"If markets are competitive, the rate of return on capital equals its marginal product, $f'(k)$ minus depreciation $\delta$"? Consider Solow model with $Y=(AL)^{1-a}K^a$. Then output per effective labor is $y=k^a$ where $k=\frac{K}{AL}$.

"If markets are competitive, the rate of return on capital equals its marginal product, $f'(k)$ minus depreciation $\delta$"?

How does market competitiveness deduce formally/mathematically the rate of return on capital equals its marginal product, $f'(k)$ minus depreciation $\delta$"? The statement seems plausible but I can't prove it mathematically. What is the formal definition of rate of return on capital here?

The relevant equations are the following.

$\frac{d}{dt}K=sY-\delta K$ where $s$ is saving rate. Assume labor $L$ and knowledge $A$ grows exponentially at rate $\delta,g$ respectively.

Reference. Romer Advanced Macroeconomics, Chpt 1, Sec 1.5

This is not proven in Romer but it is a well known result. To derive it mathematically you need to take the following steps:

First, the capital as in Romer depreciates so the evolution of capital will be given:

$$k_t = k_{t-1} + i_t- \delta k_{t-1} \tag{1}$$

where $k_t$ is the present stock of capital, $k_{t-1}$ previous stock of capital, $i_t$  is investment (where investment is equal to savings $s$) that increases the capital and $\delta$ is depreciation.

Next to increase the stock of capital the producer/investor needs to pay price $P_t$ for output that is going to be saved and converted to capital through $i_t$. Assuming that we will hold labor constant the reward that producer gets for this sacrifice is the marginal product of capital - as when labor is held constant the output $y_{t+1}$ will increase only by the additional marginal product that the increase in capital brings $f'(k)$. Moreover, we can assume that in the next period the capitalist can sell their remaining capital $(1-\delta)$ as well at price $P_{t+1}$.

Consequently the net nominal return on the investment will be: $P_{t+1}(f'(k_t) + 1 -\delta) - P_t$ and the nominal rate of return to investment will be given by:

$$ (P_{t+1}(f'(k_t) + 1 -\delta) - P_t)/P_t \tag{2} $$

This can be simplified as:

$$ (1-\pi_t) (f'(k_t) +1-\delta) -1 \tag{3}$$

where $\pi_t$ is the inflation rate $(P_{t+1}+P_t)/P_t = \pi_t$.

Whether the producer/investor is interested into the above investment depends on how other returns such as those on bonds or on debts used to finance investment behave. Lets call those returns $R_t$.

Now if we assume perfect markets, rational investor will invest into capital when the nominal return on capital is higher than the returns on bonds/debt $R_t$, but as economy accumulates more and more capital the marginal product of capital declines as it becomes more scarce (and vice versa if the return on bonds/debt is higher) and thus in competitive markets rational agents will invest up until:
$$ (1-\pi_t) (f'(k_t) +1-\delta) -1 = R_t \tag{4}$$

Furthermore, the nominal rate on return on bonds $R$ has to also satisfy the fisher equation:

$$(1+r_t) = (1+R_t)/(1+\pi_t) \tag{5}$$

where $r_t$ is the real rate of return. The fisher equation basically says that the real rate of return must be equal the nominal rate minus inflation as the above function can be approximated as $r_t \approx R_t - \pi_t$ using the fact that $\ln(1+x)\approx x$ for $x\approx 0$ (this comes from Taylor expansion). This must hold as people in the market should care about real returns and thus they should expect to be compensated for inflation when they set nominal interest rates.

Solving the Fisher equation (4) for the $R_t$ and substituting into the equation (5) which equates the nominal returns between capital and bonds/debt gives you the desired result:

$$r_t = f'(k_t)- \delta \tag{6}$$

The last equation says the real return to capital will be equal to marginal product minus depreciation.

The intuition is simple, if markets are competitive then you will invest into the capital as long as it provides higher returns than other investment. But the more you invest into the capital the smaller its marginal product and hence nominal return will be. At some point people invest into capital so much that the marginal product minus depreciation (which has to be taken into account as it decreases the value of the capital) is just equal to the real return. Also if capital would started with lower return than bonds/debt people would just invest into those until the returns will be equalized. The assumption of perfect market is important as in presence of market imperfections the equality established by eq (4) might not necesarry hold (or to be more precise it would have additional parameters which would end up in the result as well).


If renewable energy is now cheaper than non-renewable energy, why are we still worried about the climate? From a recent Our World In Data post,

For the world to transition to low-carbon electricity, energy from these sources needs to be cheaper than electricity from fossil fuels.

Fossil fuels dominate the global power supply because until very recently electricity from fossil fuels was far cheaper than electricity from renewables. This has dramatically changed within the last decade. In most places in the world power from new renewables is now cheaper than power from new fossil fuels.

