_Y(y)>0$ and $\forall \tilde{y}\neq y$,
we have that
$$
\sum_{v\in \mathcal{V}} u_i(y,v)P_V(v)\geq \sum_{v\in \mathcal{V}} u_i(\tilde{y},v)P_V(v)
$$

Additionally, one can imagine that the DM can process some information structure $S$ to update her prior before choosing an action. In such case, we can use the notion of one-player BCE in Bergemann and Morris (2013,2016) to characterise the set of probability distributions of $(Y,V)$ that are predicted by the model while remaining agnostic about $S$.

Question:

Does the narrative above (and, hence, also the Bergemann and Morris framework) assume that the DM is risk neutral? Can it be reformulated also for the case in which the DM is risk averse?

No, risk neutrality is not assumed. Risk preferences are given by the concavity of  $u_i$ which is arbitrary in this setup.


How is project value enhanced by uncertainty in market payoff? I was reading about the concept of real options valuation. It says that ROV has shown that uncertainty in market pay-off enhances the project value (V). How is that? Will it be possible to explain that with an example?

Here by 'uncertainty' we mean stochastic variability of parameter distributions.

{V = f(performance,cost,time,market requirement,market payoff) }

In general, if you own an option, holding fixed the expected value of the underlying asset and the strike price of the option, the greater the variance in the distribution of the prices of the option the (weakly) more valuable the option is. Here is a graphical example of a call option at maturity. They grey line indicates the value of the option for each value of the underlying asset where the strike price is \$5. The blue diamonds indicate the payoffs for values of the underlying at 4 and 6. If each are equally likely (p=0.5) then the underlying has an expected value of 5 and the option of \$0.5 (the lower green square). This distribution as a standard deviation of 0.5. Increase the standard deviation to 1.0 but hold the expected value constant at 5 by  changing the two underlying outcomes to \$3 and \$7 (orange squares). The expected value of this option is 1 (higher green square), so the increase in variance of the distribution of the underlying asset has increased the value of the option even though the expected value of the underlying is unchanged.

The reason for this is essentially Jensen's inequality. We can model the value of an option as a convex function $f(x)$ where $x$ is the underlying. The value of the option is $g(x) = E[f(x)]$ This implies $g(x)$ is increasing in the variance of $x$. We can see this because when we do the Taylor expansion of the function f : $$g(x) \approx E[f(x_0) +  (x - x_0)\cdot f'(x_0) +  (x - x_0)^2 \cdot f''(x_0)]$$

Because of the convexity, the second order term $f''(x_0)$ will be positive. The additive part $E[(x - x_0)^2 \cdot f''(x_0)] = E[(x - x_0)^2] \cdot f''(x_0)$ is increasing in the variance $E[(x - \bar{x})^2]$ so the value of the option is increasing in the variance.


Why is the definition of opportunity cost "the benefit of the best alternative choice" but not "the benefits of all alternative choices combined"? What is the rationale behind it?

This is because it does not make sense to compare chosen options to all other alternatives that are mutually exclusive. For example, if you have 5 choices A, B, C, D, E, you can choose one, and C is the most preferred option it does not make any sense to compare picking C to jointly picking A, B, D and E. Only one of the other options can be pick so you should use best of them for comparison


Do Keyensians believe that it's wrong for households to aggresively save money? Many Austrian/Neoclassical websites argue that Keynes and Keynesians demonise saving and encourage individual people to spend rather than save.  Someone (not an economist) recently told me that I need to feel guilty for my frugal lifestyle and that I'm being unpatriotic and should spend more money to help the economy.  But I'm not convinced that this is what Keyensians actually believe, as opposed to a straw-man used by the anti-Keynesians.  I thought the point was not that it's damaging to the economy for individual people to save their money as aggressively as they want to, but rather that if there is a paradox-of-thrift type situation during a recession, the government should take corrective action to reflate the economy.  And I also thought that Keyensians believed that a higher household savings rate is always good in the long run, and that the debate was specifically about what happens during a recession.

Is my friend right or am I?  Is it really true that there is any significant school of economists who would think that individual people saving their money prudently was bad for the economy and that such people ought to feel guilty for being savers?  Or can I carry on saving aggressively guilt-free?

The issue is more complex than your friend suggest and he is not completely right but at the same time there is a kernel of truth to what he is claiming. There are differences in the effect of savings short and medium and long run. However, to fully explain this I will have to use some math to ground the reasoning and make everything consistent.

