MRS of Perfect Complements and MRS Equilibrium Condition To preface this, I appreciate any help you offer.

In reference to 1.(d) I'm trying to understand the MRS of perfect complements. Given: utility function U(x,y), commodity pair (2,8): y/x is 4 ...> a/b which is 3, so we say the MRS at this point is undefined (Δy/Δx = Δy/0)? Given commodity pair (5,1): y/x is 1/5...< a/b which is 3, so we say the MRS at this point is 0(0/Δx)? I'm not exactly sure why this is or how to best phrase those statements if they're correct, I've looked everywhere and have not really caught any explanation.

In reference to 2.(f) Is "the expression" requested maybe MRS = (dUx/dx)/(dU/dy) = Px/Py? I'm confused by this phrase "positive values of consumption"... is this positive x and y values? How do I find this minimal combination? I'm only familiar with the approach to finding a maximal utility producing combination.

Thanks so much if you've read this far.

Your instructor seems to have a rather sloppy approach to the mathematics of microeconomics.

Let's begin with the easiest case, the bundle $(5,1)$. We have $\min\{3\cdot 5,1\}=1$. For $\delta<14$, we have $\min\{3\cdot 5, 1+\delta\}=1+\delta$ so the marginal utility of $y$ is $1$ at the bundle $(5,1)$. Moreover, for any $\delta>-14/3$, we have $\min\{3\cdot(5+\delta),1\}=1$, so the marginal utility of $x$ is $0$ at $(5,1)$.

A similar argument applies to the bundle $(2,8)$, but here the marginal utility of $y$ is $0$, and dividing by zero is not allowed. Maybe your instructor wants to hear that the MRS is infinite.

Now at the bundle $(3,9)$, increases in any commodity have zero effects, but decreases do. There is a kink and you cannot smoothly move along the curve. By any semi-reasonable standards, the MRS is undefined. But to answer the verbal question, the consumer is not willing to give up any amount of one commodity ( effect negative) to receive more of the other good (effect zero).

Now the idea that you get the optimal consumption bundle by setting the MRS equal to relative prices is complete nonsense. If prices $p_x$ and $p_y$ and income $m$ are all positive, we can still, however, still find the optimal bundle. First note that the optimal bundle $(x,y)$ wil satisfy $3x=y$. If $3x>y$, one could reduce the amount $x$ without decreasing the utility (the minimum is unchanged), but use the freed money to increase $y$, which does have a positive effect. In an optimal bundle, this is not possible. by a similar argument, we can rule out $3x<y$. So we must have $3x=y$, which means that at any optimal solution, the MRS is undefined!!! To solve for an optimal bundle, all that remains to do is plug the condition $3x=y$ into the budget constraint $p_x x+p_y y=m$. We can substitute $3x$ for $y$ to get
$$p_x x+p_y 3x=m$$ and solve for $x$, which gives us
$$x=\frac{m}{p_x+3p_y}.$$
Similarly, we get
$$y=\frac{m}{p_x/3+p_y}.$$


How printed money gets to the commercial banks? can you please clarify this for me?
Gov't need money for deficit elimination, so they issuing bonds (lets say 1T, 10 years, 2%).
Fed is buying the bonds with fresh money, so the gov't now has new 1T$ to spend.

But - I always hearing the 1T is getting to the banks, and the amount of market money is 1T x Money Multiplier.

How does the money gets from the govt to the banks?

Thanks.

Commercial banks do not "get money" from the government. They do so either from the FED or either from worldwide financial markets. And actually, banks can even buy gov's bonds, which in this case is the contrary of what you describe.

Conceptually, the FED simply buys bonds emitted by commercial banks. But using the word "buy" here is misleading. The FED is not a traditional economic agent.

First, note that the FED has two main possibilities to lower interest rates. They can do it directly by saying, "ok guys, tomorrow refinancing will be less expansive" or indirectly by injecting liquidity in the system, i.e. by proactively buying commercial banks' bonds for (almost or really) nothing, or even paying them interests for that -- which is equivalent to having negative interest rates!

When the FED does so (either directly or indirectly), commercial banks will be able to refinance themselves at a lower cost. Which means that at the end, these commercial banks will also be more, say, flexible, and thus will stimulate demands (here and there) in the national economy by lending money (here and there). If supplies stay constant, markets will reach a new equilibrium via prices rising due to these stimulated demands $\iff$ stops deflation.


How to improve the underperforming construction productivity by correcting its market failures? Globally, labor-productivity growth in construction has averaged only 1 percent a year over the past two decades, compared with growth of 2.8 percent for the total world economy and 3.6 percent in the case of manufacturing (McKinsey, 2017).

The most common market failures / problems in the construction industry are:

asymmetric information
corruption
industry fragmentation
ismatch between the agents of risk allocations & rewards
dependence on public-sector demand.

I found many organizational answers to this question: improve communication between agents, introduce shorter contracts, improve on-site execution, etc.

However, I personally think internalization would be a great choice: bringing together the agents (like Elon Musk did at Tesla). Then the agents could optimize on a vaster decision set, and the result would be Pareto-efficient. The big question is: how?

My question is: how can a decision maker improve the construction productivity? What would be the answer from an Economist point of view?

A great many things can be said on the question. But it might be worth pointing out that people have been building stuff for a very very long time, so maybe 1% productivity growth is not a bad result.

The most difficult is anti-corruption. You could compare the free press + independent judiciary + empowered investigative authorities within specified limits + free elections ... with the approach of President Xi, whose name is now entrenched in the Chinese constitution.

However, it seems that you're more interested to motivate a specific theory. So perhaps you could develop that and come back with a question that is more aligned to what you want to know. For example, you could be additionally descriptive about what it is about Tesla's operations that might have broader lessons in construction? (Note that Tesla makes cars, not ports, not bridges and not roads.)


What makes a factor valid in a factor pricing model of assets? This question is related to another question here. The question linked to asks for examples of factors commonly used in ICAPM models (intertemporal capital asset pricing model). What I want to know here is this: what is underlying principle that makes a factor a valid factor in factor models such as the CAPM, ICAPM, or APT?

This is a partial answer (see section 5.1, p. 79 of Cochrane's book, "Asset Pricing"). Suppose you wanted to treat the size of the firm as a factor. Stocks of smaller firms typically have higher average returns. You could then form a "large" holding company comprised of smaller firms. The new, large company would then have returns like a smaller company but would in fact be large and "the managers could enjoy the difference."

  What ruins this promising idea? The "large" holding company will still behave like a portfolio of small stocks---it will have their high betas. Thus, only if asset returns depend on how you behave, not who you are---on betas rather than characteristics---can a market equilibrium survive such simple repackaging schemes.

(How would I formalize this concept?)

EDIT: Also, in Financial Asset Pricing Theory by Munk (chapter 10, p.371) it says

  It is worth emphasizing that the general theoretical results of the consumption-based asset pricing framework are not challenged by factor models. ... Factor models do not invalidate the consumption-based asset pricing framework but are special cases that may be easier to apply and test. Therefore factors should generally help explain typical indiviudals' marginal utilities of consumption.

(Emphasis added) This helps to understand the role of factors in the ICAPM.


Do corporations outside of the Eurozone issue bonds in Euro? Do corporate issuers outside of the Eurozone (as in the wider EU area, or non-EU countries) issue bonds in Euros (EUR)? How relatively common is this, compared to issuing bonds in USD, if you are a non-US and non-Eurozone debtor?

Would that qualify for what's referred to as "Euro denominated" bonds, or does that simply mean that the price of the bond is listed/represented in EUR?

Note that I'm not referring to so called "Eurobonds" which is something different:

  A Eurobond is an international bond that is denominated in a currency
  not native to the country where it is issued. Also called external
  bond; "external bonds which, strictly, are neither Eurobonds nor
  foreign bonds […]

Answer by @Wecon in the comments:

  With respect to your first question: they certainly do. The BIS quarterly review of sept. 2015 (p.9) even shows that "firms in the United States increasingly issued euro-denominated debt to benefit from the low borrowing costs." This may already be a set-up for answering your second question: its (relative) importance may be highly dependent on economic circumstances


Macroeconomics two period model This is the problem I've been given.
Assume the consumers’preference over current and future consumption is given by $U(c,c')=c^\frac{1}{2}*c'^\frac{1}{2}$.
Further assume there is no government.
1) solve for the optimal decisions: $c,c′$ and $s$ ?

This is as far as I have come

$\max U(c,c')=c^\frac{1}{2}c'^\frac{1}{2}$
s.t. $c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$
$c, c',s \ge 0 $

$L=c^\frac{1}{2}*c'^\frac{1}{2}+λ(c+\frac{c'}{1+r}-y-\frac{y'}{1+r})$

(1)derv of $c$: $\frac{1}{2}c^{-\frac{1}{2}}*c'^\frac{1}{2}-λ=0 $
(2)derv of $c'$:$\frac{1}{2}c^\frac{1}{2} c'^{-\frac{1}{2}}-λ\frac{1}{1+r}=0$
(3)($λ$):$c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$

Divide equation 1 by 2

$\frac{ \frac{1}{2}c^{-\frac{1}{2}}c'^\frac{1}{2}} {\frac{1}{2}c^\frac{1}{2}c'^{-\frac{1}{2}}}= \frac{λ}{\frac{λ}{1+r}}$  = $\frac{c'}{c}=1+r$

I would appreciate if someone could give some help on how to continue so I can find the optimal decisions for $c, c', s$

Every lagrangian with two goods has the same two steps basically. First, find the optimal relationship between the two goods (often through derivatives). Second, plug that relationship back into the budget constraint to solve for each good.

This problem is no different and you have essentially only done step 1. So you just need to do step 2.

The idea is that now you have 2 unknowns ($c$,$c'$), so you need 2 equations. One equation is the relationship you found between consumption today and tomorrow from the first order conditions. The second equation is the the budget constraint (or equation 3 as you called it). Your equation 3 has still not been used toward solving for consumption.

Plugging in the equation for $c'$ that you found into your equation 3 you get:

$c + \frac{c(1+r)}{1+r} =y+\frac{y'}{1+r}$

So $c= 0.5 *(y+\frac{y'}{1+r})$

Now plug in this equation back into the relationship you found through the lagrangian, which was:

$c' = (1+r) c$.

Then you'll be done.


Why Chinese exports increased even though Yuan appreciated after the Chinese removed the peg against USD? After the Chinese central bank removed the peg against USD, yuan appreciated but Chinese exports didn't seem to be affected. They are steadily increasing. How can this be explained ?