However, In 2019, around 89% of global primary energy still came from non-renewable technologies. In fairness, New electric generating capacity in 2020 will come primarily from wind and solar, but that's for the United States. My question concerns the whole world.
So here's my question: if renewable energy is actually cheaper than non-renewable energy, why are people still worried about climate change? Doesn't basic economics predict that when a product becomes cheaper than another product it can substitute for, then people will adopt it? Unless adoption trends are extremely slow, wouldn't this news mean we have essentially already solved climate change, and all that's left is simply for market forces to replace existing non-renewable energy sources with cheaper renewable sources?
I suspect one of four things must be true,

Adoption trends are actually extremely slow perhaps due to the large fixed cost of developing new energy plants
Renewable energy cannot completely substitute for non-renewable energy, due to issues with storage. In this case, is there any price such that people would switch to renewable carbon-neutral energy sources? Also, where can I find the cost estimate for renewable energy sources, when you include costs for storage?
The cost estimate of renewable energy given by Our World In Data is incorrect
The world will probably be carbon neutral by mid-century.

To some extent, your suspicions are correct (notice some of the nuance you could add to your four main concluding points):

Renewable adoption trends are slow.

Not just slow, but there is a wide dispersion of adoption based on point of energy consumption. (e.g., Think how many gasoline vehicles there are in the world and how long it will take to replace or modify them all.)
Transaction costs are a big deal. (e.g., Maybe a company just built a new "clean" coal plant. This was not a 1-Click buy transaction on Amazon.)
Cashflow problems. (e.g., Individuals, organizations, and nations do not all have equal access to capital at low interest rates.)
There is the landlord-tenant split-incentive barrier. (e.g., Think of all the incandescent bulbs and inefficient oil heaters that sill exist in rental units. Landlords generally just want to spend as little as possible and renters often focus on the monthly rent, not what kind of heater is in the basement.)
There are regional and political pressures. (e.g., It is hard to convince all of West Virginia to abandon coal when coal mining is part of the culture.)

Intermittent renewable energy is a storage challenge.

This is a technical problem, so it is less daunting. Though the hurdles do exist to some degree, storage does not have as many of the same social or structural hurdles to implementation.

There are bound to be assumptions and approximations in every energy cost analysis.

But even if you assume the report's pricing analysis is 100% accurate, you should still be concerned about climate change.

The world will move toward carbon neutral by economic forces.

This will happen at drastically different rates for developed and developing nations without significant cooperation.

Embedded in the more nuanced arguments that you could make in points 1 and 4 is the more pithy answer by @quarague. If you've just built a greenhouse gas emitting energy generator, the chances are that it will be cheaper to continue operating it than to build a new renewable energy generator (whether or not you include the decommissioning costs, and ignoring the negative environmental externalities).

However, you have yet to mention the most important cause for concern:
The dynamics of climate change are complicated by long time delays because greenhouse gas in the atmosphere is a STOCK (the greenhouse gas emissions are a FLOW). So, even if the entire world flipped a switch today to stop all greenhouse-gas emissions, climate change would still occur because the atmosphere already has elevated concentrations of greenhouse gases, and so there are many current and future climate change effects that are baked in (sorry for the pun).

Think of filling your bathtub with water. The tub is still full of water when you turn off the faucet, just as our atmosphere will still be full of greenhouse gases even if we suddenly stop emitting them.

If you'd like to simulate this and get a better intuition for the problem, you can try the online En-ROADS interactive simulation by Climate Interactive and the MIT Sloan School of Management. And, for your reference, even if you rapidly cut carbon entirely, there are still serious consequences (e.g., temperature increase and sea level rise).


Negative Gini Coefficiente? Is it possible to have a negative gini coefficient?
In which situations is it possible?

Any help would be appreciated.

tl;dr No, it is not possible to have a negative Gini coefficient.

The Gini coefficient is the area between the line of equality and the Lorentz curve, and this area cannot be negative. Let the area of equality as in the link be denoted by $A$, and the area under the Lorentz curve be denoted by $B$, then the Gini coefficient is given by

$$ G = \frac{A-B}{B}$$

If negative income (or whatever you are measuring) is not permitted, the maximum inequality is $1$. If negative income is permitted, it can potentially be larger than $1$. In any case, the minimum for the Gini coefficient is $0$, which corresponds to total equality.


Claiming that food delivery apps are purely rent extraction I engaged in an argument with a friend about whether takeaway food ordering apps had any 'value'. I tried to persuade him that they were rent extraction enterprises but he wasn't convinced. So I want two things here. Firstly what the weak points of my argument are and secondly how I can articulate my points better.

My position was that apps are being used to provide marginal benefits to consumers who then use them at great expense to the restaurants that actually prepare the food. However, if those that benefit from the app (hungry consumers) had to shoulder the cost incurred rather than the restaurant, they would choose not to use the app. By this I mean if the money paid by the restaurant to the app firm was added as a surcharge to the customer bill, they would pick up the phone instead of using the app. Therefore because those who enjoy the limited benefits of the app would not be willing to pay to service the app, the app has no value and is a rent seeking enterprise by a tech firm to extract money from takeaway restaurants.

at great expense to the restaurants that actually prepare the food

Your premise seems to be that the restaurants are somehow losing out.

The key though is that (if done right) all three parties (the app company, the restaurant, and the consumer) should benefit. The reason is that through economies of scale (delivering for multiple restaurants) and perhaps other factors, the app company is able to organize and execute the delivery at a lower cost than the restaurant would be able to.

One piece of evidence that all three parties benefit is that all three parties continue to use this arrangement.

Note that with certain countries and certain apps, the delivery app companies are actually losing money. Venture capitalists throw money at them hoping they'll end up as the "winner-takes-all".