Consider simple closed economy. By definition in any closed economy the total output/income $Y$ must be equal to consumption $C$, investment $I$ and government spending $G$. This is just accounting and will always hold so we have:

$$Y = C+I+G$$

Now we know from basic economics that consumption must be some positive function of disposable income $C=C(Y_d)$ with $C'>0$. This is just common sense if you have more money avaiable you will consume more goods and services in general. This is consequence of the basic non-satiation assumption - more is always better on which most economics builds on. To make things simple let us assume that the relationship is linear so we have:

$$C=c_0 + c_1Y_d$$

where $c_0$ is some basic consumption that you have to undergo irrespective of income to just stay alive and $c_1$ the marginal propensity to consume, i.e. the share of income you use for consumption as opposed to saving. Now by definition disposable income is the income after taxes $Y_d = Y-T$. So in fact our consumption function turns into:

$$C= c_0 + c_1 (Y-T)$$

Second we will hold investment constant which reasonably holds in the short run (but not necessary in the medium long run) as most investments are not very liquid so we have $I= \bar{I}$. Government spending will be treated as completely exogenous (determined outside this model) so it stays $G$.

Now to solve this model we can just substitute consumption function $C$ into the first equation which gives us:

$$Y = c_0 + c_1 (Y-T)+\bar{I}+G$$

Now solving for $Y$ using basic algebra we get the following:

$$Y = \frac{1}{1-c_1} \left(c_0  +\bar{I}+G-c_1T\right)$$

Now this last equation is all you need to understand the paradox of thrift. First consider the second part of the equation $\left(c_0  +\bar{I}+G-c_1T\right)$. This part is called autonomous spending because it does not depend on income as $c_0$ is basic consumption of necessities  $\bar{I}$ is fixed investment and $G$ and $T$ is government spending and taxation which is exogenously given so in this model it wont change.

So now you must be wondering where is the effect of saving on income? Well it hides in the first term:

$$\frac{1}{1-c_1} $$

recall $c_1$ is the share of income you spend on consumption - since you can only consume or save $(1-c_1)$ is the share of income saved.

Now its trivial to see that as $(1-c_1)$ increases the $Y$ must fall holding everything else constant. So your friend is definitely right that in the short-run saving more - meaning saving larger proportion of your income not just saving more in dollar terms - leads to lower income.

However, here we set investment constant, which is reasonable in the short run, but not in the long run. Just as a matter of accounting identity the investment must be equal private $S$ and public $T-G$ savings so we have:

$$I=S +T-G$$

So increasing saving also increases income. Furthermore, saving is also necessary for economic growth. The reason for this is that economic output depends on factors of production like capital and labor. For simplicity lets focus just on capital. We know that the more capital economy has the more it can produce so we know that $Y = F(K)$ with $Y'_k>0$. So the national income increases with amount of capital avaiable.

Now how do you get more capital? By saving! Again just by pure accounting identity the stock of capital tomorrow $K_{t+1}$ must be equal to capital today net of depreciation plus investment:

$$K_{t+1} = (1-\delta)K_t + I_t$$

We already know $I=S+T-G$ lets set taxes equal to gov. spending to simplify we get $I=S$. Now going back to the original equation on consumption since $c_1$ is marginal propensity to consume $1-c_1$ must be marginal propensity to save so we can replace $I$ by savings which are $S=(1-c_1)Y$ to get

$$K_{t+1} = (1-\delta)K_t + (1-c_1)Y$$

So in the long run you get more capital accumulation and economic growth when propensity to consume $c_1$ is small or consequently when share of income saved $1-c_1$ is large. So your friend is definitely not correct if he claims that being thrifty is always bad for the economy or that (New) Keynesians claim that.

PS: This was not part of your question but note that while some pundits still use words like Neoclassical and Keynesian as synonyms for free market pro saving economists and pro more government regulation spending economists respectively, this distinctions no longer hold. The current mainstream in economics is  Neoclassical synthesis which is combination of Neoclassical micro models and Keynesian macro models as well as some other ideas. So nowadays words like Keynesian or Neoclassical dont carry the same meaning. For example, Mankiw is a New Keynesian but he is small government, pro saving, conservative/libertarian while Stiglitz work is neoclassical and he is pro big government, pro spending, liberal. The reason for this is that both the neoclassical and keynesian models are highly nuanced and the pro-spending vs pro-saving debate depends heavily not just on science but also on personal values, the ways how a person discounts future benefits of economic growth and all this discussion gets more complex once you include inequality and philosophical discussions around that.


What effect would removing the lowest denomination currency have? Here in New Zealand our lowest denomination coin is 10c after the 5c was removed about 10 years ago. I think generally the public is pretty happy about the move.

It's quite a common argument to remove the penny from circulation in the United States. The main argument being that pennies are expensive to produce, are virtually worthless, and tie up time when they're given as change. The argument is that they don't actually serve the purpose of currency very well in terms of easily transferring value, because their value is so small for the effort of using them.