Simple answer: Drawing on Ricardian model I can say the Chinese have not lost their comparative advantage in producing many types of manufactured goods. Yuan, on its way to appreciation, has to walk a few more miles to affect the comparative advantage that the Chinese possess.


Do institutions buy debt with credit? I am reading Money Power book, from 1932 (http://www.worldcat.org/title/money-power-and-human-life/oclc/3783026). The author explains how the War Loans in the UK worked.

He says that the government offered a bond at, say, X% interest rate. Then, banks offered some of their clients a loan of credit, at a cost of X-1%, which had to be tied to such bond. This is, clients would "buy" the bond with credit money. This was a sound business for the client, which would make 1%, and the bank, which would receive X-1% almost risk-free (bond was a collateral).

It occured to me, that this practice could be quite common. Is it? For example, say a poor country is selling a bond, which pay high interest rates. Then I could go to the bank and ask them for a loan, backed with the bond as collateral, and use that money to buy that loan. The bank would agree to do so, and we would both benefit.

Is this legal? Is it cheating? Does this happen?

This is extremely common in institutional markets, and is called a "repo" transaction - short for Repurchase agreement

In a repo transaction, you "sell" the bond to the bank and agree to buy it back again at a fixed price in the future (regardless what the actual market price may be).  This is the same business model as pawnbroking.  If you fail to buy the bond back, the bank gets to keep the bond.  The reason it is done as this sale-and-repurchase, rather than collateral (which is economically identical) is that it makes handling a default much simpler.  If you default on a collateralised loan, the lender still has to apply to take full ownership of the collateral, which can delay matters (such as under chapter 11 protection).  In a repo transaction, the bank already legally owns the bond and has it in custody, so it is much simpler.  It is worth noting, that although they have bought the bond from you the repo agreement (GMRA - "General Master Repo Agreement") specifies that the original owner retails "full economic interest" in the bond - that is they get any interest payments from the original bond.

That said, this is most commonly done for Treasury bonds in highly rated countries (US, UK, Germany etc.).  Since the bond is collateral for the loan banks are less keen to lend you for bonds that are risky, as should the collateral become worthless, then the loan becomes unsecured.  For this reason repo transactions in EM bonds or Corporate Bonds tend to have a significant "haircut".  The haircut is the difference between the value of the bond and the amount borrowed.  So in your example, a repo transaction of a poor country bond the bank may take a 20% haircut, so to buy \$10m of the bond, you would borrow \$8m and pay for \$2m yourself.  The logic here is the same as when you mortgage your house, the bank won't usually lend you the full value of the house.

As you might have guessed from my example using millions, repo transactions are done between banks and institutional investors, not the man-on-the-street.

For extensive details see: ICMA Guide to Best Practice in Repo


What is the name of the economics free trade "punnett square" I remember doing these in High School.It was a 2x2 square, where you could work out the math to show that countries were better off focusing on their efficiency/competitive advantage, and that free trade made both better off.

Not looking for an analysis of if the logic is right or wrong or whatnot, just what is the name of the square (assuming it has one)

Perhaps you mean an Edgeworth box?
Here is an example demonstrating Ricardo's theory of comparative advantage using an Edgeworth box:

Source: Policonomics

This shows that through trade both Portugal and England can have more wine and more cloth after opening up trade between the countries. It also shows that even if Portugal is more productive than England at producing both commodities then trade is still improving (that is what makes it comparative and not absolute advantage).

Alternatively, perhaps you mean a 2x2 table like this which shows the labor needed to produce each good:

Source: KALYAN CITY LIFE

To my knowledge this table has no special name.


U.S. Treasury defaulting on its bond obligations to China I was listening to a podcast featuring Michael Lewis (author of Flashboys and Moneyball), who mentioned a black swan scenario where the administration would threaten to selectively not pay its Treasury bond obligations to China.  If this were to be, I understand it would undermine confidence in the global economic system, but I am unable to anticipate exactly what the concrete results would be.

What would an administration threat of a default trigger?
What exactly would a default trigger?

the administration would threaten to selectively not pay its Treasury
  Bond obligations to China

  What exactly would a default trigger?

Selective default would be ineffective and it could backfire.

It would be ineffective because, at any point in time, a treasury bond has the same value regardless of who holds it. If the USA disavows its obligations toward China, clearly the top holder of US Treasury securities, China might opt to get rid of its holdings. China's move would predictably disrupt the bonds market due to the arbitrage opportunities that will ensue.

Selective default would backfire in two ways. Most certainly, the Treasury would attract fewer funds in the next round(s) of US bonds because the market would be "flooded" with the bonds of which China is getting rid.

The other way in which it could backfire is that selective default against China will make other countries wary of they themselves becoming the target of a subsequent selective default. Since no creditor likes the prospect of its holdings being singled out for default, there is a realistic chance that each country would decide to dump its holdings as well, and lead to a domino effect. That reaction is likelier now with the increasing efforts to replace the US as the "official" currency for international settlements (such as in oil or currency markets).


Is an income tax always more favourable for consumers compared to ad valorem/quantity tax? I'm studying the optimal choice of consumers with regards to taxation.
I read that for consumers, income tax is generally (for Cobb-Douglas preferences) preferred compared to ad valorem tax:
If the budget constraint is generally $p_1x_1+p_2x_2=m$, then an ad valorem tax $t$ will change it to $(p_1 + t)x_1+p_2x_2=m$.
If the income tax is the same amount, we get $p_1x_1+p_2x_2=m-R*$, where $R*=tx_1$ is the amount of taxes for both variants.
Now, the following figure demonstrates how an increased price of good 1 will lead to a preference of an income tax over ad valorem tax:

I wonder if this is the case for all indifference curves or if there are certain indifference curves where consumers might prefer an ad valorem tax. I doubt this, but I think the chapter in H. Varian lacks a mathematical (algebraic or analytical) explanation for this which I feel would help me understand.

You question can be answered using a revealed preference argument.
Let $B = \{q \in \mathbb{R}^n_+| p' q \le m\}$ be some budget set of a consumer (i.e. $B$ gives all possible bundles that the consumer can choose).
Let $q^\ast$ be the optimal choice from $B$, i.e. the bundle that optimizes the utility. Then for any other bundle $q \in B$, it must be that $u(q^\ast) \ge u(q)$.

proof: The proof is simple, if, towards a contradiction, we would have that $q \in B$ and  $u(q) > u(q^\ast)$, then this would contradict the assumption that $q^\ast$ was optimal in $B$.

Now, let's apply this to the two budget sets for your question.
The first budget set has a lump-sum tax equal to $R$.
$$
B^1 = \{q \in \mathbb{R}^n_+ | p' q \le m - R\}.
$$
The second budget set has an ad-valorem tax equal to $t$ for some good, say $i$.
$$
B^2 = \{q \in \mathbb{R}^n_+| p' q + t q_i \le m\}.
$$
Let $q^{1}$ be utility maximising, bundle in $B^1$ and let $q^2$ be the optimal, utility maximising, bundle in  $B^2$.
In addition we assume that the revenue of the ad-valorem tax equals the revenue for the lump-sum tax:
$$
t q_i^2 = R.
$$
We would like to show that $u(q^1) \ge u(q^2)$, so the lump sum tax always gives at least as much utility as the ad-valorem tax.
Using the above revealed preference argument, it suffices to show that $q^2 \in B^1$, which means that $q^2$ was a feasible option when $q^1$ was chosen.
Indeed, we have:
$$
q^2 \in B^2,\\
\iff p' q^2 + t q^2_i \le m,\\
\iff p'q^2 \le m - t q^2_i,\\
\iff p' q^2 \le m - R.\\
$$
So $q^2 \in B^1$.
Remark: notice that we did not impose any restrictions on the utility function. The only thing that we assumed was that the consumer was able (from any budget set) to pick an optimal bundle.


If North American stock markets are semi-strong efficient, why are indices like S&P500 expected to grow? I'm narrowing this question on Reddit's r/investing to North America. But don't hesitate to discuss other Five Eyes (FVEY)  – Australia,  Canada,  New Zealand,  UK,  US. I'm posting here because I prefer self-contained answers. I hate scrolling reddit's long-winded dialogue!
It's common to postulate that

stock market indices are semi-strong efficient, so that "all available information about a company has already been priced in".

in the long term, indices such as the S&P500 and NASDAQ100 are expected to grow, so that investing in them will usually profit you in the long term.

How are these two postulates compatible? Doesn't semi-strong Efficient Market Hypothesis mean that the market semi-strong-efficiently values X's share price, so that on average, X's share price remains steady?

News may arrive in the future and affect this semi-strong-efficient stock price, but shouldn't such news average out to 0? Because there's no reason to expect more negative news that positive news, or vice versa? If there was, such news should have already been priced into X's current share price.

As an example, let's postulate that the market behaves semi-strong-efficiently and values X's stock at 100 tomorrow. This means X's stock today should also be Net Present Value of 100, correct? If not, then X's stock is  undervalued today, and thus not semi-strong-efficiently valued by the market.

Similarly, if X's share price is expected to value 1000 next month, this 1000 should also be reflected into today's share price, so that today's share price is the Net Present Value of 1000.

This comment mentions "discount rate (risk free rate plus the equity risk premium", but Discount Rate is already incorporated into Net Present Value. Correct?

The composition of a stock market index is not fixed but changes over time. Moreover, it usually changes so as to increase/decrease the weight of good/poor performers.
For example, the S&P500 assigns weights to 500 large companies that meet various criteria, including at least a $8.2B cap.* These weights change over time. In fact, companies can even be dropped (e.g. recently Macy's) or added (e.g. Netflix, Google, and Facebook were not on the index in 2000 but are today).
So, suppose we're transported back 20 years to 2000-11-06. Consider the 500 stock prices that were included in the S&P500 on that day. It is true that the EMH would have predicted that those 500 stock prices would, on average, not have grown in the following 20 years (ignoring inflation). However, the EMH would not have said anything about the S&P500 Index, whose composition may and indeed did change.
(We can repeat this same thought experiment for the 500 stock prices included in the S&P500 today: The EMH predicts that, ignoring inflation, these 500 stock prices will not change over the next 20 years. However, the EMH says nothing about what the S&P500 will be in 2040.)
So, if we expect economic growth and creative destruction, then we also expect the S&P500 (and any other index whose composition is allowed to change) to keep growing over time, even if we do not expect any individual stock price to grow.

*See e.g. "S&P U.S. Indices Methodology, Aug 2020" PDF.