Rarely though is it the case that the consumer or the restaurant loses from this arrangement. A consumer who finds this a losing proposition can quite easily stop doing business with the app. Likewise with any restaurant.


How to derive formula for marginal probability of choosing nest in nested logit model? I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n choosing alternative j can be factored as the probability of individual n choosing nest k, multiplied by the probability of individual i choosing j conditional on having chosen nest k.

As I understand we are decomposing the decision process into two models: upper and lower. In the upper model decision maker chooses a nest and in the lower - alternative within the nest. Say utility of individual n choosing alternative j in the nest k is

$$ U_{njk} = W_{nk} + Y_{nj} + \epsilon_{nk} + e_{nj} $$

where $e_{nj}$ is EV I with scale parameter $\lambda_k$ and $\epsilon_{nk}$ is such that composite error term is EV I with scale parameter 1.

The lower model is trivial, it is a simple logit. However, the upper model is not clear for me. The expected utility of individual n from choosing nest k is

$$ EU_{nk} = W_{nk} + \lambda_kI_{nk} + \epsilon_{nk}$$

where $\lambda_kI_{nk}$ is expected utility that n will get from choice within the nest. And the marginal probability of choosing nest k is

$$P_{nB_k}=\frac{e^{W_{nk}+\lambda_kI_{nk}}}{\sum_{l=1}^K e^{W_{nl}+\lambda_kI_{n}}}$$

My question is how does this probability have a logit form if $\epsilon_{nk}$ is not extreme value? Or is it? Because as far as I understand the sum of two extreme value variables is not an extreme value.

Thank you!

As it turned out, my previous logic was wrong. Here is how it should be done.

Marginal probability of choosing nest $k$ is
$$P_{nB_k} = P\left[\max_{j\in B_k} U_{njk} \geq \max_{j\in B_s} U_{njs}, \forall s\neq k \right]\\
= P\left[W_{nk}+\epsilon_{nk}+\max_{j\in B_k}(Y_{nj}+e_{nj}) \geq W_{ns}+\epsilon_{ns}+\max_{j\in B_s}(Y_{nj}+e_{nj}), \forall s\neq k \right]$$

Then as $e_{nj}$ is iid $Gumbel(0,\lambda_t)$, $\max_{j\in B_k}(Y_{nj}+e_{nj})$ is iid Gumbel with location parameter $\lambda_kI_{nk}$ and scale parameter $\lambda_k$. The Gumbel distribution is preserved over linear transformations so
$$ \max_{j\in B_k}(Y_{nj}+e_{nj})=\lambda_kI_{nk}+\xi_{nk}$$
where $\xi_{nk}$ is iid $Gumbel(0,\lambda_t)$. Substituting back to marginal probability, get
$$P_{nB_k} = P\left[(\epsilon_{nk}+\xi_{nk})-(\epsilon_{nk}+\xi_{nk}) \geq (W_{nk}+\lambda_kI_{nk})-(W_{ns}+\lambda_sI_{ns}), \forall s\neq k \right]\\
=\frac{e^{W_{nk}+\lambda_kI_{nk}}}{\sum_{s=1}^Ke^{W_{ns}+\lambda_sI_{ns}}} $$

the last equality follows from the fact that $\epsilon_{nk}+\xi_{nk}$ is iid $Gumbel(0,1)$ by assumption on $\epsilon_{nk}$.


Translog cost function Can anyone help to estimate total cost under Translog cost function. I have input output and input cost data. It would be highly appreciated if someone can provide a software demo to estimate the same is given.

This article might be of help: https://ageconsearch.umn.edu/bitstream/122709/2/sjart_st0160.pdf


Prove the uniqueness of steady state I have a difference equation
  $$
p_t^{1-\alpha}=\alpha\sigma \left(y-p_t-\frac{(\sigma p_{t-1}^\alpha+b)p_t^{1-\alpha}}{\alpha\sigma} \right)
$$
  where $\alpha \in [0,1]$ and everything else is $>0$.

I need to prove that this equation has a unique steady state.

This is what I have done so far;

Simplified the expression to write it in the closed form as follows;
$$
p_{t-1}=\left[\frac{\alpha y}{p_{t}^{1-\alpha}}-\alpha p_{t}^{\alpha}-\frac{a+1}{\sigma}\right]^{1/\alpha}
$$
Substituted $p_{t-1}=p_t=\overline{p}$ in the closed form, this gave.
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
I'm stuck here. How can I prove that $\overline{p}$ has a unique solution?

Rearranging the steady state equation
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
we get
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\frac{a+1}{\sigma}.
$$
As $\alpha \in [0,1]$, the left hand side of the equation is increasing in $\overline{p}$ and the right hand side is decreasing. At least one of these is strictly monotonic because $\alpha$ cannot be $0$ and $1$ at the same time. Hence at most one solution is possible.


Incentive Compatibility Construction Rule Suppose we are interested in constructing an incentive compatibility constraint for an agent to announce his state truthfully.

Consider a simple case of the state space being $S=\{H,L\}$ and his payoffs to be $\pi_L,\pi_H$.

For the player to announce the state and payoff truthfully, is it correct to set the IC as:

$$\pi_H\leq\pi_L.$$

If it is correct, why is it so, and if it isn't, what is the correct way of setting up the IC and why? Is there a general rule of thumb in constructing an IC? Any reference would be also helpful.

