What is state space representation for DSGE modeling

This question is too broad as it stands. There is a longer answer already, but I think it is possible to deal with a core part of the question easily.

The concrete question is what is a state-space representation, if possible with some of the intuition, its uses in economics (control theory)

My background is in control systems theory, where the notion of a state space representation developed. Optimal control theory, including the Kalman filter, was developed mainly in the 1960s. Economists have kept using the core ideas, but migrated off to their own standards of mathematical presentation.
A state space model is a canonical representation of a mathematical system. The idea is to take a model derived from underlying dynamics, and re-cast it into a standard format, so that existing tools can be applied to it.
In control theory, the standard canonical form of a nonlinear state space system is:
$$
x(t+1) = f(t, x(t), w(t)),
$$
$$
y(t) = g(t, x(t), w(t)),
$$
$$
x(0) = x_0 \in R^n,
$$
with:

$x(t)$ an n-dimensional state vector,
$w(t)$ a vector of all external variables,
$y(t)$ a vector of measured variables.

A fairly typical simplification is that the system is linear and time invariant, where the description collapses to:
$$
x(t+1) = A x(t) + B w(t),
$$
$$
y(t) = C x(t) + D w(t).
$$
Typical examples of external variables that appear in $w$ include:

variables used to control the system (typically fixed by feedback rules),
variables that disturb the state variables,
noise that effects measurements.

The vector $w$ is often partitioned into groups corresponding to these cases, and the $B,C,D$ matrices are correspondingly partitioned.
Once we have the system in this format, we can then apply existing tools. A typical example is applying the Kalman filter to infer the state variable $x(t)$ from directly measured variables. Numerical toolboxes will give results based on the $[A,B,C,D]$ representation.
In economics, the variable $r^*$ is typically inferred by the use of a Kalman filter (E.g.,  see New York Fed r* page). This variable roughly corresponds to what was called “the natural rate of interest”, and cannot be measured directly. Instead, it needs to be inferred from measured variables. This example underlines why we need to differentiate between the state vector, and the variables that are directly measurable. The state vector includes all variables need for the dynamical description of the system, and not just variables that are measured.
(Note: control theory uses a somewhat unintuitive definition of “observable.” A variable is observable if its value can be inferred from measurements - even if it is not directly measured. Since observability is a mathematical requirement for the Kalman Filter to converge, this needs to kept in mind when reading documentation prepared by control engineers.)
As a final technical note, state space representations may not always exist. A key example in control theory are linear continuous time systems with a time delay. It would require an infinite number of states to represent a time delay, and so engineers are forced to either use frequency domain representations, or an approximation.


The division of the economy into sectors is arbitrary, meaning there is no universally accepted division of the economy. It is contextually dependent.


Given a rational $\succsim$ over a finite set $X$, show that there exists $x \in X$ such that $x \succsim y, \forall y \in X$

You are almost there, but I think you are starting in the wrong place. I think the key is as follows: we want to prove the claim for set $X=\{x_1,\ldots,x_n\}$.

Start with some $X'\subseteq{X}$ for which the claim is true. Such an $X'$ exists because the claim is true for the set $\{x_1\}$ (by completeness and reflexivity).

If $X'=X$ then we're done. If $X'\neq X$ then we proceed with induction:

Denote by $\bar{x}$ an $x\in X'$ such that $\bar{x}\succsim y, \forall\ y\in X'$.

Now construct a new set $X''=X'\cup\{x_i\}$, where $x_i\in X\backslash X'$. Because $\succsim$ is rational (preferences are complete) over $X$, we either have $x_i\succsim \bar{x}$ or $\bar{x}\succsim x_i$ or both. Thus, if the claim holds for $X'$ it also holds for $X''$.

We can repeat this inductive step until $X''=X$, which must happen after finitely many iterations because $X$ is finite.


Lagrangian: when to discount budget constraint?

The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".

So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).

The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.


Why is fractional reserve banking allowed?

I recommend a careful read of the Wikipedia page for Fractional-reserve banking because it seems you are confused about how fractional reserve banking works.
Banks only lend out money they have under fractional reserve banking and, in fact, strictly less than they have. Those funds are a mix of deposits, funds from other debt issuance, and bank equity. They don't lend out funds they don't have and they don't simply credit accounts at will in the way you suggest. When they credit a borrower's account with funds, they have to debit those funds from elsewhere on their balance sheet. So, lend-able funds go down as borrower's funds go up, and then the resulting loan is a new asset of the firm. While they do create money this way, they do so at the expense of these lone-able funds so the loaning out doesn't create new assets for the bank.


(i) $A$ & $B$

If player 1 play $A$ with probability $p$ and $B$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+4(1-p) = 4$
$E$ is $6p + 2(1-p) = 4p + 2$
$F$ is $6p + 4(1-p) = 2p + 4$

Since payoff from playing $F$ is more than the payoff from playing any other strategy for player 2, he will always choose $F$ in response to any mixed strategy of player 1 where 1 play $A$ with probability $p$ and $B$ with probability $(1-p)$. Therefore, $F$ is the best response of player 2 if 1 play a mixed strategy $A$ with probability $p$ and $B$ with probability $(1-p)$. However, given the strategy $F$ of player 2, player 1's best response is to play a pure strategy $A$. Therefore, there is no Nash equilibrium in which player 1 mixes between $A$ & $B$.

(ii) $A$ & $C$

If player 1 play $A$ with probability $p$ and $C$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+6(1-p) = 6 - 2p$
$E$ is $6p + 4(1-p) = 2p + 4$
$F$ is $6p + 0(1-p) = 6p$

Since payoff from playing $E$ is more than the payoff from playing $F$, he will never choose $F$ in response to any mixed strategy of player 1 where 1 play $A$ with probability $p$ and $C$ with probability $(1-p)$. Therefore, $F$ is never a best response of player 2 if 1 play a mixed strategy $A$ with probability $p$ and $C$ with probability $(1-p)$. Now let us consider the following cases for $p$ :

$0 < p < 0.5$ In this case player 2 has a unique best response and that is to play $D$, in response to which player 1 would like to play a pure strategy $B$, and not a mix of $A$ and $C$.
$0.5 < p < 1$ In this case player 2 has a unique best response and that is to play $E$, in response to which player 1 would like to play a pure strategy $C$, , and not a mix of $A$ and $C$.
$p = 0.5$ In this case player 2 is indifferent between playing $D$ and $E$, and if 2 mixes between them appropriately i.e. by choosing to play $D$ with probability $0.5$ and $E$ with probability $0.5$, it will leave player 1 with two best responses $A$ and $C$ and therefore, 1 can now choose to play $A$ and $C$ with equal probability. So, we've found a mixed strategy Nash Equilibrium in this case: Player 1 choose to play $A$ with probability 0.5 and $C$ with probability 0.5, and player 2 choose to play $D$ with probability 0.5 and $E$ with probability 0.5.

(iii) $B$ & $C$

If player 1 play $B$ with probability $p$ and $C$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing

$D$ is $4p+6(1-p) = 6 - 2p$
$E$ is $2p + 4(1-p) = 4 - 2p$
$F$ is $4p + 0(1-p) = 4p$

Since payoff from playing $D$ is more than the payoff from playing any other strategy for player 2, he will always choose $D$ in response to any mixed strategy of player 1 where 1 play $B$ with probability $p$ and $C$ with probability $(1-p)$. Therefore, $D$ is the best response of player 2 if 1 play a mixed strategy $B$ with probability $p$ and $C$ with probability $(1-p)$. However, given the strategy $D$ of player 2, player 1's best response is to play a pure strategy $B$. Therefore, there is no Nash equilibrium in which player 1 mixes between $B$ & $C$.


Resources to learn about banking and financial services industry

Assuming you want to study finance and banking from an economic perspective then common introductory texts are

Exchange Rates and International Finance from Copeland
Financial Markets and Institutions from Mishkin
Money, Banking and Financial Markets from  Cecchetti & Schoenholtz

Depending on whether you are interested in macro or micro perspective you will want to build upon the above with

The microeconomics of banking from Freiax
Intermediate Financial Theory form Danthine, Jean-Pierre
International Macroeconomics and Finance: Theory and Empirical Methods by Nelson C. Mark

The above-mentioned three intermediate/graduate  books also presuppose some knowledge of econometrics, microeconomics and mathematics so you might want to also first study that before delving into these.


Let $x_i(p, w_i)$ be consumer $i$'s (Marshallian) demand and $y_j(p)$ be firm $j$'s profit-maximizing supply at price $p$.
A competitive equilibrium is a price $p$ such that
$$
\sum_{i=1}^I x_i(p, w_i(p, e_i)) = \sum_{j=1}^J y_j(p).
$$

By strict concavity of $u_i$'s, consumer demands are functions, rather than correspondences. Assume also that $y_j(p)$ are functions.
There can be multiple goods---e.g. consumption good/time(labor)/etc.
Agent's wealth $w_i$ may depend on price $p$ and endowment $e_i$---e.g. in an Edgeworth box where demands are Slutsky demands at the endowments---or wealth transfer $w_i$ can be specified as part of equilibrium, subject to
$$
\sum_i w_i = p^T (\sum_j y_j(p) + \sum_i e_i),
$$
i.e. aggregate wealth equals total value of goods.

