Marshallian demand help Consider a consumer with a Cobb-Douglas utility function over three goods of the form $u=(x_1+z_1)^\alpha \cdot (x_2+z_2)^\beta \cdot (x_3+z_3)^\gamma$ with $\alpha, \ \beta, \ \gamma, \ z_1, \ z_2, \ z_3>0$, and $\alpha+\beta+\gamma=1$. The consumer has income $m$, and pays prices $p_1, p_2, p_3$ for the three goods.

Find the Marshallian demands for goods 1, 2 and 3 for this consumer, using the
Lagrangian method.

Hint: I won't give a full answer since this is a very specific question, but I can give you a long outline here. A Lagrangian is set-up with an objective function (in this case utility) plus or minus (depending on Kuhn Tucker constraints, which we won't worry about) a constraint function (in this case, income) which is set to zero and then multiplied by the Lagrange multiplier, which we will denote $\lambda$.

So we set up the Lagrangian function. (Plug in the utility function yourself.)

$$\mathcal{L} = u(x_1, x_2, x_3) - \lambda (p_1x_1 + p_2x_2 + p_3x_3 - m)$$

Solve for the partial derivatives $\frac{\partial \mathcal{L}}{\partial x_1}$, $\frac{\partial \mathcal{L}}{\partial x_2}$, and $\frac{\partial \mathcal{L}}{\partial x_3}$ and set them equal to zero (You can also solve for the partial of $\lambda$, but it just gives you back the budget constraint, which I skip to). For example, you get for your first derivative:

$$\frac{\partial \mathcal{L}}{\partial x_1} = \alpha (x_1 + z_1)^{\alpha - 1} - \lambda p_1 = 0$$

Once you solve the three partials, the easiest thing for you to do is solve for $\lambda$ with respect to the other variables for all three partials, and then figure out how to substitute them into the budget constraint. You should solve for each $x$ in terms of the prices, maybe some $z$'s, the greek letters $(\alpha, \beta, \gamma)$ and income $m$. (I tried working some of it out; enjoy your algebra.)


Notation? Inverse of demand function I'm looking through The Investment Decisions of Firms by S.J. Nickell, and I've come across some notation that I don't quite understand. Any clarification would be very welcome.

Nickell first assumes that firms face a downward sloping demand curve:

$$
z\{p(t)\}\beta(t) \textrm{ where } z_p<0.
$$
He says:

  The demand curve itself thus has a constant shape defined by
  $z\{p(t)\}$ and is shifted up and down by the time function
  $\beta(t)$.

So $z$ is some kind of functional form whose first derivative in $p$ is negative.

Now comes the part that confuses me. He assumes that firms adjust output according to demand.

  As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving
  for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$

How did he get this? $F$ equals quantity demanded. Fine, it's the assumption. But why is $p$ the inverse of $z$?

Then he derives these partial derivatives, which are also confusing me because he brings $z$ back in:

  $$
\frac{\partial p}{\partial K} = \frac{F_K}{z_p \beta(t)} \quad\textrm{and}\quad
\frac{\partial p}{\partial L} = \frac{F_L}{z_p \beta(t)}.$$

Like I said, any help is most welcome. I think what's mostly confusing me is the inverse function bit. I can see how he gets the partial derivatives otherwise.

Thanks.

As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving
  for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$

I am pretty sure that what is meant here is that there is a $p(t)$ trajectory for which
$$
z\{p(t)\} = F\{K(t),L(t)\} / \beta(t)
$$
(see first equation)

Given the function $z$, this depends only on the right hand side of the equation, hence let us denote the trajectory $p(t)$ as a function of this, i.e.
$$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$
A different notation would be
$$
p(t) = z^{-1}\left( F\{K(t), L(t)\}/\beta(t) \right).
$$
This second notation is useful because all the partial derivatives follow from this notation and the chain rule.


When can a person be called rich? What is the common definition for economic richness of a person?
When can you say John Doe (hypothetical name) is rich (objectively)?

Strictly economically speaking, no philosophical debate.

There is no commonly-accepted definition. Different countries and different organizations within a country will have different definitions.

For example, the African Development Bank defines the rich to be those with "per capita consumption levels" of >\$20 per day (2005 PPP).

That would be around or below the poverty line in most developed countries. For example, the 2018 US federal poverty line for an individual is \$12,140 per annum or \$33.26 per day.


Calculating Per Capita This is in reference to https://fred.stlouisfed.org/series/M2
My question is how can i calculate the M2 value on per capita basis? Do i just divide it with the population by year?

Yes if you would want to calculate it per capita basis then you would just divide by population of the country/monetary union for which you are calculating it in the same time period.
Per capita literary means per head, and whenever any quantity is calculated per capita terms whether its gross domestic product, gross national product, gross value produced, income and so on they are calculated as a ratio of total of the aggregate to population.
This being said I never seen this being done in research and I could not find any reference in google scholar as well. I was able to find this blog that does it although not for any research purpose. When we calculate aggregates on per capita basis its usually for some reason, for example per capita income gives us better idea about levels of material welfare in a country than just looking at total income, with money supply that is not really the case.


Confusion about multiple information Sets I'm confronted with the following situation:

First of all nature decides which of two games is played. Then each of two players make a simultaneous move, but only player I knows which game is played.

How can I display this information in extensive form?

The root will be the nature, which decides whether Game I or Game II is played. Now we have two additional layers for each game. Now I would use an information set to display that the moves are taken simultaneously, but how can I display that only player I knows about which game is played?

You could represent the game in extensive form like this:

The dashed lines enclose player 2's information set. This encompasses all of player 2's nodes because player 2 observes neither nature's nor player 1's choice. Player 1's information sets are the two singleton nodes because player 1 knows which branch nature has chosen.


Proving consistency of quantile regression estimators I have asked the question on the statistic section on stack exchange, but no one was able to give me an answer. I think this is actually is a question that touches econometrics so I am going to ask it here (I would like to use the answer to this in an econometric context):

I am a bit confused. The assumptions that have to be fulfilled so that OLS estimators are consistent (and efficient) are fairly straightforward.

I am currently trying to prove consistency of quantile regression (QR) estimators.

I have found the following lecture notes: https://eml.berkeley.edu/~powell/e241a_sp10/qrnotes.pdf

There are 4 assumptions listed (page 3) for the proof of consistency of the QR estimators. The first one being that the data $x_{t},y_{t}$ given $t=1,2,..,n$ has to be i.i.d. (independent and indentically distributed).

In the case of OLS the data only has to be covariance-stationary. To my understanding the above mentioned assumption of i.i.d. data rules out the possibility of for example autoregressive processes, since in that case the data points are not independent from one another. This in turn puts quite restrictive boundaries on the possible applications of QR.

Am I missing something here? Could someone clarify the assumptions for consistency of the QR estimators for me?

Thank you!

I can give you literature. An authoritative book on the subject is Koenker, R. (2005). Quantile regression. Cambridge university press.. In there we read (p.126)

  4.6 QUANTILE REGRESSION ASYMPTOTICS UNDER DEPENDENT CONDITIONS    Relaxation of the independence condition on the observations is
  obviously possible and offers considerable scope for continuing
  research. Bloomfield and Steiger (1983) showed the asymptotic
  normality of a median autoregression estimator for a model in which
  the observations were assumed to be stationary and ergodic martingale
  differences. Weiss (1991) considers models under α-mixing conditions.
  Portnoy (1991) considers a considerably more general class of models
  with “m-decomposible” errors, a condition that subsumes m-dependent
  cases with *m* tending to infinity sufficiently slowly... More
  recently, Koul and Mukherjee (1994) and Mukherjee (2000) have
  considered the asymptotic behavior of both the primal and dual
  quantile regression process under conditions of long-range dependence
  of the error process. This work relies heavily on the theory of
  weighted empirical processes developed by Koul (1992).

The section continues with a presentation of the properties of Quantile regression for AR, ARMA, and ARCH models.


Is it possible to tell anything about market structure if I have annual price data of the industries? I have three goods:

beer
carp (fish)
museum ticket

Is it possible to tell anything about market structure if I have annual price data of the industries? Here's the time-series plot on a single graph:

I would like to give advice whether a firm should entry the market or not. Is is it possible to decode price data like "the incumbent firms must have been playing a Bertrand's price competition game at the carp market"? Or do I need further details to say anything about the market structure?

Is it possible to apply IO theory if I have only price data of an industry? I really need some clues, please.

That that we have linear supply and demand functions, pretty much the simplest case such that quantity supplied of good $i$ is:
$$ Q_{s,i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t}$$
and quantity demanded is:
$$ Q_{d,i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$

In equilibrium  $Q_{i,t}=Q_{d,i,t}=Q_{s,i,t}$ so substitute and solve:
$$ Q_{i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$
$$ P_{i,t} = \frac{\alpha_{0,i,t}-\beta_{0,i,t}}{\beta_{1,i,t} - \alpha_{1,i,t}}$$

As you can see, the price depends on supply and demand. Unfortunately, shifts to the level or slope of either the supply or demand functions both affect the price. Therefore, in  general, you can't tell anything about the supply function from the behavior of the price. And if you can't tell anything about the supply function I don't see how you'd know anything about the IO of the producer industries.


A doubt on FIML assumptions In Hayashi's Econometrics, page 529, he states one of the assumptions we need for the FIML estimator.

My doubt is in the third line of point 1). He says that the vector $(y_{t1},...,y_{tM},\mathbf{z}_{t1},...,\mathbf{z}_{t1})$ are elements of $(\mathbf{y}_t,\mathbf{x}_t) $. How is that possible? If $\mathbf{y}_t=(y_{t1},...,y_{tM})'$, then it means that $\mathbf{x}_t=(\mathbf{z}_{t1},...,\mathbf{z}_{t1})$.

