Why does savings equal investment (scenario)? Scenario 1: There's an economy of two people (Joe and Amanda). Joe buys a 500 dollar car from Amanda (which she made herself from raw materials in her back yard). Amanda takes the 500 dollars and hides it under her bed. Savings = 500 dollars, Investment = 0 dollars. What am I missing?

Scenario 2: Same Amanda and Joe. Joe gets 500 dollars from the Federal Reserve (they just gave it to him). He puts it in a bank. The bank decides to only lend out 250 dollars of it. Amanda invests in inventory. S = 500, I = 250. Right?

These are two cases where investments do not seem to equal savings. How can these cases be explained?

To clarify the first scenario: you assume Joe can buy this car. However, Joe has to either earn an income (and therefore, produce something himself) or he has to dissave. In the first case, Joe has produced 500 output and earned an income of 500. He spends his income on a car, but Amanda saves the money. As is clarified in other answers, Joe's output goes to inventories (because not purchased by Amanda), which are counted as (unplanned) investments. S = I = 500, C = 500, Y = S + C = 1000. You mention in a comment that most consumers do not produce something. In fact, they do. At work, when earning their income (with which they buy the car), they created added value (this is the production (500) that goes to the inventory when Amanda does not buy it). Think of the income they earn at work as their part in the created added value, their contribution to real production. Consider the second case when Joe does not work.

In the case Joe dissaves to buy the car instead of working and using his income (he just drives around, without producing), 500 savings have to be deducted from your final result for the first scenario, so that S = I = 0, C = 500, Y = S + C = 500.

In the second scenario, forget about Joe's 500 for a while. Assume Amanda invests 250 (one car) in inventory, which means she has produced 250 (one car) first (and sold nothing), so that I = Y - C = 250, C = 0. Consumption is zero because Amanda sold nothing and Joe bought nothing.

An income can only be earned when value added was produced. Therefore, the  500 Joe has received cannot be seen as an income, and since only income can lead to consumption or saving, Joe had neither consumed nor saved. The 500 Joe received is an expansion of money supply (and has no real interpretation) and will increase prices proportionally (assume V and Y to be constant in the fisher equation (MV=PY)). The 250 that Amanda produced has now increased in value, but in real terms her investment in inventory will still refer to the 250 (one car).

This actually illustrates the importance of distinguishing nominal and real variables in the economy. The macro-economic identity you refer to is only about real variables.


How do I derive Hessian of this function and check for concavity? The function is $f(K,L)= AK^{a}L^{b}$ on the set of points $(K,L)$ with $K\geq 0$ and $L\geq 0$, assuming $A>0$
How do I find the Hessian and check for concavity?

As @Herr K. stated, the beginning point is being able to take a derivative. The Hessian is a matrix equivalent to a second order derivative sometimes denoted as $\nabla^{2}$. Start by finding the gradient, $\nabla$ which is a vector of first order derivatives of every variable in the function. In your case, this would be
$$ \nabla=
  \left[ {\begin{array}{cc}
   \frac{\partial{f}}{\partial{K}}  \\
   \frac{\partial{f}}{\partial{L}} \\
  \end{array} } \right] =
  \left[ {\begin{array}{cc}
   aAK^{a-1}L^{b}  \\
   bAK^{a}L^{b-1} \\
  \end{array} } \right] $$
The Hessian, would be the next (second) derivative of each of the above with respect to both variables and it takes the form:
$$ \nabla^{2}=
  \left[ {\begin{array}{cc}
   \frac{\partial^{2}f}{\partial{K}^2} & \frac{\partial^{2}f}{\partial{K}\partial{L}} \\
   \frac{\partial^{2}f}{\partial{L}\partial{K}} & \frac{\partial^{2}f}{\partial{L}^2} \\
  \end{array} } \right] $$
Using whatever this would be in your case (left to you to compute) you can take the determinant of the resulting matrix and, with values for your parameters $a$ and $b$ (or at least their signs like with the other variables and parameters) you can determine whether the determinant is positive or negative and determine the concavity. If it is positive definite or positive semi-definite, this would imply either strict or non-strict convexity, respectively. If you don't know how to take a determinant it is simply dont by using a matrix say

$$ A=
  \left[ {\begin{array}{cc}
   a_{1} & a_{2} \\
   a_{3} & a_{4} \\
  \end{array} } \right] $$
and then taking its determinant using the operation (without explanation for sake of brevity):
$$ A=
   {\begin{vmatrix}
   a_{1} & a_{2} \\
   a_{3} & a_{4} \\
  \end{vmatrix} } = a_{1}a_{4}-a_{2}a_{3} $$
Using this with the above, you should be able to figure the rest out and figure out whether or not its concave! (hint: for $a,b\geq 0$ and $a+b\leq1$ it is concave)


Define and characterize equilibria of the following game Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)=c_1+c_2$, where $c_t$ is the consumer's period $t$ consumption. Period 2 prices are given exogenously by $P$. Assume that $P$ is random and takes value in $\{\bar{p},\underline{p}\}$ with equal probability where $\bar{p}>\underline{p}>0$. It is common knowledge that the firm knows $P$, and the consumer knows $P$ with probability $\alpha$. The timing of the game is as follow. The firm chooses the first period price $p$ to maximize its profits in period 1. After observing $p$, the consumer decides how to allocate his income between $c_1$ and $c_2$.

1) Define an equilibrium for this game

2) Characterize the equilibria.

I am having difficulties in formally setting up dynamic games and I would therefore appreciate if someone could check out the solution I provided below, and most importantly whether the definition of equilibrium I provide (strategy sets, strategies, optimality conditions, etc) is correct, and if not why?
Thank You

SOLUTION:

1) DEFINITION OF EQUILIBRIUM

\begin{equation} \begin{split}
&p \in \mathbb{R}_+\text{ is the price chosen by the firm in period 1}\\
&a_U \in [0,1] \text{ is the share of income the Uninformed consumer allocates in period 1}\\
&a_I \in [0,1] \text{ is the share of income the Informed consumer allocates in period 1}\\
&\Theta = \{I,U\} \text{ is the type of consumer: Informed, Uninformed}\\
& S_I = \{ a : \{\underline{p},\bar{p}\} \times \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the I. consumer}\\
& S_U = \{ a : \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the U. consumer}\\
& S_f = \{ p : \{\bar{p},\underline{p}\} \to \mathbb{R}_+ \} \text{ is the strategy set of the firm}\\
&\mu_U(p) = Pr(P =\bar{p}|p) \text{ : consumer of type U belief that the second period price is } \bar{p}, \text{ given } p\\
&\mu_I(P) = Pr(P =\bar{p}|p) \text{ : consumer of type I belief that the second period price is } \bar{p}, \text{ given } p\\
\end{split} \end{equation}

An equilibrium of the game is a triple of strategies, $s^*=(a^*_I, a^*_U,p^*)$ for each type of consumer and the monopolist, together with a system of belief $\mu$ such that: for each player $i$, $s_i^*$ is sequentially rational, and the beliefs are determined by the strategy profile and Bayes rule whenever it applies. \

In particular for each price $p$, the consumer of type $i \in \{U.I\}$ solves:
\begin{equation} \begin{split}
&a_i^*(p) = \text{ argmax } \sum_p \big[ ( \frac{a}{p} + \frac{1-a}{\bar{p}} )\mu_I(P) + ( \frac{a}{p} + \frac{1-a}{\underline{p}} )(1-\mu_I(P)) \big]
\end{split} \end{equation}

Also, in the case of the Informed consumer, we know that
\begin{equation} \begin{split}
 &\mu_I(P) = Pr(P =\bar{p}|p) =
\begin{cases}
1 &\text{ if } P =\bar{p}\\
0  &\text{ if } P = \underline{p}\\
\end{cases} \\
\end{split} \end{equation}

given the consumers optimal strategies and $P$, $p^*(P)$ maximizes the monopolist first period profits:

\begin{equation} \begin{split}
&\pi(P)^* = \text{ argmax } \alpha [a^*_I(p) p] + (1-\alpha) [a^*_U(p) p]\\
\end{split} \end{equation}

2) CHARACTERIZATION OF EQUILIBRIA

for any realization of $P$ in period 2, we have that
\begin{equation} \begin{split}
a_I(p)^* =
\begin{cases}
1 &\text{ if } p <  P\\
0  &\text{ if } p >P\\
\in [0,1] &\text{ if } p =  P \\
\end{cases} \\
\end{split} \end{equation}

\begin{equation} \begin{split}
a_U(p)^* =
\begin{cases}
1 &\text{ if } p <  \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\
0  &\text{ if } p > \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\
\in [0,1] &\text{ if } p = \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}
\end{cases} \\
\end{split} \end{equation}

To define the optimal strategy of the monopolist consider the following cases, where $\pi$ denotes the monopolist first period profits:

a)Suppose $p=\bar{p}, \forall P$, then By Bayes rule $\mu_U(p) = 1/2$; also suppose that off the equilibrium path $\mu_U(p) = 1/2, \forall p \neq\bar{p}$ ; then the monopolist profits are given by:

\begin{equation} \begin{split}
\pi(P) =
\begin{cases}
\alpha \bar{p} a_I^*(\bar{p}) &\text{ if } P=\bar{p} \\
0  &\text{ if } P =  \underline{p} \\
\end{cases} \\
\end{split} \end{equation}
This is not an equilibrium strategy because the monopolist can deviate and increase its profits when $P=\underline{p}$ by setting, for example, $\hat{p} = 1/2\bar{p} + 1/2\underline{p}$; this will give him a profit of $\alpha a_U^*(\hat{p})  >0$.
Similarly it is possible to prove that no-matter what the off the equilibrium path are, there is always a profitable deviation for the monopolist. \

b)Suppose $p=\underline{p}, \forall P$, then $\mu_U(p) = 1/2$ and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall p \neq\underline{p}$ ; then the monopolist profits are given by:

\begin{equation} \begin{split}
\pi(P) =
\begin{cases}
\underline{p} &\text{ if } P=\bar{p} \\
\underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) ) &\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
this is not an equilibrium strategy because $\exists \epsilon > 0 $ such that monopolist will deviate and increase its profits when $P=\bar{p}$; for example by setting  $\hat{p} = 1/2\bar{p} + 1/2\underline{p} -\epsilon>\underline{p}$ which will give him a profit of $\hat{p}> \underline{p}$ .

c)Suppose $p=\underline{p}, \text{ if } P= \underline{p}$, $p=\bar{p}, \text{ if } P= \bar{p}$ and $\mu_U(p) = 1, \forall p > \underline{p}$ then

\begin{equation} \begin{split}
\pi_1^*(P) =
\begin{cases}
\bar{p}(\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})) &\text{ if } P=\bar{p} \\
\underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) a_U^*(\underline{p})) &\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
this represents an equilibrium as long as  $a_I^*(\bar{p}) = a_U^*(\bar{p}) = 1$; otherwise the monopolist can set a price $\hat{p}< \bar{p}$ and can achieve a profit of $\bar{p} - \epsilon >\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})$

d) Suppose $p = 1/2\bar{p} + 1/2\underline{p}, \forall P$ then $\mu_U(p)=1/2$ and and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall  p \neq ( 1/2\bar{p} + 1/2\underline{p}) $
\begin{equation} \begin{split}
\pi_2^*(P) =
\begin{cases}
(1/2\bar{p} + 1/2\underline{p})  (\alpha  +(1-\alpha) a_U^*(p)) &\text{ if } P=\bar{p} \\
( 1/2\bar{p} + 1/2\underline{p} ) ( 0 + (1-\alpha)a_U^*(p)) &\text{ if } P =  \underline{p}\\
\end{cases} \\
\end{split} \end{equation}
This is an equilibrium strategy as long as there are no profitable deviation; notice that the monopolist can set $p=\bar{p}$ when $P=\bar{p}$ and given $\mu_U = 1/2$ its profits would equal $\alpha \bar{P}a_I^*(\bar{p})$. Also, when $P=\underline{p}$ the monopolist could set $p=\underline{p}$ and get a profit of $\underline{p}$. So we need to insure that neither deviation is possible.

e) ?