Linear Expenditure System of Demands, Derivation Help This problem I am working on comes out of--surprise--the Mas-Colell book for graduate micro (3.D.6). I think I have correctly used the FOC of the Lagrangian of the utility maximization problem to derive the consumer's Walrasian demand. My answer does not match the book. We are given that

$u(x) = (x_1-b_1)^\alpha (x_2-b_2)^\beta(x_3-b_3)^\gamma$
(and then from the first part we can say that $\alpha + \beta + \gamma = 1 $ WLOG.)

The solutions say that we take a monotonic transformation:
$\ln(u(x)) = \alpha \ln(x_1-b_1) + \beta \ln(x_2-b_2) + \gamma \ln(x_3-b_3)$
and then I set up the Lagrangian of this:
$ \mathcal{L} = \alpha \ln(x_1-b_1) + \beta \ln(x_2-b_2) + \gamma \ln(x_3-b_3) - \lambda(p_1x_1 + p_2x_2 + p_3x_3 - w)$
and the FOCs are:
$\frac{\alpha}{x_1-b_1} - \lambda p_1 = 0$
$\frac{\beta}{x_2-b_2} - \lambda p_2 = 0$
$\frac{\gamma}{x_3-b_3} - \lambda p_3 = 0$
Solve for the x's:
$x_1 = \frac{\alpha}{\lambda p_1} + b_1$
$x_2 = \frac{\beta}{\lambda p_2} + b_2$
$x_3 = \frac{\gamma}{\lambda p_3} + b_3$
Which leads us to:
$$x(p,w) = (b_1, b_2, b_3) + \left(\frac{\alpha}{\lambda p_1},\frac{\beta}{\lambda p_2},\frac{\gamma}{\lambda p_3}\right)$$
This is not what the book got, so I used Walras law to get the desired result:
$p \cdot x = w$
$\implies w - (b \cdot p) = \frac{1}{\lambda}(\alpha + \beta + \gamma)= \frac{1}{\lambda}$
$$\implies x(p,w) = (b_1, b_2, b_3) + (w - (b \cdot p))\left(\frac{\alpha}{p_1},\frac{\beta}{p_2},\frac{\gamma}{p_3}\right)$$
which is the book's solution.
So my questions are:

Did I do the derivation right? How should I know to take a log transformation of the original utility function? Is there any particular information that is supposed to tip me off?

I am not sure I understand your question. $\lambda$ is a Lagrange multiplicator which has a value in the optimum. It is not a parameter and hence you cannot leave it in your solution.
 When solving the Lagrangian the optimal solution has the form $(x,\lambda)$ and in addition to your conditions
$$x_1 = \frac{\alpha}{\lambda p_1} + b_1$$
$$x_2 = \frac{\beta}{\lambda p_2} + b_2$$
$$x_3 = \frac{\gamma}{\lambda p_3} + b_3$$
it also has to fulfill
$$
(p \cdot x - w) \lambda = 0.
$$
If $\lambda \neq 0$ this also means $p \cdot x = w$. $\lambda$ cannot be equal to zero because it is in the denominator of your optimal solution for $x$:
$$x(p,w) = (b_1, b_2, b_3) + \left(\frac{\alpha}{\lambda p_1},\frac{\beta}{\lambda p_2},\frac{\gamma}{\lambda p_3}\right)$$
and hence you were correct to use the Walras law or budget constraint.

For more on optimality conditions see the Karush-Kuhn-Tucker theorem.

About $b_1, b_2$ and $b_3$: I think this is supposed to be a bliss point.


If the marginal cost is equal to 1, how does that imply marginal cost is equal to marginal benefit? The function below is a utility function simplified after subject to an implied participation constraint.
$$
E\left(\pi_{n}\right)=e^{*}-E\left(s^{*}\right)=e^{*}-c\left(e^{*}\right)
$$ where $ \pi_n $ is net profits, s is salary, $ e^*$ is optimal effort and $ c(e^*) $ is the personal cost (disutility) of that effort

The function is maximized for $ e^* $ satisfying the first-order condition
$$
c^{\prime}\left(e^{*}\right)=1
$$
The text goes on to state that at the optimum, the marginal cost of effort, $c^{\prime}\left(e^{*}\right)$, equals the marginal benefit, 1.

Q: Where did marginal benefit just come from? I could be just forgetting some basics, but I don't recall MC = 1 as an implication that MC  =  MB?

You have to look at this from the derivation of the profit equation.

From the equation $\mathbb{E}(\pi_n) = e - c(e)$, you can see that the marginal benefit of increasing $e$ is equal to 1. That is, for each extra $e$ you put in, you get that exact amount back in terms of expected profits. The $e$ term is the benefits, and $\frac{de}{de} = 1$ is then the marginal benefits. The same goes for marginal cost with the $c(e)$ term.

Follow up

In any objective function, labeling things "benefits" or "cost" could be tricky. Here, one could assume, from the functional form, that the function looks like $\pi = TR - TC$. It's then quite natural to label $TR = e$ and $TC = c(e)$. The benefit is the revenue you get, and cost is, well, cost.

It could very well be that, actually, $TR = e + f(e)$ while $TC = c(e) + f(e)$, where $f(e)$ could be any well-behaved function. The "correct" marginal benefits would then be $1 + f'(e)$, while marginal cost would be $c'(e) + f'(e)$. At the end of the day, though, you'd still have

$$1 + f'(e) = c'(e) + f'(e) \qquad\Leftrightarrow\qquad 1 = c'(e)$$


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

Only ROI is provided. If there is a way to calculate, please say so.

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

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

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

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

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


How do we define an efficient tax in microeconomics? I am currently working through problems to study for an upcoming exam. I am not seeking a solution per se. I am looking at the intertemporal choice model. I am looking at two periods where consumption in each is normal. Let $y_1, y_2>0$ be fixed endowments and $p_1=p_2=1$ be the prices of consumption in periods $1$ and $2$, respectively. Denote consumption by $c_1$ and $c_2$. Using the Slutsky Decomposition
$$\frac{\partial c_t}{\partial r} = \frac{\partial h_t}{\partial r} + \frac{\partial c_t}{\partial W_1}\frac{(y_{1}-c_{1})}{1+r}$$
where $W_1 = y_1 + \frac{y_2}{1+r}$ is the net present value of wealth, $r$ is the interest rate, and $(y_1 - c_1)$ represents savings if positive, and borrowing if negative. The individual has access to capital markets in this respect.

We are given that for a drop in the interest rate, the person's consumption for $t=1$ does not change which (w/o proof) implies that the person is a saver.

The next part of the question states

  suppose the drop in interest rate is due to the introduction of a
  capital gains tax without deductibility of capital losses. Given the
  person's behaviour above, is the tax efficient? Prove your answer
  graphically.

So this gives us a situation where only the portion of the resource constrain such that $c_1<y_1$ (individual is a saver). This gives a piece-wise resource constraint. Since there is no compensation mentioned, I assume the welfare measure to be used is Equivalent Variation (EV). My question is, how would we define whether a tax is efficient?

Would it be efficient if the Deadweight Loss is equal to zero? Or is it related to how large the tax revenue would be relative to EV? The answer to this question will give me enough information to be able to display this graphically.

When there is a deadweight loss associated with a particular policy (e.g. a tax) the policy is said to be inefficient under the Pareto Efficiency definition. In the above circumstance, the capital gains tax could have an equivalent increase in government revenue as a lump-sum income tax but would cause a larger decrease in the welfare of those saving/investing. This implies that there is a way to make at least one party better off without making another worse off. This is what the differential between EV and tax revenue is capturing.

The tax is efficient $\iff$ $DWL=0$

under the Pareto Efficiency defintion.


Term for perfect symmetric asymmetry Let $x \in [0,1]$ denote some state (e.g. market share).
Let $i \in \{1,2\}$ denote an agent (e.g. firm).
Im considering a model where payoffs $F_i(x)$ are perfectly invertible in the sense that the payoff functions can be mirrored along the line $x = \frac{1}{2}$ with $F_1(x) = F_2(1-x)$. Take $F_1(x) = x$ and $F_2(x) = 1 - x$ for instance.
Is there a fixed term for these kind of games. It has somewhat a zero sum flavor, because preferences are perfectly opposing. Firm one wants increase its market share, and firm 2 wants to decrease the market share of firm 1 in the same manner.
I can think of Hotelling or a linear city. But is this the most generic characterization?
Or, even better, is there a mathematical term for this kind of functions which are mirrored along a fixed line?

Edit: Example Consider the following advertising differential game defined by $(x,u_1,u_2) \in [0,1] \times \mathbb R_+ \times \mathbb R_+$
\begin{align}
&v_1(x_0) = \max_{u_1}\int_0^\infty{e^{-t}(x - u_1^2/2)dt}\\
&v_2(x_0) = \max_{u_2}\int_0^\infty{e^{-t}(1 - x - u_2^2/2)dt}\\
\text{s.t.}\quad & \dot x = u_1 - u_2
\end{align}
$x$ is the market share of firm 1, and $u_i$ is the respective advertising effort.
A stationary feedback Nash equilibrium $(\phi_1(x), \phi_2(x))$ solves a coupled system of Hamilton-Jacobi-Bellman equations
\begin{align}
&v_1(x) = \max_{u_1}\{x - u_1^2/2 + v'_1(x)(u_1-\phi_2(x))\}\\
&v_2(x) = \max_{u_2}\{1 - x - u_2^2/2 + v'_2(x)(\phi_1(x)-u_2)\}
\end{align}

It turns out that $(\phi_1(x), \phi_2(x)) = (1,-1)$ is the unique Nash equilibrium with associated values
\begin{align}
&v_1(x) = x - \frac{1}{2}\\
&v_2(x) = \frac{1}{2}-x\\
\Longrightarrow \quad &v_1(x) = v_2(1-x).
\end{align}

How would one characterize the class of games in general? Symmetric antagonistic?

I may propose the following definition:

  Let $(\phi_1(x), \phi_2(x))$ solve
  \begin{align}
&v_1(x) = \max_{u_1}\{F_1(x,u_1,\phi_2(x)) + v'_1(x)f(x,u_1,\phi_2(x))\}\\
&v_2(x) = \max_{u_2}\{F_2(x,\phi_1(x),u_2) + v'_2(x)f(x,\phi_1(x),u_2)\}.
\end{align}
  A differential game is symetrically antagonistic if
  \begin{align}
v_1(x) = v_2(\overline x - x) \quad \forall x \in X
\end{align}
  where $\overline x := \max X$.