The question I have is, what economic effect would removing the smallest denomination currency have?

Things to consider are:

Immediate rounding on smaller items (eg. newspapers).
Less piggybanking as pennies are no longer lost or stuck in collection jars.

I guess the question is whether removing lowest denomination coins has an inflationary effect.

The practice of rounding to the smallest currency unit is called Swedish rounding. I know of a few papers that discuss likely economic concequences.

Eliminating the Penny from the U. S. Coinage System: An Economic Analysis Lombra (2001)

Time to Eliminate the Penny from the U.S. Coinage System: New Evidence Whaples (2007)

Pennies, Pricing, and Rounding: Is All the Relevant Analysis in? Lombra (2007)


Why do we change the base year? Never, never, understood this.

I know what Nominal and Real GDP is, I know how the base year calculations work, my question still remains.

The idea behind real GDP that makes most sense to me is the one Smith came up with, which was the first conception of a GDP anyway. The point of the GDP seems to be that the amount of stuff people consume is a decent indicator of how well the country is doing. It's hard to give a small figure that can be grasped quickly of the amount of stuff people consume in units of xyz goods so we just give them weights equal to a price and add them together. So the weights clearly seem to indicate how much more "consumptiony" one good is compared to the other, I guess the prices are a decent indicator of how "important" one good is relatively (debatable).

So we take these weights (prices) in one year and say "we'll use these" from then on. So an increase in the GDP can only ever indicate people consuming more stuff.

But if we keep changing base year prices, the Real GDP get's a boost despite people not consuming more stuff, it's even possible people consume less stuff and the Real GDP still increases, only because people are throwing more money at goods, probably because the central bank threw more money into the economy.

So what exactly is this change in base year supposed to capture?

The change in base year keeps the dollar values meaningful to contemporary readers.  If we still used, say, 1900 as the base year for US GDP, the dollar values for today would be so tiny they would essentially be meaningless 'GDP units', which would make it harder for people to visualize what the numbers really mean.

Note that, when the base year changes, historical GDPs go up, not just going-forward GDPs.  So the 1900 US GDP, in 'constant' 2010 dollars, is about 10x as high as the NGDP for that year, to reflect inflation over the 20th century; but that number does a much better job of conveying to readers living in 2016 what conditions in the US in 1900 were really like than using 1900 dollars would.


Prove that $u$ is a utility function for $\succsim$ If X is finite, define this function $u : X \rightarrow  \mathbb{R}$ by $u(x) = |\{z\in X:z \prec x \}|$. Prove that $u$ is a utility function for $\succsim$.

Is it sufficient to prove that the relation is transitive and complete?
By Lemma:  If $\succsim$ has a utility function, then it is transitive and complete.

You're asked to prove that $u(x)\ge u(y)\;\Leftrightarrow\;x\succsim y$ for any $x,y\in X$, where $u(x)=|\{z\in X:z\prec x\}|$, i.e. the utility of $x$ is measured by the number of other alternatives that rank strictly below it.
Since $X$ is finite, let's suppose without loss of generality that $X=\{1,2,\dots,N\}$ where $N$ is some finite number.

I'll prove the case in which there is no indifference among the alternatives, say, $1\succ2\succ\cdots\succ N$. I'll let you finish the proof by establishing the case where there are indifferences among subsets of alternatives.

Step 1. Establishing $u(x)>u(y)\;\Rightarrow\;x\succ y$.

Suppose $u(x)>u(y)$. By the definition of $u$, the number of alternatives strictly worse than $x$ is larger than the number of alternatives strictly worse than $y$. If $y\succsim x$, this would simply contradict the previous statement. Hence, we must have $x\succ y$.

Step 2. Establishing $x\succ y\;\Rightarrow\;u(x)>u(y)$.

Suppose $x\succ y$. Since we assume no indifference among alternatives, the set of strictly worse alternatives to $x$, $\{z\in X:x\succ z\}$, must contain more elements than the set of strictly worse alternatives to $y$, $\{z\in X:y\succ z\}$. In other words, $|\{z\in X:x\succ z\}|>|\{z\in X:y\succ z\}|$. Therefore, we obtain $u(x)>u(y)$ as a result.

Taken together, steps 1 and 2 demonstrate that $x\succ y\;\Leftrightarrow\;u(x)>u(y)$ for any arbitrary $x,y\in X$.


Why did the Dollar weaken after interest rate increase? Yesterday the Fed increased the interest rate of the Dollar yet it's value decreased against various currencies (Source).