Aggregating Consumers
Under the assumption that $u_i$'s are strictly concave (and differentiable, say), one can define a representative consumer via a linear social welfare function:
$$
u_{rep}(x) = \max_{\sum_{i} x_i \leq x}\sum_i \frac{1}{I} u_i(x_i).
$$
Then $u_{rep}$ is also strictly concave. Let $u_{rep, l}$ denotes marginal utility with respect to good $l$, then by the Envelope and Lagrange Theorems,
$$
\frac{u_{rep, l}}{u_{rep, k}} = \frac{u_{i, l}}{u_{i, k}}, \;\; \mbox{for all $i$}.
$$
At a competitive equilibrium price $p$, the MRS of $u_{rep}$ at the aggregate
demand $\sum_i x_i(p)$ is the ratio of prices:
$$
\frac{u_{rep, l} (\sum_i x_i(p),w_i)}{u_{rep, k}(\sum_i x_i(p), w_i)} =  \frac{p_l}{p_k}. \quad(*)
$$
By concavity of $u_{rep}$, the FOC $(*)$ implies that the aggregate demand at $p$ is the optimal bundle for the representative consumer $u_{rep}$ at $p$:
$$
x_{rep}(p, \sum_i w_i) = \sum_i x_i(p, w_i).
$$

As you already started to say in the question, this is a general construction of a (local) representative agent under concavity assumption. By the First Welfare Theorem, CE is Pareto. Under concavity of $u_i$'s, Pareto allocations must arise from a linear SWF, which implies existence of a representative consumer.
In this specific instance, consumers receive equal social weights $\alpha_i = \frac{1}{I}$ because equilibrium wealth is equal across agents. In general, $\alpha_i = \frac{1}{\lambda_i}$ where $\lambda_i$ is consumer $i$'s Lagrange multipler. In particular, $\alpha_i$ would be increasing in agent $i$'s share of aggregate wealth.
If $u_i = u$ is homothetic, with possibly different levels of wealth across agents, one can simply take $u_{rep} = u$ trivially. (Concavity assumption not needed.)
Gorman aggregation, alluded to in the comments, is a slightly different context, I believe. Gorman aggregation says that if all wealth expansion curves are parallel at all wealth distributions, then aggregate demand is a function of aggregate wealth and one has a global representative agent. The notion of Gorman aggregation is independent of equilibrium or Pareto efficiency. Here in your question, the local representative agent depends on the initial CE allocation---e.g. different endowments distributions would lead to different social weights.

Aggregating Firms
As pointed out in the comments, to aggregate firms one just uses the CRS assumption and take $F_{rep} = F$. For a CRS firm, doubling the input means output also doubles, with the same marginal products. In equilibrium, the firm still makes zero profit. For any one of the $J$ firms, aggregate production $\sum_j y_j(p)$ would also be optimal at the same equilibrium price $p$.
At the given equilibrium price $p$, $p = \nabla F(y_j(p))$, the gradient vector of marginal products of firm $j$ for all $j$. By the CRS/homogeneous-of-degree-one property,
$$
\nabla F_{rep}(\sum_j y_j(p)) = \nabla F(y_1(p)) = p,
$$
which tells you $y_{rep}(p) = \sum_j y_j(p)$.
Putting it together, given CE with allocation
$\{(x_i(p))_i, (y_j(p))_j \}$ and price $p$,
the allocation $\{\sum_i x_i(p), \sum_j y_j(p) \}$ and price $p$
is a CE for the economy with one agent $u_{rep}$ and one firm $F_{rep}$.


Why are many railroad rights-of-way held in easement rather than in fee simple?

Though these are only educated guesses (contract law is not my forte) I'd suppose that there might be a few reasons why this is:

It's best able to handle the competing interests of the parties involved. Historically, the US Federal Government was more willing to grant public lands to railroads (particularly in the little-settled west). However, due to shifts in the general public's will over the later part of the 19th century (stemming, apparently, from an 1871 decision Hash v. United States, which granted more explicit rights over the land to railroad companies) it became more difficult to either deed massive tracts of land to the railroad companies, or lease it to them at below market-competitive rates. (Note- much of this taken from this source.) Ultimately, this leads to a hold-up problem. The country needed railroads to cover the vast interior to ensure development, but without the earlier subsidies, laying tracks for thousands of miles was prohibitively expensive. Therefore, rights-of-way allow for a sort of middle ground, allowing cheap access to the land for the beneficial railroad lines, without giving them as large of subsidies as previously done.

It should also be noted, as in this source, that these agreements are the result of an incredibly tangled history, where many different private players, in addition to state and federal governments, all had vested interests in the outcomes. Newly formed territories and states needed the railroads to allow for their development; as linking commerce with reliable, cheap and relatively fast transportation was shown to be incredibly useful with the 1825 completion of the Erie Canal. Therefore, states had incentives to lower the development costs for railroads as much as possible. However, much of the burden of the cost fell to the federal government (who owned the lands being deeded), which was influenced by other individuals hesitant to provide such lucrative benefits for private corporations. Thus, the rights of way agreements came from a compromise-position.

Note, this source puts the major reason for the shift in agreement structure as federal dissatisfaction with how the railroads were handling their earlier deeded land. Instead of selling the granted land for commercial purposes, many railroads held onto the land (assuming it would only become more valuable), which delayed development.

Finally, by granting these easements, governments were able to ensure other public utilities used the same general routes (see page 1406)- something that railroads could have foreclosed had they owned the land outright.


Why can't I use TMoLM for this bliss point problem?

It all depends on whether you treat the budget constraint as an equality or inequality constraint. These are two different problems, with two different solutions in this case.

One version of the problem (rewriting the objective in the form suggested by denesp, and dropping the constant, for clarity) is
\begin{align}
\max~&-4(x-4.5)^2 -2(y-1.5)^2\\\text{s.t. }&5x+7y=40
\end{align}
With this equality constraint, the optimum is $(x,y)=(4.78,2.29)$ as you found using the method of Lagrange multipliers and Wolfram alpha. It's true that you would prefer consuming less and staying at the bliss point, but if we assume that the budget constraint is an equality then you aren't allowed to consume less.

Another version of the problem is
\begin{align}
\max~&-4(x-4.5)^2 -2(y-1.5)^2\\\text{s.t. }&5x+7y\leq40
\end{align}
with an inequality in the budget constraint. Here, as you point out, it's clear that the optimum is $(x,y)=(4.5,1.5)$, because that's the consumption profile that globally maximizes the utility function (and it obeys the budget inequality).

Assuming that the budget constraint is an inequality is usually more appropriate (presuming that you can always "throw away" extra wealth).

When solving constrained optimization problems where some of the constraints are inequalities, we need a generalization of the basic method of Lagrange multipliers called the Karesh-Kuhn-Tucker (KKT) conditions, which has probably not been covered in your course. You can read about this in more detail in the linked wikipedia article, but the basic idea is simple: we can still set up the Lagrangian, where we write the constraint as $g(x,y)=5x+7x-40\leq 0$ and then subtract $\lambda g(x,y)$ from the objective:
$$\mathcal{L} = -4(x-4.5)^2 -2(y-1.5)^2 - \lambda(5x+7y-40)$$
We can then equate the partial derivatives to 0 ($\partial\mathcal{L}/\partial x = 0$ and $\partial\mathcal{L}/\partial y=0$) to get two conditions
\begin{align}
\frac{\partial\mathcal{L}}{\partial x}&=0 \Longleftrightarrow 8(x-4.5)=-5\lambda\tag{1}\\
\frac{\partial\mathcal{L}}{\partial y}&=0 \Longleftrightarrow 4(y-1.5)=-7\lambda\tag{2}
\end{align}
The question, then, is how to pin down $\lambda$. You're probably used to finding $\lambda$ by seeing which value causes the budget constraint to hold with equality. In this case, however, that implies $\lambda<0$. This is not allowed for an inequality constraint under the KKT conditions, which require $\lambda\geq 0$.** Since $\lambda$ turns out to be the marginal utility of wealth, this constraint is tantamount to saying that more wealth (which relaxes the inequality) shouldn't hurt us.

The KKT conditions also have a requirement called complementary slackness, which states that either the inequality must bind (holding with equality) at the optimum, or else $\lambda$ must be 0. (Since $\lambda$ measures the "cost" of the constraint, this is logical: if the constraint isn't binding, then the cost must be 0, and it shouldn't be affecting your local optimization problem.) We already saw that if the budget constraint holds with equality, we get $\lambda<0$, which isn't allowed, so complementary slackness implies our only remaining option is $\lambda=0$. Plugging this into (1) and (2) above, we get $(x,y) = (4.5,1.5)$, which is the solution you already found intuitively.

Usually, there is no need in introductory micro to use the full KKT conditions, because we work with utility functions that are monotonic, where you'll always want to exhaust your budget constraint. Simple concave quadratic utility with an interior bliss point, like in this case, is an exception where we need a very simple case of the KKT conditions to do the optimization properly.

**Note: of course, if we add $\lambda g(x)$ rather than subtract it to the Lagrangian, the signs reverse. I've seen many different conventions, and I'm going with the one that makes $\lambda$ nonnegative. The Wikipedia article uses the opposite convention.


Where does the extra money come from when a new country joins the Euro and converts its currency?

Yes the European Cenral Bank issues the extra quantity of money (as for the actual printing there are factories that print euros in many member countries of the EU, under the control of the ECB).

Regarding why this does not cause inflation, it is because alongside the additional quantity of money comes the additional quantity of output of the new member, and the additional transactions related to this additional output that the additional quantity of money is called upon to facilitate.

Use the traditional "quantity theory of money"

$$PQ = VM$$

$M$ increases but $Q$ also increases, so $P$ does not have to increase.

Of course it is much more complex and complicated and gradual in a real world situation, but the fundamental idea is the above.


The definition of a monopoly is one company having the exclusive control of a good or service. Since there are 2+ companies it does not fit the definition of a monopoly.

A duopoly is a situation where 2 companies control the majority of the market. It sounds like this describes your situation.

A more correct term may be oligopoly which describes a market where a small limited number of entities controls the majority of the market. If there is a 3rd company which controls the remaining 25% market share this may be the most correct term but otherwise a duopoly is probably the correct definition.


Understanding the elasticity of demand for a monopoly

From the sounds of the question, the firm isn't operating at optimal efficiency. As you note, this firm is not on the elastic portion of the demand curve because $|e|=|\frac{.05}{.10}|=|0.5|<1$.

As an employee of this firm, you should suggest that they continue to adjust the price until they transition to the elastic portion of the demand curve. Once they do this, the firm should notice that their marginal revenue becomes positive.

I think your confusion stems from (1) your ability to think critically and (2) a misunderstanding that its the marginal revenue which monopolies want to be positive. Questions such as these are usually designed to capture a person's basic understanding of a concept, you pulled together multiple concepts..."too smart for the question." As for marginal revenue, this can be negative, but a firm that wants to maximize profit should adjust their business practices if marginal revenue is negative.


Independence axiom of lottery when $\alpha \ge 1$

To understand why $\alpha$ must be constrained in $(0,1)$, one has to contemplate the meaning of the expression

$$\alpha L$$

when $L$ is a "lottery". How is a lottery denoted mathematically? Authors do not agree on that: for example, the way Jahle and Reny define a lottery (a "gamble" in their terminology), a lottery can be written as a vector whose elements are bivariate vectors themselves:

$$L=\{(p_1,w_1),...,(p_n, w_n)\}$$

where $p_i$ are probabilities, and $w_i$ are quantitative outcomes.