A game where the payoff function $F_1,F_2$ are such that for any two strategy profiles $s,s'$
$$
F_1(s) > F_1(s') \Leftrightarrow F_2(s) < F_2(s')
$$
is called an antagonistic game. Sounds like what you want is a symmetric antagonistic game?


Price elasticity of demand always increases with price? Is there a closed-form continuous demand function whose price elasticity of demand decreases with the price?

There are many functions where absolute value of elasticity is decreasing and which are continuous closed form.
One example of such function would be:
$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p}, $$
which is continuous closed form  and decreasing in price in its absolute value.
Original Answer:
Originally I interpreted the request too mathematically I am leaving it here since other answer references it.
Yes for example consider the following demand function:
$$Q = e^{-p}$$
price elasticity of demand is given by:
$$EL = \frac{dQ}{dp}\frac{p}{Q} = -e^{-p} \frac{p}{e^{-p}} = -p$$
So the demand function has elasticity that is always decreasing in price $p$. The function has closed form, and it is continuous.


Free data on bankruptcies Does anyone know of a free source for bankruptcy data? I'm specifically interested in the companies' balance sheets before they filed for bankruptcy.

There is a private database: http://www.bankruptcydata.com/ which appears to have all bankruptcy filings logged. It has a subscription but appears to offer a free trial. To access public court filings in general which would include bankruptcy filings you'll ned access to PACER which is the government's database for all public court filings although it does require an account and charges a fee for each document (https://www.pacer.gov/). In general public bankruptcy filings are likely not going to be offered for free publicly given that it appears there is no official government body which gathers this information.

Your best bet to find balance sheets before bankruptcy is to look at a company's 10K or 10Q SEC filings which includes balance sheets and other required data for publicly traded companies to report to the SEC quarterly. 10K reports are the year end reports and 10Q reports are quarterly reports. The SEC has a useful database for these reports called EDGAR where you can look them up: https://www.sec.gov/edgar/searchedgar/companysearch.html.

Note that the SEC filings only apply to publicly traded companies. Non-publicly traded companies have no requirement to disclose their balance sheets and that information likely isn't available to you.


Why has Apple increased app store prices in the UK? Apple have recently announced an increase in app store prices, ostensibly due to the weakening of the pound following the Brexit vote and its fallout.

Why has apple done this when the marginal cost of production of an (already developed) app is zero? Will this not reduce income for these apps?

Since payments to app developers are denominated in USD it does make sense to raise prices. The pound sterling has decreased in value relative to the USD by about 30 percent since May 2014, which I'm selecting as the arbitrary peak for comparison. So if they kept the same prices as before the recent slide, an app would make about 30% less relative to 2014. Even relative to Jan of 2016, the app would make about 14% less in USD.

As I'm involved with a few IT projects I can tell you that the marginal cost of production is never zero for a "finished product". In technology there is no such thing as a finished product. You can't put it in a box and deliver it and then say to the customer that it's their problem now (well you could but it wouldn't keep you in business very long).

I think it is perfectly reasonable to adjust app prices to reflect what appears to be a new normal exchange rate.

Edit 1:
On the marginal cost of an Apple Store app:

The question states that the cost of one additional sale is zero for an app. I disagreed in my above answer, but much too strongly and incorrectly.

The marginal cost of some apps is equal to or close to zero. Usually, these apps take the following pricing structure: a free version with advertising, and a paid version. Then there is another class of apps that benefit so strongly from economies of scale and maturity that their MC is near zero (Facebook, Spotify, ect.)

They do make up the vast majority of apps. This means that for all intents and purposes, the marginal cost of an app is zero.

But I would argue that some apps with small but enthusiastic user bases likely do have some marginal costs and set their prices near zero anyways because of consumer expectations. Each user generates bug reports, which generates cost, and small development teams are disproportionately affected by this situation.

It is one reason why pricing software is so difficult.


Collusion and number of firms How would you answer the following question?

You work for a CEO of a large firm. He says to you, "In my experience collusion is less likely to be sustained as the number of firms in the market increases. Demonstrate this using a model of Bertrand Competition."

Iet's say we have n identical firms and an infinite horizon of time.

The n firms sustaining the collusion, will find optimal to fix the same price $p_m$ where $p_m$ is the price of the monopoly level and we define $\frac{\Pi^m}{n}$  as the profits each firm is obtaining by sustaining the collusion in each moment t.

Now, of course each firm can betray the others by fixing a price lower than $p_m$, namely $p_m-ε$, where ε is small, and by doing so, the firm will capture the entire demand because in this market the firms are doing the Bertrand competition. In other words, the firm by betraying the others, will get almost π_m at the time T=t. We will also assume that in all t > T no firms will make any profits, because they will punish the firm, by fixing the price in Bertrand competition.

The firm will defect if:

$π_m/n + δπ_m/n + δ^2π_m/n.... < π_m+0+0....$

Where δ is the discount factor.

This can rewritten as:

$(\frac{π_m}{n})(\frac{1}{(1-δ)}) < π_m$

We can now see that if n, the number of firms, increases then the profits by sustaining the collusion will decrease, so the above inequality will be more likely to be true. This means that a firm has less incentives to sustain a collusion when there are too many participants, because the profits will be divided among too many firms and the punishment will be seen as less heavy.


Approaches in demand analysis What is the difference between Engel Curve and the system approach of demand analysis?

In general the demand for a certain good (say from a consumer) can be written as a function of the prices of all available goods and the total amount of money that the consumer has available.
Take the setting of two goods, $q_1$ and $q_2$ with prices $p_1$ and $p_2$ and total income $y$. Then the demands can be written as:
$$
q_1 = d_1(p_1, p_2, y)\\
q_2 = d_2(p_1, p_2, y).
$$
If you look at Engel curves, you consider prices as fixed, so the only thing that is allowed to vary are the income levels. In this case, you can simply write:
$$
q_1 = e_1(y),\\
q_2 = e_2(y).
$$
If you want to estimate the full demand system (i.e. the functions $d_1, d_2$), you have to make sure that you have variation both in income levels ($y$) and (relative) prices ($p_1$ and $p_2$), so either time series data for a given individual or panel data for a group of individuals.
If you do not have price variation (e.g. because you are using a cross sectional dataset and everyone faces the same price, or if prices do not change over time) then you can only estimate the Engel curves $e_1$ and $e_2$.


Can the theory of the second best mathematically justify labor unions in some scenarios? An EPI page says:

Some economists and policymakers might express unease at the view that the downsides of one deviation from “competitive” markets (either labor market frictions or market concentration or some other source of employer power) should be countered by introducing another market imperfection (e.g., unions or a binding minimum wage). But this unease is unwarranted. The “theory of the second best” clearly argues that once markets depart at all from perfect competition, efficiency may well be increased by further departures. For example, in the case of monopsony power in low-wage labor markets, legislated minimum wage increases can potentially move wages closer to efficient levels and increase employment.

However, looking at the theory of 2nd best... it's actually not that clear cut that one can simply infer that "doing anything to counter" a distortion is actually going to be 2nd best. In a 50-year retrospective Lipsey himself seems to have nothing to say about labor markets and his theory.
Further, as explained "for dummies" in an article in the Economist (which actually quotes from the above paper, but I'll just quote the more accessible, "dumbed down" explanation from the magazine itself):

Suppose that I told you that in the absence of the necessary conditions for teleportation, the next best thing is to forget all about the conditions for teleportation and instead fly at near the speed of light. Would you find this helpful if what you actually had was a Toyota and a half-tank of gas? Many "second-best" policy recommendations are a bit like that: the ideal market is a fantasy, so here is an ideal government to fix things. Obviously, this is not very helpful. [...]
The upshot is that in practical situations, as opposed to theoretical models, we do not know the necessary and sufficient conditions for achieving an economy-wide, first-best allocation of resources. Achieving an economy-wide second-best optimum allocation looks even more difficult than achieving the first best. Without a model of the economy’s general equilibrium that contains most let alone all of the above sources, we cannot specify the existing situation formally and so cannot calculate the second-best optimum setting for any one source that is subject to policy change. This is an important point since much of the literature that is critical of second-best theory assumes that economists know a distortion when they see one and know that the ideal policy is to remove the distortion directly, something that is necessarily welfare-improving only in the imaginary one-distortion world.

Lipsey himself says:

Are there general policy rules for piecemeal improvements?
My answer to the question in the [above] heading is "no".

So is the EPI "shooting from the hip" here when it claims that 2nd best justifies minimum wage (and possibly even labor unions, although they're not as explicit about that entailment.) Is there any theoretical work that proves that labor unions achieve a 2nd best in some scenario?

Quoting your quote, emphasis altered by me:

The “theory of the second best” clearly argues that once markets depart at all from perfect competition, efficiency may well be increased by further departures. For example, in the case of monopsony power in low-wage labor markets, legislated minimum wage increases can potentially move wages closer to efficient levels and increase employment.

It seems to me the quoted Economist summary misrepresents this quite a bit! No one is claiming that unions or state intervention will yield the second best (which is a somewhat vague concept - what are the exact information constraints here?), merely that they may be welfare increasing.
To expand on this a bit: if we don't know where "the competitive equilibrium" wage would be, but people agree that it is in the [8,40] dollar range, then instituting an 8 dollar minimum wage policy would not have a detrimental effect and would be welfare improving in cases when the current wage is below \$8. But there is no claim of optimum, not even in expected value.
Mathing it up:
$$
\begin{align*}
S(p) & = p \\
D(p) & = a - p \ \text{ where } \ a \sim \text{U}[16,80]
\end{align*}
$$
so
$$
p^* = \frac{a}{2} \sim \text{U}[8,40].
$$

The paragraphs above are the gist of the answer, but I can also produce a hiperbole on par with the Economist. Suppose that I told you that due to variance in quality some lollipops manufactured by a certain company are detrimental to one's health. The Economist seems to argue, that someone proposing the idea of quality control is promising not that they will catch some of these poisonous lollipops but all of them. And since this is clearly not possible, as quality control itself has varying quality, it is pointless to even try to improve the situation. But quality control may be justified even if some of the subpar lollipops are discovered. In fact quality control may be justified even if it supervised by a nonexpert and hence only catches half as many subpar lollipops as possible, as stopping some bad effects are preferential to stopping none.
It occurs to me that given the difference between the marital ideal and the status of the average marriage it is rather fortunate The Economist does not give marital advice.