We can therefore characterize the following (Perfect Bayesian) Equlibria:

1ST EQUILIBRIUM)
\begin{equation} \begin{split}
&a_I(p)^*  = 1, a_U(p)^* = 1 \\
&\pi_1^*(P) \text{ defined as above } \\
&\mu_U(p) = 1 \text{ both "on and off" the equilibrium path } \\
&\text{(that is, if the monopolist deviates } \\
&\text{from the equilibrium strategy } \mu_U(p)=1 \text{ still holds} \\
\end{split} \end{equation}

SET OF EQUILIBRIA)
\begin{equation} \begin{split}
&a_I(p)^* =1, a_U(p)^* \in[0,1] \\
&\pi_2^*(P) \text{ defined as above } \\
&\mu_U(p) = 1/2 \text{ both "on and off" the equilibrium path } \\
&\text{(that is, if the monopolist deviates } \\
&\text{from the equilibrium strategy } \mu_U(p)=1/2 \text{ still holds} \\
& \text{Also, The following two conditions below hold:}\\
&\alpha \bar{P}a_I^*(\bar{p}) < (1/2\bar{p} + 1/2\underline{p})  (\alpha a_U^*(\bar{p}) +(1-\alpha) ) \\
&\underline{p} < ( 1/2\bar{p} + 1/2\underline{p} ) (\alpha a_U^*(\bar{p}) +0)\\
\end{split} \end{equation}


What prevents a government from issuing debt to finance riskier investments? Germany has a 10Y bond yield of -0.336%. Meanwhile, Brazil has a 10Y bond yield of 6.415%.

Why doesn't the German government issue bonds in order to buy Brazilian bonds? It seems a no-brainer win. Actually, the German government could also issue bonds to invest in the stock market. Sure, there is a risk involved, but on average seems a winner.

Here are some hand-wavy arguments to suggest that the "no-arbitrage" rule applies here:

Bond-buying at the scale that nation-states would do it would drive up prices.  So you'd see yields on Brazilian bonds fall even if coupon rates don't move.
Germany would have to finance such a play by issuing yet more bonds, which would drive German bond prices down and therefore German bond yields up even if coupon rates don't move.
Credit markets would likely punish Germany for taking that kind of risk, pushing German bond yields up further because coupon rates would rise.
Ironically, the piling-in of high-quality foreign money into Brazil could serve to stabilize it, which would again lower Brazilian yields because coupon rates would fall.
and 6. Movement in foreign exchange markets that would further close the gap. (credit to @Gio in a comment above).

The net effect: the arbitrage opportunity you've spotted would quickly vanish.  In essence, your plan would amount to Germany volunteering to prop up the Brazilian economy in exchange for being made worse-off.


How much do real-estate prices correlate with apartment rental prices? I've been trying to find sources that mention this, and have tried to find graphs of data so I can compare myself, but I haven't been able to. Do real-estate sale prices correlate positively, negatively, or not at all with apartment rental prices? A more nuanced answer than that is also appreciated.

In theory there should be a direct positive correlation (especially in countries where house prices are a part of the index which is used to define rental rates), but in reality there is a lot of variation in the results for this question depending on the country, possible effects of different laws, the business cycle, whether new housing is being built or not, etc.

For Italy, Manganelli, Morano and Tajan (2014) in "House prices and rents. The Italian experience" study response functions and conclude:

2)  Rents depend on real-estate prices, the short-term interest rate and the differences between actual and expected GDP with a one year delayed reaction.

On the other hand, Ambrose, Eichholtz and Lindenthal (2013) in House Prices and Fundamentals: 355 years of evidence, Journal of Money, Credit and Banking find that when housing prices stray from equilibrium growth, rental rates do not react the same way even if it takes decades for the prices to return to steady growth.

For a recent study on U.S. data (model built to match trends from 1960-2013), I'd check Gelain and Lansing (2014): House prices, expectations, and time-varying fundamentals, Journal of Empirical Finance


Set of "rationalizable actions" What does it mean if a question asks me to find the set of "rationalizable actions" for a given game?

A strategy is rationalizable if it survives an iterated elimination of strictly dominated strategies. That is first you eliminate all strategies that are never a best response independent what action the other player chooses. Then you only consider the strategies that survive and repeat this step: eliminate all strategies that are never a best response independent what action from the set of surviving strategies the other player chooses.
The set of rationalizable strategies is then a set that only contains best responses to some rationalizable strategy. Therefore this is a larger set than Nash equilibria, which are best responses against each other.


Critiques to Mankiw, Romer, Weil (1992) paper on Economic Growth I'm searching for bibliography on the critiques to the Mankiw, Romer, Weil (1992) paper.

If possible, could you give references to each critique?

Any help would be appreciated.

Here are three:

Klenow, P., & Rodriguez-Clare, A. (1997). The neoclassical revival in growth economics: Has it gone too far?. In NBER Macroeconomics Annual 1997, Volume 12 (pp. 73-114). MIT Press.

Klenow, P. J., & Rodriguez-Clare,A. (1997). Economic growth: A review essay. Journal of monetary economics, 40(3), 597-617.

They pose certain conceptual issues as regards entangled causalities (in the economics' sense, not in the econometrics' sense).

All are freely downloadable.

Bernanke, B. S., & Gürkaynak, R. S. (2002). Is growth exogenous? taking Mankiw, Romer, and Weil seriously. In NBER Macroeconomics Annual 2001, Volume 16 (pp. 11-72). MIT Press.

This paper argues that one of the most important implications of any Solow framework is that long-run growth depends only on exogenous technological progress, and that MRW do not test in the right way this implication (the authors do it their way and reject this implication).

But I don't think you will find papers with deep statistical-econometric testing/validation of the model.


Why does the violation of a preference axiom not invalidate the discipline of economics? There are many examples of the preference axioms of consumer theory being violated. I feel like there are very few cases where at least one of these axioms isn't broken, so is any theory or model created under these assumptions representative of the real world?

The violation of the preference axioms is frequent in the real world, which questions the usefulness of neoclassical model results. This is something that most neoclassical economists will say that it is just "noise", or has a minimum effect, while there is limited empirical data to sustain this - maybe some lab work (controlled experiences), but not real world observations that I know of. The consensus in most policy making institutions and economic universities it is, still (and unfortunately), that neoclassical assumptions are what should be used.

You can read Transaction cost economics for a different perspective which doesn't assume the preference axioms, from Oliver Williamson, Nobel Prize in 2009. It refutes the preference axioms, stating that there is bounded rationality in every choice: humans have limited reasoning and searching capabilities.

Here's a summary of his work on TCE:

  Williamson’s contributions to the field of Transaction Cost Economics complement, and extend, those of Coase. First, Williamson started with an explicitly behavioral assumption of human behavior (bounded rationality). Second, he recognized that transacting parties sometimes behave opportunistically and take advantage of their counterparties. Finally, he identified features of transactions (e.g., specificity, uncertainty, frequency) that cause markets to fail; and hence, are likely to lead certain transactions to be organized within firms (hierarchies) rather than markets.

Source


How does the taylor rule make the MP curve upward sloping? I know it's because $\bar n \tilde Y$  is added in but isn't the MP curve chosen by the government?

In effect, the desired real interest rate $R_t$ is chosen by the government (via changes in the nominal interest rate). But it is chosen based in the Taylor rule. The "standard" IS-MP-PC model (as in Jones's Macroeconomics textbook), is as follows:

IS:

$$\tilde{Y}_t = \bar{a} - \bar{b}\left(R_t - \bar{r}\right)$$

PC:

$$\Delta \pi_t = \bar{v}\tilde{Y}_t + \bar{o}$$

MP (Taylor rule):

$$R_t = \bar{r} + \bar{m}(\pi_t - \bar{\pi})$$

In the latter equation, the central bank/government choses $R_t$ according to deviations in inflation from the target. As inflation increases, the real interest rate increases.

Now, according to the Phillips Curve, inflation responds positively to demand shocks ($\tilde{Y}_t$). In consequence, the real interest rate responds positively to $\tilde{Y}_t$. This is why, in the IS-MP diagram (with axis $R_t$ and $\tilde{Y}_t$), the MP has a positive slope.


Neo Keynesian Modelling and the Lucas Critique Some people from the more extreme microfoundations/internal consistency camp keep bashing the Neo Keynesian model for the Calvo pricing being not micro founded and hence the whole model not being immune to the Lucas critique.

However, one could argue that we don't  observe inflation rates far away from the steady state in the US, and, as long as this was true, the lack of micro foundations was minor.

Which argument dominates here? Are there other arguments to be made around NK and the Lucas critique? What is the literature's conclusion?

Before gravity was conceived as a physical phenomenon and studied by Isaac Newton, gravity was not microfounded. So before Newton, modelling the real world should assume away gravity, because it was not microfounded. We should instead create models into which humans could walk on air.

When a phenomenon appears stable, while attempting to understand and decipher it (to micro-found it), we should be wise enough to accept its existence and include it in our models, of course in an "ad hoc" manner since we haven't decipher it yet.

Whether firms "cannot" or "choose optimally not to" change their prices everyday, makes little difference, as long as we haven't settled on the optimization framework into which such an optimal choice could come about. It makes little difference because in the absence of knowledge it can be seen as a semantic issue: I can re-interpret "cannot" as "an optimizing agent cannot do what is suboptimal -it goes against every inch of its existence".