But MasColell, Whinston and Green, define a lottery as a vector containing only the probabilities:

$$L=\{p_1,...,p_n\}$$

and so the "Lottery Space" is a vector space, including vectors each containing only probabilities.

But in both cases, the authors make clear that an expression like $\alpha L$, when the time comes to translate it into a mathematical operation, denotes a multiplication of only the probabilities linked with $L$ by another probability, $\alpha$. This is implementing the reduction of compound lotteries to simple ones. JR describe it as "the decision maker cares only about effective probabilities of each outcome". MWG call it the "consequentialist premise", that allows them to work only with simple lotteries.

So it is not valid to consider $\alpha$ outside $(0,1)$, because it is defined as a probability (the open bounds are used in order to avoid triviality in the statement of the Independence axiom).

Also, the above imply that $\alpha$ does not directly interact with the quantitative outcomes of the lottery/gamble...

...which points to a (perhaps interesting, perhaps not) research direction: What can we say (if anything), if we start tweaking the outcomes linked with a lottery? Will "attitude towards risk" get in the way and prevent us from drawing any general conclusions? Or not?


Can the endowment point be the competitive equilibrium?

The endowment can be an equilibrium allocation. But as is written on page 518, if indifference curves are smooth, which we will assume in the following, they must be tangent to the offer curve at the endowment. So if the offer curves cross at the endowment, the indifference curves must cross too. But if the indifference curves at the endowment cross each other, the endowment cannot be Pareto efficient. But every equilibrium allocation is efficient, so the initial endowment will not be an equilibrium allocation if the offer curves cross each other there (they would need to be tangent.)


Why are oil prices increasing in India despite countervailing trends in the international market?

To repeat EnergyNumbers's comment as an answer to this question, the following chart from the St Louis Federal Reserve shows the increase in the crude oil price in US Dollars in the first half of 2016.

Other factors also affect local diesel and gasoline prices and timings, but this increase will have contributed to the local change.


Some confusions about instrumental variable (econometrics)

Endogenous variables are correlated with the error terms and z is
  correlated with endogenous variable.

  Doesn't this imply that z is correlated with error terms?

No it doesn't. For mean-centered variables for simplicity, we have for linear models,

ENDOGENEITY: $E(xu) \neq 0$

RELEVANCE : $E(xz) \neq 0$

The above conditions do not necessarily imply non-Validity ($E(zu) \neq 0$).

A very simple example to show that we may have a valid instrument.
Assume $x, u$ are centered on their mean, $E(x) = E(u) = 0$, and that $x$ has a symmetric distribution, $E(x^3) = 0$. Assume that  $E(u \mid x) = ax$.
Then

ENDOGENEITY

$$E(xu) = E\big[E(xu\mid x)\big] =  E\big[xE(u\mid x)\big] = aE(x^2) \neq 0.$$

Consider the candidate instrument $z = x^2 + v$, with $v$ independent from $u$, but $E(xv) \neq 0$. We have

RELEVANCE
$$E(xz) = E(x^3) + E(xv) = 0 + E(xv) \neq 0$$.

VALIDITY

$$E(zu) = E(x^2u) + E(vu) = E\big[E(x^2u\mid x)\big] + E(v)E(u)$$

$$= E\big[x^2E(u\mid x)\big] + 0 = E\big[x^2a x\big] = aE(x^3) = 0.$$


Your computations are OK, but this is a trick question

The total quantity set by the colluding firms and the quantity chosen by  a monopolist are not the same in this question. By setting $MR = MC$ you are solving the profit maximization problem a monopolist instead of the profit maximization problem of the colluding firms.

The reason for the difference is that the marginal cost is increasing over the entire domain of the production function. Suppose the monopolist wanted to produce $Q=10$. The cost would be $5 \times 10^2 = 500$. If each of the colluding firms produced 5 the total cost of producing 10 units would be $5 \times 5^2 + 5 \times 5^2 =250$. I.e. the two colluding firms have smaller diseconomies of scale than the monopolist.

The most efficient way to produce any quantity for the cartel is to split production between the colluding firms equally. In this case the marginal costs of the cartel members are equal. Shifting a unit of production from firm 1 to firm 2 will make producing that unit more expensive, as the marginal cost increases in $Q$.

Viewed through the  $MR = MC$ argument that you used, the cartel of two colluding firms has access to a lower cost function unavailable to the monopolist: $C(Q) = 5\big[\big(\frac{Q}{2}\big)^2 + \big(\frac{Q}{2}\big)^2\big] = 5\frac{Q^2}{2}$.

Thus the cartel's profit function  is $\pi (Q) = (1400 - 5Q)Q - 5\frac{Q^2}{2}$. $MR(Q) = 1400 - 5Q - 5Q = 1400 - 10Q$, $MC(Q) = 5Q$
and $MR(Q) \overset{!}{=} MC(Q) \Rightarrow 1400 = 15Q \Leftrightarrow Q^* = 93\frac{1}{3}$. The total profit of the cartel is $\pi(Q) = 65333\frac{1}{3}$. The profit of the individual cartel member is $\frac{\pi(Q)}{2} = 32666\frac{2}{3},$ which is larger than the Cournot equilibrium you correctly obtained.

The monopolist, as described in your solution has a profit function of
$\pi(Q) = (1400 - 5Q)Q - 5Q^2$ For him $MR(Q) = 1400 - 5Q - 5Q = 1400 - 10Q$, $MC(Q) = 10Q$. And $MR(Q) \overset{!}{=} MC(Q) \Rightarrow 1400 = 20Q \Leftrightarrow Q^* = 70$. I.e. the monopolist stops producing earlier, as its higher $MC$ prevent it from making profits at larger quantities.


How to find the discount rate at which two deals are equally attractive?

Presuming the question is about inter-temporal choices and corresponding discount rate (not central banking).
It is rather easy to prove existence of such discount rate.

At one extreme, when discount rate is 100% (carpe diem mode), the person in question (N) will only look at the immediate value and expense. Dealer A effectively sells the car for \$2,000 (in present terms), while Dealer B sells it for \$4,000. So N will choose A.

At the other extreme, when discount rate is 0% (immortal person without any propensity for investment, perhaps?), N will just sum up all the value and expense streams. So A effectively sells the car for \$20,000 (both in present terms and in nominal terms), while B for \$19,500. So N will choose B.

With discounting function being continuous, it follows that somewhere between 100% and 0% there is a discount rate where A and B have the same present value.

PS: I am not sure if I should have posted this as a comment, but it's too long, and actually proves existence of the answer. I may expand this to include the actual solution if the question is updated to specify whether equal payments or equal principal payments or some other amortization schedule is used.


GDP data for the baltic states from 1988-1994

The Penn World Tables for those countries goes back to 1990. Some of their indicators may even go back further but I haven't checked.

Access to them is found here


References for particular definitions of risk and uncertainty

Knight's 1921 essay was not written in formal mathematics (and trying to formulate a direct translation into modern mathematics may be quite problematic). Since Knight's time, a formal decision theory literature has developed which makes distinctions that are at least reminiscent of Knight's.

Lars P. Hansen (2012), writes "Motivated by the insights of Knight (1921), decision theorists use the terms uncertainty and ambiguity as distinguished from risk." Hansen references Gilboa et. al. (2008) who write:

  In economics, Knight (1921) is typically credited with the distinction
  between situations of "risk” and of “uncertainty.” In his formulation,
  “risk” designates situations in which probabilities are known, or
  knowable in the sense that they can be estimated from past data and
  calculated using the laws of probability. By contrast, “uncertainty”
  refers to situations in which probabilities are neither known, nor can
  they be deduced, calculated, or estimated in an objective way.

Gilboa and Schmeidler (1989) introduced a max-min utility theory where instead of simply maximizing expected utility, agents solve a max-min problem where to model ambiguity aversion, the objective is minimized over different priors. This work has been expanded on by Hansen and Sargent in their work on robustness.

Sliding away from Knight and into the broader topic of decision theory, it would be mandatory to reference Leonard Savage's classic, Foundations of Statistics where he introduces the notion of subjective probability.

Your particular example (in more modern decision theory)

You have a Bernoulli likelihood function for winning the lottery.

To a frequentist statistician, $p$ is a scalar value, a parameter (albeit unknown). There is only one possible outcome for $p$ and so there's no randomness.
If you're a Bayesian in the spirit of Savage, you're willing to extend the tools of probability to model uncertainty in your own mind; you will treat $p$ as a random variable!

If we put a prior on $p$ (eg. the beta distribution is a conjugate prior to the Bernoulli likelihood), we can compute posterior probabilities and make decisions based upon standard expected utility. In terms of how we behave though, there's no difference between a 20% probability that's objective vs. subjective.

Under the ambiguity aversion, max-min model though, we may maximize our control variable over expected utility, taking into account that it will then be minimized over multiple priors: we choose control variable $x$ to maximize utility and then (after observing our choice $x$) a mean guy chooses the prior to minimize utility.

References

Hansen, Lars P., 2012, "Challenges in Identifying and Measuring Systemic Risk", NBER

Hansen, Lars P. and Thomas Sargent, 2001, "Robust Control and Model Uncertainty," American Economic Review

Gilboa, Itzhak and David Schmeidler, 1989, "Maxmin Expected Utility with Non-unique Prior," Journal of Mathematical Economics

Gilboa, Itzhak, Andrew W. Postlewaite, and
David Schmeidler, 2008, "Probability and Uncertainty in
Economic Modeling", Journal of Economic Perspectives

Savage, Leonard Jimmie, 1954, Foundations of Statistics


Optimal choice for a weird leontief function

One needs to go case-by-case and arrive at a utility function with branches. To get you started, if $x_1 < x_2/2 \implies \min(2x_1,x_2) = 2x_1$, but then also $\min(x_1,2x_2) = x_1$. Therefore in this case, $u(x_1,x_2) = 2x_1$. etc

There are two other intervals to consider as regards the relation between $x_1$ and $x_2$, so in all you will obtain a utility function with three branches. The middle one will be a $\max$ operator, and it can usefully be split in two, resulting in four branches.


There is the Kaitz-Index, which relates the minimum wage to the average hourly wage, thus facilitating cross-country comparisons on the minimum wage level. The OECD has compiled it here. There you'll also find other metrics comparing minimum wage levels across countries.