Calculating mean variance portfolio with risk aversion parameter I want to calculate the classic mean variance portfolio (Markowitz) with a risk aversion parameter $\gamma$.
I have the following problem where I want to maximize:

  $max(x_t) \ \ x_t^T\mu_t - \frac{\gamma}{2}x_t^T\Sigma_tx_t$

Where $\mu$ = mean

$x_t$ = portfolio at time $t$

$\Sigma$ = sample covariance matrix

w(t) is the vector of relative weights in the portfolio, calculated as follows:

$w_t=\frac{x_t}{1_N^Tx_t}$

1_N is a vector of ones (N elements)

My concern is that in the literature I have seen only solution for the above maximization problem where the risk aversion parameter drops out:

$w_t=\frac{\Sigma_t^{-1}\mu_t}{1_N\Sigma_t^{-1}\mu_t}$

I'm wondering at which point do we account for the risk aversion parameter during the optimization?

Here's how you would account for risk aversion.

First of all, I think this problem is usually set up using weights as the variables (see here http://www.math.ku.dk/~rolf/CT_FinOpt.pdf, page 141). If you want to use dollar weights, you will have to transform the result.

The optimization problem is:

$$ \max_w \quad  w'\mu −\frac{\gamma}{2} w'\Sigma w$$

Weights add up to 1:

$$ w'e = 1 $$

Where you have $N$ assets, $w$ and $\mu$ are $N\times1$ vectors, and $\Sigma$ is an $N\times N$ matrix. $e$ is an $N\times1$ vector of ones.

The Lagrangian for the problem is:

$$ \max_w \quad  w'\mu −\frac{\gamma}{2} w'\Sigma w + \lambda (w'e - 1)$$

To optimize, we take the gradient of the objective function, and set each element equal to zero. So we will have an $N\times 1$ vector equation:

$$0 = \mu - \gamma \Sigma w + \lambda e$$

This can be solved for $w$:

$$w = \left(\gamma \Sigma\right)^{-1} \left(\mu + \lambda e\right)$$

Plugging this back into the constraint, we get:

$$ \left(\left(\gamma \Sigma\right)^{-1} \left(\mu + \lambda e\right)\right)'e = 1$$

The transpose reverses the order of the terms, and has no effect on the symmetric $\Sigma^{-1}$ matrix:

$$ \left(\left(\mu' + \lambda e'\right)\frac{\Sigma^{-1}}{\gamma} \right)e = 1$$

This can be simplified to:

$$ \mu' \Sigma^{-1} e + \lambda e' \Sigma^{-1} e = \gamma$$

Solving for $\lambda$, we get

$$ \lambda = \frac{\gamma - \mu' \Sigma^{-1} e}{e'\Sigma^{-1} e} $$

So the overall formula for the weights is:

$$w = \left(\gamma\Sigma\right)^{-1} \left(\mu + \frac{\gamma - \mu' \Sigma^{-1} e}{e'\Sigma^{-1} e} e\right)$$


What is the function of capitalism? To my understanding, all systems have a function.  For instance, the function of the respiratory system is to supply oxygen to the body.  The function of the digestive system is to break down and disseminate nutrients.  The function of a mass transit system is to facilitate large scale public transport.

Capitalism is defined in Merriam-Webster as,

  an economic system characterized by private or corporate ownership of capital goods, by investments that are determined by private decision, and by prices, production, and the distribution of goods that are determined mainly by competition in a free market

So clearly, capitalism is a system, but I can't glean from that definition its purpose.

What would most economists consider the function of capitalism?

Please be clear to separate, if you will, this system's function from the functions of other economic systems in your answer.

Capitalism is defined as an economic and political system in which a country's trade and industry are controlled by private owners for profit, rather than by the state.

It functions as an allocation mechanism which generally yields the most optimal results under the Fundamental theorems of welfare economics and prevents issues such as Tragedy of the commons which a communist system would fail to deal with (unless all resources were heavily policed and regulated with knowledge of every consumers utility functions).

Under a pure capitalist system, government intervention in the market place is viewed as harmful.

However according to mainstream economic theory, there is some benefit to have from government intervention when dealing with  externalities, market saturation and responding to Business Cycles.


Quasi-Linear Functions I understand that quasi-linear functions have a general form

$U(x_1,x_2,...,x_n,y) = f(x_1,x_2,...,x_n) + y$

and that for a quasi-linear function, the income effect with respect to the other variables of the function ($x_1,x_2,...,x_n$) are all $0$, i.e., income has no effect on the consumption of those goods.
I also read online that a if a preference is quasi-linear, then indifference curves are parallel.

However, I came across this function, $U(x,y)=e^xy$, where the income effect with respect to $y$ is $0$, when the price of $y$ changed. Is this also a quasi-linear function, since IE is $0$? Plotting the graph of this function, I see that the indifference curves are actually not parallel.

$U(x,y)$">

The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $x$.

The curves you map maintain horizontal distance regardless of $y$. That is because the non-linear variable here is $y$, not $x$. The curves are still shifted, but along the other axis.

Taking the logarithm of the utility function you get
$$
x + \ln y
$$
which is more clearly quasi linear.


How to find the profit-maximizing output level? Table 11 In this post, step 5 states the profit-maximizing output level is quantity 5. But in this case, $p=28$, $MC=30$, $p\neq MC$. Why it is the profit-maximizing output level?

Step 4 states the output level where price equals the marginal cost is the output level that maximizes profits. If so, both $q=4$ and $q=5$, the profit is $\$40$. Why we don't choose $q = 4$ as the profit-maximizing level?

Actually 4 in this case would be completely valid answer since the net profit at both quantity 4 and quantity 5 is equal  $\Pi(q=4)=\Pi(q=5)=40$. I would even argue 4 is potentially better answer.

I think that the reasoning of whoever made that textbook was that given the profits are equal whichever quantity has marginal costs closer to price should be preferred.

In this case the marginal costs at $q=4$ are 17 and $q=5$ are 30 and 30 is just more closer to 28 than 17. Again, I would not say that necessarily the best answer to discrete problem like this, but my guess is they want to teach to always look at point where price is the closest to MC since in optimum it should be equal.

I would not read too much into this exercise. At least this chapter of that online resource is at high-school or at best very simple undergraduate level. At such low level often rigor is sacrificed for the sake of making the material not too difficult in order to not discourage young students. In your future courses/self-study you will get more precision and nuance.


Why is CRRA utility often used in macroeconomics DSGE model? As title says, why is CRRA utility often used in macroeconomics DSGE model? That is, the form of

$$u(c_t) = \frac{c_t^{1-\sigma}}{1-\sigma}$$

I cannot find any theoretical background around this..... After all, there is often no risk involved in DSGE models..

Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable growth rate (see also this post), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time.

So one would want a model that admits a constant growth rate at its steady-state. In the benchmark deterministic/continuous time "representative household" model, the Euler equation  takes the form

$$r = \rho - \left(\frac {u''(c)\cdot c}{u'(c)}\right)\cdot \frac {\dot c}{c}$$

This is the optimal rule for the growth rate of consumption. The rate of pure time preference $\rho$ is assumed constant. The interest rate $r$ has its own way to become constant at the steady state. So in order to obtain a constant consumption growth rate at the steady state, we want the term

$$\left(\frac {u''(c)\cdot c}{u'(c)}\right)$$
to be constant too. The Constant Relative Risk Aversion (CRRA) utility function satisfies exactly this requirement:

$$u(c) = \frac {c^{1-\sigma}}{1-\sigma} \Rightarrow u'(c) = c^{-\sigma} \Rightarrow u''(c) = -\sigma c^{-\sigma-1}$$

So

$$\frac {u''(c)\cdot c}{u'(c)} = \frac {-\sigma c^{-\sigma-1} \cdot c}{c^{-\sigma}} = -\sigma $$
and the Euler equation becomes

$$\frac {\dot c}{c} = (1/\sigma)\cdot (r-\rho)$$

Barro & Sala-i-Martin (2004, 2n ed.), extend the required form of the utility function when there is also leisure-labor choice (ch. 9 pp 427-428).
These fundamental property extends to the case of stochastic/discrete time.

XXXX

To compare, if we have specified a Constant Absolute Risk Aversion (CARA) form, we would have

$$u(c) = -\alpha^{-1}e^{-\alpha c} \Rightarrow u'(c) = e^{-\alpha c}\Rightarrow u''(c) = -\alpha e^{-\alpha c}$$ and the Euler equation would become

$$\dot c = (1/\alpha)\cdot (r-\rho)$$

i.e.here we would obtain a constant steady-state growth in the level of consumption (and so a diminishing growth rate).


What will giffen goods and Veblen goods peform in general equilibrium model? I'm a beginer of microeconomics.
According to consumer behavior theory, a consumer possesses a set number of money as budget constraint at first. And their is negative income elasticity of demand and extremely positive income effect for giffen goods. In contrary, the man who demand Veblen goods will increase his quantity demanded if price increase.
But in chapter general equilibrium, I was informed that their is no money in this model, and the 'number' of price are used to indicate relative price, not absolute one (I learned that from Intermediate Microeconomics:A Modern Approach by Hal R. Varian, and a textbook self-complied by the school). I am confused that if their is no money,  are the changes of income and price samely relative in partial equilibrium model? Or the 'number' of them are absolute? How can we witness giffen effect and Veblen effect in general equilibrium model?

In general, what you are referring to is what we call numeraire pricing. This means that the prices, as you mentioned, are relative to one of the goods (you can pick any good as a numeraire, which is how we measure prices through the whole economy).
The reason for this is Walras' Law states (basically) that since we have a bunch (say $N$) goods in the economy, and say we have a bunch (again, $N$) markets in the economy, then solving all of these will leave one redundant equation. This applies in particular to general equilibrium because there are essentially more equations than variables, and hence you have one (or more) redundant variable(s) at the end.
To get a little more technical if you want: if money is something that we can consume, then we would insert cash as an additional good and make that our numeraire. Thus we can have "money," in the sense that it has a value as a bubble, and we can include it as a good in our utility.
To answer your questions about Veblen/Giffen effects, we can still observe these effects, they have nothing to do with money. These effects in their essence are talking about goods that we buy more of when the price increases. Even if this price is in terms of "money" or a numeraire, the point is that we buy it when it's more expensive to us, so we push other things that we might need out of the way in order to buy this.
A quick example (don't think too deeply about it, it's just an illustration) would be college in America, using, say, $1 McDonald's McChicken sandwiches as a numeraire. An argument could be made that each year more students attend college yet the price in terms of McChickens goes up each year ((i.e., consumption increases, as the price goes up). This could be described by the Veblen effect, where we maybe don't want to be perceived as someone with "only" an associate's degree or high school degree (not that those are bad, but it's just the way things are in America). If I measured in dollars or McChickens, the Giffen or Veblen effect doesn't change.