According to this article it should be the other way around :

  In economic theory, if the interest rates in one country increase, then the currency value of that country will increase as a reaction. If the interest rates decrease, then the opposite effect of depreciating currency value will take place.

Source

So why did the value of the Dollar decrease?

It is correct that higher rates should strengthen the dollar... IF

the increase in interest rates is unexpected and
all else equals.

I haven't followed this particular rate hike, but here is a basic explanations why the dollar might weaken after an increase in interest rates:

The rate hike was widely expected and already priced in by the market, As a result, it didn't have much of an effect on the exchange rate. Keep in mind that the dollar can fluctuate quite significantly even on days without a Fed decision.  The move you refer to might be one of these fluctuations and have nothing to do with the rate hike. Or a small move due to the rate hike might have been overshadowed by a bigger move due to a different factor.

It's worth keeping in mind that it's not the events themselves that matter for exchange rates (or stock prices), but how these events turn out relative to expectations -- markets are forward-looking.


Simple Keynesian model It is given that economy is at equilibrium at $ Y_0 $ = 1000. If Government undertakes a fiscal change so that tax rate $t$, increases by 0.05 and government spending increases by 50, will the budget surplus go up or down?

I tried doing it this way:

$$Y = C( Y - tY ) + I + G$$

Differentiating the above equation I got

$$dy = \frac{di+ dg - c'y dt}{1- c'(1-t) }$$

Where $c'$ is mpc, $t$ is tax rate.

Also,

$$BS = ty-G- TR$$

So,

$$dBS = tdy +ydt-dg-dtr$$

Now, $Y=1000, dt = 0.05, DG= 50$. Plugging these first into $dy$ and then using $dy$ in $dBS$, I got the following:

$$dBS = \frac{50t(1-c') }{(1-c'(1-t) }$$

Therefore the budget surplus goes up.
Is my method correct? Also, what could be the economic intuition behind this?
This is not a homework assignment. I am currently preparing for an upcoming entrance exam and was trying out a few back questions from dfs for conceptual clarity.

Your method is correct. This however does not have to be the case in general. Your have a budget surplus simply because your government spending does not rise too much relative to your tax revenue collection.

Consider the following with MPC = 0.3, t = 0.2 and Y = 1000 and dt = 0.5:

If you increase spending too much, you will end up with a budget deficit.

There could be other factors as well, for example, the rate of taxes you have in the economy when the policy changes are made. For the same parameter set mentioned above, with dG and t free, you have (t on vertical axis and dG on horizontal axis):

There are some dG,t pairs which generate surplus/deficit/balanced budget.


Resources to derive economic forecasts What publicly available resources are there, with sample code, that can be used to build my own macroeconomic model? A search on Github shows that some people have posted code, but I think we can do better.
Ultimately, I am looking to use this model to forecast key economic indicators such as components of GDP and the labour market.
I am method agnostic, so would be happy to see VARs, DSGEs and machine learning examples.

For forecasting with VAR in R there are some good tutorials on econometrics with R. This tutorial from Justin Eloriaga also helped me when we had to make VAR for our quantitative macro assignment. PenState also has good sources for econometrics, here are their sources for VAR.


What is virtual valuation? I’m reading through a textbook in auction design when it describes a term, virtual valuation

$$\phi_i(v_i) = v_i - \frac{1 - F(v_i)}{f(v_i)}$$

where $f$ is the pdf of a bidders valuation and $F$ is the corresponding cdf.

It is described as the price you want to charge minus the cost of not knowing $v_i$. Can someone explain how the cost was derived from the probability distributions? Note I’m not an economics student by training.

Suppose you face a single buyer whose willingness to pay, $v$, is distributed according to $F(v)$. If you charge a price $p$, he will buy if and only if $v>p$, leaving you with expected revenue of $$r(p)=\Pr(v>p)p=[1-F(p)]p.$$

Let's maximise revenue by computing an FOC:

$$r'(p)=1-F(p)-F'(p)p=0.$$

We can rearrange this as

$$\phi(p)\equiv p-\frac{1-F(p)}{F'(p)}=0$$
i.e., the 'virtual valuation' should be zero.

If we return to $r'(p)$ and think about what the individal terms mean, we can see where the "cost" you ask about comes from: a unit increase in price causes an extra unit of revenue in the fraciton $1-F$ of the time that the buyer is willing to buy, but reduces the liklihood of him buying by $F'$. This is the fundamental trade-off that a seller who doesn't know the buyer's willingness to pay must make.

If you know anything about standard monopoly theory then this setup should be quite familiar. Usually, when we look at a profit-maximising monopolist with demand $D(p)$ and zero marginal cost we solve
$$\max_p D(p)p\iff \underbrace{D'(p)p-D(p)}_{\text{marginal revenue}}=0.$$
In an auction-like setting, the 'demand' $1-F(p)$ is simply the probability of the buyer being willing to pay $p$.