Nevertheless, bashing for micro-foundations of an observed regularity (and it is a prevailing observed regularity across time, space, cultures and economic systems that firms don't changes their prices continually), is a good thing, in that it keeps up the pressure to finally find those microfoundations.
I just hope the bashers don't argue in addition, that in the meantime, we should only be modelling people as walking also on air -sorry, be modelling firms only as being able to change their prices continually.

And to send the ball back, I have never seen in my life any convincing "micro-foundation argument" for the assumption that prices are perfectly elastic.


Transformation Function In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (p.135):

$\begin{cases}Max\text{ }p*y\\s.t.\text{ }F(y)\leq0\end{cases}$

Where y is production vector and F(.) is a transformation function.

The only information I have found about this function states that $F:R^L\to R$ and that production set of a firm could be represented as $Y=\{y\in R^L:F(y)\leq0\}$. Also if $F(y)=0$ it is named transformation frontier (p.562)

So I am interested in what concrete are these transformation function and transformation frontier. Intuitively it must be related to some feasibility restrictions but I whant to have some full and strickt explanation.

Will be very greatfull for help!

I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was looking for a concrete example of how a production function and a transformation function might be related. So I am providing a simple example.

Suppose we have a firm that produces a single output, $ q$, using a Cobb-Douglas production function of the following form:

$$ q\le f(x_1,x_2)=x_1^\alpha x_2^\beta $$

That is, $ f(x_1,x_2) $ describes the maximum amount of $ q$ that can be produced by a firm with this technology. So the production possibilities set, $ \mathbf Y$, is the set of all vectors $ \mathbf y=(-x_1,-x_2,q), \mathbf y\subseteq \mathbb R^3 $ (in this case - clearly could be extended to $ \mathbb R^n $), such that the inequality above holds.

$$ \mathbf Y=\{\mathbf y|q\le f(x_1,x_2)\} $$

Note that the vector has negative signs on $ x_1$ and $ x_2$ because they are inputs, while the output, $ q$, is positive.

An equivalent way to represent this is as a transformation function, $ F: \mathbb R^3 \to \mathbb R$, where

$$ F(\mathbf y) = q-x_1^\alpha x_2^\beta $$

Using this notation,

$$ \mathbf Y=\{\mathbf y|F(\mathbf y) \le 0 \} $$

Where $ F(\mathbf y) \le 0 $ implies that the first inequality is satisfied.

This is fairly simple to interpret when there is only one output; however the concept of a transformation function is useful because it allows us to think about production technologies that might use multiple inputs and produce multiple outputs.

I hope this is useful!


Where does bank’s Tier1 capital lie Banks need to maintain certain minimum capital as per Basel norms but where does this money lie. If it lies in a bank account, what would happen if that bank goes bankrupt?

Say, SBI keeps its Tier1 capital in HDFC, what would happen if HDFC goes bankrupt? Or if RBI keeps it with itself, can it loan it out as per fractional reserve system?

“Capital” is not an asset (as that would be on the right hand side of the balance sheet) they are long maturity instruments on the right hand side of the balance sheet.

Common equity is one major component. It’s the residual you get after you subtract all other items on the right hand side of the balance sheet from total assets - which is what forces the balance sheet to balance. (There are presumably some accounting details I am skipping over.)

If we simplify things, it’s how much more its assets are worth than its liabilities, and so it cannot get put anywhere. All that can happen is that assets it owns (a loan to another bank) can lose value, and so the credit loss reduces equity (and hence capital).

(I think perpetual preferred shares are the other major component of Tier 1 Capital. These are issued by the bank in question, and so the failure of another bank has no effect on them.)


Do discontinuous preferences imply no continuous utility function? I am trying to think of a preference relation that can be represented by a utility function but such that there does not exist a continuous utility function.

I know that you can represent continuous preferences with a discontinuous utility function, but I am not sure if the opposite is true. I am struggling to show that NO such continuous utility function exists.

My thoughts are that if you can define something with discontinuous preferences then maybe you can use this to imply that there do not exist any continuous utility functions. Note that Lexiographic preferences will not work because I am interested in a preference that can be represented by a utility function (albeit discontinuous).

The easiest way to prove it is using the 'old' definition of continuity.
$\succ$ is continuous iff whenever $x\succ y$, there exists neighborhoods of $x$ and $y$, $B_x, B_y$, such that all $z\in B(x)$ and $z'\in B(y)$, $z\succ z'$.

Suppose $x\succ y$. Because $u$ represents $\succ$, $u(x)>u(y)$. Let $2\epsilon=u(x)-u(y)$. Because $u$ is continuous, there exists some $\delta>0$ such that for all $z\in B_{\delta}(x)$, $u(z)>u(x)-\epsilon$. Similarly, for all $z'\in B_{\delta}(y)$, $u(z')>u(y)+\epsilon$. But then for all $z\in B_{\delta}(x)$ and $z'\in B_{\delta}(y)$, $z\succ z'$ as required.


Who were the economists who made lots of money in the financial markets? I prefer to learn economic theories from economists who made lots of money from the financial markets and were aided by their economic ideas. Who were they? What are the notable economic theories did they formulate?

At university classes, I always learned that the only economist who ever became rich was Ricardo. However, Ricardo was already rich before becoming an economist.

This article seems support this claim. It also gives an explanation for the lack of rich economist:

  If becoming wealthy is your goal in studying economics, you may be disappointed.  Although most economists make a good living, few have become rich from their knowledge of economics. In fact, if economists had some secret for making money in, for example, the stock market, they would likely be using those secrets to their own financial advantage [...] In short, economics won’t necessarily make you richer, but it may keep you from making some decisions that would make you poorer.


What is the point of silent auctions? A while back, someone decided to auction off something unique and valuable, as a silent auction: everyone would submit one bid only, and whoever bid the highest would win.

I submitted a bid that I thought would be higher than most people would pay, but I didn't win.  He posted the final result, though, and the winning bid wasn't much higher than mine.  If I had known it would go that high, I would have bid higher, and other people probably would have as well, which means he ended up leaving money on the table.

That makes me wonder: what is the point of running a silent auction if it can so easily be shown that it brings in less money than an auction in which people can openly bid against each other?

But perhaps this is not always the case. The price might have been so high that you would not have been willing to bid above it. In fact it might have been so high that the no one but the highest bidder would have given that bid. Knowing this the highest bidder could decrease his bid. In fact in English auctions (non-silent rising bid auctions) the final price is the second highest valuation of the unique good. In the auction you describe (a sealed bid first price auction) the final price is usually somewhat random based on what people believe about each others valuations and each others bidding strategies. It can happen that the final price is higher than it would have been in an English auction because one of the bidders overvalued what his competition would be willing to pay. So the English auction is not clearly advantegous.

It has been shown that given some rather strict conditions the English auction and the sealed bid first price auction have the same expected yield for the auctioner. This is known as the "Revenue Equivalence Theorem".

In practice it is somewhat agreed upon that in rising price auctions bidders experience something called 'auction fever'. Basically being outbid encourages them to raise their own bid. This is true even if the auction was designed in such a way that revealing the true highest valuation by all bidders would be an equilibrium move. (Auction by proxy, the auction system eBay and several other sites use is such an auction type.)

So it seems that in practice you might be right and rising price auctions might be better in expected value for the seller. This is just speculation but in your case you I think you might merely be experiencing remorse because you have undervalued your competitor's bid. I have been there :)
 (By the way this could encourage you to overbid on your next silent auction, to the benefit of the seller. Such overbidding might give him higher than English auction payoffs. Again, which system is better is not always clear.)


One bidder in 2nd price auction in second priced auctions, the highest bidder wins the auction and pays the price of the second highest bidder. What happens if there's only one bidder? That bidder would have won the auction, but does the winner pay his/her own bid? Or does the winner pay zero, imagining a second bidder with zero value bid?

Thanks

The purpose of a 2nd price auction (Vickrey Auction) is to encourage bidders to bid their true maximum willing payment for a good rather than the lowest price they expect will win the auction. The reason for this is because they pay the value of the unknown second bid they don't have to worry about bidding to high because the highest bid will always pay the exact lowest they would have had to pay in order to obtain the product and don't have to worry about overbidding.

For a auction to be called an auction there must be multiple bidders. If I know that I am the only bidder I would always just bid $.01 and get everything for free. To prevent this and other types of bidder collusion all auction houses have a minimum starting bid for a product usually determined by the seller. If no one is willing to pay more than the minimum then the product simply wont be sold. Individual rule for vickrey auctions differ depending on the venue but I imagine if only one bid is made then the price would default to the minimum bid set before the auction.


Does the Italian national debt really go all the way back to Mussolini's excessive prewar spending? The WWI & WWII debt is  quite well defined in the UK for example and quite shocking how many years it took to pay off.

Somebody told me that the Italian debt crisis really goes all the way back to Mussolini's excessive prewar spending and in 2011 Berlusconi defended Italy's debt crisis as he argued it went back to the war and was just the way the country runs.

What is the true reason? Given that a country such as Russia managed last year to pay off all the Soviet debt it seems quite strange that a G7 economy like Italy could have a debt going back to Mussulini's era starting in the twentys.

Obviously a countries debt is made up of many things such as public spending and in all countries but chronically in some corruption. But there often is an isolated or small number of decisions, policy or event that accountants for the beginning of the problem.

P.S is this question perhaps for the history stackexchange?

Some historic/economic background is summaried here: https://en.wikipedia.org/wiki/Economy_of_Italy_under_fascism

Debt Crises are the inability to pay interest on the debt a country has already taken up to that point. There are many theories, about how such crises can develop, however all have one thing in common. You need debt to have a debt crisis.

What is debt? Debt is when a country spends too much compared to how much it earns (in case you are familiar with simple econmic terms, that is G (gov. spending) exceeds T (gov earning — $ \Delta M$ could be taken into account, is however negligible for most countries, as money printing is usually not the way to finance debt.)

"War is expensive", means that during war-time there are HUGE governmental expenses to production of weapons and what not.
However, to say that the debt crisis is due to WW expenses one would have to look at how debt levels of Italy have developed after the world wars.

I have found a graph in my book and saw that in the 1970 Italy had a debt ration $(\frac{B}{Y})$ of 40 percent, which in today's terms is not much, rather below average.

In the years 1980 to 1990 Italy took up a lot of new debt, peaking at a debt level of ca. 120% in the early years of the 90's. From there debt has been 'steady' with increases to 130% in 2014. Thus, one can say. No. The debt crisis is not due to war expenses, when looking at it from this purely expense-tracking perspective...


Why can't countries immediately stop the deficit? Governments often say things like "we will reduce the deficit by 2025".