More complex than simple and multiple regressions? I am currently in an Econometrics class that requires us to write a research paper that showcases our skills in regression/ modeling.  What is slightly more complex than simple and multiple regressions that I may be able to learn and utilize in my research project?  I realize that the model depends on the research question, but I'd like to learn different types of modeling anyhow.

Thanks.

Welcome to the wonderful world of econometrics! Most introductory econometrics courses will extend Ordinary Least Squares (OLS) by considering binary outcomes models such as the logit and probit.

Whilst OLS is typically restricted to modelling continuous outcomes bound between $-\infty$ and $\infty$, in research one will often come across data where this is not the case. The most elementary extension is the binary (yes/no) outcome data.

This data takes the form of a yes/no (coded 1/0). Clearly, OLS runs into several problems. First and foremost (for your purposes) is the fact that you can predict outcomes outside of the range $[0,1]$. The simplest solution: logistic regression.

Logistic regression estimates the probability that $y=1$ conditional on your explanatory variables. It does this creating a mapping of your regression equation $\beta_0 + \beta_1 x$ on to the space $[0,1]$ by considering:
$$ \ln \left( \frac{P(y=1)}{1-P(y=1)} \right) = \beta_0 + \beta_1x $$
Depending on what data you can find (and what software you have available - for example EViews, R, Stata and SPSS all have built in routines for logistic regression) this might be an interesting way to proceed.


Does the VCG algorithm work for any bundle valuation function? Assume we have $n$ buyers of $|M|=m$ items in a combinatorial auction.
So as we know, we can use the Vickrey-Clarke-Groves (VCG) algorithm to allocate bundles of items to buyers as best as possible. However, each player $i$ has a value $v_{ij}$ for every single item $j$ and a value $v_i(S)$ for each bundle of items $S \subset M$.

My question is as follows: can we always use the VCG algorithm to find the best solution to this problem, even when the $v_i(S)$ function is non-additive, for example if $v_i(S) = \max\limits_{k \in S} v_{ik}$? (So in this case, player $i$ values a bundle of items as much as its most valued item, no matter how many items are in the bundle)

In theory, the answer is yes.
In practice, the answer is no, because it is computationally intractable.
My take-away from talking to computer scientists was that determining a winner and computing the transfers are NP-hard problems. See, e.g., this write-up by Kirk Pruhs.


Show that $x(p,w)=w\cdot x(p,1)$ with homothetic preferences Someone gave me a proof of this, but I am not sure if it is correct.

Let $B(p,w) = \{x: p\cdot x \leq w\}$ (the budget set). Then:

\begin{align}
x(p,w) &= \arg \max_{x\in B(p,w)} u(x)\\
&=\arg \max_{\alpha x\in \alpha B(p,w)}  u(\alpha x) \\
&=\arg \max_{y\in B(p,\alpha w)} u(y) \\
&=\frac{1}{\alpha} \arg \max_{y\in B(p,\alpha w)} u(y) \\
&=\frac{1}{\alpha} x(p, \alpha w)
\end{align}
Where the result follows from taking $\alpha=\frac{1}{w}$.

Is this proof correct (I am not sure of the middle three equalities)? Where is homotheticity used?

EDIT: A monotone preference relation $\succsim$ on $X= \mathbb{R}^{L}_{+}$ is homothetic if all indifference sets are related by proportional expansion along rays; that is, if $x \sim y$, then $\alpha x \sim \alpha y$ for any $\alpha \geq 0$.

Also, recall a continuous $\succsim$ on $X = \mathbb{R}_{+}^{L}$ is homothetic iff it admits a utility function that is homogenous of degree one; $u(\alpha x) = \alpha u(x)$.

An indirect proof. Suppose
$$
x(p,w) = w\cdot x(p,1)
$$
does not hold. This is equivalent with stating
$$
U(x(p,w)) \neq U(w\cdot x(p,1)).
$$
(To be precise: $x(p,w)$ and $x(p,1)$ may be set valued. In this case we are talking about two elements at least one of which is not included in both sets.)

Case 1.
$$
U(x(p,w)) > U(w\cdot x(p,1))
$$
As $U$ is homothetic
$$
U(x(p,w)) =  U(w \cdot \frac{1}{w}\cdot x(p,w)) = w \cdot U(\frac{1}{w}\cdot x(p,w)).
$$
Using this we have
$$
w \cdot U(\frac{1}{w}\cdot x(p,w)) = U(x(p,w)) > U(w\cdot x(p,1)) = w\cdot U(x(p,1))
$$
and thus
$$
U(\frac{1}{w}\cdot x(p,w)) > U(x(p,1))
$$
However as $\frac{1}{w} \cdot x(p,w)$ is clearly an element of $B(p,1)$ this is impossible as $x(p,1)$ gives maximal utility in that budget set.

Case 2.
$$
U(x(p,w)) < U(w\cdot x(p,1))
$$
As $w \cdot x(p,1)$ is clearly an element of $B(p,w)$ this is impossible as $x(p,w)$ gives maximal utility in that budget set.

Thus we have proven that
$$
U(x(p,w)) = U(w\cdot x(p,1))
$$
which is equivalent with
$$
x(p,w) = w\cdot x(p,1).
$$


Why is an inflation rate of 2% optimal? Learning from different sources I got the impression that the European Central Bank targets at a inflation rate of 2% (not sure along which time scale actually).

Can it be explained by general principles why a (long-term) inflation rate of 2% seems optimal?

Or does it depend on the current economic situation, and in some scenarios an inflation rate of 0%, 1%, or even 5% or deflation would be optimal (in the short or mid run)? But in the long run always 2%?

In an article in the Times, Jon Moulton stated:

  "The Bank [of England] has had this [2%] inflation target for more than a
  decade yet it lacks any theoretical basis."

EDIT: Just in case you are interested, I have a theory about optimal inflation. If you wish money to be neutral - i.e. be a Veil over Barter then you would like money to behave as closely as possible to ordinary "stuff" that gets bought and sold. One property that the bulk of goods have is that they decay over time, i.e. they rot, rust, become obsolete etc etc. If you consider the average decay rate of "stuff" to be X% then inflation should be such that money loses its value at a rate of X% too. I'd say that the decay rate of "stuff" is generally greater than 2%, my feeling is that it is more like 5% or so.


Where can I find historical (very old) USD printing data? Where can I find historical/very old USD printing data? Charts can be useful, but I want numbers. It is for a project.

The Bureau of Engraving and Printing maintains public records on their web site, but that data only goes back to 1980. Therefor your best option might be to write or call them and ask for information for the time period you're interested in. If they are not forthcoming, you can call back and try your luck with a different staff member. If that doesn't work, you can resort of filing a Freedom of Information Act request. According to this page the average turnaround time is 23-31 days depending on the complexity of the request.

There is also this site, which has information on the US money supply from 1959 - present, but I am not sure it is presented in a way you'll find useful for your project. I am also unsure if it meets your definition of "very old."

Good luck!


What exactly people mean when they say "X is a $Y billion industry"? I'm reading the "Personal MBA" book right now and it says that "Yoga is $8 billion industry worldwide". What does it mean? And how can one calculate the amount of money in some industry?

Considering the economic of world as a whole it could simply refers to that total spending by people on yoga might be \$8 billion. As in an economy total spending is the total revenue of a sector. For this particular scenario it simply means that yoga as an economic variable have transaction of \$8 billion.
Regarding the calculation we can de assured that no exact and absolute calculation could be made on national level how could it be calculated exactly at global level. However the estimated aggregated spending on yoga sector nation could be determined. Which further on processing might result to provied aggregate global spending on yoga as a figure, which is tentative not exact figure.


Do I understand correctly that taxation-based government expenditures allow GDP to grow? Suppose (it's an example) GDP= $1 000 000, Marginal Propencity to Consume(MPC)=0.25, that borrowing money is an absolute taboo for our government and that said government wants to increase GDP.

What can the government do to increase GDP of its country? Well, it can increase government spendings. The expenditures multiplier is equal to 1.33, meaning that for each \$1 spent by the government the GDP increases by \$1.33. But with borrowing out of question our government will have to tax population first in order to raise the money. But additonal taxation will decrease our GDP. Will our expenditure multiplier be big enough to compensate for effects of our tax multiplier? Yes, because the absolute value of the tax multiplier will always be less than the absolute value of expenditure multiplier, no matter what MPC we will have (except if we will have MPC=1, but in this case we will get divizion by zero error), consequently meaning that EM-TM always >0. In our particular example we will lower GDP by 0.33 for each dollar that we taxed, but at the same time increase by 1.33 for each dollar that we spent, meaning that taxation and consequent spending of N dollars will give us +N dollars to GDP, 1.33x-0.33=x(1.33-0.33)=x*1=x

1.Is my understanding of given model correct?

2.If it's correct, then why don't government tax and spend tax money much much more actively than they are doing today? It looks like it would be a "cheat code" for unlimited exponential growth. Maybe there are other factors that prevent it, but that our simple model ignores?

In your comment you say you want the answer in context of the classic textbook model for goods market/output equilibrium.

Following the Blanchard macroeconomics textbook lets consider closed economy so the output will be given by:

$$Y=C+I+G$$

where $Y$ is output/income (they must be always equal), $C$ consumption, $I$ investment and $G$ gov. spending.

The consumption will be given by:

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

where $c_0$ is the part of consumption that does not depend on income and $c_1$ is marginal propensity to consume (MPC) and $T$ are taxes.

In this basic model investment is just fixed $I=\bar{I}$

Solution to the model (i.e. autonomous spending) based on the set up above is given by (the solution is derived by substituting $C$ in second equation for its place in first one and then solving for $Y$):

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1T ]$$

Now since you say that we take it for granted that government runs balanced budget $G=T$ because borrowing is "taboo" we can substitute the equality between taxes and gov. spending so we have:

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1G ] = \frac{1}{1-c_1}[c_0+ \bar{I} +(1-c_1)G ]$$

Now to clearly see the effect of G spending lets just set $c_0=\bar{I}=0$ (this is not necessary but it cleans the equation). So we have:

$$Y =  \frac{1}{1-c_1}(1-c_1)G = G$$

Hence there is no multiplier on Government spending if budget is balanced. So without debt government cannot increase output in any way (at least not in this simple model).