I don't think I'm understanding the 'english' here.

Any help would be appreciated.

I think I understand now. If we look at example 8.1 on the previous page, the $y_{tm}$ variables may be the same, which would allow for some of the $\mathbf{z}_{tm}$ variables to be endogenous. And we pool all those endogenous variables (both the $y_{tm}$ and of the $\mathbf{z}_{tm}$) in one vector,  $\mathbf{y}_{t}$, and all the remaining ones (exogenous) in the $\mathbf{x}_{t}$ vector.


Estimating the second derivative of function from optimizers Consider the following optimization
$$x^*(s) = \max_{x\in X} \big(\,f(x)-sx\,\big)$$
where $f$ is assumed to be a strictly concave function and $X$ is an interval constraint, e.g $X = [0,b]$. We do not know the exact function $f$.

Assume that we can provide a parameter $s$ and obtain the corresponding (unique) optimizer $x^*(s)$.

My goal is to estimate $f''(x)$ from a collection of parameters $(s_1,s_2,\ldots)$ and associated optimizers $(x^*(s_1),x^*(s_2),\ldots)$.

How would one go about doing this? I was thinking of applying the envelope theorem or using finite differences in some way but I'm not quite sure how to proceed.

The first order condition of the maximization problem is
\begin{equation}
f'(x)-s=0\iff f'(x)=s
\end{equation}

We can then replace $x$ by $x(s)$ because this is the optimal value given $s$. Since this is true for every $s$, we can differentiate with respect to $s$ which yields
\begin{equation}
f''(x(s))x'(s)=1
\end{equation}

Which can be rewrite as
\begin{equation}
f''(x(s))=\frac{1}{x'(s)}
\end{equation}

Then $x'(s)$ can be estimated by finite differences which would give $f''(x)$.


Difference in interpretation of log-lin coefficients with binary variable Consider the model:
$$y_i=\beta_0+\beta_1D_1+\beta_2D_2+u_i$$
Where $D_1=\{0,1\}$ and $D_2=\{0,1\}$ are binary (dummy) variables, and $y_i$ is a continuos variable in levels.
In this model, for example $\beta_1$ is interpreted as the gap between the category when $D_1=1$ and $D_1=0$. Besides, this gap is in absolute terms, say in terms of the units of the $y_i$ variable.
Now, consider the similar model but dependent variable is in logs:
$$\ln (y_i)=\beta_0+\beta_1D_1+\beta_2D_2+u_i$$
I would thought that, for example, $\beta_1$ is the gap between the mentioned categories but in relative terms, say a proportion or percentage. Say in average category $D_1=1$ receives $\beta_1*100$ percent more (or less) of $y_1$ than $D_1=0$ in average.
But I found another approximation in my class for this gap, and it takes the gap starting with the fact that the relative gap:
$$Rel. gap\; between\; [D_1=1]\; and\; [D_1=0]=\frac{y_1-y_0}{y_0}*100\% = \left(\frac{y_1}{y_0}-1\right)*100\%$$
Where $y_1=y(D_1=1)$ and $y_0=y(D_1=0)$
Taking this into account we can get:
$$\ln (y_1)-\ln(y_0)=\beta_1 \implies \frac{y_1}{y_0}=e^{\beta_1}$$
Replacing this into the relative gap equation specified earlier we get:
$$Rel. gap\; between\; [D_1=1]\; and\; [D_1=0]=\left ( e^{\beta_1} -1 \right)*100\%$$
My question is: What is specifically the difference between those two approximations? And even if one of those perhaps is wrong, because the resulting gaps that you can obtain are way different. Moreover, which one should one use with academic purposes.
Thanks!

As you can see from the plot, they are not the same. Infact, the 'rule of thumb', i.e. your first formula, only holds in approximation. The error gets larger for large coefficient values.


What are the applications of complex numbers in modern economics? I wonder what the most interesting applications of imaginary numbers in mainstream economics are. I read about the unit root test (Dickey-Fuller test) in time-series analysis; and I studied System Dynamics about phase portraits that were determined by complex numbers. So maybe the maths of these numbers are necessary for econometrics and for more sophisticated macroeconomic modelling.

To sum up, what are the most important application of complex numbers in economics? Could you give me some examples, models relevant to complex numbers?

I will be grateful if you can send me this information.

You might refer to this answer to a question similar to yours.

Financial economics often use the complex plane to visualize duration and the internal rate of return, to which this resource proves very helpful.


Federal Reserve capital injections What were the specific conditions that the Federal Reserve attached to the capital injections forced upon the major Wall St. banks during the Great Recession? How and why do they differ from normally legal and binding regulations?

First, two clarifying points:

The capital injection program, known as the Capital Purchase Program (CPP), was authorized by Congress as a part of the Troubled Asset Relief Program, and run by the US Treasury, not the Federal Reserve. However, the Federal Reserve Board and in particular the Federal Reserve Bank of New York (FRBNY, primary regulator of the major dealer banks) were both involved in the design of the program.
The program was voluntary, not mandatory, though the heads of the nine largest institutions were certainly strongly encouraged to participate, as described by Timothy Geithner, then-President of the FRBNY, in Stress Test:

  As we developed our scripts, I pushed hard to make the capital investments sound as close to mandatory as possible. We couldn't force participation, but we could make it sound inevitable. I expected some firms to resist taking new capital with government strings attached. But in a financial crisis, no firm is ever as well capitalized as it thinks. Lehman thought it had plenty of capital until it didn't.

Having made that clear, we can turn to your question, which is about the "government strings" that Geithner mentions. These took three primary forms:

Dividends and warrants. These were in the authorizing law. Quoting the Treasury's CPP page:

  Most financial institutions participating in the CPP pay Treasury a five percent dividend on preferred shares for the first five years and a nine percent rate thereafter. In addition, Treasury received warrants to purchase common shares or other securities from the banks at the time of the CPP investment. The purpose of the additional securities was to enable taxpayers to reap additional returns on their investments as banks recover.

The ability to nominate up to two members to the board of directors of a participating institution, should the institution miss six quarterly payments to the Treasury, per the CPP contracts.
Limits on executive compensation. These were in the authorizing law that Congress passed (titled the "Emergency Economic Stabilization Act of 2008"), and they're well-described on the law's Wikipedia page:

  If the Treasury purchases assets directly from a company, and also receives a meaningful equity or debt position in that company, the company is not allowed to offer incentives that encourage "unnecessary and excessive risks" to its senior executives (that is, the top five executives). Also, the company is prohibited from making golden parachute payments to a senior executive. Both of these prohibitions expire when the Treasury no longer holds an equity or debt position in that company. The company also is given "clawback" permission; that is, the opportunity to recover senior executive bonus or incentive pay based on earnings, gains, or other data that proves to be inaccurate.

  If the Treasury purchases assets via auction, and that purchase exceeds $300 million, any new employment contract for a senior officer may not include a golden parachute provision in the case of involuntary termination, bankruptcy filing, insolvency, or receivership. This prohibition only applies to future contracts; golden parachutes already in place will remain unaffected.
  In either scenario, no limits are placed on executive salary, and existing golden parachutes will not be altered.


How buying bonds indirectly from the government prevents the central bank from financing government deficit? From Krugman's macroeconomic textbook (highlighting is mine):

"In an open-market operation the Federal Reserve buys or sells some of
the exist- ing stock of U.S. Treasury bills, normally through a
transaction with commercial banks—banks that mainly make business
loans, as opposed to home loans. The Fed never buys U.S. Treasury
bills directly from the federal government. There’s a good reason
for this: when central banks lend directly to the government, they are
in effect printing money to finance the budget deficit. As we’ll see
later in the book, this can be a route to disastrous levels of
inflation."

But how does this indirectness prevent the central bank from printing money to finance the budget deficit? If the government can order the central bank to buy its bonds, then it can also order the central bank to buy its bonds indirectly, during its open market operations. I fail to see how this indirectness of buying bonds serves as a safeguard against abuse of of money printing press by the government.

Even though central banks are part of the government they are in most developed countries independent, this is similar to way how courts operate. For example US government has Judicial Branch, where the supreme court justices are part of the government and appointed by president (the same way as chairman of Fed is) but this does not mean government can order supreme court or Fed around. Government could wait for Fed's chairman term to expire and appoint new one but it is expected that these appointees are technocrats and thus its not guaranteed they will act according to party line (i.e. line of the party currently in power).
In principle government could ask ('order') Fed to buy its bonds, however Fed could (and should) say no if it conflict with its dual mandate of price stability and full employment. Or they could say yes if they believe that is consistent with their mandate. This being said as Krugman points out in his book Fed wont go this route in normal times in order to keep it's 'distance' from the government. This really has no economic rationale but rather political one. The more entangled Fed (or any other central bank) gets with the government the more likely it is that there will be pressure for politicization of the institution.
This idea is made explicit by Fed in its FAQ, where it states:

Conducting transactions in the open market, rather than directly with the Treasury, supports the independence of the central bank in the conduct of monetary policy.


Macroeconomic model of the open economy - NCO and loanable funds Can anyone explain to me why a capital flight from a country increases the demand for loanable funds in that country. Here's the standard explanation.
https://policonomics.com/lp-open-economy-loanable-funds/
In this explanation the NCO shifts left (lower NCO) & the demand curve for loanable funds goes right (increases at every level of r).
I get I+NCO = demand curve. But if I talk it through, then my mental modle always comes up with -
capital flight, no one wants the currency  less people want the loanable funds so at any R demand is less, hence a shift to the left of the  demand curve - not right.
I have a second problem scenario which I can't make fit the model. Gvt encourages investment throught tax credits. Demand curve must shift right in the loanable funds market, but surely NCO would shift to the left (or is therejust a move leftwards along the curve?)...thus according to the textbooks shifting the D loanable funds left not right.
None of the textbooks really explain why a capital flight means people want more "loanable funds".
And if a country becomes more attractive to invest in the demand for loans in that country must go up, not down.
I just can't get my head around this 3 market model.
What's wrong with my thinking.