Why can't they simply eliminate the deficit immediately by spending less than the taxes they receive?

Surely if a country is in debt, it should try and run a surplus so it can start paying off it's debts and reduce its interest payments.

Is there some reason why they can't do this?

Some countries have very strict fiscal rules that basically force them to keep on spending more, or reduce the government's ability to cut spending (Brazil is a good example). Other countries have looser rules, but can't cut because of political pressures.

Remember acute spending cuts mean less money for hospitals, schools and many other services that rely on public money. It's often a very impopular measure.

That's why economists often favor clear, well-defined spending rules. Such as establishing a primary surplus target during times of economic growth, a maximum deficit during recessions and maybe a debt/GDP ratio ceiling.


How was the Cobb Douglas function derived? In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.
How we get the formula $f(K,L)=AK^aL^{1-a}$? How we get that $K$ must be elevated to some factor "$a$" and this must be multiplied by $L$ elevated by some factor "$1-a$"? What is the proof of this formula? I can't find an answer to this questions i search it a lot.
Update.
I found an article that shows how to deduce it, but i don't understand some steps, could someone help me? Here goes the link:
https://www.studocu.com/en-gb/document/kings-college-london/mathemtics-for-economists/lecture-notes/cobb-douglas-revision/4229598/view
In these terms, the assumptions made by Cobb and Douglas can be stated as follows:

If either labor or capital vanishes, then so will production.
The marginal productivity of labor is proportional to the amount of production per unit of
labor.
The marginal productivity of capital is proportional to the amount of production per unit of
capital.

Solving.
Because the production per unit of labor is $\frac{P}L$
, assumption 2 says that:
$$\frac{∂P}{∂L} = α\frac{P}L $$
for some constant α. If we keep K constant($K = K_0$), then this partial differential equation
becomes an ordinary differential equation:
$$\frac{dP}{dL} = α\frac{P}L $$
This separable differential equation can be solved by re-arranging the terms and integrating both
sides:
$$\int \frac{1}P \, dP = α\int \frac{1}L \, dL$$
$$ln(P)=α*ln(cL)$$
For example here, from where it comes the constant "c"?, then following:
$$ln(P)=ln(cL^α)$$
$$P(L,K_0)=C_1(K_0)L^α$$
where $C_1(K_0)$ is the constant of integration and we write it as a function of $K_0$ since it could
depend on the value of $K_0$.

What is the proof of this formula?

There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production function but I seen many others as well.
What you provide below in your update is not so much of a proof that production has to be Cobb-Douglas rather its a proof that if we make some specific assumptions about production (which although quite general might not always necessary hold actually) we get function that is Cobb-Douglass.

$ln(P)=α∗ln(cL)$
For example here, from where it comes the constant "c"?,

the $c$ is an integration constant. Whenever, you have indefinite integrals you have to add some constant $c$ to the solution because constants get eliminated during differentiation we can never know if there was or wasn't some constant previously so after integrating we always add $c$.
In this case when you integrate this separable differential equation the solution would actually look like:
$$\ln(P)= \alpha (\ln(L) + C) \implies\ln(P)= \alpha \ln(cL) | C= \ln(c) $$
(actually the variables should even be in absolute values - but since the function is only defined for non-negative values of $P,L$ and $K$ we can omit them).


How to solve a pricing competition for complementary goods I am preparing for an exam and I have come across this question in my textbook but I am not quite sure how to solve it so would really appreciate some help!

  Suppose that company L produces left shoes and company R produces right shoes. If L charges $p_L$ for a left shoe, and R charges $p_R$ for the right shoe, then the price for a pair of shoes is $p=p_L + p_R$. The quantity of pairs of shoes purchased is determined by the demand function $q = 100 - p$. The cost of production is $c>0$ per shoe. Both firms choose prices simultaneously and independently of each other. Formulate this situation as a game (specify the players, strategies and payoff functions).

So obviously there are two players in the game, company L and company R. In terms of strategies I am slightly confused but I assume they are the following three:
- $p_R < p_L$
- $p_R = p_L$
- $p_R > p_L$

Please can you tell me is this is correct?
Secondly, I am extremely confused by the payoff. I understand it would be optimal for both firms to price their shoes at the same level. And I know that to calculate the payoff we need to calculate profit.

Profit = $(100-p)\cdot q - c\cdot q$
Where I am stuck is whether it is necessary to sub in the expression that we have for q?

Finally, I believe that the payoff when prices are equal is just half the total profit for each firm. However, I have absolutely no idea what the payoff would be when the prices don't align. Any hint would be really useful! Thank you!

Here are some hints in the spirit of helping you prepare for the exam by thinking about the solution yourself:

You are right about the two players.

Your formulation of strategies is incorrect. Think of a strategy as a complete set of instructions that a player writes down before the game begins. These instructions are handed to a lawyer and must contain the information that the lawyer needs to play the game on the player's behalf and in accordance with his wishes. In particular, the instructions can't ask the lawyer to use information he doesn't have access to and must tell the lawyer how to deal with any circumstances that might conceivably arise. Put yourself in the shoes of the lawyer: you are supposed to choose a price on $R$'s behalf. What is the set of instructions that you would understand and have enough information to implement? Note that "$p_R<p_L$" doesn't pass this test because

it would require that the lawyer knows the value of $p_L$ before the game starts.
it doesn't pin down which $p_R$ the lawyer should choose. Even if he knew that $p_L=10$, there are many choices of $p_L$ that satisfy $p_L<p_R$. The poor lawyer would be very confused!

The payoffs from a game are a function that takes as an argument (i.e. as an input) a strategy for each player and spits out as the result the corresponding payoff. So $L$'s payoff function will look something like this $$\underbrace{(\text{L's strategy},\text{R's strategy})}_{\text{input}}\mapsto \underbrace{\text{payoff for $L$}}_{\text{output}}$$.

Now, imagine that I have asked $L$ and $R$ what their strategies are and have them written down here in front of me: $p_L=\ldots,\ p_R=\ldots$. Your job is to give me a rule that I can use to take this (and only this) information and calculate what each player's payoff will be. That rule is the payoff function.

Once you found the right payoff function, the question of what happens when the two prices are not equal will answer itself because the function will output the payoffs for any valid input strategies.


Impact of auction systems that allow 'sniping' I'm a bit perplexed that eBay has auctions that have a fixed time ending, and so winners are often determined by who can 'snipe' the auction, by putting in the winning bid at the last possible microsecond.

Often specialised software is used to achieve this.

An alternative system, like New Zealand's Trade Me auction site uses, is that the auction is auto-extended by two minutes, if a bid is put within the last two minutes of the auction.

What this means in practise (from what I've seen in auctions I've particpated in) is that an auction might be extended by 10-30 minutes due to 'last minute' bids between two or more bidders.

Now, it should be noted that the auto-extending dynamic doesn't completely negate the sniping dynamic. For example if Bob sees that Alice has put a bid on a item for \$1000, which is due to close in two days, and Bob is willing to spend \$2000 on that item, it still makes sense for Bob to hold off bidding on the auction until closing time. Alice may not be paying attention\be at her computer (though trade me allows auto-bids) at closing time, and by bidding at the last minute, he reduces the window in which she can outbid him.

This auto-extending system seems like it evaluates the market value more genuinely than the sniping system.

Ebay is surely aware of this dynamic, and it must be a conscious decision to keep the sniping-allowed system.

My hypothesis is that systems that allow sniping encourage higher selling prices, as bidders pre-emptively increase their bids as to stay out of the sniper's willing-to-bid range.

Firstly, it's right to say that the hard ending rule on ebay seems to be behind sniping. Here's a figure from Roth & Ockenfels' AER paper:

On eBay (fixed end time) bids arrive much later than on the now defunct Amazon auction platform (which used an extendable end time).

One explanation (see here and here) for this is that people want to hide their private information. If you think that you discovered a 'hidden gem' then the last thing you want to do is start bidding and let everyone else know that it's valuable! Note, from the figure, that sniping seems to be more acute in antiques auctions, where private information is likely to be more important.

The other leading explanation is that some bidders do not understand that they can use the automatic proxy system to take care of the bidding, and 'naively' sit incrementing their bid manually. Sniping is then a way to take advantage of such bidders' poor reaction speed. An experiment by Ely & Hossain that appeared in AEJ Micro verified that sniping indeed gives you an advantage over such naive bidders. This marketing Science paper by Zeithammer and Adams also provides evidence that manual incremental bidding is common and that it is therefore wrong to think of eBay as a simple implementation of a second price auction.

So, two perspectives on what is going on here: concealment of private information and exploitation of naive bidders. Both of these have one thing in common: they imply that Sniping leads to lower prices, not higher.

So why does eBay persist in the use of such a mechanism? Only they know. But here is what I suspect:

eBay is a two-sided market. It must attract both sellers and buyers. Usually, optimal pricing in a two-sided market involves subsidizing one side of the market to get them onto the platform before milking the other side. But buyers already pay $0 to eBay; what if a bigger subsidy is needed? One way to implicitly subsidize buyers and attract them to the platform is to choose a mechanism that results in a lower selling price. In the end, if the result is indeed to attract more bidders, this might not even be that bad for eBay. Bulow and Klemperer have shown that adding bidders to an auction is almost always more profitable for the seller than redesigning the auction but keeping the number of bidders the same.


(Modified comparative advantage model)Can trade with a country that only has one industry be mutually beneficial? Suppose there are two countries, China and Japan. Suppose there are also only two goods, namely food and bronze mirrors. China produces both food and bronze mirrors, while Japan produces only food. In order to produce 1 unit of food China has to give up 0.3 units of bronze mirrors, while Japan needs to give up 0 units of bronze mirrors due to absence of bronze mirrors production.
In order to produce 1 bronze mirror China has to give up 3.33 units of food, while Japan can't sacrifice anything for 1 bronze mirror because it doesn't produce bronze mirrors in the first place.

Given this situation, can it make sense for China to start specializing in producing and trading bronze mirrors with Japan?

Normally we would compare opportunity costs for both industries in both countries in order to conclude who has comparative advantage in what industry and then make countries specialize and trade accordingly, but in this example this will work only with one country. In this example Japan has comparative advantage in food production because it's opportunity cost of producing food is zero. So Japan should specialize in producing food. BUT, it's unclear how can we justify trade where China specializes in producing bronze mirrors. After all, it's incorrect to say that China has COMPARATIVE advantage in producing bronze mirrors (how can there be any comparison if there is nothing to compare with? Besides, from more mathy point of view, attempt of such comparison will fail due to division by zero).