The intuition is that of course if government just takes money from left pocket to the right there cannot be any increase in output. Since in this model all agents have the same MPC and hence all spending including the Gov. spending has the same multiplier. If you take 10 euros from consumers and just spend it as a government you are not creating any additional output. you are just reducing $C$ - reduction of which would also be affected by multiplier, by the same value you are increasing $G$ which is affected by the same multiplier. The effects must cancel each other out. Only if you would allow for heterogeneous agents where people have different MPC government could increase output in the short term by redistributing money from people with low MPC to people with high MPC, but the output cannot be raised "exponentially" and this would only be one time increase - i.e. increase in level or short run growth. I would recommend reading the full chapter 3 from Blanchard et all (2010). Macroeconomics an European perspective, that deals with this in greater detail.


What are the consequences of having negative GDP growth? Many countries are facing negative GDP growth due to coronavirus.
I am curious to know what will the significant consequences of having negative economic growth?

Broadly speaking GDP is measure of output economy produces and from economic perspective output is equal to income. Assuming we are talking about real growth not just nominal one a main consequence of negative economic growth is that on average people's real incomes will be lower and their material standards of living worse.
Under spending approach to GDP (which is equivalent to income approach), GDP in closed economy is given as:
$$Y = C+I+G,$$
where $Y$ is the output/income, $C$ is consumer spending, $I$ is investment and $G$ is government spending.
So another way of thinking about consequences of negative growth is that negative growth will make left-hand side of this equality ($Y$) lower. If $Y$ declines then economy will have to either:

consume less - which directly affects well being of consumers.
invest less - which affects the rate at which economy can grow.
government will have to spend less - which means that government will have to cut provision of some of the public goods or other spending it does.

In either case it will lead to some worse material standard of living for the particular country.
Also the above are only the most direct effects. I did not touch upon any potential distributional questions. Economic growth empirically has a first order impact especially on the welfare of poor (see  Barro & Sala-i-Martin Economic Growth and sources cited therein). There are also further implications, it is not much of an overstatement that almost everything in economy depends on what the economy can produce, but exploring this to more full extent would be beyond the scope of stack exchange post.
As a further reading I recommend any macro textbook or even just introductory economics textbook (my concrete recommendations would be either Blanchard et al. Macroeconomics a European Perspective or Mankiw Principles of Economics), any of them will explore significance of output to economy from many different sides.


The Principle of Optimality and the Bellman Equation Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality?

$$\ max \Sigma\beta^s U(C_t)$$

Subject to the following resource constraint:

$$C_t + K_{t+1}= F(K_F, E_F,S_t)$$

Where:
$$E_t = F_E(K_E,E_2,R_t, R_{t+1})$$
$$S_t = \Sigma(1-d)E_t$$
$$R_{t+1}=R_t -E_t$$
And the Bellman equation is given:

$$V_t (K_t,R_t,E_t)= max_{C_1,K_2,E_1,E_t} {U(C_t)+\beta V_{t+1}(K_{t+1},R_{t+1},E_{t+1}) + \lambda_2 (F_2 (K_2,E_t-E_1)-E_t)}$$

This problem is taken from eq. (27) and eq. (39) of the following textbook: https://are.berkeley.edu/~traeger/pdf/KarpTraegerDraft.pdf.
The specified parameters in the model are:

Consumption $C_t$
Capital $K_t$
The energy sector $E_t$
The carbon stock $S_t$
The finite resource $R_t$

Would genuinely appreciate being able to see how this is conducted on this type of model.

Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:
Assumption 1: $\Gamma(x)$ (our set of feasible values) is non-empty for all $x\in X$.
Assumption 2:$\lim_{t\rightarrow\infty}\sum_{t=0}^\infty \beta^t F(x_t,x_{t+1}) $ exists for all $\tilde{x}\in \Pi(x_0)$ ($\Pi(x_0)$ being the correspondence of $x$ considering an initial $x_0$)
Assumption 3:$|V^*(x)<\infty|$ for all $x \in X$.
Assumption 4: For any $x_0$, there exists a plan $\tilde{x}\in \Pi(x_0)$ such that $u(\tilde{x})=V^*(x_0)$.
In your story here since $\beta<0$ we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian:
$$\mathcal{L}=\sum_{t=0}^\infty\{\beta^tu(c_t)+\sum_i\lambda_{t,i}(F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}\}$$
Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.
Hope this helps
Edit: After looking at your addendum im guessing the proceedure would be the same
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]\}+\max_{c_{t+1},...,c_{T}}\{\sum_{t+1} \ [\beta^{t+1}u(c_{t+1})+\sum_i\lambda_{it}[F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]]\} $$
or
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]+V(k_{t+1})\}$$


Why didn't the money printing by the US Federal Reserve since 2008 lead to inflation? http://www.telegraph.co.uk/finance/comment/liamhalligan/8484530/Americas-reckless-money-printing-could-put-the-world-back-into-crisis.html

  America's reckless money-printing could put the world back into crisis
  Last week, Ben Bernanke suggested that the US base interest rate will
  stay close to zero for an "extended period". It's been there since
  December 2008.

Today, the opposite has happened. Deflationary forces are strong in the U.S. Europe is even worse given the negative yield. Why has deflation prevailed despite money printing by central bankers?

Because you can't push on a piece of string.

As in the Great Depression of the 1930s, we've collectively got massive personal debt, depressing demand, moving us towards deflation, which pushes up debt, and so it goes.

We've also known since the 1930s that monetary expansion in and of itself cannot stimulate demand under these conditions: it can only create space for demand to expand into.

Add in the international fashion for fiscal contraction, a demand-suppressant, and that's how we get QE without broad-money expansion and without inflation.

We do have potential inflation. And at some point that potential will be realised. But not while there's this huge overhang of personal debt. It might get worn down over a long period of deflation, through defaults. Some other inflationary shock might inflate some of the debt away. And enlightened policy-makers might engage in a large exercise in personal-debt cancellation.


Why does socialism not work? There are many arguments that socialism doesn't work. It's becoming an increasingly popular topic as congresspeople such as Bernie Sanders and Alexandria Ocasio-Cortez push for policies that are considered to be socialist.

However, the articles about why socialism fails generally cite multiple different reasons, and there seems to be no clear argument.

The only reason why something would fail practically when it should work theoretically is that it is making incorrect assumptions about the world.

What assumptions does socialism make in order to work theoretically, and why aren't those assumptions accurate reflections of the world?

(Please note that this question takes the standpoint that there are incorrect assumptions in the theoretical basis of socialism, and is trying to identify weak points in this theoretical basis. It may be true that socialism does work, or that there's a version of socialism that works - this question is specifically asking about what parts might not work. Similar questions may be asked about capitalism, or other economic theories.)

Socialism doesn't work because an elite does not like to give up their privileges easily. After it took a wave of bourgeous revolution over a number of centuries to force the aristocracy to give up their powers and to enter a power-sharing agreement with the bourgeois which led to the decline of feudalism.

Marx suggested that this movement would not stop with the bourgeois, but would then go on to animate the so called third estate.  This indeed did occur over the 20C, but in the final analysis, not successfully and this has led to some people to say that Socialism does not work. The other side of this thesis is that it was simply strangled at birth, and this is what the Trumans policy of containment amounted to.


Without knowing the Slutsky equation and income/substitution effect, how can I show a certain good is inferior or Giffen? Say I've got a function $x_1(p_1,p_2,m)$ where $p_1, p_2$ are the prices for good 1, good 2 respectively and m is the income.
Now, I haven't heard of the Slutsky equation yet nor the income/substitution effect.
How do I show for this function that it is an inferior/normal and ordinary/Giffen good?
My initial thought was to simply use the partial derivate and to check whether it's greater or less than zero, since
$\frac{\partial x_1(...)}{\partial m} \gt 0$ should be true for normal goods, $\frac{\partial x_1(...)}{\partial m} \lt 0$ for inferior goods, $\frac{\partial x_1(...)}{\partial p_1} \gt 0$ for Giffen goods and $\frac{\partial x_1(...)}{\partial p_1} \lt 0$ for ordinary goods.
Is this correct?
Is this all I can do to formally show that the good is normal/ordinary? Is this sufficient?
Also, if I can show that the same function for $x_1(m)$ is homogeneous of degree 1 (i.e. homothetic), i.e. $tx_1(m)=x_1(tm)$, this means that the partial derivative $\frac{\partial x_1(...)}{\partial m}$ is homogeneous of degree 0, so it would be a constant. Could I thus conclude that the good is strictly normal/inferior (is that a term?), i.e. it doesn't change if the income changes?

yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.
In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as
$$
x_1(p_1, p_2,m) = x_1(p_1, p_2,1) m
$$
where $x_1(p_1, p_2,1)$ is the unit income demand function which is always greater or equal to zero.
From this, it follows that homothetic preference can never lead to Giffen goods (but to show this, you do need the Slutsky decomposition).


What is state space representation for DSGE modeling I'm beginning with DSGE modeling, and a mathematical representation (perhaps trivial for most of the people that are more with this topic) is the space-state state representation of a dynamical model, like depicted here (s.2), or similar documents.
I have already read the Wikipedia article about this topic, and I get the general idea (in this case applied to engineering). But I want a further explanation in how this applies in DSGE modeling, and if possible some literature (preferably economics-focused). The concrete question is what is a state-space representation, if possible with some of the intuition, its uses in economics (control theory), and some key insights about the notation.
Thank you!

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

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

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

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

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

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

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


Why do banks keep vault cash instead of depositing all reserves at the central bank? In what way to commercial banks benefit from keeping some of their excess reserves in the vault as opposed to depositing all of their excess reserves with the central bank (assuming the central bank pays IOER)?

What sort of proportion of a banks reserves are kept in vault VS at the central bank?

Thanks!

"Vault cash" is the money that a bank will keep on premises (which is usually kept in their vault) to deal with their day-to-day cash needs.

The bank must have a certain amount of cash on hand in order to deal with these types of every day transactions.  "vault cash" is considered to be part of a bank's reserves, and banks will usually be required to have a certain amount of "vault cash" on hand.

Bank reserves are essentially an antidote to panic. The Federal Reserve obliges banks to hold a certain amount of cash in reserve so that they never run short and have to refuse a customer's withdrawal, possibly triggering a bank run.

The required bank reserve follows a formula set by the Federal Reserve Board's regulations that are based on the amount deposited in net transaction accounts. These include demand deposits, automatic transfer accounts, and share draft accounts.