As @brunosalcedo suggests in the comment, the agent's payoff, $\pi(a,s)$, should depend on i) the realized state $s$ and ii) the announced state $a$. Note that the announcement $a$ should in general be a function of the realized state $s$, i.e. $a:S\to S$. Incentive compatibility should thus be
\begin{equation}
\pi(s,s)\ge \pi(a',s),\quad\forall a'\ne s.
\end{equation}
In words, incentive compatibility requires that the agent's payoff when his announcement agrees with the state (i.e. the state is $s$ and his announcement is also $s$) be no lower than his payoff when his announcement disagrees with the state (i.e. the state is $s$ but his announcement is $a'\ne s$).

Check out the following book chapters for more detail:

Mas-Colell, Whinston, and Green (1995) (aka MWG) chapter 14.C discusses IC in the context of moral hazard
MWG chapter 23.B introduces IC for truthful implementation
Krishna (2010) chapter 5 is similar to the previous MWG chapter, but draws out more implications of IC.


Should I use current prices or volumes estimates? I am writing an empirical paper with National Account's data from the OECD. The analysis of the data consists basically in calculating the cross-correlations among countries of GDP, investment, consumption etc, over a period from 1960:1 to 2015:1.

The problem is that there is a huge amount of measurements available in OECD's website and I don't know which one to choose.

The OECD's website suggests that:

  To extract the historical series, the measures to be selected are: CARSA or VOBARSA for data expressed in national currency (respectively in current prices or volumes estimates) and CPCARSA or VPVOBARSA for data expressed in US dollars and GPSA or GYSA for quarter-on-quarter and year-on-year growth rates.

I think that might be adequate for me.

But I still don't know if I should choose between the data in current prices or in volumes estimates.

What are the differences between the two approaches? Which implications would I be faced with by choosing each of them?

If you are conducting empirical exercises, you should use real values i.e. volumes. If the data has time dimension, it is even more imperative to deflate the variables. The reason for this suggestion is following:

Take output data, over time output changes and usually around a rising trend. The trend growth of output could be due to rising prices instead of volume of output. The following image shows the effect of inflation on output:

The nominal GDP grows at a relatively higher rate than real GDP. The economy appears to have grown, in fact, much of this growth is due to rising price levels, not due to positive growth in volume of output. Using the data from the graph, you can see that

nominal GDP in 2012 is 1.3 (=16000/12500) times that in 2005, that's
a 28% increase in GDP.
real GDP grew by 8% only.
in the same period, prices rose by 18.5% (=1.28/1.08-1).

This shows that the volume of GDP, real output, did not grow as fast. The nominal GDP, on the other hand, gives us false impression that the economy is growing at a faster pace.

If you are using the data in statistical or econometric exercises, you should use volume data in constant prices.

And, if you are comparing performance across countries, you should convert the data to international dollars using purchasing power parity (PPP). I think OECD database offers PPP based macro-variables


Some doubts about netput vectors I have started studying producer theory on my own and there are some confusions.

We know that a production plan is $y=(y_{1},y_{2},y_{3}....y_{L})$ where $ y_{i} $ is an output if its greater than $0$ and input if it's less than $0$. The production set $Y$ is such that any $y \in Y$. The vector $y$ is also called a netput vector.

1. Take free-disposal: if $y \in Y$ and $y^{1} \leq y$
, then $y^{1} \in Y$. How are we comparing the vectors $y^{1}$ and $y$ here? Are we taking the cartesian length(norm) of both vectors and comparing? (or does it mean that each component of $y^{1}$ is smaller than the corresponding components in $y$) ?

2.Non-increasing returns to scale is given by:if $\forall y\in Y \implies ay\in Y$ for $a\in[0,1]$, then $Y$ shows non-increasing returns to scale.This  would mean that vectors smaller than $y$ are also in $Y$. The way I knew it, decreasing returns to scale means output changes less than proportionately to change in inputs. How does these two ideas relate?

3.What is the difference between the transformation frontier and an isoquant? I am having a hard time vizualizing them. What would the frontier and isoquant look like if we draw $Y\in R^{3}$ using level curves for outputs and the two axes for the two inputs?

I hope I explained my questions clearly.

Thanks

We say that $x \le y$ when $x_i \le y_i$ for each $i=1,\ldots,n$.
If the production set can be represented by a production function $F$, and $F$ is homogeneous of degree $r < 1$ ($F(\lambda x) = \lambda^r F(x)$ for $\lambda >0$), then the production set satisfies the non-increasing returns to scale property you state. Moreover, if $F(0)=0$, and $F$ is concave, the production set also satisfies non-increasing returns to scale. Hence, you can think of the definition as a generalisation of the intuitive understanding you have.
An isoquant takes the output level as given and specifies all the input combinations that give you exactly the given level of output. The transformation frontier is the frontier of the production possibilities set. The production possibilities set takes the inputs as given and specifies all the output combinations that are possible for the given inputs. You can think of the transformation frontier as the output bundles that are not wasteful (i.e. efficient).