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

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

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

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

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


Are there real world examples of a shoe event horizon? This is a famous quote from Douglas Adam's The Restaurant at the End of the Universe:

  Many years ago this was a thriving, happy planet – people, cities, shops, a normal world. Except that on the high streets of these cities there were slightly more shoe shops than one might have thought necessary. And slowly, insidiously, the number of the shoe shops were increasing. It’s a well-known economic phenomenon but tragic to see it in operation, for the more shoe shops there were, the more shoes they had to make and the worse and more unwearable they became. And the worse they were to wear, the more people had to buy to keep themselves shod, and the more the shops proliferated, until the whole economy of the place passed what I believe is termed the Shoe Event Horizon, and it became no longer economically possible to build anything other than shoe shops. Result – collapse, ruin and famine.

Are there real world examples from history where a society's economy has become so mono-industrialised that it collapses? Does this situation have an alternative (more serious) name in economics?

I'm not sure if this will be a constructive question. A potential edit could be to ask if something like Ireland's potato famine is an example of the shoe event horizon.

The dutch disease is when an economy is so dominated by a single export industry that the success of that industry drives up the currency and makes the exports of other industries un-competitive. This drives some of the firms making other goods out of business.

Continuing this cycle can lead in the short term to even greater concentration of the economy in that industry.

If later the export industry experiences a significant shock, the broader economy also experiences that shock. This happens for two reasons:

In a diversified economy some firms are experiencing positive shocks when others experience negative ones. In a concentrated economy it is less likely that the other industries will collectively experience an sufficient offsetting shock.
A negative shock to the primary exporting industry is likely to depreciate the currency which should help other industries but shrinking capacity from the expansion will diminish other industries' ability to generate sufficient exports to offset the shock.

Further, if the industry responds by boosting output (like sometimes happens when oil prices fall and producers become desperate) this can make export prices fall further still, worsening the shock.


Yield and interest rate? Are they the same thing?

Is yield the annualized return rate?

Why when yield rise, yearly return increases but price falls?

They are related, but not the same thing.

The yield associated with an asset is the ratio of the (usually annual) return produced by that asset to its price: $$\text{Yield (%)}=\frac{\text{Annual Return (\$)}}{\text{Price of Asset (\$)}}.$$

Simple case: when yield = interest rate

An interest rate is an example of a yield. For example, suppose a bank account pays an interest rate of x% p.a. That means that \$100 deposited in the account today would become \$(100+x) in a year's time. The annual return is therefore \$x. We can plug these numbers into our formula to calculate the yield: $$\text{Yield}=\frac{\$x}{\$100}=x\%.$$ In this example the yield is just the interest rate, so the two coincide.

More complicated case: yields and asset prices

Things get a little more complicated when the interest rate is fixed but the price of the asset can change over time.

For example, the government issues bonds when it needs to borrow money. Each bond has an initial price and a set annual return (known as the coupon payment) that does not change over the entire life of the bond. Suppose that we have a bond that initially costs \$100 and has a coupon payment of \$5 (meaning that you get \$5 for each year that you hold the bond). We can check the bond's coupon yield: $$\text{Coupon Yield}=\frac{\$5}{\$100}=5\%.$$
This is, again, just like an interest rate.

Now, though, suppose that banks start offering an interest rate of 10% on an ordinary savings account. Suddenly, your bond's 5% doesn't look so great! But the \$5 coupon payment can't change—it is permanently fixed when the bond is created. People are therefore going to want to try to sell their bonds to get their money back so that they can put it into a bank instead. As per normal supply & demand logic, lots of people selling bonds is going to make the price of bonds fall. So now you can buy a bond for, say, \$80. That means the yield has changed!
$$\text{Current Yield}=\frac{\$5}{\$80}=6.25\%.$$
Alas, people are still going to be reluctant to buy the bond because it still has a lower yield than the bank account—so the price is going to have to fall even further. Eventually, the price of the bond falls to \$50, at which point $$\text{Current Yield}=\frac{\$5}{\$50}=10\%.$$ Now the bond and bank account have the same yield (meaning that someone with money to invest would get the same return from either). People are now indifferent between holding the bond or a bank deposit and equilibrium is restored.

In practice, things are a little more complicated because people compare the returns on bonds to all kinds of assets—stocks, real estate, etc. But the principle is the same: the price of the bond adjusts to ensure that the yield on the bond is in keeping with the risk-adjusted return available elsewhere.

A final note: yields where there are no interest payments

The last important reason why we distinguish yields from interest rates is that some assets don't have any interest rate to speak of, but it nevertheless makes sense to talk of a yield. One example is stocks—there is no interest payment on such assets, but the board may decide to return some of the firm's profits to investors in the form of a dividend. A dividend is an amount that is paid to investors for each share that they hold. We can use the dividend to calculate a yield just as though it were an interest rate:$$\text{Dividend Yield}=\frac{\text{Dividend (\$)}}{\text{Stock Price (\$)}}.$$


Ordinally Separable Utility Representation Let $X_i$ be a separable, compact, Banach space.
Definition: A weak order $\succeq$ on $X=\prod_{i=1}^NX_i$ has an ordinally separable representation if there exists $u_i: X\rightarrow \mathbb{R}$ and $W:\prod_{i=1}^Nu_i(X_i)\rightarrow \mathbb{R}$ strictly increasing in each coordinate such that \begin{equation}
U(x_1,...,x_N)=W(u_1(x_1),...,u_N(x_N))
\end{equation}
represents $\succeq$.
Definition: Given a weak order $\succeq$, an index $i\in I$ is separable if for all $x,y,z,z'\in X$, $x_iz\succeq y_iz\iff x_iz'\succeq y_iz'$, where $x_iz$ means $x_i$ in the i-coordinate of a vector $x\in X$ and z otherwise. A weak order satisfies singleton separability if all $i\in I$ are separable.
In Decision Theory by Tomasz Strzalecki chapter 3,, there is the following theorem without proof:
Theorem: Suppose that $\succeq$ is a weak order on $X=\prod_{i=1}^NX_i$. The ordering $\succeq$ satisfies singleton separability and continuity if and only if it is has an ordinally separable representation with continuous functions $u_i$ and $W$.
He claims that the proof is quite hard. Does anyone know how to prove this or have a reference?
Remark: The assumption that $X_i$ is separable, compact and Banach is not essential here, but let's just make the spaces $X_i$ as nice as possible for the time being.

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$  is a connected separable topological space by a theorem of Eilenberg.  A proof of Eilenberg's theorem is given in Debreu's book Theory of Value. Debreu assumes the domain there to be Euclidean, but the proof easily generalizes to the present setting. One can also use the assumption that each $X_i$ is a second-countable topological space (for metrizable spaces implied by separability). Debreu showed that continuous utility representations exist in that setting too, but the proof is considerably harder.
Define $\succeq_i$ on $X_i$ by $x_i\succeq_i y_i$ if $x_iz\succeq y_iz$ for some $z$. Clearly, this defines a continuous weak order and the $z$ used does not matter by singleton separability. There exists a continuous utility function $u_i:X_i\to\mathbb{R}$.
Next, we can show that $x_i~\sim_iy_i$ for $i=1,\ldots,N$ implies $x\sim y$. By repeated applications of singleton separability we get $(x_1,y_2,y_3,\ldots,y_N)\succeq (y_1,y_2,y_3,\ldots,y_N)$, $(x_1,x_2,y_3,\ldots,y_N)\succeq (x_1,y_2,y_3,\ldots,y_N)$, and so on, until we have $(x_1,x_2,\ldots,x_{N-1},x_N)\succeq(x_1,x_2,\ldots,x_{N-1},y_N)$. By transitivity, $x\succeq y$. Similarly, $y\succeq x$, and, therefore $x\sim y$.
Let $U=\prod_{i=1}^N u_i(X_i)$, a product of (possibly unbounded) intervals. Define $\succeq^*$ on $U$ by $u\succeq^* u'$ if for some $x,y\in X$ one has $u=\big(u_1(x_1),\dots,u_N(x_N)\big)$, $u'=\big(u_1(y_1),\dots,u_N(y_N)\big)$, and $x\succeq y$. By what we have just shown, $\succeq^*$ contains, together with $u_1,\ldots, u_N$, the same information as $\succeq$. It is also obvious that $\succeq^*$ is strictly monotone on $U$. If a utility representation $W:U\to\mathbb{R}$ of $\succeq^*$ exists, it must be strictly increasing. We are done if we can show a continuous such representation exists. Using any of the mentioned utility representation theorems, it suffices to show that $\succeq^*$ is continuous.
So let $u\succ^* u'.$ There exist $x,y\in X$ such that $u=\big(u_1(x_1),\dots,u_N(x_N)\big)$, $u'=\big(u_1(y_1),\dots,u_N(y_N)\big)$, and $x\succ y$. Since $\succeq$ is continuous and by the definition of the product topology, there exist open neighborhoods $V_i\ni x_i$ and $W_i\ni y_i$ such that $x'\in\prod_{i=1}^N V_i$ and $y'\in\prod_{i=1}^N W_i$ implies that $x'\succ y'$.  Let $V$ be the interior of $\prod_{i=1}^N u_i(V_i)$ and $W$ the interior of $\prod_{i=1}^N u_i(W_i)$. Then for $u''\in V$ and $u'''\in W$ one has $u''\succ^* u'''$, so $\succeq^*$ is, indeed, continuous.