How to calculate change in GDP using valued added? Suppose that Roots buys 5 million dollars in leather produced this year in Alberta and 1 million dollars in zippers produced last year in Toronto to make 9 million dollars worth of leather jackets in Montreal this year. They sell 7 million dollars worth of jackets this year.  What is the total contribution to Canadian GDP this year contained in these jackets? (million means million dollars)

A) 5 million   B) 6 million C) 7 million    D) 8 million    E) 9 million

Here the answer is D), which I don't quite understand. We calculate contribution to GDP through value added. In my opinion, the final sale, which is 7 million dollars, should be the sum. Could someone explain?

You're dismissing the value added by the companies producing the leather and the zippers. The value added method calculates the total value added to goods and services at each step of production.

The value added by Roots is 9-5-1=3 (total value of final products - value of resources) and then you add the value of produced leather in Alberta (5) and zippers in Toronto (1).

3 + 5 + 1 = 9.

All produced goods and services are accounted for with this method, not just the ones sold to consumers (the jackets were produced even if they weren't sold), so the 7 million figure is irrelevant in this.


If Trickle Down Economics Works, Then Shouldn't "Trickle Up" Economics Work as Well? A lot of controversy in economics on the political stage comes from trickle down economics and whether it even works or not. The idea is that if we give sufficient tax cuts to corporations, they'll use that money to expand and create more jobs and just overall bettering the economy for everyone. My question is, why can't that work the other way around? If you give the working class just as much in tax cuts, they'll eventually spend that money to where it does end up at an American corporation. And in my opinion, this way is more capitalist, because the consumers decide what businesses get what percentage of their tax cut, instead of just multiplying the corporation's wealth.

As you learn more about economics you should gradually come to distinguish the vocabulary of economics from that of politics (which is also not the same as political science). When politicians or political activists speak, they use a variety of terms that are very vaguely defined, and sometimes outright derogatory. They are not actually meant to convey exact economic concepts or ideas, but more like feelings. "Trickle down economics" is an example. "Neo-liberism" is another example. You won't find any formally trained economists discussing "trickle down economics" or "neo-liberist" policies simply because these are not properly defined concepts, i.e. they don't really mean anything specific. In order to discuss productively with economists and generally any scholar from a discipline that is based on the scientific method (biology, physics, statistics, etc.), it is important to acquire the right vocabulary. The dictionary or an encyclopedia is usually a good place to start.

P.S. it is not a coincidence that controversy tends to stir around topics that are not discussed in scientific terms - it is difficult for people to agree on ideas if they can't articulate them :)


Does Modern Monetary Theory (MMT) provide a useful insight into how to manage the economy? According to advocates of MMT, the primary risk once the economy reaches full employment is inflation, which can be addressed by gathering taxes to reduce the spending capacity of the private sector. Deflation is not seen as a risk because the 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).
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?
https://en.wikipedia.org/wiki/Modern_Monetary_Theory

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.


How do I construct the score process of a Markov model and verify that it is a Martingale? The following is a specific question that is useful for demonstrating a general idea.

Consider the following autoregressive model:
$$
X_{t+1} = \alpha_0 + \beta_0 (X_t - \alpha_0) + W_{t+1},
$$
where $-1 < \beta_0 < 1$ and $W_{t+1}$ is distributed as a normal with mean zero and variance one.

How should I go about constructing the bivariate score process associated with the parameters $\alpha_0$ and $\beta_0$? How can I verify that it is a martingale?

Progress:

I would begin by constructing the log-likelihood process as follows (conditioning on $X_0$):
$$
\ell_t(\theta \mid \textbf X) = -\frac t2 \ln(2 \pi) - \frac 12 \sum_{j=1}^t
   (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0))^2.
$$
Then, the score process can be constructed as
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0)) \\
 \sum_{j=1}^t (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0)) (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Is this correct? How do I proceed?

The derivation of the score process is correct. To verify that the process is a Martingale, recall the definition. It becomes clear that if we substitute $W_{t+1}$ back into the equation
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t W_j \\
 \sum_{j=1}^t W_j (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Because $W_{t+1}$ are Normal with mean 0 and variance 1 (I'm assuming they're iid), then
$$
E[s_{t+1} \mid s_t ] = s_t + E \left [
\begin{matrix}
(1 - \beta_0) W_{t+1} \\
W_{t+1} (X_t - \alpha_0)
\end{matrix} \middle |
s_t
\right ]
= s_t
$$
and we are finished.