Global Macro Investing textbooks and reading list

There are quite a few books in the market on global macro strategies, the following could be useful:

Bernad Baumohl (2012) The Secrets of Economic Indicators: Hidden
Clues to Future Economic Trends and Investment Opportunities
Greg Gliner (2014) Global Macro Trading: Profiting in a New World Economy

Also try reading Burstein's book


What is the difference between conditional and unconditional risk premia?

In case someone finds this helpful, I'll answer my own question.

Risk premium, say for example on some risky assets such as stocks, is the expected return on the risky asset less the risk-free rate.

However, empirically we have that this risk premium on stocks is much larger than the riskfree rate and also that it varies quite a bit more. So we introduce the idea of conditional vs. unconditional, which speaks to this time-varying aspect.

Conditional risk premia means risk premia conditioning on time-$t$ information (so with a time-$t$ subscript as in conditional moments, conditional variance, etc.). So it is the conditional expectation of the return of a risky asset over the risk free rate, where the conditioning is done on the agent's available information at time-$t$. Although it depends on the model, this is an object that is in general time-varying.

Unconditional risk premia just means risk premia without conditioning on time-$t$ information. Generally, this is not time varying.

A good reference for learning about conditioning information on asset pricing models is Chapter 8 in John Cochrane's book "Asset Pricing".


Your conclusion is correct here. Since at the current FX rate you can buy the burger cheaply at an equivalent rate of 2 EUR in the US. So basically, if you live in Germany and want to buy a Big Mac, you are better of buying the Big Mac from the US since via the FX rate you will still have 1 EUR left(if you started with 3 EUR) and hence this will increase the demand of the USD in the foreign exchange market and hence

USD should appreciate in the future => USD in undervalued => EUR is overvalued.


Often a good answer consist the most of shedding light on some aspects of the terms of the question itself. I hope a couple of remarks could be useful.
About causality, the fact is that current deficit implies reserves exit from USA, which only can be supported by reserves entering, otherwise resulting in a diminishing stock of reserves, eventually exhausting it. A capital-account superavit --net inflow of capital-- represents an inflow of reserves, in that sense financing trade deficit (see Economic Report of the President, 2004). Letting aside causality --which can be very complex and call for other matters-- both imbalances are bond to present themselves simultaneously. In other terms, the fact that trade deficit and capital superavit pop up strongly correlated doesn't mean a causality. I suspect --sorry again for the lack of evidence-- that there's no causality there indeed. Instead, those paths suggested in the question about rates of interest or foreign exchange rates could be more promising sources of explanations, probably besides productivity and cost structures (again, see Conclusions in Economic Report of the President, 2004). It would be an example of third -unknown- causal factor.
About repaying the debt with capital inflows, we have to keep in mind that those capital inflows are new debt. For example, see Kirabaeva, Koralai & Razin, Assaf. (2010). Composition of Capital Flows: A Survey, NBER Working Paper Series. Aside with debt-instruments, capital inflows also comprise equity-instruments, i.e. Foreign Direct Investments and equity purchases. Depending on the context, even the last ones could also be considered national 'debt', as it represents foreign ownership of 'local' firms.
Interestingly, the "US's control to decide what to do with the capital we get" reflects the conflation in an unique agent "US" which in practice is a set of --millions of-- firms, consumers, authorities... There is no central control on the employment of all those flows, even though all those agents are acting, and controlling, in the USA, in that sense all them constitute "US's control".


Can the Fed raise the M1 money supply permanently w/o helicopter money?

I am going to just discuss the situation for the monetary base, not M1. It is possible that the monetary base and M1 could move in opposite directions.

The following three instruments are the main components of the monetary base.

Currency in circulation: dollar bills, and coins. Currency holdings are the result of voluntary decisions by the private sector, and the Fed has very limited ability to influence those decisions.
Required reserves of banks held at the Fed. In order to increase this, loan amounts outstanding must increase. One can argue that Fed decisions influence borrowing decisions, but it is not something that they can easily directly control. (We would need to go back to the Volcker Fed to see a historical attempt to do this; its effectiveness has been debated. Sorry. no reference handy.)
Bank settlement balances in excess of required reserves (excess reserves). These balances went from near zero to being quite large as a result of “quantitative easing.”

The Fed cannot drive excess balances negative, as that would imply that some banks are short reserves - which is not allowed by regulators. However, it is free to make excess reserves as positive as they wish, as quantitative easing proved.

They do this via:

buying something;
or lending against something (which includes repurchase agreements, or repos).

That “something” is usually Treasury securities, but there are classes of private debt securities is can buy or lend against (as was done in the Financial Crisis). There is no other choice; the Fed does not have a legal mandate to dump money from helicopters (or whatever). So the answer to your question is that they essentially need to buy enough Treasury bills/bonds to increase the monetary base, unless they want to lend against other private debts.

I am not able to give a legal opinion, but it is unclear that the Fed has the ability to not demand payment for matured bonds/bills. Even if they did so, it would just be an irregular loan to the Treasury. (Since Fed profits are returned to te Treasury anyway, there is no point not taking the contractual Treasury payments.)


Definition 1. Excess supply is supply minus demand.

Definition 2. Excess demand is demand minus supply.

Example 1. A baker posts a sale price of \$2 per loaf of bread.

At this price, he is willing to sell up to 300 loaves of bread (per day), but consumers are willing to buy only 200.

We say that quantity supplied and quantity demanded (at the price of \$2) are 300 and 200 and write $$S(\$2) = 300\text{ and }D(\$2) = 200.$$

Excess supply is then $$ES(\$2) = S(\$2) − D(\$2) = 300 - 200 = 100.$$ (If we'd like, we can also say that excess demand is −100.)

Example 2. A baker posts a sale price of \$1 per loaf of bread.

At this price, he is willing to sell up to 100 loaves of bread (per day), while consumers are willing to buy 250.

We say that quantity supplied and quantity demanded (at the price of \$1) are 100 and 250 and write $$S(\$1) = 100\text{ and }D(\$1) = 250.$$

Excess demand is then $$ED(\$1) = D(\$1) − S(\$1) = 250 - 100 = 150.$$ (If we'd like, we can also say that excess supply is −150.)

In the real world, the baker may not know precisely how many loaves of bread he's willing to sell at each and every price. Nonetheless, if it is the case that QS obviously greatly exceeds QD (or vice versa), he will either notice this and respond accordingly or go out of business.


With the right keywords (and author searches) from earlier papers, I actually found some academic output already; this might not be peer-reviewed yet, i.e. preprints:

McKibbin and Fernando (2 March 2020) The Global Macroeconomic Impacts of COVID-19: Seven Scenarios

  examines the impacts of different scenarios on macroeconomic outcomes and
  financial markets in a global hybrid DSGE/CGE general equilibrium model.
  The scenarios in this paper demonstrate that even a contained outbreak could significantly
  impact the global economy in the short run. [...]
  These results are very sensitive to the assumptions in the model, to the shocks we feed in and
  to the assumed macroeconomic policy responses in each country.

And that's already cited by Wang et al. COVID-19's Impact on China's Economy Based on Data of Spring Festival Travel Rush

  this paper starts with the analysis of the daily railway passenger volume data during the Spring Festival travel rush, and establishes three time series to model the railway passenger volume, GDP in the first quarter model, and GDP in the last three quarters respectively. The forecast results indicate that: (a) Affected by the epidemic, China’s economy will lose 4.8 trillion yuan in the first quarter of 2020, which is an expected decrease of 20.69% and a year-on-year drop of 15.60%; (b) China's expected GDP growth rate for the full year of 2020 will reduce from 6.50% to 1.72%. However, there are some positive factors not taken into account in these models, which means that the forecast results may be underestimated. With the global spread of the epidemic, the instability of the world economy will in turn impact China’s economy.

A more recent (March 30) paper, Greenstone and Nigam that emphasizes/quantifies the benefits of the lockdowns, in the US:

  Using the Ferguson et al. (2020) simulation model of COVID-19’s spread and mortality impacts in the United States, we project that 3-4 months of moderate distancing beginning in late March 2020 would save 1.7 million lives by October 1. Of the lives saved, 630,000 are due to avoided overwhelming of hospital intensive care units. Using the projected age-specific reductions in death and age-varying estimates of the United States Government’s value of a statistical life, we find that the mortality benefits of social distancing are about \$8 trillion or $60,000 per US household. Roughly 90% of the monetized benefits are projected to accrue to people age 50 or older. Overall, the analysis suggests that social distancing initiatives and policies in response to the COVID-19 epidemic have substantial economic benefits.

JP Morgan has put out some estimates already, although they don't detail their methodology as much as a peer-reviewed publication would (have to).

  The U.S. economy is projected to contract by 14% in the second quarter, after experiencing a 4% contraction in the first quarter, before recovering to 8% and 4% growth in the third and fourth quarters. Euro area GDP will suffer an even deeper contraction, with double-digit declines of 15% and 22% in the first and second quarters, before rebounding by 45% and 3.5% in the third and fourth quarters.

  “There is no longer doubt that the longest global expansion on record will end this quarter. We now think that the COVID-19 shock will produce a global recession, as nearly all of the world contracts over the three months between February and April,” said Bruce Kasman, Chief Economist at J.P. Morgan. Initially, the expectation was the novel-recession may generate limited labor market damage, but J.P. Morgan Research is now forecasting the unemployment rate for developed markets as a whole will rise 1.6 percentage points in the next two quarters.

  “The rise in unemployment will be sharper in the U.S. than in the Euro area. Most immediately, U.S. initial jobless claims should spike above 400,000 in the coming weeks,” said Michael Feroli, Chief U.S. economist at J.P. Morgan.

McKinsey has a somewhat spread-out (March 25) presentation, but a couple of slides with their scenarios/estimates I could fish from there:

Likewise, PWC has a March [white]paper mostly focused on Australia... but which does come with a global map attached of their estimates of GDP losses, alas in absolute figures only (and I think over one year based on what they say on the previous slide):

Before the COVID-19 outbreak, there has been a niche academic domain of estimating the economic impact of pandemics already, typically using a CGE model. Results depend of course on the assumptions used. E.g. Smith, Keogh-Brown et al. had a series of papers approximately a decade ago. Quoting from a later one that covered more than one European country:

  Results suggest GDP losses from the disease of approximately 0.5–2% but school closure and prophylactic absenteeism more than triples these effects.