I did not encounter this kind of example before but I think comparative advantage would still apply. In Ricardian model opportunity cost can be expressed as a ratio of outputs $a$ for food $f$ and bronze mirrors $b$ as $a_f/a_b$. If Japan can create A units of food but 0 bronze mirrors the opportunity cost would be given by $A/0$ now this is undefined in mathematics, because limit of the expression would not give you the same value from both sides, but since we are working only with positive quantities (presumably output cant be negative) we could evaluate the limit only from positive side which will give you $\infty$ meaning that no matter how much you sacrifice food you will get no mirrors.

The rest of the problem could be solved in this way, the equilibrium price that should be $a_{f}^{'}/ a_b^{'} < P_f /P_b < a_f/a_b$  would just have the upper bound equal to $\infty$. The exact quantities demanded would be given by the assumptions on individual demands in countries etc. Even in case this would not be mathematically rigorous I think you could carry out virtually the same analysis with just assuming some small infinitesimal production possibility for Japan and examining how the solution behaves as it goes to 0.


Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc. I am solving old problems from various qualifiers from different universities to prepare myself for an upcoming test. I came across this and wanted to ask if anyone can confirm my answers?

My answers:

** I use $\succeq$ to denote "at least as good as".

(a) A certainty equivalence, in general, is the amount of money $c(F,u)$ so that:

$F \in \Delta(\mathbb{R})$, $\delta_{C(F,u)} \backsim F$ $\equiv u[C(F,u)] = U(F)$

Here, I've never seen something like this and so it is my best guess:

$ \forall$  $x<M $ , $C(F,\sqrt(x))= u^{-1}[\int\sqrt(x)dF$]
$ \forall$ $x\geq M$, $u[C(F,\sqrt(M))]= [\sqrt(M)] \implies C(F,u)=x$

Since $u()$ isn't 1-1, thus not invertible, over $x\geq M$ I eventually just decided my result above was true?

Or should it be something more like:

$u[C(F,u)] = F(M)\sqrt(x) + [1-F(M)]*\sqrt(M)$

(b) I know that the certainty equivalent is less than or equal to the expected value of $F$ iff an agent is risk averse.

I think that is the same as saying $u[C(F,u)] \leq u(\int x dF)$ , $\forall F \in \Delta(\mathbb{R})$

(c) An agent is risk averse if and only the agent's preferences are represented by a concave utility index $u(.)$  and so this agent is risk averse since:

$x < M$ $\implies u(x)=\sqrt(x)$, which is clearly concave.
$x\geq M$ let ${x_1,x_2} \subset [M,\infty)$ and let $\alpha \in [0,1]$ Then $\alpha*x_1 + (1-\alpha)*x_2 \in [M,\infty)$

Now, note that $u(x_i)=\sqrt(M)$, for $ i=1,2,3$

and so $$u(x_3) \geq \alpha u(x_1) + (1-\alpha)u(x_2)$$

$$\to u(\alpha x_1 + (1-\alpha)x_2) \geq  \alpha u(x_1) + (1-\alpha)u(x_2)$$

(D) Again, I am not sure about this. All i know about F.O.S.D is that for two lotteries F,G, then for all EU money monotone preference:
 $$F \geq_{FOSD} G \iff F \succeq G $$

Any and all help is appreciated.

Suppose random variable $X\sim F$.

Certainty equivalent of the lottery $X$ is the constant $c$ that solves:

  $u(c) = \mathbb{E}(u(X))$

Because $u(x) = \min(\sqrt{x}, \sqrt{M})$ is concave, by Jensen's inequality:

  $u(c) = \mathbb{E}(u(X)) \leq u(\mathbb{E}(X)) $

Agent is risk averse because $u$ is concave.
If agent prefers lottery $X\sim F$ to lottery $Y\sim G$, it is not necessary that $F$ FOSD $G$. Consider the following two lotteries:

  $X = \frac{M}{2}$ with probability 1.

  $Y = M$ with probability $\frac{1}{2}$ and $Y = 0$ with probability $\frac{1}{2}$.

Since $u$ is concave, lottery $X$ is preferred over lottery $Y$. This because $X$ gives the expected value of the lottery $Y$ with probability 1. Also, $F$ does not FOSD $G$ because $F\left(\frac{M}{2}\right) = 1 > \frac{1}{2} = G\left(\frac{M}{2}\right) $.


Connection between saving and investments-Y=C+I+G+NX I understand
Y-G-C = I+NX = Savings . The equation that we see everywhere in economic textbooks. I see the mathematical logic.
BUT
Given that I = capital investment in GDP accounting surely Savings are what is not spent after you haven't spent anything on capital goods etc. So in my head
Y-G-C-I = NX = S and infact why not take NX over to the other side!
What "mistake" am I making here?
I can regurgitate the textbooks and get the marks but I just don't understand it for myself from first principles.

The $I$ is actually not just capital spending. In some models that might be true, for example, in Solow growth model we will assume that all investment is capital spending but $I$ is not just that.
For example, $I$ would include both purchases of new houses (although argument could be made they are capital goods, they would not satisfy more narrower definitions requiring that capital being factor of production), or investment in inventory for example (see more detailed explanation in Blanchard et al. Macroeconomics: a European Perspective, pp 41-43).
Also, spending on capital good is literary considered saving in economics. In economics saving has more broad meaning than what ordinary people just consider 'saving'. For example, in economics when you buy stocks you are also saving even if you spend money doing so.  In economics by definition the portion of income that is not consumed is saving (see ibid. Macroeconomics: a European Perspective pp 569-570).
PS: As pointed in the comments, you should have minus in front of $C$ so the equation should be: $ Y-G-C = I+NX$.


Which database language and database management system should an economics student learn? (Provided that one has to be learned) I will have to pick essentially any DBS, learn how to use it and use it during a period of 2-3 month. I do not possess any noteworthy background knowledge, so I will be starting from scratch independent of my choice.
Since I am hoping to become an academic economist, I was wondering whether a particular language or specific DBS is particularly widely used in economic research or is otherwise especially useful for economic research.
Since I could not find any articles/statistics on this topic otherwise, I thought I would ask here.

First, some clarity. The standard language used for interfacing with a DBMS is SQL. However, its syntax and functions slightly vary according to the DBMS system you are interfacing with.
The most popular ones are MySQL and PostgreSQL. The choice between them is mostly indifferent. In my personal experience, I found that building a MySQL database from scratch is slightly easier, even if PostgreSQL is more flexible.
If you are a complete beginner, first I suggest you to start by learning SQL syntax and best practices, and if you need to build your own database by choosing between one of these two.


How do I test if a given policy was successful? I have some data for medical R&D and sales, my professor has asked me to check if a policy implemented in 2016 has made any effects on R&D and R&D/Sales data.
I tried to plot growth rates over years, but he says we don't need growth rates, and it has to be solved just by mean R&D expenditures each year. (Mean over multiple companies). How do I go about significance testing of policy change?

Let us assume a balanced panel of companies $i=1,...,N$ observed in time periods $t=1,...,T$.
Let $s_{it}$ denote R&D expenses for company $i$ time $t$ and let
$$\bar s_t :=\frac{1}{N} \sum_i s_{it},$$
denote the time $t$ average.
Consider now a simple regression model
$$(1)\ \ s_{it} = \beta_0 + \beta_1I[t\geq 2016] + e_{it}$$
where $I[t\geq 2016]$ is indicator for the time being after policy implementation. I assume the policy is intended to stimulate R&D investments by companies and hence we assume $\beta_1>0$ if the policy has effect.
Using the normal equations for POOLED OLS estimator it is easy to show that
$$\hat \beta_0 = \frac{1}{NT_1} \sum_i \sum_t s_{it}I[t< 2016],$$
where $T_1$ is number of time periods before treatment. Hence $\hat \beta_1$ is roughly the average R&D expense level BEFORE treatment/policy implementation.
If model (1) is true then $plim \ \ \hat \beta_0 = \beta_0 = \mathbb E[s_{it} \lvert t<2016]$ which is the expected level of R&D expenses for any company for any year before treatment.
You can also show that
$$\hat \beta_1 = \frac{1}{NT_2} \sum_i s_{it} I[t \geq 2016]-\frac{1}{NT_1} \sum_i \sum_t s_{it}I[t< 2016]$$
If model (1) is true then $plim \ \ \hat \beta_1 = \beta_1 = \mathbb E[s_{it} \lvert t\geq 2016]- \mathbb E[s_{it} \lvert t<2016]$ which is the expected level of R&D expenses for any company for any year AFTER treatment.
You could plot the series $\{s_t\}_{t=1}^T$ and get - if model (1) is true - something like this

However something else may be happening over time and while you do not have any control group of companies not affected by the policy implementation you can still potentially do better than estimate model (1).
Consider the following plot

where in each time period going from $s_t$ to $s_{t+1}$ there is a small increase in R&D let us call it $\lambda$. Then in the year of treatment the increase is slightly smaller.
The blue and read line are found using the estimators above and it is clear that because they do not take the time trend into consideration they overestimate the effect. The data used for the plot was generated after the model
$$s_{it} = \beta_0 + \lambda \cdot t + \beta_1 I[t\geq 2016] + e_{it},$$
which includes the time trend.
Here in this model the effect is immediate. But the effect could also happen in the years after the treatment instead of only in the year of treatment hence you could have a model like
$$s_{it} = \beta_0 + \lambda \cdot t + \beta_1 I[t\geq 2016] + \beta_2 I[
t\geq 2017]+ e_{it}.$$
The take away is that you should do as the professor says and calculate the time averages $s_t$ and plot them and look at them. Paying particular attention to what goes on around in the year treatment and in the years close to but still after treatment (based on the principle that effect comes after cause and takes some time be fully realized)
I R to simulate data and generate the plot by means of the following lines of code:
# simulate panel of RD
library(data.table)
set.seed(1)
N <- 50 #Number of firms
T <- 16     #Number of time periods
TT <- 9 #policy implementation year (treatment year)

b_0 <- 5
b_1 <- 2

# Simulate model (1)
s <- b_0 + b_1 * (1:T >= TT) + rnorm(N*T)

# Make dataset
id <- rep(1:N,each=T)
time <- rep(1:T,N)
I <- as.numeric(time>=TT)
mydata <- data.table(id,time,s,I)

# Compute time averages
time_agg <- mydata[,.(s_t=mean(s)),by=time]

#Plot time averages
plot(time_agg$s_t,ylim=c(0,10),pch=20,xlab="time",ylab="R&D")
points(1:TT,rep(b_0,TT),type="l",lwd=1,col="blue")
points((TT):T,rep(b_0+b_1,T-TT+1),type="l",lwd=1,col="red")
savePlot("plot1.jpg",type="jpg")

# Simulate model (2)
s <- b_0 + b_1 * (1:T >= TT) + 0.5*1:T + rnorm(N*T)

# Make dataset
mydata <- data.table(id,time,s,I)

# Compute time averages
time_agg <- mydata[,.(s_t=mean(s)),by=time]
model <- lm(s~I,data=mydata)
hat_b_0 <- coef(model)[1]
hat_b_1 <- coef(model)[2]

#Plot time averages
plot(time_agg$s_t,ylim=c(5,16),pch=20,xlab="time",ylab="R&D")
points(1:TT,rep(hat_b_0,TT),type="l",lwd=1,col="blue")
points((TT):T,rep(hat_b_0+hat_b_1,T-TT+1),type="l",lwd=1,col="red")
savePlot("plot2.jpg",type="jpg")


References on the sustainability of the public debt I am an student who does some research on sustainability of public debt.