Does "I" in Y = C + I + G + X - IM refer to the domestic investment? We know the national income identity: $Y = C + I + G + X - IM$.

From many resourses, $I$ refers to domestic investment. For example in this link: https://opentextbc.ca/principlesofeconomics/chapter/23-4-the-national-saving-and-investment-identity/

However, in Mankiw's book, it says that $I= I^d + I^f$, the summation of domestic investment + foreign investment, as shown in

And actually in later pages of Mankiw's book, "I" is referred to domestic investment at many places also.

What does "I" accurately refer to?

I see your confusion that could also apply to $C$ and $G$ because Mankiw starts from
\begin{equation} Y= C^d+I^d+G^d+X~~~~~  (1) \end{equation}

$C^d$, $I^d$, and $G^d$ have been subjected to the same ``bit of manipulation'', which is to add and substract the foreign part such that we get the same equality (1).

So the idea here is just that $I$, after the manipulation, is essentially equal to $I^d$, the domestic investment, because $I^f$, the foreign portion, has been substracted from $I$ and appears now in import expenditures ($IM$). In other words $I$, after manipulation, is reinterpreted as domestic investment.

\begin{equation} Y= C + I +G +X -IM ~~~~~  (2) \end{equation}

I think it safe to say that calling $I$ domestic investment is here an "abuse of language"!

The most important is that identities (1) or (2) are consistent with the way we calculate GDP using total value added.


Proving there are infinitely many strictly dominant strategies My attempt:

$(0, 1/3, 2/3)$ works as a strategy that strictly dominated $L$.

To prove that there are infinitely many of them:

Let $A_n=(0,1/2+\epsilon_n,1/2-\epsilon_n)$ where $\epsilon_n=1/n$ for $n=1,2,3,...$. We can see that for $n > 10$, $A_n$ strictly dominates L. So, since $n$ diverges, there are infinitely many $n$ for which $A_n$ is a strictly dominant strategy.

I chose $n> 10$ since $2<2.5-5\epsilon_n \Leftrightarrow 0,1> \epsilon_n$ and $1/n<0,1$ for $n>10$. The other inequalities don't require $\epsilon_n$ to be restricted.

Would this be a correct way to prove this?

That is a fine proof. Though you probably want to show your last statement "The other inequalities don't require $\epsilon_n$ to be restricted". It is fairly obvious, but won't hurt to be super clear.

Perhaps a simpler proof would be that any convex combination of the two strategies already found will also work. That is $\lambda(0,1/2,1/2)+(1-\lambda)(0,1/3,2/3)$ also strictly dominates $L$ for any $\lambda\in(0,1)$. So, in fact, there are not only countably many strategies that strictly dominate $L$ but uncountably many of them.

(Idea of the proof:) Notice that this holds because strict dominance is given by a set of inequalities (3 in this example) so if you have two strategies that satisfy those inequalities, any convex combination of these strategies will necessarily also satisfy the inequalities since expected utility is linear in probabilities. I'll leave the formal proof for you to write, but hopefully the logic is clear.


Determining the Relative Price of Trade I'm pretty sound with the concept behind comparative advantage, but still don't get how to calculate the exchange rate in terms of quantities of each products being exchanged, e.g. 1 apple for 2 cars etc.

Specifically: when two counties, each having two separate comparative advantages in producing two separate goods, wish to exchange these goods, how will the exchange quantity/rate be calculated?

If I understand correctly the question, the trick to calculate the equilibrium (relative) price of trade in the Ricardian model of comparative advantage is to specify demand once the opportunity costs have been determined. Prices of internationally traded goods, like other prices, are determined by supply and demand.

Consider two countries, two goods (computers and textiles), and given unit labor requirements such that in equilibrium the world price of computers will be between 0.5 and 2 textiles per computers. The opportunity costs define the bounds of equilibrium relative prices of trade (0.5 and 2), while the structure of demand determines the equilibrium relative price (p).

For instance, p=1 can be an equilibrium price such as one country will specialize in computers and the other in textiles. This price will be determined by the interaction of the relative supply and the relative demand of computers and textiles. This interaction is discussed in the Chapter 3 of International Economics: Theory and Policy of Krugman, Obstfeld and Melitz. An interesting video with a concrete example can be found here.


Human Capital and Investment into Children Famous labor economist James Heckman made the argument that given money for investing into educating people, we should invest into young children, and pretty much only that age group, because they have the most periods of time to compound human capital over time, and arguably get the most benefit from it.

http://jenni.uchicago.edu/papers/Heckman_Masterov_RAE_2007_v29_n3.pdf

Obviously it's a pretty strong argument, and I was wondering what sort of research in the literature you might find important that takes the stance against this sort of idea. My intuition tells me that we'd want to smooth investment into all sorts of age groups of people for some sort of externality reason.

You ask for studies which I don't have, but I can think of a few reasons:

Diminishing returns - if a child has a decent home and parental support you can give them a good education at reasonable cost. To get a child from a broken home to the same point requires much more intervention - and cost.
Moral hazard - there are many hard-working families just above the breadline. The parents go to great lengths to ensure their children have a decent home. It may be that they get no support for all this effort - but if they simply stop making an effort, the child then gets massive support from school.
Need for high achievers - for society to achieve (cure cancer, explore space, win wars, etc.) there is a need for high achievers. If the education system focuses entirely on getting under-achieving children up to average, this comes at the cost of high achievers.

I'm from the UK, where there is an extensive welfare system and extra investment in education in deprived areas, called the Pupil premium. Although, not nearly as much as suggested by the paper you reference.

Many people do not support the pupil premium. There is a large "squeezed middle" in the UK, not poor enough for extra support, not rich enough for private schools, and this group sees extra support for deprived areas as unfair.


Why does the Balance of Payments have to (more or less) equal zero It seems to me that the non-reversible transactions included in the current account, and the reversible transactions included in the capital and financial account, are unrelated, and should therefore be able to fluctuate independent of one another.

Imagine an economy where each person bought a huge number of items online from overseas. The balance on current account would be negative... but the capital and financial account unaffected, meaning they no longer add to zero.

What am I missing?

It was also really strange for me at first. I asked from many people who might happen to know the answer. However, they were all not 100% satisfactory. I think, Kenny LJ's answer is 99.5% correct but i feel that it's still missing something. My explanation is to add to his/her answer something i think very very subtle, but hopefully important, one.

Think you have an organization and record the purchase in your balance sheet. You will purchase the good as an asset with an debit entry. Also, you will credit your cash with the same amount.

When these two entries come to the BOP, the asset will be in CA and the cash will be in FA. Why the money should be recorded in FA? When a country starts to hold a foreign currency /it was well before and independently of your purchase/, it's recorded in BOP as an investment to the foreign country. Now when you pay with the money of the foreign, you stop your investment and make it a physical good. So your stake with the other country doesn't change, but its form. So the balance is not affected in the end.

If you work with other examples, it'll become clearer and clearer. Let me give some.

Imagine you export something to a foreign country. How will it be 0 in BOP? An export debits your goods and increases the CA. If you get a foreign currency from your foreign partner for the payment, it'll be credited as investment to the other country as you hold its currency which will credit or decrease the FA.

Imagine you borrow from a foreign country, it'll be an increase in FA as it's a foreign investment to your country. But in which currency will you hold your investment proceedings? You may start holding that country's currency. This will be recorded as your country's other investment to the other country which will be decrease in FA.

If you sell above investment money into your country's central bank and your central bank holds it as cash, your country's other investment's increase will be undone which will result in positive FA from the investment /it'll look like the foreign country invested positive amount in net/. But the official reserve account of the country will increase with the same amount which will also make the BOP 0.

I hope it helps.


How Are The Interest Rates Affected By Inflation Changes? I have been reading about how an increase in interest rates causes inflation to decrease as there is less money in circulation. However; I have never seen an article explaining how interest rates are affected by rising or falling inflation (The opposite).Could someone explain to me how interest rates are affected by a changing inflation rate?

I would note this similar question on interest rates and inflation.
I would summarise the answer there as follows.

Short-term interest rates are set by the central bank (in most developed countries; things are different in places with currency pegs).
Longer-term bond yields should roughly equal the expected path of the short-term rate (plus a term premium). One can debate this point, but it is a fairly standard assumption.

The answer therefore ends up: what is the effect of inflation on how the central bank sets policy? This is the subject of monetary economics, and is an open-ended topic that cannot easily be answered here.

In practice, we often see the central bank raising rates as the expected rate of inflation rises. Since they are worried about expected inflation, they do not automatically react to every wiggle in measured inflation. For example, a central bank may not react to an oil price spike. (Reactions varied during the 2008 oil price spike, with the ECB hiking, and the Fed not.)


Why do we need base money at all? For example, we could imagine an arrangement like this:

There are currently \$0 in the economy.
A business takes out a \$50 loan. The bank sets there bank account to \$50 - \$50 debt = \$0 net.
The business now pays it workers \$40 to make stuff (through the bank).
The workers in turn pay back \$45 to the business to buy stuff (through the bank).

(We can imagine there being a lot more steps)

Etc...
We can use bank notes instead of currency

What is the purpose of base money then?

TL;DR You have defined a de facto currency with that bank as a central bank.  It is generally considered better to have everyone using the same currency.

In theory, what you suggest can indeed work, and has been used for period of time in various places.  Legally speaking the Bank of England doesn't mandate a minimum reserve, and so in theory a bank could operate with zero reserve (The EU mandates 2%, the US 10%).  In practice each bank must agree their own reserve ratio with the BoE, and the bank would likely laugh if someone said "zero".

You system in essence works, and all you have done is created a currency tied to that bank (lets call it Bank Z).  The problem is that no-one can actually pay anyone who isn't a customer of Bank Z (for more on electronic transfers see: https://economics.stackexchange.com/a/6170/119).  Paying someone who banks with Bank Z is easy, just move the money from your account to theirs, but if they don't have an account we can't give them any money - because we don't have any.