So they went as high ~10% GDP losses. While they did not envisage government mandated lockdown(s), they modelled "prophylactic absenteeism" in response to ~3% death rate media reports as the most severe case. An earlier paper of theirs, on UK only is open-access in the BMJ. Quoting their assumption from there

  almost everybody in the population will know someone who has died once the mortality rate reaches one death per 300 people, triggering prophylactic absenteeism.

The same authors (Keogh-Brown and Smith) also published in 2008 a fairly detailed post-hoc account of how economic predictions for SARS fared in hindsight... which generally wasn't all that good, meaning that most models overestimated the economic impact of SARS. It cites positively an 2003 ADB paper of Fan that hedged its bets on two different scenarios (early containment being one of them).

Another paper Verikios et al. (2015) in a similar vein but attempts a global estimate (GTAP CGE), and also discussed prior/related work:

  The two pandemic scenarios modelled give quantitatively and qualitatively different
  results. [...] Our results show that the peak GDP effects on the world economy of an influenza pandemic are in the range 0.06 to 1.01 per cent in the peak year depending on the nature of the pandemic. Our results are smaller than the only previous estimates of the global economic effects of influenza pandemics by McKibbin and Sidorenko (2006). They estimate reductions in global GDP of between 0.8 and 12.6 per cent depending on the severity of the pandemic. Our results are not directly comparable to McKibbin and Sidorenko as they did not model a continuum of pandemic scenarios across the virulence-infectiousness continuum. Rather, they assume a given reproducibility of the virus (infectiousness) and vary virulence across their four scenarios. Further, they implicitly assume multiple waves for all their scenarios as they presume that
  new infections continue to occur for greater than one year. This is likely to characterise only more extreme pandemics (such as the 1918–19 flu) rather than milder pandemics. Differences in the range of influenza pandemics modelled by us and McKibbin and Sidorenko are justifiable given the uncertain nature of future pandemics. Our results thus build upon the findings of McKibbin and Sidorenko by extending the estimation of potential GDP estimates over a further range of potential pandemic threat scenarios.

  [...] Our results show that the global macroeconomic effects of an influenza pandemic may be significant but are also likely to be short-lived. The largest economic impacts of an influenza pandemic are driven by reduced international tourism, due to risk-modifying measures by households and travel restrictions imposed by health authorities, and lost workdays, due to illness or formal social distancing measures designed to contain the virus. This is consistent with the work of Keogh-Brown et al. (2009, 2010). [...]

Verikios seems to work for KPMG Australia nowadays and they have released a COVID-19 model that is fairly detailed in explaining its assumptions, for example it has separate pre- and post-stimulus arms/scenarios, but it mostly covers the Australian case, in particular only models the government response of that country.

KPMG uses a NiGEM (New-Keynesian) macro model, and they've basically tweaked some parameters of this using estimates of shocks in four areas: reduction in productivity due to additional sick leave by workers, reduction in productivity due to additional carer’s leave by workers, incremental investment premium, capacity utilisation adjustment (i.e. supply chain disruptions). The first two were informed by fairly detailed labour & health statistics, the latter two seem a bit more arbitrary to me, e.g. they added a flat 1% investment premium.

I was hoping the IMF would have something more concrete than this, but I guess
they are weary of advancing anything more detailed yet:

  The COVID-19 pandemic would cause a global recession in 2020 that could be worse than the one triggered by the 2008-2009 global financial crisis, but world economic output should recover in 2021, the IMF said on Monday.

  Georgieva issued the new outlook after a conference call of G20 finance ministers and central bankers, who she said agreed on the need for solidarity across the globe. [...]

  The outlook for global growth is negative and the IMF expects “a recession at least as bad as during the global financial crisis, or worse,” Georgieva said.

  Earlier this month, Georgieva had warned that 2020 world growth would be below last year’s 2.9 percent rate, but stopped short of predicting a recession.

  Georgieva on Monday said that a recovery is expected in 2021, but to reach it, countries would need to prioritize containment and strengthen health systems.

  “The economic impact is and will be severe, but the faster the virus stops, the quicker and stronger the recovery will be,” she said.

The CBO has now posted its own assessment for the US economy. It disclaims that it is "highly uncertain, especially for later periods", but basically, for the second quarter of 2020 it predicts 10 percent unemployment and the economy to contract by "more than" 7 percent.

The WTO Secretariat has posted this self-explanatory chart on April 8:


Philosophy books for decision theory and economics idea

Your question is a little bit unclear. But still, if you wish to delve into the philosophical side of economical topics/question/phenomena, you can check the Stanford Encyclopaedia of Philosophy's article on Philosophy of economics. It might be a good place to start and refine what you are looking for.


Is VCG mechanism applicable in reverse auction? If so, how?

In general, VCG is also applicable to reverse auction settings. VCG is not even restricted to auction settings and can be used quite generally, see wikipedia for an introduction. If you want a deeper treatment, I recommend Tilman Börger's book (it used to be fully online, maybe there are still copies flying around). In reality, there are often problems associated with VCG, see, e.g., Ausubel and Milgrom writing about "The Lovely but Lonely Vickrey Auction."
The general idea of VCG is that each agent's transfer is their own externality.  In your setting, you cannot simply do that because there is a budget restriction on transfers. Papers with such a knapsack constraint in economics are Ensthaler &Giebe and Jarman &Meisner (JET 2017), who use clock auctions as in Milgrom & Segal (JPE 2020).


What are the advantages of using euro for Greece?

The main argument for joining (and then remaining) in the Euro for Greece was to limit inflation.
Before Greece joined the Euro, the inflation was almost always in double digits, often exceeding 20% or event 30%. The reason for this was that the Government was running a huge budget deficit, which caused an over-supply of Drachmas. The expectation of inflation was also a factor causing inflation, since people were expecting the value of the Drachma to fall quickly, so they converted any extra funds into other "strong" currencies (mainly US dollars, Deutschmarks, or Swiss Francs), causing the value to fall further as a positive feedback loop.
The introduction of the Euro was indeed successful in limiting inflation, since now:

people now stopped converting their money to foreign currencies
The value of the currency was determined by the entire eurozone economy, so that the local deficit wasn't immediately reflected in higher currency supply and inflation directly (but still did indirectly).
The rules of the eurozone in theory required all countries to run a deficit of only up to 3%.

Of course as we all now the third point wasn't followed by Greece, which meant that the money supply in the Country was increasing more quickly than in the rest of the eurozone, which meant that the inflation in Greece was still much higher that the rest of the eurozone.
Since the value of the Euro itself was stable, this inflation was mischaracterised as "economic growth". But the reality was that the costs for people and business were still increasing year-on-year (though much slower than before), which meant the economy was becoming less and less competitive, so the balance of payments was very negative. Since the currency was no longer floating, this could only be paid by continually increasing the national debt.
After the fail of Lehman Brother Bank, there was increased scrutiny in the underlying fundamentals for new loans by the internation money markets, so Greece became insolvent. The eurozone then created a so-called "rescue package" that allowed Greece to pay the eurozone banks the money they had lent, by loaning new money to the Greek government. All this money didn't benefit Greece at all, the only purpose was for the big eurozone banks to not go also insolvent due to an imminent Greek default. The Greek people are and will be paying back this money for more than 50 years.
So the argument for joining and staying in the Euro was almost entirely driven by the desire to limit inflation. Whether that is actually a good or a bad thing is debatable. As always there are winners and losers with both high and low inflation.
The people that benefit from high inflation are:

Workers with high bargaining power (both public and private sector), that could demand pay-rises to match the inflation.
People with access to the banking system, since they could borrow drachmas and convert them to real estate, and then pay back the loans with currency that had much reduced value. The instability of the currency also made real-estate attractive, since its value tends to follow inflation, so it was a no-lose proposition. As a personal anecdote, when my parents took a loan for their house in the early 80's, they paid about 60% of their income for the monthly loan payment. When they eventually paid back the loan in early 2000's it was less than 5% of their monthly income, despite high interest rates.
People that had income or capital abroad where able to retain the value of their assets, whereas everybody else was losing money due to inflation, so their relative purchasing power was increasing quickly.

The losers with high inflation are all the rest of the people who couldn't benefit from the above bullet points:

Workers with low bargaining power saw their income be reduced year on year since it didn't follow inflation (Pay cuts are always much tougher to swallow than pay increases below inflation)
People that could not get a loan saw their rent become higher and higher every year, while their income didn't catch up to the increase.
People that had money in Drachmas were losing money year on year (This is a huge amount of money, just consider the amount of currency in peoples pockets for an entire country).

Finally, although in the question you mention that being in the eurozone is not synonymous with being in the EU, politically there was the argument to be in the "center" of Europe, being involved in as many EU projects as possible, since EU membership was seen as beneficial both politically and economically.


Why do government jobs provide better benefits than industry jobs?

I'll describe in turn five potential explainations:

Risk aversion interacting with the tax code
Is this true when we control for worker attributes?
Lack of incentives for cost control
Solving turnover, training and other HR problems
Political economy: unions and politicians

Risk aversion interacting with the tax code

I've heard it argued (but not empirically substantiated) that because government positions generally offer very low risk of job loss they attract risk adverse people.

All other things equal, relatively risk adverse people like relatively generous insurance and have a greater precautionary savings motive against negative shocks. Because retirement savings, health insurance, and to a lesser extent, life insurance have tax advantages when purchased through the employer, this risk aversion would interact with the tax advantage to induce a high level of retirement savings, health, and life insurance. Disability insurance is not tax advantaged, but employer-run plans do help solve some of the adverse selection problems associated with stand alone disability plans and so they too have an incentive to be employer provided and more generous in government positions for similar reasons.

Is this true when we control for worker attributes?

That all said, I'll echo @rocinante's point that among some sub-populations of workers, it isn't clear that government benefits are particularly extensive. My reading of the Are government employees overpaid? literature is that highly educated or skilled government workers do not typically appear overpaid (in a Mincer regression residual sense) relative to their private sector counterparts, and this includes measures of income and benefits. And elite financial and IT firms are often famous for their lavish benefits of meals, profit sharing, and low cost health care. However, part of the story there is that wages are higher in those fields and benefits are generally increasing in wages (if only because so are the tax advantages). It may be surprising to hear this but the average education and age of a public sector worker is substantially higher than in the private sector, explaining some of the higher wages and higher benefits.