What mechanisms are at work when the market falls dramatically one day and then rises just as dramatically the next? We've seen this scenario several times over the past few weeks:  The major stock market indexes will plummet one day and then rise dramatically (though perhaps not to the previous days starting price) on the following day.  Yes, in some of the cases there was a policy pronouncement or some such that might have had some effect, but usually not.

One would think that traders, if they were doing a normal risk/benefits analysis, would not change strategies so abruptly.  Is there some non-obvious benefit to abruptly "betting" one direction and then the other, or is some sort of mechanical market force at work?

In short...human nature! (I'm only half-kidding). Of course, mechanically, what drives these abrupt changes is simply a large amount of buyers one day...and a large amount of sellers the next.

A great metaphor you might like is Mr. Market. Also this video of Warren Buffett talking about Mr Market. It's just a metaphor used to show how capricious, and seemingly irrational, the market can be.

I don't think there's some philosophically deep mechanism at play (other than human nature). Maybe you could make the argument that new information comes out that justifies a 5-10% swing...but that argument seems flimsy to me.


Who was the first economist to write about central banking? I am interested in the origin of a central banking theory. I've heard that Karl Marx was the first author who wrote about central banking system. Was he really the first one? Or was there any other author who had written about this topic before Karl Marx?

By central banking I mean system where all currency in some country is provided by one institution. This institution has a legal monopoly and no other institution can create its own currency. Every subject in the country can use only the currency created by the central bank.

Commercial banks have existed long before any institution akin to a modern central bank was established (the first "central bank" seems to have been the Bank of Sweden, created in 1668, albeit this was a for-profit, private institution). Because of the important role of banking in financing trade, wars, and the recurrence of crises, inflation, and convertibility problems, many politicians, economists, and other individuals wrote about issues related to money theory and banking. For instance, two prominent books about monetary theory are John Law (main book in 1705) and Fernando Galiani (main book in 1751). Later on, both Adam Smith and David Hume wrote about banking. According to this source Adam Smith argued that:

the banking system is guided by the invisible hand just like any other branch of  industry.  Smith  argued  that  the  profit  motive  would  ensure  that  the  supply  of  money is self-regulating even after the development of privately issued banknotes, and that  a  significant  savings  in  social  capital  results  from  using  paper  instead  of  precious  metals   as   circulating   currency.   In   this   way   Smith   attempted   to   show   that   free   competition   in   the   business   of   issuing   currency   directs   self-interested   market   participants to unintentionally promote the social interest.

In sharp contrast, David Hume opposed for-profit, fractional reserve banking. According to this article:

Hume claims the following about the kind of bank he prefers: ‘‘No bank could be more advantageous, than such a one as locked up all the money it received, and never augmented the circulating  coin,  as  is  usual,  by  returning  part  of  its  treasure  into  commerce’’  ([1752]  1985, pp. 284–5). Hume recognizes that a bank that does not lend cannot stay in business by itself: ‘‘It would have no profit from its dealing’’ ([1752] 1985, p. 285). Therefore, rather than allowing the bank  to  earn  a  profit  by  lending  out  money  or  by  charging  fees,  Hume  suggests  a  purely
subsidized public bank, where ‘‘the state bore the charges of salaries of the directors and tellers of this bank’’ ([1752] 1985, p. 285)

A bit later, amid the Bullionist controversy (regarding whether paper money should be convertible to gold or not), a myriad of writings appeared (see link). For example, in 1801, Walter Boyd wrote a letter to the primer minister William Pitt advocating for full convertibility, in order to avoid high inflation associated with private banks incentive to expand paper money under limited convertibility.
A notorious economist at that time was Henry Thornton, who wrote in 1802 the book "An Enquiry into the Nature and Effects of the Paper Credit of Great Britain", partly in response to Boyd's views. Apparently, some (?) consider Thornton as the founder of central banking.
Later on, Ricardo also wrote about central banking. In particular, he argued in favour of central bank independence (yet, remaining a for-profit institution). This is a quote from 1824 (emphasis mine):

It is said that Government could not be safely entrusted with the power of issuing paper money; that it would most certainly abuse it ... There would, I confess, be great danger of this if Government – that is to say, the Ministers – were themselves to be entrusted with the power of issuing paper the hands of Commissioners, not removable from their official situation but by a vote of one or both Houses of Parliament. I propose also to prevent all intercourse between these Commissioners and Ministers, by forbidding any species of money transactions between them.
The Commissioners should never, on any pretense, lend money to Government, nor in the slightest degree be under its control or influence ... If Government wanted money, it should be obliged to raise it in the legitimate way; by taxing the people; by the issue and sale of exchequer bills; by funded loans; or by borrowing from any of the numerous banks which might exist in the country; but in no case should it be allowed to borrow from those who have the power of creating money.