Which papers are essential to understand what determines the sustainable debt threshold, and the sustainable level of public debt?

A big debate started after a flaw had been found in the conference paper "Growth in a Time of Debt" by Reinhart and Rogoff. There are several replications and many articles in the media and blog posts. Reinhart and Rogoff have the book "This Time is Different" on the topic and several well published articles, cf. here. Research on the topic continues to be published in top journals, e.g. "Is There a Debt-Threshold Effect on Output Growth?", Review of Economics and Statistics 99:1, March 2017. You should look at the references and also at which papers have cited the ones you find important.


Modeling job application process in labor market I need to model the job application process by job seekers. Specifically a job seeker when looking for a job, comes across a job and then decides whether to apply for it or not. I want to model this application decision. Assuming that job seeker has already come across a job, I want to model what are the factors (of the job/employer) that will make this person apply or not apply to the job.

I will appreciate any references that have done similar modeling or any suggestions in this direction. I have been reading papers from job search theory but I can't figure how to use search theory to build my model. The search literature seem to argue that only factor that may affect person's decision to apply is the wage offer from current job versus an outside option. However, in the absence of wage offer; for example person looking at job description and thinking whether he should apply to this job or not, is the question that I'm really after. I guess this is still a search question, it's just that I want to model for other factors (expectation about wages, job condition, employer, my own taste towards the job/employer etc) in the application decision. Appreciate any help!

Chapters 6-8 (especially 8) in Modern Labor Economics: Theory and Public Policy, a textbook by Ehrenberg and Smith, cover the topics concerning labor supply. I guess the parts in Ch. 8 that discuss hedonic wage theory would be most relevant to your inquiry.


What effect would converting the lowest denomination note to a coin be? In New Zealand our lowest denomination note is a $5 note.

For perspective, a can of coke costs $1.50 - $2.00, a Big Mac costs around $5.

The idea of a $5 coin is on the cards.

The question is - what effect does turning a note in to a coin have?

I imagine there would more of a piggybank effect (coins being lost, stuck in collection jars).

Are there any studies on what's happened when notes have been replaced by coins?

The usual rationale for replacing notes with coins is that small-denomination notes and coins circulate heavily, and notes need to be replaced much more often than coins. However, as an example, the costs of transitioning to a dollar coin (think vending machines, etc.) in the U.S. would be significant enough to make the change not worthwhile, according to a FRB staff working paper. See "Costs and Benefits of Replacing the \$1 Federal Reserve Note with a $1 U.S. Coin" for details on their approach, which covers the issues pretty well.


What changes the structural rate of output/unemployment? To me the structural rate of unemployment was defined by $$\frac{1}{1+m} = F(u_n,z)$$ where $m$ is the markup and $z$ a catch-all variable. We could equivalenty get this in terms of $Y_n$ by letting $u_n = 1 - \frac{Y_n}{L}$, $L$ being the labor force.

Let $L$ now rise. What happens to $u_n$ and $Y_n$? From the equation, if $L$ rises but everything else must remain constant, then $Y_n$ must rise in order to still satisfy the equation, by the same amount. But then $u_n = 1 - Y_n/L$ is unchanged? Does this make sense? Why is the natural rate of unemployment unchanged when we just increased the amount of people willing to supply labor?

EDIT: The markup is the one that firms use to set prices, with respect to wages, $\text{prices} = \text{wages} \cdot (1+m)$, F is a function which is negative in $u_n$ (more $u_n$ means smaller $F$-value given constant $z$) and positive in $z$, and $Y_n$ is the structural rate of output.

Let $L$ now rise. What happens to $u_n$ and $Y_n$?

From the equation, if $L$ rises, everything else equals, then $u_n$ will rise (that is, the fraction will be smaller).

$Y_n$ will also rise and that is a fairly reasonable conclusion - Extra work force will lead to more outputs.
Now, the misleading conclusion is in regard to the markup. Consider now the labour market. When demand (work force) increases and all else remains equal (supply side - firms offering job opportunities), wages will lower and thus that setup will have a new equilibrium - with lower wages and more work force.

This is my interpretation of your question, although the answer may not be 100% accurate, or an answer that actually answers your question.


Efficient social allocation awards item to the highest valuer in a sponsored search auction. Whom is it optimal for? In mechanism design, for example in the case of a single item auction, we often assume that a socially efficient outcome is to allocate the good to the bidder with the highest valuation of that good. In the case of a sponsored search auction, we want the items (slots on the sponsored search space) to be handed out in such a way that the most valuable slots go to the bidders with the highest valuation of the same slots.

In the sponsored search auction case, whom is the allocation optimal for? The bidders, or the web user, or both? If it is optimal for the web user, why?

The optimality concept used here is usually Pareto-optimality, a.k.a. Pareto-efficiency.

  Pareto efficiency or Pareto optimality is a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off.

So a Pareto-efficient allocation is not necessarily 'optimal for' anyone. It just means that given the current allocation no one can be made better off without cost to others.


Do companies have non compete agreements between each other? Do companies also know for a better lack of words armistice? When searching for non compete agreements everything that comes up are agreements between companies and employees, but do whole companies also have non compete agreements with other companies? This is akin to price fixing, but would include more, since competition is not only restricted to price. I wonder if companies which are not allowed to merge because of monopoly laws have such non compete agreements. What are such non compete agreements called? Because right now this term seems to only apply to contracts between companies and employees.

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


How do I calculate the services sector as a percentage of GDP? Could anyone explain how I should calculate the share of services sector as a percentage of GDP?

I get the numbers for US GDP from Table 3: https://www.bea.gov/system/files/2019-10/gdp3q19_adv_0.pdf

The GDP in 2018 (numbers in billions of $) was 20,580.2 and services accounted for 9,633.9, which gives us 46.8% of total GDP. However, there are many resources that say services make up about 80% of the US economy, including Statista  or Wikipedia (under the 4.1 Nominal GDP sector composition section).
Note that I use the numbers for 2018, Statista only up to 2016, and Wikipedia  for 2015-16, but that shouldn't make so much of a difference.

What am I missing in my calculations?

The table you are looking at is a GDP expenditure measure.  But this ignores the fact that services are involved in the costs of goods purchased, and that goods are involved in the cost of services purchased.  For example, if you buy a cake in a supermarket, the price reflects not just the food manufacturing industry, but also agriculture, transportation, retail trade and many more.

To consider the importance of different industries in the economy, better analysis comes from the GDP income measure considering the value added (essentially wages and profits) for each industry. So looking at table 5 of another BEA release on GDP by industry, and stripping out duplicated values for sub-industries, you get something like

                                                                     2018
                                                                $ billion
Gross domestic product                                           20,580.2

--Private industries--
Agriculture, forestry, fishing, and hunting                         166.5
Mining                                                              346.6
Utilities                                                           325.9
Construction                                                        839.1
Manufacturing                                                     2,321.2
Wholesale trade                                                   1,212.2
Retail trade                                                      1,126.9
Transportation and warehousing                                      658.1
Information                                                       1,067.7
Finance, insurance, real estate, rental, and leasing              4,301.6
Professional and business services                                2,579.4
Educational services, health care, and social assistance          1,792.5
Arts, entertainment, recreation, accommodation, and food services   860.6
Other services, except government                                   437.2
--Government--
Federal                                                             790.8
State and local                                                   1,753.8

If you add up the first five rows (Agriculture through to Manufacturing) you get \$3,993.3bn (19.4% of total GDP), while if you add up the other rows you get \$16,580.8bn (80.6% of total GDP)


Imperfect Substitutes and Utility Funcitions The utility function for perfect substitutes is defined as U(X,Y) = aX + bY. If the two goods X&Y are imperfect substitutes what would be their utility function?

In general, any utility function that produces non-linear and downward sloping indifference curves will feature imperfect substitutability between goods. (By considering only goods, we're implicitly assuming "more is preferred to less", or positive marginal utility, for each good.) In other words, look at the marginal rate of substitution and see if it depends on the quantity of either good; if it does, then there is imperfect substitutability.
Examples

Quasi-linear utility: $u(x,y)=v(x)+y$, where $v(x)$ is non-linear in $x$ (e.g. $v(x)=\sqrt x$)
Additively separable utility: $u(x,y)=g(x)+h(y)$, where $g,h$ are non-linear functions
Cobb-Douglas: $u(x,y)=x^\alpha y^\beta$, where $\alpha,\beta$ are constants
Leontief (perfect complement): $\style{text-decoration:line-through}{u(x,y)=\min\{\alpha x,\beta y\}}$ [Leontief utility function features nonsubstitutability rather than imperfect substitutability.]
CES utility: $u(x,y)=\left(\alpha x^\rho+\beta y^\rho\right)^{1/\rho}$, where $\rho\ne1$. Note that the class of CES utility functions includes Cobb-Douglas (as $\rho\to0$), Leontief (as $\rho\to-\infty$), and linear utility (as $\rho=1$) as special cases.

This lecture note by Simon Board has examples and indifference curves corresponding to the cases above.


how to explain the term “Compensated law of demand” in a plain way? The term “compensated law of demand” I met in advanced microeconomics textbook, which appeared in “revealed Preference “

The compensated law of demand is a tool we use to analyze the decomposition of substitution and income effects. We take a price change which changes the relative price ratio (the slope of our budget constraint in a simple two good example) which leads to a new tangency point on a new 'rotated' budget line. For simplicity, if we consider the tangency condition:
$$
|MRS| = \frac{MU_1}{MU_2} = \frac{p_1}{p_2}
$$
Then as $p_1$ rises, for example, the value of our MRS defined as the slope of our indifference curve would rise at the optimal point. So our new tangency would be higher up on this new lower indifference curve. What we would then do is ask the hypothetical: Under the new prices, how much income would leave the consumer just as well as off before the price changes? This is the idea of compensated demand. What would we do in practice is essentially shift the budget line up (or down) until it is tangent to the original indifference curve. This gives us the pure substitution effect abstracting away from the change in purchasing power.