Lack of incentives for cost control

So while it may well be true that, as @kbelder says, public sector employers have less of an incentive to hold down benefits, they have less of an incentive to hold down wages too, at least in conventional wisdom government employee benefits are more different from private sector benefits than the wages are, so that leaves us with a remaining puzzle beyond poor cost containment incentives.

Solving turnover, training and other HR problems

@rocinante's point about turnover is also a good one. We can think of the employee-employer relation as a sort of investment that pays returns while employment persists. Any situation where the initial employer investment (e.g., in training, background checks, customized equipment) is large relative to the immediate payout is one in which the employer is going to want to incentivise the employee to stick around until that investment is repaid. Pensions, which back-load compensation have a powerful incentivizing effect towards retaining staff under these circumstances.

Similarly, when positions require idiosyncratic human capital (extensive expertise and training not useful for other employers), this makes the employee strongly value continuing the employee relationship, and if government employers particularly require this sort of expertise (say maintaining ICBMs) then employment protection benefits may be another efficient way to attract staff without paying higher wages.

Political economy: unions and politicians

Finally, two more reasons from a political economy angle. First, unions may not just get better benefits because they have more power and influence but also because the people who run unions wish greater power and influence. To those ends, higher wages give them less than retirement and benefits programs which require wealth and staffing to be run by the union. Second, benefits, particularly defined benefit pensions, can be opaque in their true costs. Since politicians have to answer in elections to the public, HR strategies that please government employees (like high defined benefit pensions) but appear to be low cost to the public (again like high defined benefit pensions) therefore are relatively attractive to the politicians that administer the government.


Taxes and GDP, where do these come in in the formula?

In this equation:

GDP = C + I + G + NX

Taxes come from a combination of C+I but are offset by an increase in G.

Any government taxation which is not spent is an investment, so remains part of I.


Confusion about the EMP approach to perfect complements. Solved UMP but struggling with EMP

You already got your answer in the math forum, from the economics point of view maybe it could help that you can use the relation between the expenditure function and the indirect utility to get your solution.
Your indirect utility of the UMP is $v(p,w)=\frac{m}{p_x+p_y}$, this provides the expenditure function for the EMP $e(p,\overline{u})=\overline{u}\ (p_x+p_y)$. Now the hicksian demand is $h(p,u)=(u,u)$. This is the case of course of the utility $u(x,y)=min(x,y)$.


Which values to assign to a quantified dummy variable

It's generally not good practice to go from categorical/dummy variables to quantitative variables, as csilvia noted, because categorical/dummy variables have less information than their quantitative analogs. None of the variables in your dataset are quantitative except for math, reading, writing, and total, which I'm assuming are outcomes of interest, not inputs.
If your teacher wants you to include quantitative variables, I'd go back to the original dataset and see if there's anything quantitative included there rather than trying to make a categorical variable fit as a quantitative one.


Does it matter if there's simultaneous determination between two dependent variables?

In your concrete example, I think 2SLS is something worth considering because the error term u (of $y$, i.e. of tax collection) is plausible in theory to be affected by presence of cartels in ways that make it more unreliable in reporting (e.g. via increased corruption). Of course you can/should test this endogeneity.


Neoclassical economic profit and growth theory versus marxian

If there is perfect competition, then yes, you will get zero economic profit, but even then, technology does no poof out of nowhere. Otherwise, you might find very different results.

Let's think of the nature of competition through product differentiation. (We'll get into technology later.) Imagine two firms making their own widgets in Cournot competition, but instead of the normal price function $P = X - q_1 - q_2$, we have

$$P = X - q_i - d\cdot q_{-i}$$ where $-i$ denotes the other firm that is not firm $i$, and $d \in (0,1)$, $1$ being no product differentiation, and $0$ being completely different products sold by the two companies.

If $d = 0$, then the firms aren't really competing since their products are completely different and are acting as monopolists, making economic profit. If $d = 1$, then the firms are competing through regular Cournot and still making profit (probably).

In Bertrand, we can consider product differentiation as part of determining how much of the market you will steal away by undercutting price. The way a normal Bertrand model works is that you set price and then produce (as opposed to Cournot, where you set production, which affects the price). If you have a lower price, and there is no product differentiation, you steal away the whole market, so each firm will push price down to the price under perfect competition, where both firms split the market.

With product differentiation though, you don't lose the whole market when you are undercut, since your product is still a bit different. With full differentiation $(d=0)$ then you get the same result where each firm act as monopolists.

Under each of these models, technology doesn't "just happen", but is reflected through lowering the marginal costs of the firm, which is essential for the firm to be competitive against its rivals. The less technologically advanced the firm is, the more of the market they will lose out on. Economic profit here can be used to invest into research and development or just straight up "technology" to reduce the cost of capital usage over time or other such things like that.

Even in perfect competition with no economic profit, the reason there is no economic profit will partly be because firms have to constantly update their technology to remain competitive, regardless of how differentiated the products are (as long as $d > 0$).


I am a financial economist, so I will attempt to answer what I think you are asking.  A market order isn't one concept, it's many.  Each place that accepts trades operationalizes it differently.  It also depends upon whether it is a small order or a large order.

For a small order, the order result will clear at the bid if it is a sell order at the bid and at the ask if it is a buy order.  Only in that case will the outcome be the same everywhere.

For a large order, it depends upon where the security trades at and it can depend upon the internal policies of which broker-dealer you use.  For example, if you place a large market order on a stock that trades on the NYSE then what you are actually doing is instructing your broker to go to the floor of the exchange, to a place called the "post," and continue to bid until you are the high bidder, if buying or the low bidder if selling, until your order is filled.

On the other hand, a large market order on the NASDAQ will be filled up to the stated size limit.  After that, it depends upon a number of things.  For example, if there are multiple market makers then decisions to fill or not fill is made by each maker depending on orders they have, their inventory needs and what risks they will take and for what fee structure.  Because a market order essentially says "I will pay any price," the makers have liberal pricing powers on large market orders.  Also, the processing broker can sometimes cross the order with other orders they have pending from their own customers and so it never really makes it to the market itself, other than for record keeping purposes.

Outside of this, there are order crossing services in what have been nicknamed "dark pools," which are available to the largest financial institutions but not to ordinary traders. Different rules are in use for things other than stocks and for stocks traded in non-US exchanges.

To provide an example, imagine you own $\$1,000$ and also own 100 shares of ABC which is currently trading at 9.90 bid and 10.10 ask.  It trades on the NYSE.  A decision to sell 100 shares at market would increase cash to $\$1,990$ and reduce shares to zero.  Since you may or may not be permitted to short sell the stock depending on your contract with your broker and the availability of shares to borrow, a decision to short sell 100 more shares of ABC would increase cash by the new bid, which for our purposes will be 9.85 per share, for $985 and add one hundred shares on the short side in the margin account.  The cash will also be segregated and held as collateral on the long side of the margin account.

A large market order for a thinly traded stock could take a month or more to fill.  For a "pink sheets" stock it could take years.  For a heavily traded stock, it could take less than a second.

It is important to note that the recording of a trade for customer accounting purposes may not match market records.  Consider a person who sold 100 shares of ABC for 9.65 per share.  They record an increase in cash of $965 and a decrease in shares by one hundred shares.  The "tape" may not record it this way if it is part of a large buy order.  In that case, if the weighted average price was 10.10 then the market would see one large order for 10.10 per share, though it may also be the case that nobody actually received that amount of money.  It is also the case that the time of order completion will not match.


How to get details on all the models used for evaluating ACA?

In terms of actually getting it passed as legislation, the only evaluation that has any real significance is the one done by the CBO, as that's the official scorecard for Congress and debates that come up on the floor. I have no idea what the specific models they use are called, but I do know they are very linear and simplistic (by design) compared to other mainstream models. It's definitely bound to be a fruitful search if you look there.


How does burning a huge amount of cash money affect the monetary supply?

It would seem logical that this would result in reverse-inflation or lower prices, but this is actually not the case (at least over time).

The banking system has a "supply and demand" market for reserves.  When say 1 million dollar is withdrawn and not redeposited, the banking system in the aggregate will bid more for current reserves to replace the ones that were lost.  The Fed does not like to see changes in the rates the bankers lend each other reserves and if it thinks this exchange rate is higher than the targeted Fed Funds rate, it will sell REPOs to primary dealers in an attempt to move this rate down.

So in the end, the amount of money will roughly be the same (as near as I can tell).  What will change is that the Fed will have new assets it didn't have before.  These assets in turn will accrue interest, and profits from the Fed get turned over to congress.  So in a round about way, burning money is a transfer of wealth from the individual to the public.

In the same way, counterfeiting does not actually increase the money supply in the US over time.  New fake dollars are deposited, drive down reserve rates which results in the Fed selling assets to absorb the new money.  Less Fed assets = less taxpayer money.  So counterfeiting hurts, but just not in the way most suspect.


If all quantities produced rise by 10 percent, and all prices fall by 10 percent, which of the following occurs?

It helps to write out things as (simple but) proper equations. Denote production as $Y$ and prices as $P$

It is given that

$$Y_{t+1} = 1.1 Y_t \\
P_{t+1} = 0.9 P_t$$

Nominal GDP is given by $P_t Y_t$

$$ P_{t+1}Y_{t+1} = 1.1 Y_t \cdot 0.9 P_t  \\
= 1.1\cdot 0.9 \cdot Y_t P_t$$

So nominal GDP changed by $1.1 \cdot 0.9 = 0.99$ - it dropped by one percent.

So which answer is it?

I assumed that the initial value is base value for the description of the relative changes. If, instead, we have that the larger value is base for the relative change,

$$0.9 Y_{t+1} = Y_t \\
P_{t+1} = 0.9 P_t$$

we get that
$$ P_{t+1}Y_{t+1} = \frac{1}{0.9} Y_t \cdot 0.9 P_t = Y_t P_t  $$

nominal GDP does not change.


Why a bank never combine all benefits/promotions and issue a single Credit Card

Should P&G drop all their brands and only sell one shampoo and one deodorant and one toothpaste?

Volkswagen group own Skoda, Volkswagen and Audi (plus some additional brands - but lets exclude those for this example). Each brand has several series/models that are comparable between brand "levels". E.g Skoda Octavia = Volkswagen Passat = Audi A6.. Essentially the same car (sure not exactly the same car, but comparable in size, equipment, material etc.) - but different prices.

Would it be more consumer-friendly to skip the different brands and just offer one car?
Same thing with credit cards. Different cards offer different things and a eligible for consumers with different income/wealth levels.