(The quote is from here, but it belongs to Ricardo's 1824 pamphlet "Plan for the Establishment of a National Bank", which can also be found as an appendix on this book)
Meanwhile, others consider the "founding father" of central banking to be Walter Bagehot, a "famous" editor of The Economist (you might have noticed that there is still a column under his name) that in 1873 published the book "Lombard Street: A Description of the Money Market", where he advocated for the role of the Bank of England (still a private, for-profit institution) as a lender of last resort in the case of a financial crisis. Despite of this, he also advocated for a free banking sector. A quote (from here):

I shall have failed in my purpose if I have not proved that the system of entrusting all our reserve to a single board, like that of the Bank directors, is very anomalous; that it is very dangerous; that its bad consequences, though much felt, have not been fully seen; that they have been obscured by traditional arguments and hidden in the dust of ancient controversies.
But it will be said — What would be better? What other system could there be? We are so accustomed to a system of banking, dependent for its cardinal function on a single bank, that we can hardly conceive of any other. But the natural system — that which would have sprung up if Government had let banking alone — is that of many banks of equal or not altogether unequal size. In all other trades competition brings the traders to a rough approximate equality. In cotton spinning, no single firm far and permanently outstrips the others. There is no tendency to a monarchy in the cotton world; nor, where banking has been left free, is there any tendency to a monarchy in banking either.

For further references, check the book "Monetary Theory and Policy from Hume and Smith to Wicksell: Money, Credit, and the Economy".


An explanation of the formula for the present value of an annuity I'm reading a book that gives three formulas. In all of these formulas:

$x$ is the amount of money to be received in the future
$n$ is the number of years
$r$ is the interest rate.

The first is the present value of a lump sum: $PV = x / (1 + r) ^n$. Pretty self explanatory. Suppose I wanted $100$ dollars in $20$ years and could invest money at a $5\%$ rate of return, what amount of money would I have to invest today to have $100$ dollars in $20$ years? $100 / (1.05) ^{20} = 37.69$. Alternatively, if I have $100$ dollars today and it depreciates at the rate of $5\%$ a year for $20$ years, it would be worth $37.69$ dollars in $20$ years.

The second formula is the formula for the present value of a perpetuity: $PV = x / r$ ($x$ is the amount of money to be received at the end of each year forever. So getting $100$ dollars at the end of each year forever would be the same as getting $2000$ dollars today, assuming I can invest that at the rate of $5\%$ per year.

The third formula appears to combine the two: the formula for the present value of an annuity: $PV = (x / r) - ((x / (1 + r) ^ n) / r)$ ($x$ is the amount of money received at the end of each year for $r$ years).

Notice that the left hand side is exactly the formula for the present value of a perpetuity. The right hand side is the present value of a lump sum divided by $r$.

I'm trying to understand the third formula. How would someone derive it? Why do we subtract the present value of a lump sum from the present value of a perpetuity? I understand intuitively that an annuity must have a lower present value than a perpetuity. If $r = 20$, there needs to be some way to indicate that payments stop after $20$ years. I'm just not sure how this is incorporated into this formula.

Take your second statement: $\dfrac{x}{r}$ is the present value of a perpetuity paying $x$ starting from now (first payment in year $1$)

Now consider the present value of a perpetuity paying $x$ with first payment in year $n+1$.  This is the value in year $n$ of $\dfrac{x}{r}$, discounted back to now so dividing by $(1+r)^n$ to give $\dfrac{x}{(1+r)^n r}$

So the present value of an annuity paying $x$ for the first $n$ years is the difference between these, namely $\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$

An alternative approach would be to consider the sum of the present values of the $n$ payments $$S= \dfrac{x}{(1+r)} + \dfrac{x}{(1+r)^2} + \cdots + \dfrac{x}{(1+r)^{n}}$$ and you might calculate this using the following to allow cancelation  $$(1+r)S - S = x -  \dfrac{x}{(1+r)^n}$$ which since $(1+r)S-S=rS$ becomes $$S=\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$$

It may be worth noting that the second term tends to zero if $n$ increases, so the value or cost of a long-term annuity approaches that of a perpetuity


Creation of Money - Hypothetical Situation Imagine you sell me some consulting services. I don't have any cash, so I give you a $100 IOU instead. You now have a $100 Note Receivable on your Balance Sheet.

My question is: Did we just create money?

No, you didn’t. All money is debt, but not all debt is money.

There are two easy ways of thinking about this. The first is to think about the properties of money: unit of account, store of value, medium of exchange. Focusing on the latter two:

Is the money you owe to me a good store of value? No offense, but I’d rather have cash or keep it in an insured bank account. So no, your debt to me isn’t a good store of value.
Can I use it as a medium of exchange (i.e., can I trade your debt to me to someone else as payment for something)? Not likely, so no, it’s not a medium of exchange.

The second is to do the accounting exercise and compare it to the case where a bank makes a loan. I’ll bold all the things that count as money in the hands of people in the real economy. In this case, it’s:

  Me

  My assets: \$100 owed from you
  My liabilities: \$0

  You

  Your assets: \$0
  Your liabilities: \$100 owed to me

Now compare the case where I deposit \$100 into the bank and the bank lends out \$50 to you (I’m going to ignore bank capital to make it clearer):

  Me

  My assets: \$100 owed from the bank (bank deposit).
  My liabilities: \$0

  Bank

  Bank assets: \$50 cash + $50 owed from you
  Bank liabilities: \$100 deposit owed to me

  You

  Your assets: \$50 cash
  Your liabilities: \$50 bank loan

You’ll notice that in the first case, no money has been created, because I can’t pay anyone with your promise. In the second case, I have \$100 of money, because I can pull \$100 out of the bank (again, ignoring bank capital here) and pay someone with it, and you have \$50 of money because you have \$50 in cash (though you also have an outstanding loan for \$50), so the amount of “money” in the real economy has increased.