Since you are reading from an advanced microeconomics textbook I imagine you are familiar with Marshallian and Hicksian demands, these concepts are very closely related to how we compute this hypothetical budget constraint. Sadly that is the simplest way I can think of explaining it as it is an 'advanced' concept. In practice, we don't usually use the tangency condition to arrive at these results but it helps in simplifying the concept.


Collar strategy for options trading I am new to investing on the market so I am doing some research and studying. From what I can tell, the purpose of a collar strategy for options trading is to protect your gains you've made on the underlying stock.

If you want to protect your gains, why would you not simply buy an OTM put?

Is the purpose of writing a covered call in a collar strategy to offset the premium cost of the put?

Edit: To clarify, why would you write a call to possibly cap your gain and lose your shares if your goal is to protect from loss?

A collar strategy is only in part to protect your gains made on the underlying stock. The collar strategy consists of buying protective puts and at the same time writing covered calls.

You are right in saying "If you want to protect your gains, why would you not simply buy an OTM put?". This is known as the Protective Put and if you simply want to protect your gains, this is the way to do it.

When you also write a covered call, you once again got it right that it is usually used to offset the premium costs of the put.

However when you combine the two strategies (creating an equity collar), your view of the market would be that it is moving sideways. I.E. you are expecting the asset prices to not rise above your written call's strike price, and you are protecting your asset prices in case it falls below your put's strike price. And in the meantime, you are still holding on to the asset earning dividends while the covered call is financing the protective put.

On a side note, this question seems to be more in tune with the Quantitative Finance Stack Exchange community.


What is the exact relationship between the efficient-market and random-walk hypotheses? The efficient-market and random-walk hypotheses are often discussed together as formalizations of the idea that "you can't consistently beat the stock market". But in my experience, they're often used interchangeably, and I've never seen anyone carefully explain the difference. (For example, Wikipedia says that A Random Walk Down Wall Street "argues that asset prices typically exhibit signs of random walk and that one cannot consistently outperform market averages. The book is frequently cited by those in favor of the efficient-market hypothesis." The Wikipedia article on the EMH says "there is a very close link between EMH and the random walk hypothesis" and the one on the RWH says that it "is consistent with the efficient-market hypothesis", but neither spells out a precise relationship.)

What is the exact distinction? Is either hypothesis a special case of the other? Put another way, could one imagine a stock market in which either hypothesis holds but the other does not? The EMH page says "Samuelson published a proof showing that if the market is efficient prices will show random-walk behavior", which seems to imply that the EMH is a special case of the RWH - so that one could imagine a stock market in which RWH holds but not EMH, but not the opposite. Is this correct?

They don't have anything to do with each other.

Use the superscript $e$ to denote "anticipated value" in whatever way it is formed. Then the Efficient Market Hypothesis states

$$X^e_{t+1,market} = E(X_{t+1}\mid I_t)$$

In words, the value anticipated "by the market" equals the true expected value conditional on all information available at time $t$. So first and foremost, the EMH is about what is the predictor that reflects the anticipation of "the market", and it asserts that it is the conditional expected value based on all information available (to state the obvious, there are many other predictors of future values. We can use the median, the mode, etc etc etc).

By the properties of the conditional expectation we have that

$$X_{t+1} = E(X_{t+1}\mid I_t) + u_{t+1},\;\;\; E(u_{t+1}\mid I_t)=0$$

Therefore the EMH claims that the value anticipated by the market, is off-target only by something ($u_{t+1}$) that cannot be predicted as anything other than zero, given the information available. But that is what it means to "use the information efficiently": to extract from it everything that it can possibly tell you.

The "cannot beat the market" assertion is reflected in the part of the EMH that "the market" forms an expectation as though it was a single mind having available, again, all information. This implies that while "inside", "private" information may matter for what individual agents do and accomplish, if we somehow "average/combine" over those individuals each having a different information set, we will obtain the conditional expected value we would have obtained if we have collected all information in one Information set.

Nowhere in the above do we mention, or need to mention, what is the Data Generating Mechanism for $X$. It can be anything.


Breadwinners by class I have a hypothesis that single-income families greatly increase as you get into the mid to upper middle class bracket. And that 2+ income sources are common in lower classes and also the extremely rich.

I can't verify this though because I can't find any research that looks into how many single income households there are by class. Instead, I see mostly gender connections.

Does anyone know of any research that could be helpful here?

According to the third row in this table from the American Enterprise Institute (they credit the Census Bureau but I could not find the same indicators in their data)

your hypothesis is incorrect: the Mean Number of Earners per Household is increasing in income for each Quintile.

I am surprised by the average number of earners in the highest quintile, but I think it is not very surprising that the poorest people are poor because they have trouble finding stable employment.


What does commitment in adverse selection mean? In this slide deck, p15. it says
"The revelation principle requires that the principal can fully commit to the terms of the contract. If this is not the case, an indirect mechanism, which allows for some commitment, may strictly outperform any direct revelation mechanism, which allows for no commitment."

It sounds contradictory, because the first sentence sounds like "For revelation mechanism to outperform indirect mechanism, the principal must be fully committed." But then why does direct revelation mechanism allow for no commitment in the later part of the paragraph?
What exactly does "if this is not the case" mean here?
What exactly does commitment mean here?

Think about an auction, where the designer is selling good and trying to sell it to the person that values it the most while collecting as much revenue as possible.

A direct mechanism means that the seller asks buyers for how much they value the good and based on that decides who gets the good and how much they pay. Suppose that the designer uses a second-price auction so it tells the buyers "tell me your valuation and I'll give the good to the person with the highest valuation, but this person will only pay the second-highest valuation.

If the seller has commitment power (for example if they can write down the rules in a binding contract, etc), this auction achieves exactly what the designer wanted to do, maximizes revenue, allocates the good to the person who values it the most, and it is incentive compatible. However, notice the temptation that the designer faces:

Suppose the highest valuation is $\$2,000$, and the second-highest valuation is $\$1,200$. If the designer lacks commitment power, the designer might change the rules after learning the valuations and charge a lot more than $\$1,200$ for the good. However, if the bidders know that the seller lacks commitment power they will not truly reveal their valuations and there is no guarantee anymore that the auction will obtain good results.

Now keep assuming that the seller has no commitment power, but consider the following mechanism that is NOT a direct mechanism. Suppose the seller runs an ascending price auction. That is, bidders yell out how much they want to pay and can outbid each other as much as they want. Whoever's bid is the highest gets the good and pays what their last bid. Now the seller does not know exactly the valuation of the buyers, and arguably is no longer tempted to change the rules in the middle of the auction.

In conclusion, if there is no commitment, a not-direct mechanism can out perform a direct mechanism.


What does it mean when R&D is "3% of GDP"? When we talk about R&D, does it mean that the final goods of R&D amount to 3% of GDP? Or does it mean that a specific nation is spending an amount of money on R&D that is equivalent to 3% of GDP?

Expenditure, which can be thought of as a form of investment that does not involve the production of physical capital.

  First, the satellite account provides estimates of expenditures on R&D that are designed to be used in conjunction with the national income and product accounts measures. Second, it treats R&D expenditures as a form of investment, recognizing the role R&D plays in adding to knowledge and in developing new and improved processes and products that lead to increases in productivity and growth. Third, it provides estimates of the stock of knowledge capital.

[...]

  Expenditures on R&D can be viewed as generating future income and product. With this view, a case is made for treating them as investment, paralleling the treatment of business expenditures on structures and durable equipment, and for recognizing a stock of intangible capital, just as there is a stock of tangible capital.

Source: BEA’s R&D satellite accounts.


list of exotic production functions? Standard production functions are Cobb-Douglas, CES, Leontief.

The most exotic production function I have seen is the Ethier production function.

I am wondering whether there is a book/list of other exotic production functions. i.e. production functions with heterogeneous capital, integrals, etc.

I could not find it doing a google scholar search.

Belo (2010) uses a production function
$$
Y_{j,t} = \epsilon_{j,t} F^j(K_{j,t})
$$
where a competitive producer $j$ is allowed to choose the state-contingent productivity level $\epsilon_{j,t}$, subject to the constraint set
defined by the following analytically tractable CES aggregator:
$$
E_t\left[ \left( \frac{\epsilon_{j,t+1}}{\Theta_{j, t+1}}\right)^\alpha \right]^{1/\alpha} \leq 1.
$$
In Cochrane's  2016 essay, "Macro-Finance", Cochrane mentions this specification as a potentially important ingredient for understanding the interaction of macroeconomics and financial markets:

  The second ingredient, I think, is to enrich the production technology
  so that consumer investors can shape the riskiness of the
  technological opportunities they face. Belo (2010) and Jermann (2013)
  have recently explored specifications of technology that allow such
  choices.


In economic modelling, how are we sure the linearized equilibria behave similarly to original equilibria? The following problem is in the context of continuous time, although I suspect something could be said about discrete time also.

Let's assume that we have the following equation:

$$\dot x=f(x)$$

where $x(t)\in \mathbb{R}^n$, for $t\in I\subset \mathbb{R}$.

I'm reading a book on Ordinary Differential Equations. In this book, they state that one should verify some conditions on $f$, usually $f\in C^2(E\subset \mathbb{R}^n)$, to be sure that the linearization of $f$ behaves in a qualitatively similar manner (ensures existence of a $C^1$-diffeomorphism).

For example, consider

$$\begin{aligned}
  \dot x &= -x -\frac{y}{\ln \left(\sqrt{x^2+y^2}\right)}\\
  \dot y &= -y -\frac{x}{\ln \left(\sqrt{x^2+y^2}\right)}\end{aligned}$$

with $f(0)=0$. This $f$ is only $C^1$ not $C^2$(transform to polar coordinates$(\theta,r)$, and then see $\frac{d^2 \dot \theta}{dr^2}$ tends to infinity as $r$ goes to zero). The linearization gives a stable node, but according to a more general definition, we have a stable focus for the original non-linear system.

However, whenever I see any kind of linearisation being done (DSGE or Growth Theory), not once do I see any concern related to this... Maybe, there's something, which I'm missing, that makes economic diff. equations already satisfy this. Or am I wrong, and this simply isn't, and shouldn't be a concern to theoretical macroeconomists?

Any help would be appreciated.

Many macroeconomists have a somewhat cavalier attitude towards checking the validity of linearization methods, but definitely not all of them. For an example of the approximation issues being taken seriously, see Appendix A.3 on "Log-Linearization and Determinacy of Equilibrium" in the book Interest and Prices by Michael Woodford.