What we could do is give them an IOU - "if you open an account with us, we will credit it with Z\$X".  This would be a bank note for Bank Z.  Of course the value in that note is tied to our faith in that bank.

Suppose now we imagine Banks X, Y and Z operating in our economy.  What happens when Xavier (who banks at Bank X) want to pay Yvette (who banks at bank Y)?  Well neither bank can change the customer account of the other, so both customers are given IOUs (bank notes) and take them to the other bank.  Should the other bank accept these notes and increase they account accordingly?   Well that depends...

If both banks are both well used then sure, why not.  But suppose instead that 98% of people bank with Bank X, and 1% with bank Y - whose IOU would you rather have?  It is much easier to spend Bank X money, than Bank Y money, so that in itself could cause bank X money to command a premium.

If X\$100 = Y\$101 then clearly I will charge you more Y\$ for fixing your roof that X\$.  This can quickly spiral and before you know if Y\$ are worthless - they have suffered hyper inflation and been inflated away.

You have effectively created a state with multiple currencies, and in such situations, one currency would be expected to fall into disuse and become worthless.  Historically countries have often actively defended their currency from the use of external currencies by imposing limits on how much currency can be brought in.  It can be a real problem if your people are trading using dollars instead of your national currency, for the same reason.  If dollars are more useful, then why use local currency, and so it decreases in value.

Your set up is particularly fragile, because the banks are merely lightly regulated corporation, not the government.  The government can mandate that all government workers are paid in local currency - in effect it can force itself to have a large number of customers.  It is not a coincidence that the central banks are the bankers to the treasury of their respective countries.

As a government this is also a pain - you have multiple currencies, and you don't control any of them.  Your economy is not in your control, which has cause problems in the past.

Gold backed?  Assets?

I deliberately haven't talked about this part, as you don't really address it, but I suspect many will think first about it.  The backing and assets of the bank are really secondary to this discussion.  Their is nothing to stop all our banks saying that their dollars are pegged to the value of gold.  This simply means that you can, on demand, exchange your IOU for gold.  So long as nobody does this then you are fine.  Equally, even if a few do, you can buy they gold with your own IOUs and so that is still fine.

Any assets your bank has, can be used to "back" your currency and add strength to it, even though you have no "reserve" with any central bank.  This will help you.  Bank Y could respond by advertising that it has tons of gold in the vault and owns lots of houses and farmland, so it will be more likely to be able to honour it's IOUs that Bank X can.  This could turn the tide back in your favour, as people lose faith in X\$ and switch to Y\$.

Remember, even if your Bank Z starts with zero assets, it will make a profit in Z\$ over time, which can be used to buy things and so it will accumulate assets with time.  None of these need become liquid reserves.

Examples?

Your case is a bit too extreme, but near examples are worth a look

The biggest example that springs to mind is the Free Banking Era (aka "Wildcat Banking") in the US from about 1837-1862.  Andrew Jackson had "defeated" the Second US central bank and believed that banks could just be allowed to run things on their own.  The average life of a bank in that era was 5 years, and each bank's currency was worth a different amount, depending on the regard with which that bank was held.  These banks did not run without reserve, for the reasons given about, but they were free to choose how much and how to do it.  The only reserve requirement was having enough to convince your customers to use you...

A more modern example is "points".  Plenty of businesses award "points" for various activities, which can be redeemed for goods and services.  If you think about it, these points are like a constrained version of exactly your situations.  Usually points can only be exchanged with the issuer, but if you added the possibility to transfer points from one person to another, then you have a basic currency operating as you described.  The key feature here, is that these points are not backed by anything, and cannot be converted to gold or "hard currency", only airline flight, or groceries or whatnot.


Consumer utility optimization question I am having diffculty with this question:

Can we proceed using Lagrange here since the utility function is convex, or do we have to treat them as perfect substitutes? If latter or any other method, please explain.

My attempt:

$$L= U(X,y) +λ(y-p1-p2)$$.

By using Lagrange, I got these FOCs:
$$ α Ax1^{(α -1)} - λp1=0$$& $$1-λp2=0$$.

So, is it correct to use Lagrange method here?

I'll offer a little guidance to get you moving on the problem your own.

You've set up the Lagrangean incorrectly.

The maximization problem you have looks like:

$$\max \ U(x_1, x_2) \\
\text{s.t.} \ p_1 x_1 + p_2 x_2 \leq y $$

The Lagrangean should read:

$$\mathscr{L} = U(x_1, x_2) - \lambda (p_1 x_1 + p_2 x_2 - y)$$

The Lagrangean you offered had utility as a function of $X$ and $y$, but you can see your utility function does not have $y$ as an argument in it. So this is just a notation error to keep track of. As for the part with the Lagrangean multiplier, you can imagine that if the inequality constraint is binding, then $p_1 x_1 + p_2 x_2 - y = 0$, so in that case, maximizing the Lagrangean is just maximizing utility minus $0$, but it's a special $0$ that helps to incorporate the budget constraint into the problem. (As for the non-binding case, you can think about what is happening to the optimization problem yourself.)

Your first order conditions are correct though. So that tells me you've mostly made some typing errors. To answer your question about whether the Lagrangean is okay here, the Lagrangean will work as long as the optimum is an interior solution. So if you suspect either $x_1, x_2$ will be equal to zero, then you may have to break out Kuhn-Tucker's theorem. So think about under what conditions Lagrange's theorem holds.


Extensions of Nash equilibria to games with infinite strategies In the Jehle and Reny textbook (which I should add I have not read much of beyond a few sections of interest), a theorem stating that there is always a (mixed) Nash equilibrium in finite strategic form games is proven. The book assumes that all players have the same number of actions available, but it's not difficult to imagine how this might be extended to the case where this isn't true.

What I'm interested in, however, is whether there is some extension of this to games, particularly those where there may be infinite choices. For instance, there's clearly no equilibrium in a game where a player wins by picking the highest number, but if we have, for instance, the same game, but where the number must be within the interval $[0, 100]$ (or any interval that contains its upper bound), the best response functions "converge". Similarly, I would also suspect that there need to be "well-behaved" cost and demand functions in competition models to get "good" results.

As such, I have two questions:

Is there any sort of well-defined setting in which a game with infinite strategy choices will have a Nash equilibrium?
What would relevant reading for this be?

Yes, there is such a setting. The result is that

  If each player's strategy space is

  convex
  compact

  and if payoffs are continuous then there exists at least one Nash equilibrium (possibly in mixed strategies).

This holds even when the set of possible actions is uncountably infinite. If one additionally assumes that payoffs are quasiconcave then the best-response correspondence will be convex even when we restrict attention to pure strategies so that we are then guaranteed to have at least one equilibrium in pure strategies in such a game.

I believe the original reference here is

I.L. Glicksberg. "A further generalization of the Kakutani fixed
point theorem, with application to Nash equilibrium points."
Proceedings of the American Mathematical Society, 3(1):170–174, 1952.

The treatment in Glicksberg's paper, though, does not  seem very accessible. A good starting reference is more likely to be section 1.3 of Fudenberg & Tirole's book "Game Theory".


Best resource to learn Microeconomics fast? I'm going to be honest, I have 2 full days to study for an Introductory Microeconomics exam final, and I am starting from scratch. The course was given online and the assignments were very easy to complete since we had 3 tries and the questions repeated.

I have no background in micro but my math is very good for this type of course (completed Calculus 3 and Combinatorics), and I am looking for a resource to learn the material quickly.

I have looked online and haven't really found one. The book we use in class is full babbling analysis of graphs that I already understand and random examples that don't cover the material efficiently. I am looking for a sort of summary that covers the essentials.

I know that I should have studied earlier, but I still want to try my best to have, at least, a passing grade.

Any recommendation will be appreciated.

Thank you in advance.

It would help a lot to know at what level this class is exactly. You mention calculus, however most classes titled like yours ("Introductory Microeconomics") tend to be very light on math. I'm going to assume that your class is of the learning-introductory-concepts variety and not of the taking-partial-derivatives-to-maximize-utility variety. If I'm wrong please correct me (or ignore this answer).

You have two options:

1) Obtain a virtual copy of an intro micro textbook (I can vouch for Mankiw's Principles of Microeconomics) and read it cover to cover.

2) Use a (probably) less-reliable resource such as this or the link you posted in your answer.

Of course, I choose these options with the assumption that your textbook is as bad as you say it is. I would still recommend you go over the course material to determine what your instructor considers important.

Important note: at this level of economics, knowing how to do things (calculate elasticity, do simple math in simple word problems about total cost and average cost, etc) is not nearly as important as understanding the concepts being taught. You say you've taken Calculus 3; expect basically the opposite of your Calc 3 final for this final. Of course, you should know the equation for, say, the elasticity of demand. However, the much more important thing is understanding what elasticity means and how to apply the concept to novel real-world problems.

Here is my off-the-top-of-my-head list of concepts you need to master.

Competition. How does profit get driven to 0 in a competitive market? What happens when demand increases in the short run? What then happens as time goes on?
Monopolies. Understand why it is in a monopolist's best interest to produce less than the efficient amount. Understand why a monopolist faces a downward sloping MR curve while a competitive firm faces a flat MR curve. Be able to show your understanding on a graph.
Understand the difference between an increase in demand (ie, an outward shift in the demand function) and an increase in quantity demanded. Same for supply. Be able to show it on a graph.
Understand what happens to quantity supplied and quantity demanded when a binding price floor is in place. Be able to show it on a graph.
Firm cost and revenue graphs. How are ATC, MC, MR, P, AVC, and AFC related? Does the answer to this depend on the structure of the market? Why are supply curves upward sloping while demand curves are downward sloping? What are sunk costs?
Opportunity cost. what is opportunity cost? If firms in a competitive market see 0 economic profit, how is it that real-world firms achieve positive accounting profit?
Specialization and comparative advantage.
Price discrimination.
Deadweight loss. What is it and why do taxes create it. Be able to show it on a graph.
Externalities, public goods, and Pigovian taxes.
Elasticity. Bonus points for understanding in terms of calculus. Understand perfect elasticity and perfect inelasticity. Show it on a graph.