How is monetary policy fair in the current economy? I've just read Money Creation in the Modern Economy, an article published by the Bank of England.

This article brings about a lot of questions in my mind. This article talks about money being created through commercial bank loans, and that central banks only have the authority to set interest rates or to employ quantitative easing in order to stimulate the economy. Loans being repaid should theoretically destroy the money that was created and balance the books, but inflation grows in that economy as banks earn interest (and in this case, the interest is actually the money that never existed, assuming that they destroy the rest of the money paid back).

So, my questions is:

How is this fair for every other constituent in the economy who has earned money the hard way, by working and not by generating it so freely like interest returns on loans do for commercial banks?

How is inflation fair, as populations increase and every time someone gets a bank loan, new money is added to the economy over time as they pay back that loan, and everyone else's currency is de-valuated?

Consider country X which has one constituent named Roger who works at a commercial bank in X and, coincidentally is also the president of X. He gets a loan from his bank for 100 dollars (with an interest rate of 5%), however, in his economy, only 100 dollars of money exists as actual currency and he has 0 dollars in his bank account. He will never be able to pay off his interest rate of 5% without getting another loan. His bank or state will coincidentally not be able to pay him his wage over time without printing new money from a central bank or giving his bank a loan, or, perhaps a bail-out. All this interest will do is inflate the economy, and you can only fight interest with interest.

Now apply this concept to our current economic system on a macro level, and it feels to me like it will ultimately fail as a system, because inflation can only combat inflation for so long until the price of every asset reaches infinity.

The description you're providing of how interest works is based on a couple of fairly common misunderstandings about how the system actually works, so lets clear that up first.

Fractional reserve banking and the gold smith banking systems preceding it, are a result of the development of an accounting technology called double entry book keeping in the 13th century.

Double entry book keeping grossly simplified. Each transaction must consist of a debit and a credit tuple, occurring in two separate books. Asset books are traditionally written on the left, and liability and equity on the right. On the right hand side (liability and equity side), credit adds, debit subtracts, and its the opposite on the left hand side, debit adds, credit subtracts. Just remember the right hand side is what you think it should be, the left hand side is the opposite.

When a bank makes a loan, it performs the book keeping operation:

[debit loan, credit customer account].

This creates the loan, and it creates money in the form of the matching deposit. When  loan capital is repaid, the operation is:

[credit loan, debit customer account]

and this removes the deposit money that was created.

So how does interest get paid? Well in your example, Roger gets a loan for 100, lets say it's a 10 month, simple interest loan (easier math.)

The first month, Roger repays 10 capital, and 5 in interest. That's:

[credit loan 10,        debit Roger account 10]

[                       debit Roger account 5,
                        credit bank interest income account 5]

The bank's interest income account is also classified as a liability, although not strictly as a deposit account. Once the bank has recognised that income, it can then use it to pay expenses. Let's assume Roger works for the bank, and the Bank pays Roger a monthly salary of 5. That just reverses the operation above, ie.

[                       debit bank interest income account 5,
                        credit Roger account]

The next month, Roger pays another 10 in capital, and pays 5 interest, and gets the interest back in salary. This continues until the loan is repaid.

The key concept is that in an economy, money is exchanged and individual units of money are constantly circulating through the economy and being re-used, as in the example above. It's a flow system essentially, and this applies to interest payments just as much as payments for food, etc.

Money in a fractional reserve monetary system today also predominantly takes the form of liability bank accounts, rather than physical currency, and most banking operations these days operate as transfers from bank account to account. The writers who concentrate on physical currency in describing this system seem to miss this point, and consequently get key points incorrect.

This doesn't mean that there isn't a very challenging accompanying inflation problem. Generally over time, banking systems tend to create more new loans at a slightly higher rate than they are repaid, and so the system is in a more or less continuous state of growth. There's no reason why it couldn't be regulated to be stable - but the problem is nobody at this time knows what that would do to the economy. The best guess at the moment is that a low but positive rate of expansion is optimal.


How to calculate cost of equity? So i have this question on leverage and cost of capital:

Leverage and the cost of capital Gamma Airlines has an asset beta of 1.5. The risk-free interest rate is 6%, and the market risk premium is 8%. Assume the capital asset pricing model is correct. Gamma pays taxes at a marginal rate of 35%. Draw a graph plotting Gam- ma’s cost of equity and after-tax WACC as a function of its debt-to-equity ratio D/E, from no debt to D/E = 1.0. Assume that Gamma’s debt is risk-free up to D/E = .25. Then the interest rate increases to 6.5% at D/E = .5, 7% at D/E = .8, and 8% at D/E = 1.0. As in Problem 21, you can assume that the firm’s overall beta (βA) is not affected by its capital structure or the taxes saved because debt interest is tax-deductible.

The answer sheet is this ( only a part of the table is shown)

..

I understand how they got to the RA and how it doesn't change according to the capital structure.

I also know how to calculate D/V from D/E.

The problem is that i dont seem to understand which formula I should use to calculate RE.

If , for example, i use this formula

RE=RA+(RA-RD)D/E for D/E equal to 0.10 i get

RE=0.180+(0.180-0.06) · 0.10 = 0.192 which is not 0.1941.

Then which formula should i use ?

R=rf+B(rm-rf)?