Defining a Competitive Equilibrium for a two Period Model Consider an economy that lasts for two periods populated by a representative consumer/worker, a representative firm, and a government. After these two periods, the economy ends.

The question asks me to
- Define a competitive equilibrium for this economy.
- Show the demand and supply curve in a diagram with variables (r, Y ).
- Show the effect of an increase in the current capital stock K.
- Compare the equilibrium path with the case in which workers have a preference shock that reduces the disutility from work h − l in period one. Preferences are back to normal in period 2.

My Problems I am so confused: what are the steps to defining a competitive equilibrium? Do I just rewrite what competitive equilibrium is from the lecture notes, or do I have to define one from the problem above?

Also, there are a lot of variables with no numbers. I am very confused to how to approach a question with no numbers.

A competitive equilibrium is specific to the economy you're modeling. Your economy has three markets:

Capital market for $K$
Labor market for $L$
Goods market for $Y$

Furthermore, there are several optimization problems / budget identities that have to hold.

The household has a consumption/savings and a leisure/labor decision
The firm has a labor and a capital choice
You need to ensure that the government's budget constraint holds.

In a competitive equilibrium, you need to clear all the markets, and any quantity that is chosen, needs to be done so optimally, given everything that's happening in your economy. Here, you have two periods which need to be in equilibrium at the same time. If I were grading your homework, I would expect you to write the above for minimum points. You should then show what the conditions/equations are for each of the bullet points.

Example: Multi-Period Non-Equilibrium

For example, the household will chose savings in period 1, given what he expects to do with those savings in period 2. If you'd expect him to do something stupid with the savings in period 2 (say, throw them away), then it would be optimal to save nothing at all. In this case, you would be at an equilibrium in period 1. However, your choices for period 2 would be non-optimal, hence we do not have a 2-period equilibrium.


Solving a two stage game by backward induction: which is the equilibrium notion? Take a two-stage game with complete information and simultaneous actions in each state:

(1) Player 1 and 2 simultaneously choose action $a_1\in A_1$  and
$a_2\in A_2$ respectively.

(2) Player 1 and 2 observe the outcome of the 1st stage
$(a_1, a_2)$, then simultaneously choose action  $a_3\in A_3$ and
$a_4\in A_4$ respectively.

Payoffs are $u_i(a_1, a_2, a_3, a_4)$ for $i = 1,2$.

Suppose that I do the following (typically called backward induction):

(A) find the functions $a^*_3(a_1,a_2)$  and $a^*_4(a_1,a_2)$ such that

$$
\begin{cases}
a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
\end{cases}
$$

(B) find $a_1^*, a_2^*$ such that
$$
\begin{cases}
a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
\end{cases}
$$

Which equilibrium notion I'm I applying?

Basically, in (A) I find the pure strategy Nash Equilibrium of stage 2, for each possible action played in stage 1; in (B) I find the pure strategy Nash Equilibrium of stage 1, given the solution from (A).

How is $\{a_1^*, a_2^*, a^*_3(a_1, a_2), a^*_4(a_1, a_2)\}$ called?

It is called a subgame perfect equilibrium.


Was factory beef farming (in the USA) the result of government-subsidized corn? In The Omnivore's Dilemma, Michael Pollan writes that industrial beef farming, in the form of concentrated animal feeding operations (CAFOs), is dependent on low corn prices for its cost-effectiveness. He also implies that corn prices low enough to justify CAFOs were only possible because the US agriculture policy artificially depressed the price of corn.

Pollan attests that the first CAFOs began in the 1950s. But he also spends a good deal of time in the previous chapter discussing how Earl Butz, under President Nixon (that is, in the 1970s), was instrumental in ending the New Deal corn price stabilization policies (a buffer-stock scheme) and replacing them with corn supply subsidies (direct payments to farmers).

If that's the case, then how did the first CAFOs manage to be profitable, or even seem like a good idea, if corn was not to become artificially cheap for at least another decade? Or were CAFOs always profitable, but received a tremendous boon when corn prices were later driven down by policy?

This is as much a question about the timeline as it is about the underlying economics, so I'd prefer to see an answer based in some kind of scholarly source.

Also I'd appreciate it if someone with more than 150 rep could create an "agriculture" tag for this question. I had wanted to add "prices" and "subsidies" tags as well.

Was factory beef farming (in the USA) the result of
  government-subsidized corn?

Answer A) The answer is negative, according to

MacDonald, J. M., & McBride, W. D. (2009). The transformation of US livestock agriculture scale, efficiency, and risks. Economic Information Bulletin, (43). USDA.

Although not a historical-statistical account, they write (p. 21, bold my emphasis)

  "Two recent reports argue that Federal commodity programs reduce prices for a particular input, purchased feeds, and that those
  reductions drive structural change (Pew Commission on Industrial Farm
  Animal Production, 2008; Union of Concerned Scientists, 2008).
  Specifically, they assert, drawing on Starmer and Wise (2007), that
  policy encourages increased production of feedgrains, that the
  increased production substantially reduces feedgrain prices, and that
  the buyers of purchased feed benefit from lower feed prices. While
  producers of homegrown feed have been the direct recipients of
  commodity payments, the reports argue that the payments have not been
  large enough to offset lower commodity prices, so that commodity
  programs have largely benefited large-scale animal feeding operations
  at the expense of smaller diversified crop and livestock farms.
  However, the size-related differences in production costs that are
  summarized in figures 6-8 cannot be attributed to the effects of
  commodity programs. In developing cost-of-production estimates, ERS
  prices homegrown feed at its market value—that is, the price that
  homegrown feed would have drawn as feed in regional feed markets—and
  does not attempt to estimate the actual cost of producing the feed. To
  the extent that larger operations realize lower feed costs in ERS
  estimates, it is because they use less feed per cwt of production and
  not because purchased feed costs them less.
  But even if commodity policies reduce purchased feed prices, there’s no reason why that should alter the size structure of livestock farms.
  No technological barrier prevents small farms from replacing homegrown
  feed with purchased feed: if purchased feed prices are lower than the
  costs of growing feed, then small livestock operations can simply buy
  feed and realize the same savings as large farms. A difference
  between the price of purchased feed and the cost of homegrown feed
  does not explain why feeders build large rather than small
  operations."

-__________________________________________________________________

Answer B) The answer is positive according to

Starmer, E., & Wise, T. A. (2007). Living high on the hog: factory farms, federal policy, and the structural transformation of swine production. Tufts University.

They report the following estimations, among others: The feed costs for Hog represent ~$60\%$ of operating costs of CAFO's. That in the period $1997-2005$ feed subsidies meant that feed was sold at a price $26\%$ below production cost. Hence, the savings in operating costs for CAFO's was around $60 \% \times 26\% \approx 15\%$ (of Costs, not of Revenues). For  the top four CAFOs in terms of market share, the authors estimated that this amounted to total cost-savings of $\approx 4.4$ billion USD for the 1997-2005 period.

This paper appears to be an aggregator of previous more detailed studies, so lots of references.


If Y=K+C, why the value of a marginal unit of capital is the same of the value of a marginal unit of output? In a simple model of welfare dynamic optimisation with non-renewable natural resource where:

$$
\dot S_t = -R_t
$$
$$
\dot K_t = Q(K_t,R_t)-C_t
$$

one of the first order condition is that $P_t=\omega_t Q_{R}$, where $P_t$ is the shadow price of the natural resource stock $S$, $Q_{R}$ is the marginal product of the natural resource and $\omega_t$ is the shadow price of the human-capital stock $K$.

In the Perman et al. (2011) "Natural resource and environmental economics" this equation is explained as "the marginal value (or shadow price) of the natural resource stock must be equal to the value of the marginal product of the natural resource".
In particular $\omega_t$ is redefined as the value of one unit of output, with the justification that units of outputs and units of capital are in effect identical in this economy among the optimal path, as any output that is not consumed is added to capital.

It is this redefinition that I do not understand.

This is a single-good model, that is, output is of one good which can be either consumed or added to capital.  The dynamic optimisation problem is to maximise the discounted present value of utility, utility being a function of consumption meeting the standard conditions $U_C > 0$ and $U_{CC} < 0$.

Perman et al identify the second of two state variables as capital $K$.  Now a state variable features in the statement of a dynamic optimisation problem in a limited number of ways:

An equation of state, in this case $dK_t/dt = Q(K_t,R_t) – C_t$.
An initial condition, in this case $K_{t=0} = K_0$.
A terminal condition, in this case by implication that (assuming an infinite
planning horizon) $K_t → 0$ as $t → \infty$.

Suppose that instead we were to specify the second state variable as the stock of the single good, say $G_t$.  Would this result in any essential changes to 1-3 above, which would change the solution of the problem?  It would of course if we allow for stocks of the good destined for consumption but not yet consumed, eg being packaged, transported or on shelves awaiting purchase.  But in a simple model such as this we will probably want to ignore such complications.  Suppose therefore we also assume that, for the part of output that is consumed, consumption is immediate.    Then the stock of the good would consist entirely of capital, and we could re-write 1-3 substituting $K$ by $G$ throughout.

This substitution would not affect the substance of the problem, so the same optimal solution would be obtained whether we make the substitution or not.  In particular, the time path of $\omega_t$ consistent with the optimal solution would be the same, whether $\omega_t$ is regarded as the shadow price of capital or the shadow price of the good.  Hence along an optimal path $\omega_t$ is also the value of a unit of output of the good.


Clarification about Market Structure Oligopoly or Monopoly As per the definition i found in the internet,

Oligopoly, in which a market is run by a small number of firms that together control the majority of the market share.
Monopoly, where there is only one provider of a product or service.

Consider this situation in a country.

There are 4 main bus operators in this country. A,B,C and D and some individual bus owners.

The bus fare is controlled by a national body.
Some particular bus owners  who has AC installed in the bus, charges a 0.2$ extra per route.

In the above scenario in what classification would you put the transport sector of this country. Is it a monopoly as the price is regulated by a single body? or is it an oligopoly as the market share is dominated by these four main company.

Further more for my upcoming exam if I get a question on Market structure of this country transport system how can I elaborate?

This market is an oligopoly that is subject to government regulation.

It cannot be a monopoly because there is more than one firm. The presence of the regulating government body is a "red herring", it distracts from the main point- there are multiple firms.

It does not appear to be competitive because:

Four is subjectively few firms
Implicitly, regulation would be entirely pointless in a competitive market.
One firm has managed to differentiate themselves by adding air conditioners. Firms in perfect competition cannot be differentiated from one another.