Government spending/borrowing difference/ national debt increase disparity My background in economics is non-existant, I'm just doing some research before the UK general elections and am having some difficulty in understanding certain data.

Assuming the figures found in this and this link are accurate, the UK government spent £714 Billion in the 2014 fiscal year. They generated £619.8 billion in public revenue in the same year. I'd assume the difference between these figures, £94.2 billion, is what they needed to borrow to make up the difference. But the website lists the UK national debt to be of magnitude £1258 billion in 2014 and £1185 the year before, meaning that only a total of £73 billion was borrowed in reality.

Moreover, the HM treasury PESA (page 19) gives public sector current expenditure in 2014 to be ~£666 billion, which is over £45 billion more than what they generated from public revenue. Even if the statistic of £619.2 billion is incorrect, and the government actually generated exactly the £666 billion spent, the difference £714-£666 billion=£48 billion is less than the amount by which the national debt rose in the 2014 fiscal year. I can't get the figures to match up!

Finally, from what I've learned today, government borrowing works by having banks etc. purchase bonds which the government pay interest on, and then in a certain number of agreed years pay the purchaser back what they paid. The statistics for government spending in the second link above show £47.4 billion being spent on interest. However, I can't find any account of the original loans being paid back. That is, surely some of the bonds the government sold should mature each year, requiring them to pay the owner of the bond the original amount they sold it for?

Apologies for my economic naivety.

There are many ways of calculating public sector finances.

One approach is similar to corporate accounting and the UK Treasury publishes Whole of Government Accounts.

Those are different in construction to the Public Sector Finances statistics published by the Office for National Statistics, who follow international National Accounts methodology

In respect of your question of what happens when these bonds mature and are in effect rolled over, does not lead to a difference in treatment: in both cases the repayment is not treated as government expenditure but of finance, and if the new bonds cover the same amount as the old bonds then in effect nothing changes and Government debt stays the same.

The translation from expenditure and revenue to borrowing and changes in debt is not simple.  The ONS produced an explanatory article. Separately the ONS published this diagram as part of a bulletin last year which illustrates some of the calculations they did for the 2015/16 financial year:


Gini coefficients and median / mean income ratio Simple question: If the GINI coefficient (market incomes) goes up, is it a necessary feature that the mean income rises relative to the median?

No, since you can change the Gini coefficient by altering the distribution of incomes in the top half (keeping the total the same) without changing the mean or median income; you can also change the Gini coefficient by altering the distribution of incomes in the bottom half without changing the mean or median income.

Compare the Gini coefficients of {2,2,2,2,2,6,10,10,10,10,10} and of {1,1,1,1,6,6,6,6,6,6,26}: the former will have a Gini index of about 0.33 while the latter will have a Gini index of about 0.46 though the mean and median do not change (6 each)


How much real value has the stock market lost since Feb. 2020? In this question I'm using the Vanguard Total Stock Market Index Fund(VTSMIF), which for my purposes I find to be a sufficient indicator of the US market as a whole. (Please explain why this isn't an accurate indicator is that's the case.) I'm also using strictly the M1 money supply, as it most closely correlates to the stock market(and another source), and also because the increases in M2, M3, and MZM have been comparable over the time frame referenced.
In February, 2020 the VTSMIF was at $\\\$164.96$. At that time the M1 money supply was roughly $\\\$4,003,000,000,000.$ As of June 2020 the M1 money supply had been increased to $\\\$5,209,900,000,000.$ That's a roughly 30% increase in the money supply. At the same time, the VTSMIF had only recovered to $\\\$154.75$, or a ~6% decrease.
So we're looking at a 30% increase in money supply that, typically, directly ties to stock prices. I also don't think population increase in the time frame is a material factor, but please correct me if I'm wrong. Even if we assume no further increase in M1 money supply since June, and use today's VTSMIF value of (currently) %\$171.21$, we're looking at a ~4% increase in value, when historically we'd expect an increase in value closer to the increase in money supply.
I know there are myriad factors in play, these are unprecedented economic times, etc etc. But is it incorrect to say that despite slight upwards movement in USD value since February 2020, the stock market has actually lost a significant amount of real value, due to the drastic increases in money supply over the time frame? Assuming it's correct to state that, how would one calculate the actual loss of value?

Increases in the money supply do not affect the real value of investments unless they lead to inflationary increases in the (quality-adjusted) prices of goods and services. Which this increase has not done. If you choose to measure the value of your investment as the proportion of the M1 money supply that it represents then sure, it has been devalued, but that’s a rather pointless and futile way to measure the value of investments.