Backward Bending Supply Having a hard time connecting the two picture, why is the supply of good 1 decreasing after a certain point of p1?

Raising your wage means that you can consume more. It also means (if you want to) that you can afford to consume more leisure. However, because you have to forgo more consumption per hour of leisure than you did at the old wage, this causes you to substitute away from leisure and towards consumption (more consumption, less leisure) which causes you to supply more labor.

However, with a high enough wage, the marginal utility of consumption is very low while labor supply has gone so high that the marginal utility of leisure is very high. Eventually the wage is so high that the same wealth effect that made you work more makes you consume so much that consuming more looks less attractive than working less. This causes labor supply to drop and the supply curve to bend back.

All of this depends on the household utility function. There are some utility functions where the backward bending never occurs. For example, linear utility.


What is the name of the gain/cost ratio I wanted to know what is the formal name of the (gain/cost)x100 ratio calculation (growth rate perhaps?). Any clarification would be greatly appreciated.


How to choose instruments for GMM estimation? I have a linear regression with one dependent variable and ten INdependent variables. I want to estimate this relationship using GMM, but I need at least ten instruments.

Can I use the same dependent variables as instruments for the regression? If not, how would I go about choosing instruments in such a case?

EDIT: my research studies reported generalized trust levels in 77 nations and has covariates such as lnGDP2013 lnPopulationSize2014 dummy variables for history of legal institutions from: Germany, Scandinavia, Britain, France, Germany, a dummy variable for whether or not a nation was involved in the transatlantic slave trade and a dummy variable for whether or not a nation was colonized by Europe in the last 300 years.

I have transformed the dependent variable from a percentage TRUST = Percentage of people who answered that 'most people can be trusted' when surveyed to ln(TRUST/1-TRUST) which gives residuals whose estimated kernel density is roughly normal.

In addition I have a few other covariates such as probability of two individuals chosen from a nation being of the same ethnic group.

EDIT2: I've parsed the model down so I am using GMM to explain the impact of whether or not a nation send or received slaves during the transatlantic slave trade on reported generalized trust levels using Africa as an instrument for whether or not a nation sent or received slaves. I can argue that being an African nation in and of itself should not having a dampening impact on trust but would be correlated with the likelihood that a nation was involved in the transatlantic slave trade.

As jmbejara says, better to be more specific, but let me guess.

You have a linear model $y = \beta_0 + \beta_1 x_1 + \cdots + \beta_{10} x_{10} + u$, where $x_1, x_2, \ldots, x_{10}$ are endogenous. Then, yes, you need at least ten instruments.

Can you use the same dependent variables as instruments for the regression? No, you can't. Instruments should be exogenous (uncorrelated with $u$) and relevant (strongly correlated with the endogenous regressors). But if some of the regressors are (believed to be) exogenous, then you use those exogenous ones as instruments. But you still need extra instruments for endogenous regressors. The rule is that you need at least $k$ extra instruments if there are $k$ endogenous regressors.

Finding good instruments (exogenous and relevant) is usually difficult. I suspect it would be really hard to find 10 instruments for a regression unless you have good economic justification.

That said, again it would be better to see your specific problem.


Using metrics provided by FRED to identify the business cycle I am interested to see how the "business cycle" is associated with our firms sales. The hope is that if we can show that our sales line up with the business cycle then we can make decisions accordingly.
If, for example, our sales have historically been hurt when in a recession, then it would be good to keep a pulse on this macroeconomic factor to know when we should increase or decrease spend.
The problem is that I am not sure how to identify the business cycle. I am not an economist by trade. I have been pulling some data from FRED, but I am not sure which indicators to look at for identifying the business cycle. I have two hypothesis:

Unemployment. Record low unemployment may be an indicator for a recession. I think that this is getting slightly more convoluted these days as I think that employment takes into account people with either W-2s or 1099s, but as the people who rely solely on 1099s increase I think that the meaning of unemployment changes. Something like a gig economy effect. This makes this metric maybe less reliable than it used to be as the gig economy becomes a reality.

% difference in real vs potential gdp This metric seems to ask, 'Is real gdp exceeding potential gdp'. I don't understand how it is possible to have gdp exceed potential gdp, but I am not an economist so I don't need to worry about that ;). This does seem like an okay metric to represent the business cycle since it seems to quantify if real gdp is too high.

Edit
My above thoughts simplify to the following question:
If someone asked you to show him/her the the business cycle, what graph --- on FRED --- would you show that person?

Prediction is difficult, especially about the future. (Berra, Goldwyn, Bohr et al.)

The problem with noting that recessions often occur immediately after low periods of unemployment is that you do not now at the time that it was a low rather than a pause.

There are several occasions when unemployment appeared to stop falling but which were not immediately followed by recession.  Consider for example 1963, 1967, 1976, 1985, 1995, 2003 in your first linked FRED chart.  Once unemployment shoots up, you are probably in a recession and so do not need to predict

Potential GDP is even harder to quantify meaningfully, as it is based on "the output the economy would produce with a high rate of use of its capital and labor resources."  So if unemployment falls below levels previously thought typical before leading to inflation and high interest rates, this can lead to a suggestion that GDP is exceeding its potential; unemployment is certainly now well below what the Federal Reserve recently thought was possible.  But this may merely mean that the judgement of what was typical was wrong: interest rates are still historically low, and inflation is not accelerating, while the fiscal position is loosening

Looking at your second linked FRED chart, actual GDP exceeding potential GDP can happen several years before a recession, or immediately before, or even not happen at all.  So it is not a precise signal

So you may be correct that a recession is looming, or you may be wrong and it is several years away.  Welcome to macroeconomic forecasting


Example of Price discrimination Can anyone explain why this isn't an example of price discrimination:

The same golf club selling for $100 at Sports Authority and 80 on Amazon.
I believe that consumer's demand elasticity to price at Amazon would be higher than the consumer at certain Sports Authority. May be the idea of an Sports Authority which I've is wrong?

I came across this in a quiz in the Standford online course for Economics.

Price discrimination is when a firm charges different prices for a homogeneous product to different consumers.

In this example we see no such discrimination among consumers. This is just a case where there is competition between two firms.


Does the currency in circulation equal the annual production of goods and services? I have learned that in a country currency is printed equal to production of goods and services in a year.

I want to know if in a year a country produces rice of 10000 rs. The government will print currency of up to rs 10000. In the next year the country again produce rice of 10000 rs. Will the government again print 10000 rs. more? If yes will it not lead to excess currency in circulation?

The amount of currency in circulation does not equal the annual production of goods and services in a country. It is usually much smaller.

For example, as of July 2013, currency in circulation—that is, U.S. coins and paper currency in the hands of the public—totaled about $1.2 trillion dollars.

In contrast the U.S. GDP for the same year was about $17 trillion dollars.

You can find more information on how Fed prints currency to meet demand from this Fed article.


Why do we treat demographic variables as being exogenous? I I would like to ask about demographic variables.
In general, we usually include those variables as independent variables when we run  regressions such as wage equation or test score equation.
What I wonder is that why we take them as exogenous in general? Intuitively, I understand that those are something given to agents when they need to make a decision, so in that regard it makes sense that we say they are exogenous. However, mathematically E[Xe]=0 is the definition of exogoneity, and so I guess demographic variables can be still correlated with unobserved error terms. For example, if we consider the following equation
Y_i = X_i*b_i + e_i where Y_i is test score, X_i are demographic covariates (race, gender, age ...). Then, isn't it possible those variables are correlated with innate abilities or whatever in e_i ? But, I do not find any convincing answers why (nearly) every people take those as exogenous....

Any answers for this?

You are completely correct that mathematically endogeneity is defined $\text{cov(X,e)}\neq0$, which is true not only when it’s correlated with omitted variable but also with measurement error or reverse causation/simultaneity.

However, every single variable in economics can be subject to correlation with omitted variables or measurement error, so if you would want to use the word for any variable that potentially has this problem then it would be meaningless. So the word is reserved for those cases where this is actually a serious issue.

Furthermore, often endogeneity is used more narrowly as to referring just to simultaneity or reverse causality - variables determined by the model as opposes to variables outside the model - because that’s where the term originated.

Also some demographic variables can be definitely subject to endogeneity, but I would also say most of them aren’t. Consider race for example in a regression where you try to estimate let’s say how it affects the employment. Well you can’t pick your race (at least to my understanding technology is not there yet), so there is no possibility for simultaneity. Next race would be simple categorical variable so there is not much space for measurement error and also I am not biologist but from my understanding biologically there is only human race so social construct of race should be orthogonal on unobservables like innate ability. Similar story would hold also for sex (although you can have sex change that is usually not done on a whim but when person actually experiences gender dysmorphia so it can be treated as predetermined at birth) and age.

An example of endogenous demographics variable could be number of children in family that could have simultaneous relationship between wages/wealth and could be also correlated with some unobservables.

Sometimes it can be also context dependent. In model where you regress Y on X there can be endogeneity there but in a model where you regress S on the same X it might not be there because conditions are not satisfied. In your research you should always think hard about this issue and decide on case by case basis. But in general demographic variables  just happen to not be endogenous often when comparing to other types of variables.