Take for example MasterCard Black which many banks will offer along with a large credit (e.g 100.000$ per month) and maybe a 24/7 concierge service that can take care of e.g plane tickets, helicopter service or whatever.

Take your average Joe with a yearly salary of 30.000, will he ever be in need of a 100.000 $ credit (or other way around; could he ever repay that debt?!).. and how often will he be using a concierge service for finding business class plane tickets from New York to Paris departuring the same day? Probably never.

That is why banks are not offering all customers the same credit card, because their customers have different needs.


How to build a Social Accounting Matrix from an Input-Output Table?

IOTs are usually derived from Supply and Use tables, which have an industry by commodity format.  If you used these to derive your IOT, then you can use those to build your SAM, though note you need both the Supply AND Use tables for this. You end up with something that looks roughly like what you see on slide 7 here - with both commodity and activity (industry) rows and commodity and activity columns in your SAM.  If you have the data to support it, this is probably the preferred way to go about structuring a SAM.

However, if you only have an industry by industry IOT, you can still build a SAM from that (with additional data from the country's National Accounts or another source), and this approach is far from uncommon.  Instead of having separate activity and commodity rows and columns, you'll only have activity rows and columns, but you can still construct a SAM (though you may have to combine some of the non-IOT cells in your SAM to get the structure to work).  As an example to hand, here is a paper outlining the construction of a SAM for Scotland using an industry by industry format (as Scotland is a part of the UK, it also shows an approach to constructing a SAM using estimates when full National Accounts data outside the IOT are not available).


Is there a school of thought or concept that acknowledges that people aren't rational actors when it comes to economic decisions?

Yes this is a blossoming area of research in economics and it spans approaches from psychology to game theory, social theory, and cultural biases.

A lot of these approaches fall under the rubric of "Behavioral Economics".  Unlike classical market economics which often begins with assumptions around individual and collective rationality, behavioral economics seeks to study how economic decisions are actually made by individuals and groups (i.e. it does not take utility-maximizing rationality as an assumption but instead studies its role in animating economic decisions and systems).

"Irrational" behavior should not be confused with unexplainable or unpredictable behavior.  A simple example is why people buy lottery tickets.  This may be irrational under classical utility theory, but it is explainable and can be modeled, tested, and incorporated into larger economic frameworks like market options pricing by using behavioral economics.


In April 2019 our economy created 263000 new jobs, however, total employment went down from March to April by103000 jobs. Why this difference?

The main difference comes from the source of the data. While both figures can be found in the latest BLS report, the +263K comes from their establishment survey, which includes only non-farm jobs as reported by employers.

  Total nonfarm payroll employment increased by 263,000 in April, compared with an average monthly
  gain of 213,000 over the prior 12 months. In April, notable jobs gains occurred in professional and
  business services, construction, health care, and social assistance. (See table B-1.)

On the other hand, the -103K figure comes from the household survey and reflects a shrinkage on the workforce (due to a decrease in labor force participation).

  The household survey includes agricultural
  workers, self-employed workers whose businesses
  are unincorporated, unpaid family workers, and
  private household workers among the employed.
  These groups are excluded from the establishment
  survey

Furthermore, in the latter (household) survey the number of the unemployed decreased even more by 387K. The difference between 387K and 103K is about 284K which is a good enough approximation for the 263K from the establishment survey given the different methodologies.


How does an import driven economy function?

This is an implausible scenario for a country of any size, the problem being how the imported goods would be paid for.  An individual retailer might pay for its imported goods out of income from sales to other residents of the country.  But for the country as a whole this would not work: there has to be sufficient income from foreign sources to pay for its total imports. The only exceptions would be if the country were running down its assets, or borrowing from abroad, neither of which could continue indefinitely.

In theory the required income could come from a) re-export at a profit of imported goods, and/or b) sales of imported goods at a profit to incoming tourists. However, even with clever marketing, it seems unlikely that (a) could be achieved on any scale if it involves no physical processing to add value, in competition with foreign suppliers operating by more direct routes. So far as (b) is concerned, even if the country has natural tourist attractions (eg scenery, heritage), to bring in many tourists would require "tourism infrastructure" (transport, hotels, etc) that is not normally considered part of "retail".


tl;dr: Uncertainty & trade tariffs

detail
The economics have been assessed by a large number of economic specialists, including the very well-respected and independent Institute for Fiscal Studies.
What would Brexit (meaning the UK leaving the EU) mean?
uncertainty
Well, firstly it means years of uncertainty, while replacement trading arrangements are negotiated - typically that takes somewhere in the region of 10 years. The Doha Round of trade talks has been going on since 2001.
Uncertainty requires higher rewards to balance the risks, and that means lower inward investment, and fewer long-term deals.
trade tariffs
Free access to the European market depends on accepting some specific rules that the Brexit campaign consider to be unacceptable, such as the free movement of labour. And it requires membership either of the EU, or of the European Free Trade Area (EFTA). On leaving the EU, UK membership of the EFTA would require EU approval, and the German Finance Minister Schäuble has made it clear that he considers that that would not happen. So one thing that the Brexit campaign and the EU heavyweights should agree on is that the UK would not be a member of the European Economic Area, and that means trade tariffs. A Brexit would be an existential threat to the EU project, so it will be in the interest of its remaining members to make exit as painful as possible, to preserve the unity of the remaining 27. And EU27 trade with the UK is a very small proportion of EU27 total trade, so it's a cheap price to pay.

summary of all effects
The IFS summarises the effects, and the vast majority of specialists predict negative effects at least out to 2030. Here's their summary, from p18 of the report linked above:

It's worth noting that it's not only UK's trade with the EU that would be harmed. The UK has trade deals with much of the world because it is a member of the EU. Without that membership, it would need to negotiate its own bilateral trade deals: such negotiations will also typically take a decade. So the UK's trade with non-EU countries would be reduced too. And there's the reduced economic efficiency of having a much smaller pool of talent to draw on for the workforce. So the numbers in the table above may (depending on the scope of individual studies) include those effects, as well as the EU-trade effect.


Ramsey growth model - per capita vs aggregate variables

For per capita $c=C/L$, I guess it comes from

$$\int_0^{\infty}e^{-\rho t}\frac{c(t)^{1-\sigma}}{1-\sigma} dt = \int_0^{\infty}e^{-\rho t}\frac{(C(t)/L(t))^{1-\sigma}}{1-\sigma} dt$$

$$= \int_0^{\infty}L(t)^{-(1-\sigma)}e^{-\rho t}\frac{C^{1-\sigma}}{1-\sigma} dt$$

Normalizing initial population to $L_0=1$ we have $L(t) = e^{nt}$ so

$$...= \int_0^{\infty}(e^{nt})^{-(1-\sigma)}e^{-\rho t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt =  \int_0^{\infty}e^{-(1-\sigma)nt}e^{-\rho t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt$$

$$=\int_0^{\infty}e^{-[\rho+(1-\sigma)n]t}\frac{C(t)^{1-\sigma}}{1-\sigma} dt$$

Regarding the application of the discounte factor throughout the whole Hamiltonian, it just implies that the multipliers $\lambda$ and $\mu$ are to be treated as present-value multipliers, and not current-value multipliers. In the textbook formulation and in per capita magnitudes, we usually use current (running) value mutipliers and we totally ignore the discount factor. This affects the intertemporal first-order condition. In generic notation, denote $B$ the state variable, $q$ the mutliplier and $r$ the discount factor. Then, with present-value mutlipliers, we have

$$\frac{\partial H}{\partial B} = -\dot q$$

while if we interpret $q$ as a current-value multiplier we have

$$\frac{\partial H}{\partial B} = r q-\dot q
$$


Sensitivity/continuity of optimal solution

As no one has answered, I will give this a try.

To begin with, together with compactness, and of course nonemptiness, of $X$, which you have assumed, you need $f(x,y)$ to be continuous in $x$ in order to ensure, via Weierstrass extrem value theorem, that $$\theta(y)=\text{argmin}_{x\in X}f(x,y)\tag{1}$$ exists for any $y$. It is good, as you do, to assume strict convexity in $x$, for this gives us a unique solutions to the minimization problems, and thus $(1)$ is a singleton.

Under these assumptions, I will try to show that $(1)$ is continous in the following sense, where I am essentially using the definition of continuity in terms of limits of sequences. The fact that the argmin is a set makes it complicated to directly apply the notion of continuity of function which seldom, in elementary calculus, maps elements to some set of sets.

Theorem. Consider a nonempty compact set $X\subseteq \mathbb{R}$ and a continuous function $f:\mathbb{R}^2\to\mathbb{R}$. Furthermore, suppose that $$\theta(\bar{y})=\{z|z\text{ minimizes }f(x,\bar{y})\text{ over }x\in X\}$$  is the singleton $\{\bar{z}\}$ (to ensure this, assume strict convexity in the first argument of $f$). Then, for sequences $z_n\in \theta(y_n)$, where $\{y_n\}_{\mathbb{N}}$ is a sequence converging to $\bar{y}$, we have $z_n\to \bar{z}$.

Proof. I will give a short proof. It is by contradiction, and is a variant of the proof of Lemma 6.3.2. in Nonlinear Programming: Theory and Algorithms, 3rd Edition, 2006, by Bazaara et al.

So suppose $z_n\to\bar{z}$, $z_n\in \theta(y_n)$, and suppose $|z_n-\bar{z}|>\epsilon>0$ for all $n\in I$ where $I$ is some index set. Since $X$ is compact and $z_n\in X$, the sequence $\{z_n\}_I$ has a convergent subsequence $\{z_n\}_{I'}$ with limit $z_0$ in $X$. By assumption, $|z_0-\bar{z}|\geq\epsilon>0$ and thus $z_0\neq \bar{z}$. In addition, for $\{y_n\}_{I'}$ it holds that $$f(z_n,y_n)\leq f(\bar{z},y_n)\tag{2}$$ since $z_n$ minimizes for $y_n$, but $\bar{z}$ does not minimize for $y_n$ necessarily.

Now, use continuity of $f$ in both of its arguments, and take the limit over $I'$ in $(2)$, to get that $$f(z_0,\bar{y})\leq f(\bar{z},\bar{y})\tag{3}.$$ But as $z_0\neq\bar{z}$ and $\bar{z}$ minimized $f(x,\bar{y})$, it follows that $z_0$ is another, distinct minimizer of $f(x,\bar{y})$ over $X$. Hence, $\theta(\bar{y})$ is not a singleton. But this contradicts the fact that $\theta(\bar{y})$ was a singleton. QED.