Principal-Agent Problem with Two Agents Consider the following principal-agent problem. You are a manager in charge of two workers, indexed with $i=1,2$. Each worker is asked to perform a particular task, and each can either work hard at the task or loaf (i.e., each worker can choose his action $a_i\in\{a_h,a_l\}$).
The value to you of each task will be either $\\\$10$ or $\\\$0$. The value is determined in part by how hard the worker works, but there are also random factors not within the control of the worker:
$\bullet$ If the worker works hard (i.e., $a_i=a_h$), the probability that the value to you of his task is $\\\$10$ is $0.7$. If the worker loafs (i.e., $a_i=a_l$), this probability is $0.3$;
$\bullet$ Moreover, there is a correlation between the outcomes of the two tasks. If both workers work hard, then there is $0.6$ joint probability that each will produce a $\\\$10$ outcome;
$\bullet$ If both loaf, then there is $0.2$ joint probability that each will produce a $\\\$10$ outcome;
$\bullet$ If one loafs and the other works hard, then there is a $0.25$ joint probability that each will produce a $\\\$10$ outcome.
Each worker has a utility function $U(w,a)=\sqrt{w}-a$, where $w$ is the amount of money received, and $a=0$ for loafing and $a=0.8$ for working hard. The two workers have the same utility function, and each has a reservation utility level of $1$ (i.e., the worker's expected utility must be at least $1$ to get the worker to take the job). You are risk-neutral. The problem is as follows:

Suppose that you wish to offer contracts that will induce both to work hard: Each worker must be induced to take the contract; each worker, assuming that his fellow worker is going to work hard, must be induced to work hard. What is the optimal contract to offer?

I'm familiar with the standard principal-agent problem, but I cannot understand the situation of this problem and cannot set up the conditions for individual rationality (IR) and incentive compatibility (IC).
(For reference, this is actually Problem 16.3 in Kreps' textbook "A Course in Microeconomic Theory." Any suggestion or hint would be much appreciated!)

While it's not explicitly mentioned in the question, it seems safe to assume that the manager gets to observe separately the outcome of each worker, i.e. the value of $v_i$ for $i=1,2$. If this is the case, the IR and IC conditions depend on the exact terms of the contract.
For example, suppose the manager conditions each worker $i$'s wage on both $v_i$ and $v_{-i}$. That is,
\begin{equation}
w_i(v_i,v_{-i})=
\begin{cases}
w^{11}&\text{if $v_i=10,v_{-i}=10$}\\
w^{10}&\text{if $v_i=10,v_{-i}=0$}\\
w^{01}&\text{if $v_i=0,v_{-i}=10$}\\
w^{00}&\text{if $v_i=0,v_{-i}=0$}
\end{cases}
\end{equation}
Further, let $p_h^{11}$ denote the probability that both workers produce the high-value outcome (indicated by the superscript $11$) when $i$ works hard (indicated by subscript $h$) and the fellow worker $-i$ also works hard (we suppress the second $h$ in the subscript as we always assume that $-i$ works hard). In more compact notations,
\begin{equation}
p_h^{11}=\Pr(v_i=10,v_{-i}=10 \mid a_i=a_h, a_{-i}=a_h).
\end{equation}
Probabilities $p_h^{10},p_h^{01},p_h^{00},p_l^{11},p_l^{10},p_l^{01},p_l^{00}$ are similarly defined.
Then, the IR constraint for worker $i$, assuming worker $-i$ is going to work hard, would be
\begin{equation}
p_h^{11}\sqrt{w^{11}}+p_h^{10}\sqrt{w^{10}}+p_h^{01}\sqrt{w^{01}}+p_h^{00}\sqrt{w^{00}}-a_h\ge 1.
\end{equation}
The IC constraint for worker $i$, again assuming worker $-i$ is going to work hard, would be
\begin{multline}
p_h^{11}\sqrt{w^{11}}+p_h^{10}\sqrt{w^{10}}+p_h^{01}\sqrt{w^{01}}+p_h^{00}\sqrt{w^{00}}-a_h\\ \ge p_l^{11}\sqrt{w^{11}}+p_l^{10}\sqrt{w^{10}}+p_l^{01}\sqrt{w^{01}}+p_l^{00}\sqrt{w^{00}}-a_l.\end{multline}
From the information given by the problem, you should be able to figure out the eight probabilities.
If the wage is conditioned simply on $v_i$, or
\begin{equation}
w_i(v_i)=
\begin{cases}
w^1&\text{if $v_i=10$}\\
w^0&\text{if $v_i=0$}
\end{cases}
\end{equation}
then the IR and IC for each worker would be
\begin{align}
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a_h&\ge 1 \tag{IR}\\
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a_h&\ge 0.3\sqrt{w^1}+0.7\sqrt{w^0}-a_l. \tag{IC}
\end{align}


Have social accounting matrices and input output models fallen out of fashion? if so, why? My perception may be uninformed but 'worldwide' I believe collection of official data on input output tables of economies dont seem to be keeping up with the times. Im aware that compilation of such data sets is difficult hence the large time gaps between releases but I would think that 5 years or so would suffice. examples of countries where such data are up to date would be much appreciated

National Statistical Institutes do still compile IO tables (see http://ec.europa.eu/eurostat/web/esa-supply-use-input-tables for EU versions, although these are 5-yearly as well).  They're generally more interested in producing the Supply and Use tables (which are then transformed into input-output tables) due to their usefulness in balancing the 3 measures of GDP.