What does "the world's strongest currency" title mean for investors? Kuwaiti Dinar shows up as the most expensive in the world. Does that mean I better invest in KWD (or the safe USD) instead of my home currency (IDR)?

A little background: I am just an ordinary person without economics / finance background and hope to learn from the experts here. My home currency never fails to show up in the top 5 least valuable currencies in the world. Economic crises that come too often and stay too long make me wonder if I make the right decision to put all of my retirement savings in local equity mutual funds.

The strong currency understanding in your question does not have any interpretation that would be an economic factor that should effect investment, or any other decision. It is merely units.

It answers the question of '1 US dollar buys least of which currency?'. It also answers the question of '1 Colombian peso buys least of which currency?', or any other benchmark currency of your choice.

The Kuwait government could say that they decided to replace KWD by KWD2 and 1 KWD would equal 1000 KWD2. Then everything that has a price of $x$ KWD would now have a price of $1000x$ KWD2. Now, your US dollar would buy a lot of KWD2, but your investment decision would not be effected by this change because there has been only a change in units, and no other intrinsic differences.


Bubble and productive investments Is there some consenus in the literature as to what features characterize a bubble investment as distinct from a productive investment?

I think one thing you can think of is this:

Assuming such a thing is possible (Fama may cry never here), one might first define a productive investment as an investment in an asset that is undervalued by the market relative to its fundamental value.

The distinction then between this type of investment and a bubble investment is that bubble investments stray far above the fundamental value of an asset. Bubbles are usually made worse than they might have become otherwise whenever trend investors attempt to play bubbles for profit. And it is generally the realization of this fact (that bubbles have pushed securities beyond fundamental valuations)by market participants that crashes the bubble.

This paper on the topic is nice.


What were M.F. Chamans's calculations about how much trade would gain by the burning of Paris? In "That Which Is Seen, and That Which Is Not Seen", Bastiat (1850) writes:

What will you say, Moniteur Industriel? what will you say, disciples of good M.F. Chamans, who has calculated with so much precision how much trade would gain by the burning of Paris, from the number of houses it would be necessary to rebuild?

What were M.F. Chamans's calculations about how much trade would gain by the burning of Paris? (Also, who was M.F. Chamans?)

Auguste Louis Philippe de Saint-Chamans (1777-1860), also known as Vicomte de Saint-Chamans, was a French politician who served as the master of requests to the Council of State in 1820, Councilor of State in 1827, and the deputy for the Marne from 1824 to 1827. He also wrote a number of economic treatises. (The "M.F." is, I believe, incorrect and actually just "M." for Monsieur in the original.)
Unfortunately, Bastiat remembered Saint-Chamans's quote incorrectly -- though his critique of the idea is just as valid.
Saint-Chamans argued that the Great Fire of London (1666) in its destruction led to a net economic gain (due to spending to rebuild) of 1 million pounds (or 25 million French francs).
You can read more about this fallacy here or read Saint-Chamans's work in its original French: M. le vicomte de Saint-Chamans, Traité d’économie politique suivi d’un apercu sur les finances de la France, vol. 1 (1852).


What are some reasons for decline of georgism? I understand that Georgism was a major economic idea in the late 19th century.  The idea of a simple land-value tax and basic income derived from an equitable distribution of natural resource exploitation seem to be great ideas solving many of the challenges faced in other economic systems.  Also, Progress and Poverty was one of the most popular books on economics in the late 19th century.

Why has this seemingly great concept lost popularity?  My speculations is that it either does not benefit the ruling elites (so, politics) or that, more plausibly, the idea of basic income derived from natural resources was calculated to be unsustainable amid the population growth of the turn of the century.

Warren Samuels addresses this issue in his article "Why the Georgist Movement Has Not Succeeded: A Speculative Memorandum."

In summary he argues first that Georgism did not succeed because of the conflation of the ideas of income and productivity, leading people to view a tax on land as a tax on productivity even though the tax is meant to equate income with productivity as referenced in the question. This idea, he says, was reinforced by self-interested beliefs of land owners as well as traders in other markets (he names equities) including preference for passing property through inheritance and viewing the proposed tax system as a threat to income through speculation (such as through the stock market), since it would eliminate speculative gains in land prices. The rise of mortgages and home-ownership increased the number of people with these self-interests.

Second, Samuels addresses the fact that George was viewed by many, including mainstream economists, as a radical and admits that relative to the status quo of changes in tax policy being incremental, the proposed single tax system is radical. The Bolshevik revolution in the 1910's and subsequent red scare caused people to distance themselves from ideas that could be viewed as radical leftist.

Samuels continues on to discuss possible internal reasons, such as lack of leadership amongst George' successors, but notes that he is less confident in discussing these factors.

In addition to this article, four responses and Samuel's response to the respondents can be found in the Jul 2003 issue of American Journal of Economics and Sociology.


BreakEven/productivity/salary/employees If employees’ salaries are increased by 20%, what is the increase in productivity required to break even?

Can someone assist me in solving this question?

Productivity =  (output per period)/(number of employees)

How is productivity related to the employees salaries?

Thank you

Break even is when your [economic] profits equals to zero:

\begin{align*}
\text{Cost} &= \text{Revenue}\\
N \cdot w &= Q \cdot P
\end{align*}

where $N$ is the number of workers, $w$ is wages, $Q$ is quantity produced, and $P$ is price.

An $x$% increase in productivity is the ability to hire the same amount of workers and get $x$% more output.

Assume that $N$ and $P$ remains unchanged, a 20% increase in wages would lead to the LHS's being $L \cdot 1.2w$ instead. How would the RHS need to change to maintain equality?


Australia and Saudi Arabia - Structural Similarities I refer specifically to this article in which Andrew Critchlow paints a pretty dire view of the Australian economy.

The line which challenged my priors was:

Just like Saudi Arabia, which is now burning its foreign reserves to compensate for falling oil prices, Australia faces a collapse in export revenue.

This did not match my perception of the Australian economy at all.

What similarities does the Australian economy have with the Saudi economy? Is the comparison fair? In particular, I am interested in structural issues the economies face.

Secondly, is the quoted line accurate? My understanding was that the extraction cost for oil was very low in the gulf and the lower prices served to make other countries unprofitable before it cost them.

It's an analogy. The Saudi economy as an analogy for the Australian economy. Like any other analogy, you can stretch it to a certain point, but then it breaks.

The article sets out the ways in which the analogy holds. Stretching it further will probably break it. Let's look at how the article says the analogy holds:

both economies have a significant dependency on exports of a very small number of commodities: for Saudi, it's oil and gas; for Australia, iron and coal.
both boomed during the good times of the commodity super-cycle, both fuelled in part by huge and rapidly-rising demand from China; this flattered their domestic economies.
both now see a large drop in export revenues from falling demand and falling global prices in their key commodities; this exposes weaknesses elsewhere in each of their economies.
In Australia's case, this has caused a rapid increase in its net foreign debt; for Saudi, they're eating into their foreign reserves. So in each case, they're taking a big hit to the balance sheet, and for both countries, net foreign reserves have decreased a lot.


DMP: Two Labor Markets Consider the following extension of the standard DMP model, in which we have two labor markets with different returns, indexed $i$. The markets are completely separated and firms and workers can only go to one market. It might help to think of them as different countries, for example.

I am ultimately interested in the different waiting time of unemployed to find a job.

As usual, we match using a CRS function $M(u,v)$, such that the likelihoods of finding a job and finding a worker are given by $f(\theta) = M(u,v)/u = M(1, \theta) $, and $q(\theta) = M(u, v)/v = M(\theta^{-1}, 1)$.

The vacancy-value equation is

$$ \rho V_i = -c q(\theta_i) (J_i - V_i)$$

Free entry implies $V_i = 0$. Manipulating the equations we get

$$ \frac{q(\theta_1)}{q(\theta_0)} = \frac{J_0}{J_1}$$

which makes sense - the job-filling-rates are inverse to the value of the jobs. Say $J_1 > J_0$. Then, market $1$ is tighter, leading to a lower job-filling-rate for an individual firm.

Using the earlier definitions, I can use that $f(\theta) = q(\theta)\theta$:

$$ \frac{f(\theta_1)}{f(\theta_0)} = \frac{J_0}{J_1}\frac{\theta_1}{\theta_0}$$

Let $W_i$ denote the average waiting time of an unemployed in market $i$. As we're using Poisson rates, $W_i = 1/f(\theta_i)$:

$$ \frac{W_0}{W_1} = \frac{J_0}{J_1}\frac{\theta_1}{\theta_0}$$

Which doesn't make sense anymore. Consider again $J_1 > J_0$: A worker in market $1$ is much more valuable to firms than a worker in market $0$. However, the relative waiting time for workers in market $1$ increases in the relative value of workers in market $1$.

I don't understand this. Was my derivation wrong? Or am I not allowed to to the ceteris paribus argument while ignoring $\theta_1/\theta_0$ on the right-hand-side?

Found my mistake. I guess I'm not allowed to move the thetas to the other side.

Define $q^{-1}$ be the inverse of $q$. Then,

$$ \theta = q^{-1} \left( \frac{c}{J} \right)$$

$$ \frac{W_1}{W_0} = \frac{f(\theta_0)}{f(\theta_1)} = \frac{f \left( q^{-1} \left(\frac{c}{J_1}\right)\right)}{f \left( q^{-1} \left(\frac{c}{J_0}\right)\right)}$$

$J_1 > J_0$, hence $\theta_1 > \theta_0$ and $W_1 < W_0$.


Whom would you consider to be the father of rational choice theory? Whom would you consider to be the father of rational choice theory?
My suggestions that would qualify for major publications:
Arrow (1959), Economica
Homans (1961)
Becker (1976)
Sugden (1991), Economic Journal

Plus, would you agree that inter-temporal choice theory was derived from rational choice theory? Or vice versa?