Reference suggestion. The Theorem of the Maximum. This theorem is more general and "provides conditions for the continuity of an optimized function and the set of its maximizers as a parameter changes". It involves the concept of a multivalued function (note that the assumption of the argmin being a singleton simplified the proof of the theorem above).


Microeconomics: Calculating and Comparing MRS for Various Utility Functions

The rule for differentiating a natural logarithm is

  $$\frac{d}{dx}a\ln(x)=\frac{a}{x}$$

so
$$\begin{align}\frac{\partial{}}{\partial{}x}U_3(x,y)&=\frac{\partial{}}{\partial{}x}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&=\frac{1}{3}\frac{1}{x}\\&=\frac{1}{3x}.\end{align}$$

$$\begin{align}\frac{\partial{}}{\partial{}y}U_3(x,y)&=\frac{\partial{}}{\partial{}y}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&=\frac{2}{3}\frac{1}{y}\\&=\frac{2}{3y}.\end{align}$$

$$\text{MRS}=-\frac{\frac{\partial{}}{\partial{}x}U_3(x,y)}{\frac{\partial{}}{\partial{}y}U_3(x,y)}=-\frac{\frac{1}{3x}}{\frac{2}{3y}}=-\frac{y}{2x},$$

I guess that by following this template you should be able to work out the other derivatives and the MRSs that you are looking for.


Your understanding is wrong.

Nations accrue debt by selling bonds. They pay interest on those bonds to the bond-holders.


Static game with complete but imperfect information

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.


After a comment exchange, let's provide the answer under disrete-time formulation. The problem is now written

\begin{align}
      &\max_{\{u\}_1^T, \{y\}_1^T}\sum^T_{t=1}{F(y_t,u_t)}\\
\text{s.t.} \quad & y_{t+1}-y_t = f(y_t,u_t)\\
& y_1 = \text{given}\\
& y_{T+1}~~\text{free}
\end{align}

where $y_{t+1}$ is the value of the stock variable at the begining of period $t+1$/end of period $t$.
The Lagrangean is
\begin{align}
\Lambda = \sum^T_{t=1}{\big(F(y_t,u_t) + \lambda_t[f(y_t,u_t)-y_{t+1}+y_t]\big)}
\end{align}

Note that $\lambda_t$ is associated with $y_{t+1}$.

The FOCS are (to hold $\forall t$ in the horizon)

$$\frac {\partial \Lambda}{\partial u_t} = F_u(y_t,u_t) + \lambda_tf_u(y_t,u_t) = 0 \tag{1}$$

$$\frac {\partial \Lambda}{\partial y_{t+1}} = -\lambda_{t} + F_y(y_{t+1},u_{t+1}) + \lambda_{t+1}[f_u(y_{t+1},u_{t+1}) +1]    = 0 $$

$$\implies F_y(y_{t+1},u_{t+1}) + \lambda_{t+1}f_y(y_{t+1},u_{t+1}) = -(\lambda_{t+1}-\lambda_{t}) \tag{2}$$

which are the conditions we would have obtain if we have used the Hamiltonian formulation. To obtain the terminal value for $\lambda_t$, we set $t=T$ in $(2)$, (since the last variable in the $y$-sequence is $y_{T+1}$), and we obtain:

$$F_y(y_{T+1},0) + \lambda_{T+1}f_y(y_{T+1},0) = -(\lambda_{T+1}-\lambda_{T})$$

$u_{T+1}=0$ because there is no decision to be made in period $T+1$. Moreover $\lambda_{T+1}$ should also be set to zero, since it is associated with $y_{T+2}$ (end of period $T+1$) and we are out of the horizon. So we are left with

$$F_y(y_{T+1},0) = \lambda_{T}$$

Whether now we will get $\lambda_T=0$ or not depends on the formulation of the objective function  $F(y_{t},u_t)$: does it indeed include the state variable? If yes what is the value of its derivative with respect to the state variable when the decision variable is set to zero? Etc


Use Brouwer's Fixed Point Theorem to Prove existence of equilibrium(a) with completely mixed strategies

Rather than deprive you of the joy of solving your own problem, I will show how one can also use Brouwer's theorem to show that the function $f:\mathbb{R} \to \mathbb{R}$
$$
f(x) = \frac{33+x}{16+x^4}
$$
has a fixed point, even though its range is not compact.

Note that $f(1) = 2$ and $f(2) = 35/32$. As $f$ is monotonically decreasing over $[1,2]$, this also means that $f([1,2]) \subset [1,2]$. Also, $[1,2]$ is compact and $f$ is continuous over this set. So $f: [1,2] \to [1,2]$ will have a fixed point by Brouwer's fixed point theorem.

Similar tricks are used in game theory and general equilibrium theory. By showing that the optimal decisions by some players cannot possibly be outside of some boundaries one can frequently create a compact space over which fixed point theorems are applicable.


It looks right to me. There's an alternative (but equivalent) way to solve part D: we know that the consumer's choice will satisfy

$$ x^* =  \frac{ a(m-t)}{ p_x - s} ,\  y^*= \frac{(1-a)(m-t)}{p_y},  $$
or, with a balanced budget,
$$ x^* =  \frac{ a(m-sx^*)}{ p_x - s},\ y^* = \frac{(1-a)(m-sx^*)}{p_y}.$$

The government wants to maximize $\min\{x,y\}$, which is achieved when $x=y$. Solving
$$ x^*=y^*\iff \frac{ a(m-sx^*)}{ p_x - s}= \frac{(1-a)(m-sx^*)}{p_y}$$

for $s$ yields

$$ s^* = p_x - \frac{a p_y}{1-a}. $$


Is the CES a special case of the translog production function?

the translog is an approximation of the CES function. Let consider a 2 factors production function $A(\alpha K^\gamma+(1-\alpha)L^\gamma)^{1/\gamma}$.
First, take the log of this production function, and then approximate this to the first-order using the McLaurin series for $\gamma$.

Using Mathematica (laziness, sorry):

Series[Log[A(\[Alpha] K^\[Gamma]+(1-\[Alpha])L^\[Gamma])^(1/\[Gamma])], {\[Gamma], 0, 1}] // FullSimplify

You reach
$%\log \left(A K^{\alpha } L^{1-\alpha }\right)+\frac{1}{2} \gamma(1-\alpha) \alpha  \left((\log (K)-\log (L))^2\right)+O\left(\gamma ^2\right)$

$\ln Y=\ln(A)+\alpha\ln K+(1-\alpha)\ln L+\gamma \alpha(1-\alpha)\left(\ln K+\ln L -\ln K\ln L \right)+O\left(\gamma ^2\right)$

This corresponds to the usual translog functional form used in econometric estimation.


What are the pros and cons of eliminating corporate taxes?

The first and most important insight here is to recognize that burden of taxation (the incidence) need not fall on the entity taxed. Economist Herb Stein is famous for having quipped (on the subject of the corporate income tax):

  "...that it’s as if people think that if the government imposed a tax on
  cows, the tax would be paid by the cows."
  Are Taxes on Corporations Taxes on People?

The corporation may front the money, but ultimately that money comes from customers, employees, vendors, creditors, and equity holders. Miller and others have written down models where corporate shareholders bear the burden but it doesn't have to be that way.

So to assess the consequences of eliminating the corporate income tax, we really want to know who pays (bears the burden of) the corporate income tax. To the extent that it is the providers of capital of the firm who bear the burden, we might assume that lowering the corporate income tax would act like an income tax cut for the rich. But there are simple models where customers pay some of the corporate income tax, such as if domestic and foreign products (as well as investments) are perfect substitutes, then (immobile) labor pays all the burden of the corporate income tax. Unfortunately, the issue of incidence remains murky despite decades of work on the subject.

Regardless of who pays, it is clear that enormous resources are devoted to corporate income tax avoidance probably because corporations have so many more opportunities for tax avoidance than individuals do, due to their complexity, multi-territoriality, and general scale and scope. If eliminating the corporate income tax freed most of those resources invested in corporate taxation preparation and avoidance (on the order of \$150 billion by two estimates just in compliance costs, ignoring strategically wasted resources), resources amounting to something like half of actual corporate income tax revenue would be usable for other activities. Free up those resources and you can make everyone better off, at least in principle. You might for example create a revenue neutral increase in dividends, capital gains, and interest taxation to offset the corporate income tax cut. This could leave beneficiaries of current spending essentially unaffected while the holders of current capital in taxable accounts would be a bit worse off and tax-free accounts (retirement, charities) slightly better off from the direct effects of the tax changes. But since their compliance costs fell massively, there might be enough to make everyone better off.

The bottom line is that that the corporate income tax is a expensive way to gather tax revenue from rich people and we are not sure it even does get money from rich people. The corporate income tax can plausibly harm non-rich people in the process. Taxing the income of the rich directly, if that is the end goal, is much more efficient. The resulting efficiency gains can be split between the rich and poor. Even if you don't particularly want to tax the rich, a more efficient tax policy means  tax rates can be lowered. So there is something for a liberal and conservative economist alike to love with this tax reform.

Here are a couple of papers that suggest that higher corporate income taxes induce corporations to have more debt and less equity (greater leverage), although there is some evidence that reducing these taxes would not reduce leverage in the short run.

  ...low tax rate firms lease more, and have lower debt levels, than high
  tax rate firms

Debt, Leases, Taxes, and the Endogeneity of Corporate Tax Status

  The results suggest that taxes have had a strong and statistically
  significant effect on debt levels. For example, cutting the corporate
  tax rate by ten percentage points (e.g. from 46 to 36%), holding
  personal tax rates fixed, is forecast to reduce the fraction of assets
  financed with debt by around 3.5%

Do taxes affect corporate debt policy? Evidence from U.S. corporate tax return data

  ...we show that taxes have a first-order effect on capital structure.
  Firms increase leverage by around 40 basis points for every
  percentage-point tax increase. Consistent with dynamic tradeoff
  theory, the effect is asymmetric: leverage does not respond to tax
  cuts.

As Certain as Debt and Taxes: Estimating the Tax Sensitivity of Leverage from State Tax Changes

If I can recall papers that estimate the costs of tax avoidance and excess burden of taxation I'll add them as well. Suggestions welcome.