How to define the natural rate output I'm following Woodford's Interest & Prices to derive the microfundations for a New Keynesian model with staggereed prices.
I defined the utility function and disutility function (1.1 at page 144) as
$$ u= \frac{\xi_t C_t^{1-\sigma}}{1-\sigma}, v= \frac{h_t(j)^{1+\eta}}{1+\eta}, $$
where $\xi$ is a consumption disturbance.
Then I have defined the production function ( 1.7 at page 148) as
$$ y_t(j)= A_t * h_t(j). $$
With this functional forms we can write the real marginal cost (1.10 at page 149) as
$$ s_t(y,Y,\xi) = \frac{v_h(f^{-1}(y_t(j) /A); \xi)}{u_c(Y_t; \xi)A} \frac{1}{f'(f^{-1}(y_t(j)/A))} = \frac{y_t(j)^\eta \cdot C_t^{1/\sigma}}{A_t  \cdot\xi_t } .$$
Woodford says that with flexible prices we can write the natural rate of output as (1.14 at page 151)
$$s(Y^n, Y^n, \xi)= \mu^{-1}.$$
Therefore I get that the natural rate of output is equal to
$$ Y_t^n =( \mu^{-1} * A_t * \xi_t )^{\eta/ \sigma}. $$
However the problem with this definition is that in the steady state where the shocks($\xi$ and $A$) are zero also the natural rate of output will be zero.
Is this formulation correct? Is there another way to express the natural rate of output?

That is not correct  conclusion and there are several problems with the formulation you use.

$A$ is not a shock - it is a technology parameter of production function (see Woodford Interest and Prices pp 148). Heck, $A$ cannot even be zero. Following Woodford:

$A_t> 0$ is a time-varying exogenous technology factor

there can be shocks to $A_t$ that increase it or decrease it but $A_t$ itself is strictly positive at all times.

It is actually not $s(Y^n,Y^n,ξ)=μ^{−1}$ but $s(Y^n_t,Y^n_t;ξ)$. This might seem trivial but note $s(.)$ has only 2 arguments ($Y^n_t$ and $Y^n_t;\xi$  not 3 arguments $Y^n_t$, $Y^n_t$ and $\xi$). In addition, the solution to $s(Y^n,Y^n,ξ)=μ^{−1}$ gives you the natural rate of output not the equation itself. In this context how I read the Woodford semicolon (;)  is used to separate variables and shock parameters. Hence, it is just to indicate that the ξ is not a variable but given by the shock parameters of the economy (i.e. indicate variable conditional on vector of shocks.

Your definition of utility function is extremely non-standard, you define it as:

$$ u= \frac{\xi_t C_t^{1-\sigma}}{1-\sigma} $$
but why are the shocks $\xi_t$ even multiplied by consumption in the first place? In Woodford the equation (1.1) is given as:
$$E_0 \left[ \sum^{\infty}_{t=0} \beta^t \left( u_t(C_t, M_t/P_t; \xi_t) - \int^{1}_{0} v(h_t(i);\xi_t)di \right) \right]$$

first as you can see $C_t$ is not even conditional on shocks $\xi_t$, the real money holding is $M_t/P_t; \xi_t$ is conditional on vector of shocks.
you can't just assume $u_t(C_t, M_t/P_t; \xi_t) = \xi_t C_t^{1-\sigma}$, I mean that makes no economic sense whatsoever, why should utility be product of vector of shocks and consumption?

I don't think you will get the same result with some more reasonable utility function and take into account that the variables are conditional on the vector of shocks (e.g. $ M_t/P_t; \xi_t$ does not mean  $M_t/P_t$ has mean zero even if $\xi_t$ is given by Gaussian distribution with mean zero).


Deriving First Order Condition of Hansen and Singleton (1982) Hansen and Singleton (1982) considers the maximization of expected utility,
\begin{align*}
   \max \mathbb{E} \sum_{t=0}^\infty \beta^t U(C_t)
\end{align*}
with respect to the budget constraint,
\begin{align*}
C_t + \sum_{j=1}^J P_{jt} Q_{jt} = \sum_{j=1}^J P_{jt} Q_{j, t-1} + W_t,\ t= 0, 1, \cdots,
\end{align*}
where $C_t$ is the consumption at time $t$, $P_{jt}$ is the price of security $j$ at time $t$, $Q_{jy}$ is the amount of security $j$ at time $t$, $J$ is the number of securities and $W_t$ is the labor income at time $t$.
Then, they show that the first order condition is
\begin{align*}
P_{jt} = \mathbb{E} \bigg[ \beta \frac{U'(C_{t+1})}{U'(C_t)} P_{j, t+1}  \bigg],\ j = 1, \cdots, J.
\end{align*}
I tried to see it in the simplest case, where $t=1$ and maximize $U(C_0) + \beta U(C_1)$ with the Lagrange multiplier, but I could not get the similar first order condition. This seems to be a canonical example of non-linear GMM.

This has nothing to do with GMM, which is a statistical method for estimating parameters. This is a macroeconomics model or say a dynamic optimization problem.
From FOCs for $C_t$, $C_{t+1}$, and $Q_{jt}$, youg can get $$\mathbb{E}\frac{\beta U'(C_{t+1})}{U'(C_t)} = \frac{\lambda_{t+1}}{\lambda_t}$$, and $$\lambda_t P_{jt} = \lambda_{t+1} P_{jt+1}$$.
Then by replacing $\lambda$ you have $$P_{j t} U^{\prime}\left(C_{t}\right)=\beta \mathbb{E} \left[P_{j t+1} U^{\prime}\left(C_{t+1}\right)\right]$$.


Short term and long term price elasticity of currencies If I have 1% of the world supply of US dollars, and I decide to sell all of them to buy Euros, how much should I expect the exchange rate to change? Is there some kind of empirically determined price elasticity of demand that I could use to predict this? I've tried looking this up but I haven't been able to find it.

I'm also interested in the long run price elasticity of demand. Eg, how much should I expect my 1% USD sale to affect the price of USD in a year?

There is no way to predict that.

A "price" is just the level of the last transaction. What happens in the next transaction does not depend on the previous transaction. So if you sell dollars at 1.10 dollars per Euro, the next person to sell dollars might sell them at 1.12 or 1.08 or some other amount based on a myriad of factors. There is no way to predict what trades might occur in the future, just from knowing the trade you made in the present.


What does the following mean: "supply has positive elasticity”? What confuses me in particular is that I always thought of the concept of ‘elasticity’ with respect to something. That is, supply of x can be elastic with respect to y.
Am I missing something?

Price elasticity of supply is: $\frac{dQ_s}{dp}\frac{p}{Q_s}$. Because supply slopes upward $\frac{dQ_s}{dp}>0$. $Q_s$ and $p$ are also positive so    $\frac{dQ_s}{dp}\frac{p}{Q_s}>0$.


converting sum of consumption basket to integral In our lecture, first we had expenditure of consumption basket as $\sum_{i=1}^{n} p_{i}c_{i} $ but later my professor converted it to $\int_{0}^{1}p_{i}c_{i}di$. I want to understand how she did this? And how can I differentiate $\int_{0}^{1}p_{i}c_{i}di$ with respect $c_{i}$

This is because there’s a relationship between summation and integrals. An integral is basically a way how to calculate an area under the curve. With right sum you can do the same thing. For example, mathematically

$$ \sum_{i=0}^n f(i) \approx \int_0^n f(x)dx $$

In fact under the Reimann’s definition of integral, integrals are literary defined as a limit of a sum of function output multiplied by change of x or:

$$\int_a^b f(x)dx = \lim \sum_{i=0}^{n-1} f(\xi_i) \Delta x_i$$

Now to go back to your question you can imagine $p_i c_i$ as an area of rectangle of height $p_i$ and base $c_i$. Consequently $\sum p_i c_i$ can be considered an adding of of all areas of these rectangles and $\int p_i c_i di $ is just continuous approximation of this area.


How can increasing debt boost potential returns? May 26 2020. Why Even Carl Icahn Couldn't Save Hertz From Chapter 11 - Bloomberg

  Hertz’s debt binge began when it was acquired by private equity firms from Ford Motor Co. in 2005; the new owners quickly took out a \$1 billion dividend. Piling on debt juiced the potential returns for the owners and helped pay the inflated \$2.3 billion price tag for the Dollar and Thrifty brands in 2012, which Hertz struggled to integrate.

Doesn't the embolded sentence above contradict the one below? Wouldn't debt lower potential returns, if you can't re-pay it and the interest?

Oct 16 2006. Hertz share sale raises \$1.32 billion - The New York Times

  Hertz Global Holdings has raised \$1.32 billion in one of the largest initial public offerings of the year, less than 12 months after private- equity firms bought the company, according to an underwriter.

  The offering of more than 88 million shares, which represents about a 27.5 percent stake in the company, sold Wednesday for \$15 per share. Hertz said last month that it hoped to sell shares for \$16 to \$18 each.

  Clayton Dubilier & Rice, Carlyle Group and Merrill Lynch put up \$2.3 billion of the \$15 billion including debt that they paid for Hertz in December. The firms rattled potential investors by adding debt, raising the company's interest costs and pushing down profit.

  "The offering wasn't in high demand, or they wouldn't have had to lower the price in order to get it sold," said Kevin Miller, a partner at Alston & Bird in New York. "This thing isn't going to pop."

The first statement refers to the return on investment to the owners of the shares. After they are paid the 1 B dollar dividend  what happens to the company afterwards is not going to change that fact. Of course they would prefer that the company continue to do well. If they sell the shares immediately after receiving the dividend then they don't care.  Since the company had  1 B dollars less equity after paying the dividend it is easy to understand that in or near 2005 they would want to replace that capital with new debt and your quote suggests that is what happened. Less equity and more debt, if all else is unchanged, results in higher ROE (return on equity) which is one of the company's measures. Owners might benefit from this measure improving if the market likes it. Of course all else is not unchanged because they have to pay interest on the new debt. The actual effect on ROE depends upon circumstances. So the dividend was certainly good for the owners and what happens afterwards is potentially good but uncertain for the owners.
The part of the second statement, "raising the company's interest costs and pushing down profit", refers of course to the company's profitability position. In other words this can be interpreted as the company's return on revenue (profit margin); not the owner's return on investment.


Will an increase in interest rate cause short run aggregate supply to shift? Most business borrow for financing their purchases of raw material and capital etc. Shouldn't a interest rate rise cause cost of production to increase, therefore shifting SRAS to left? Thanks in advance.

Yes, however a supply shift as a result of interest rates can be (sticky).this is why after a stock drop, a recession can take 1 year- 18 months to occur. So when we look at economic indicators over the past year, the 10-year approaching 3% has not led to a reduction in aggregate supply. It may not be until FY 18 Q4 that we see a drop a significant drop in in supply. Supply and interest rates do not have a “flexible” relationship, but instead a “sticky” one in which supply may not drop immediately, but over time your answer is correct.