The answer is there is no single father of rational choice theory. The reason is that rational choice theory is not so much a theory but rather a coherent framework (others would call it a paradigm) resting on the key foundations of methodological individualism and instrumental rationality / self interest.

Both of these aspects played important roles in the marginal revolution in the late 19th century. Important names here are, of course, Walras, Jevons, Menger and Marshall although none of them can be regarded single father of Economics in the modern sense.

Some decades later, Robbins (1932, p. 15) famously defined Economics as

  the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.

This definition nicely summarizes much of the framework rational choice theory is embedded in. It contains instrumental rationality, in so far as that humans have goals (denoted as preferences) but only limited resources (constraints) and they choose so as to maximize their utility. It also hints at methodological individualism because it places human behaviour at the center of its investigation. Note, however, that Robbins again only summarized a research paradigm that was already well developed. (He says so himself on the same page.)

Still, I guess it is fair to say that rational choice theory as we know it today took off after WW2. Here different names come to mind, depending on which focus you want to have. Arrow/Debreu for general equilibrium theory, von Neumann/Morgenstern if you want to stress expected utility (and game theory). A bit later, Becker is a good choice if you want to stress that rational choice theory is an approach that is driven by its methodology (rational behavior, methodological individualism) and need not investigate an economic topic at all.

But yeah, asking for a single father of rational choice theory would really stress things a bit too far.


If saving equals investment, can there ever be a stock of savings? National product accounting identities imply $S=I$. Since these are flow quantities, everything saved in a particular period is invested, thus there is never an accumulated stock of savings.

Is this reasoning correct? If so, where do household savings during a particular year (say bank account balances) go?

The accounting identity assumes that anything saved is an investment, i.e. something will be done with that saving in the future. It's a bit circular, I know. It's not a lagged model, i.e. there's no explicit indexing by time of the variables, unlike (e.g.) in the Hansen–Samuelson model, which uses discrete time. There are also macroeconomic models with continuous time, e.g. Ramsey–Cass–Koopmans model.

Actually, there's a little more to be said here (as I may have committed a fallacy myself above). German economist Fabian Lindner takes issue with those (including economics textbooks authors) who use a one-good model, like Ramsey's to model financial savings:

  The fallacies loanable funds theory commits might be explainable by the mis-application of some ideas and
  concepts of neoclassical growth models – especially the Ramsey (1928), Solow (1956) and Diamond (1965)
  models – to the sphere of money and finance. Those models are routinely taught in contemporary graduate
  economics classes (Blanchard and Fischer, 1989; Romer, 1996).
  The Ramsey and Solow models are models of real investment only. Financial markets, financial
  assets and financial saving do not play any role in those models. There is only one good which, for simplicity,
  will be called “corn”. Corn has three functions: it can be consumed, invested and used as a means of
  payment since wages and interest payments are made with it. Full employment is assumed.

  Without money and other financial assets, the only way units can save is to increase their tangible
  assets, i.e. to invest. Given that full employment is assumed, corn production is always at its maximum in
  each period. If the corn is consumed, it cannot be invested; if it is invested, it cannot be consumed. There is
  thus a real trade-off between consumption and investment.

  Only under this assumption does it indeed make
  sense to talk about a limited saving fund which is increased when it is not consumed.
  However, this trade-off between consumption and investment is not a finance constraint, but a
  resource constraint. [...]

  In contrast to Solow’s and Ramsey’s model, Diamond’s (1965) full employment corn economy
  allows units to lend and borrow. However, they do not borrow and lend money but again the one good, corn.
  They face a triple trade-off: they can eat (=consume), plant (=invest) or lend their corn. Here, consumption
  are not consumption expenditures but the actual eating of the corn. When a unit wants to lend its corn, it can
  of course not eat it so that it has to restrain from consumption to be able to lend it. [...]

  But since in the real world money is normally not eaten or planted
  and keeps circulating in the economy when it is spent or lent, those models cannot be any guide for the
  analysis of a monetary economy. Specifically, what is true in a one good economy – units have to consume
  less to lend and invest more – is fundamentally wrong in a monetary economy.


Regression approximation for the rate of change in occupancy rate of residential market with respect to price I have historical data on occupancy rates for a given neighborhood, along with characteristics and other local economic variables.
I am looking to estimate the regression equation with occupancy rates as dependent variable and price, level of supply, characteristics and economic vars as independent variables.
D(p) = Rent + Median Income + Supply + # of HH

Median Income is time fixed effect.

My questions are:

How do you interpret the equilibrium price from the regression output? i.e price at which Occupancy rate is 100%. or price elasticity of demand.
Interaction term. Given that supply follows rental prices, add supply * rent as interaction term.
What other regression methods would make sense for this problem?

If the goal of the project is to model housing demand, I would suggest using the actual number of occupied housing units. Census data may calculate occupancy as an algebraic function of supply, that is, occupancy is derived using the supply variable. This may show up as a strong correlation in the model, but it doesn't capture the stochastic properties of housing demand.

If you are interested in the price elasticity of housing, just use a log-log regression. Here is a good post about equilibrium price.

What is the level of analysis? If you have panel data (historical data for $n$ neighborhoods), I would suggest looking into spatial regression.

For a given neighborhood $x$, the occupancy rate of the surrounding neighborhoods may affect demand for housing (the negative affects of vacancy "spill over" into $x$). Another example, if rent is high in the neighborhoods surrounding $x$,a shift in the demand for housing will occur, because people are making a spatial decision regarding where they will live and take into account the housing characteristics of the neighborhoods of a given area.  Therefore, to model housing demand, the occupancy rate and independent variables of surrounding neighborhoods may capture the process of housing demand in $x$.
This is refereed to as spatial dependence and it can affect the model a lot. I would recommend the software Geoda and GeodaSpace to analyze and run spatial regression models.


Is there a discrepancy between the ratio and total expenditure methods of measuring elasticity? According to the Expenditure method of measuring price elasticity of demand, if the two expenditures in comparison are constant (in spite of changes in price), the demand is considered to be unitary elastic.
However, there are several examples where this does not seem to hold true.
For instance,

Price: 10 ; Demand: 12 ; Expenditure: 120

Price: 08 ; Demand: 15 ; Expenditure: 120

Here, though the expenditure is equal, if we calculate the percentage changes in price and demand, they come to be 20% and 25% respectively. So as per the ratio (or percentage method), the elasticity would be equal to 25% / 20%, which is not equal to 1. Then, how can we conclude that the expenditure method gives the right results?

If something rises by $25\%$ then you then need a $20\%$ reduction in the new larger number to get back where you started: $1.25 \times 0.8=1$. This asymmetry means comparing large percentage changes can produced oddities

Another approach, closer to the idea of approaching the derivative as a limit, is to use logarithms and calculate the elasticity as a ratio of logarithms, something like $$\frac{\log\left(\frac{Q_2}{Q_1}\right)}{\log\left(\frac{P_2}{P_1}\right)}= \frac{\log(Q_2)-\log(Q_1)}{\log(P_2)-\log(P_1)}$$

and with your example this would give $\frac{\log(15)-\log(12)}{\log(8)-\log(10)} = \frac{\log(1.25)}{\log(0.8)} =-1$ as expected


Why can per share earnings growth only keep up with GDP growth, if no new shares are issued? I transcribe Ben Felix's speech (BSc Mechanical Engineering (Northeastern University), MBA Finance (Carleton University), CFA, CFP, CIM) starting from 8:33. I just screen shot his animation, rather than taking the time to create a GIF out of it.

But growth in aggregate corporate earnings does not directly benefit investors. It is increase in earnings per share that benefits investors. The problem here is that per share earnings growth can only keep up with GDP growth, if no new shares are issued.
Let me explain. If you own shares in a company in a rapidly growing economy, and a new company listed shares on the stock exchange, you don't benefit from the new company's economic impact. You'd need to re-allocate some of your capital to the new company, to participate in its earnings. But doing so does not increase the value of your portfolio. The economy is grown, but your portfolio has not. Over time, the effect of this slippage has been meaningful.

China might again be a good example. Much of the growth in the market value of Chinese equities has come from an increase in the number of listed companies, as opposed to price appreciation from existing listed companies.

Please see the titled question.

I don't fathom the boldened phrase. What if company B's share price rockets faster than company A's? Wouldn't company B's faster rate of share price increase increase my portfolio's value?

The question is really about industry jargon, not economics.

  Wouldn't company B's faster rate of share price increase increase my
  portfolio's value?

What he's referring to is that, at the moment company B went public, shifting your portfolio to company B stocks is simply a re-balancing of your portfolio. There is no profit from re-balancing.

(Of course, once a stock is in your portfolio, there is capital gain if stock price increases, vice versa.)

On the other hand, in the scenario where equity value increases at the moment of IPO, the value of the company increases, which contributes to increase in overall market value (what he calls "GDP growth" and "economic impact").
The difference in value is captured by investors who holds private equity prior to public listing (e.g. venture capital firms).

Each time this occurs, there is capital gain in the economy that's not captured by investor in public markets, simply because they are late to the bandwagon.
This is what he calls "slippage". He claims "over time, the effect of this slippage has been meaningful", which implies the gains of public market investors lag gains
of private equity investors.

(If one compares the historical returns of the public market---e.g. that of stock indices---and the average VC firm, the latter is probably higher. However, it's not clear whether this is still the case after adjusting for  risk. The risk profiles are very different. The typical VC portfolio may have 30 companies, with one eventually turning a profit.)

A rephrasing of the quoted statement might be:

"...per share earnings growth [for investors in public markets] can only keep up with GDP [(market cap)] growth, if no new shares are issued [i.e. no new public listings occur]..."