There are also attempts by groups such as WIOD to produce world input-output databases.

Part of the reason for the long delay in producing IO tables following the reference year is simply the very large quantity of data required to compile them, plus the time taken to investigate and resolve issues highlighted by the confrontation of the many various data sources.

Social accounting matrices are also still produced, and frequently used to calibrate things like CGE models rather than being used directly via SAM multipliers.

Most of the work in input-output modeling I see these days relates to environmental modeling, but this is by no means exclusive. The International Input-Output Association (www.iioa.org) runs a publication called the Journal of Economic Systems Research which can point you at some current work.

The UN are also working on a handbook foot IO and extensions due some time this year, which aims to provide a range of options for countries with smaller or less well established statistical institutions to construct tables, as well as more established ones.


How can trailing and forward P/E ratio of main stock markets show such huge difference I've recently found this web page, listing the average P/E Ratios of main stock markets:

http://www.zutosmoney.com/

Here I reproduce just a small extract:

INDEX - Country - Trailing PER - Forward PER
CAC40 - France - 24.3 - 13.2
IBEX35 - Spain - 21.4 - 12.7
NASDAQ - US - 34.5 - 18

I've noticed two things:

Most trailing P/E ratios are quite high: I understand that 'normal' values should be in the range of 16 - 20.
There is a very big difference between trailing P/R and forward P/R. I understand that trailing is calculated using the actual earnings of past year, and forward is calculated with the companies' annual forecasts.

Am I right concluding that:

Stock markets are probably over-evaluated, considering the earnings in 2016?
Stock markets are possibly over-optimistic regarding the earnings in 2017?

[Edit: I've corrected my conclusions after answer from Klas Lindbäck, to avoid expressing too strongly my personal concerns about the earnings of companies in 2017]

Am I right concluding that:

Stock markets are probably over-evaluated, considering the earnings in 2016?
Stock markets are over-optimistic regarding the earnings in 2017?

No, you cannot draw that conclusion a priori. You can conclude that the market expects a big increase in profits in 2017. You can have an opinion about whether the market is over-optimistic, but unless you got divine insight, or have better models than they do, you cannot conclude that their forecast is wrong. Remember that the market is a collection of people with access to roughly the same information that you have access to.


Uniform price vs. pay-as-bid auctions in energy markets These two types of auctions are most commonly used in the energy trading markets. What would be advantages and disadvantages of each? And in the end, can we expect them to deliver similar outcome?
One, quite obvious disadvantage of PAB auction is the need to "guess the clearing price". If not all generators correctly predict the MCP, more expensive generators may be scheduled at the expense of cheaper ones, resulting in higher costs for the same payment by consumers. But what can be considered as a disadvantage of uniform price auction here? And what are the advantages of each auction type?
As for the second question, I think we can expect the same (or at least very similar) outcome when we consider the perfect competition? Under perfect competition all players would know what the marginal production unit is for any given level of demand, so that under pay-as-bid they would all bid at the marginal cost of that unit - achieving the same outcome they would obtain under uniform price auction by bidding at cost. How would the outcome change if we consider (bit more realistic) case of oligopoly?

I believe "Auctions of Homogeneous Goods: A Case for Pay-as-Bid" by Pycia and Woodward answers your questions theoretically. This is quite recent and their results are striking. They also briefly discuss some empirical insights.

The  pay-as-bid  (or  discriminatory)  auction  is  a  prominent
format  for  selling  homogenous goods such as treasury securities
and commodities.  We prove the uniqueness of its pure-strategy Bayesian
Nash equilibrium and establish a tractable representation of
equilibrium bids.  Building on these results we analyze the optimal
design of pay-as-bid auctions, as well as uniform-price auctions (the
main alternative auction format) allowing for asymmetric information.
We show that supply transparency and full disclosure are optimal in
pay-as-bid, though not necessarily in uniform-price; pay-as-bid
is revenue dominant and might be welfare dominant; and, under
assumptions commonly imposed in empirical work, the two formats are
revenue and welfare equivalent.