Why do i need to calculate DV?

Somebody please explain how i can get accurate results of RE using which formula? Thanks i greatly appreciate it.

Your formula is wrong. The derivation for the cost of equity $r_E$ is actually:

$r_A = r_D(1-t)\frac{D}{V}+r_E \frac{E}{V}\\
r_E \frac{E}{V} = r_A -  r_D(1-t)\frac{D}{V}\\
r_E = r_A \frac{V}{E} -  r_D(1-t)\frac{D}{V} \times\frac{V}{E}\\
\mathbf{r_E = r_A \frac{V}{E} -  r_D(1-t)\frac{D}{E}}$

Since $\frac{D}{V}=\frac{D}{D+E}=0.09091$ when $r_A=0.1941$, then $E=10$ which means $D=0.10E=1$, making $V=D+E=11$ for use in the above.

$r_E = 0.180 \times\frac{11}{10}-0.06(1-0.35)\times 0.10 = 0.1941$


Why is so much of macroeconomic theory dedicated to economic growth? I'm an economics PhD student and throughout my macroeconomics education I've found the topic of economic growth always being discussed at great length. From an Naive perspective id expect that one of the main topics should be around the distribution of income and inequality (i.e. Bewley models) as these seem to be some of the more policy driven discussions these days (at least in the first world hemisphere).

With this in mind, why is economic growth such a core part of macroeconomic analysis?

Have you read:

Barro and Sala-i-Martin (2004, 2e), Introduction
Acemoğlu (2009), Introduction
Paul Romer (2016), "The Deep Structure of Economic Growth"
Robert Lucas (1988), "On the mechanics of economic development", Introduction:

  I do not see how one can look at figures like these without seeing them as representing possibilities. Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia's or Egypt's? If so, what, exactly? If not, what is it about the 'nature of India' that makes it so? The consequences for human welfare involved in questions like these are simply staggering: Once one starts to think about them, it is hard to think about anything else.

Ultimately what one believes is the most important topic is a subjective/normative question, but I think the above pieces make a strong case that economic growth should be it.


Link between Productivity and Labour Force? In this article from VoxEU, the author states:

  By definition growth in output per capita equals growth in labour
  productivity times growth in hours per capita. The slowdown in
  productivity growth that began 40 years ago was partly offset between
  1972 to 1996 by an increase in the labour-force participation rate of
  0.4% per year, as females and baby-boom teenagers entered the labour force. In contrast during 2004-2014 the participation rate has
  declined at an annual rate of 0.5%, and over the shorter 2007-2014
  interval at an annual rate of 0.8%. This transition from a 0.4%
  increase to a 0.8% decline accounts for a 1.2% reduction in the growth
  of per-capita real GDP for any given growth rate of labour
  productivity.

I don't understand how the participation rate influences the growth of GDP per capita. Is it because in the per capita, besides the labour force, we also take into consideration the rest of the population? And what's the link between productivity and labour force?

Any help would be appreciated.

Output per capita is $\dfrac{Y}{N}$. This can be decomposed as:

$$ \frac{Y}{N} \equiv \frac{Y}{h} \times \frac{h}{L} \times \frac{L}{A} \times\frac{A}{N} $$

where $h$ are aggregate hours of work, $L$ is number of employed, and $A$ is active population. $\frac{Y}{h}$ is aggregate labour productivity, $\frac{h}{L}$ is "work intensity", $\frac{L}{A}$ is the employment rate and $\frac{A}{N}$ is the participation rate.

The above equality can be turned into growth rates, which means in approximation, the growth of output per capita is the sum of the growth rates of each of the subcomponents.


Microeconomics of GDP Macroeconomists tend to study GDP in terms of macro variables.

Microeconomists study general equilibrium (among other thinga of course).

Is there theory about the relation between general equilibrium and GDP? Where should I look for this?

Macroeconomics nowadays tend to require microfoundations. As a result, general equilibrium are more of a topic for macro than micro analysis. The dynamic stochastic general equilibrium (DSGE) model and its variants underlie much of macroeconomic analysis nowadays.

Standard texts:

Stokey and Lucas (1989) Recursive Methods in Economic Dynamics
Ljungqvist and Sargent (2004) Recursive Macroeconomic Theory


Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the simple case of $U(x,y) = \ln(x) + \ln(y)$ s.t. $p_x x + p_y y = I$. I get that the Hicksian and Marshallian demand functions are $x^* = \frac{p_y}{p_x} y = \frac{I}{2 p_x} $ respectively. Plugging these in gives me this for the decomposition:

$$ \frac{\partial x^*}{\partial p_x} = - \frac{p_y}{p_x^2} y - \frac{1}{2 p_x} \frac{p_y}{p_x} y $$

which is definitely not zero. Am I misunderstanding?

You interpreted the passage you read as essentially saying that "with log preferences, the Marshallian demand for the good does not depend on its price."
That doesn't sound very plausible, don't you think?

But the passage clearly refers to an intertemporal choice problem, it is about log-preferences over a single good and over time, it does not relate to the Slutsky decomposition of the static Marshallian demand function.

I would suggest to continue reading the notes up to page 14.


Extending the Muth-Mills Model to Different Income Groups? I am exploring the generalized treatment of the Muth-Mills monocentric city model covered in Brueckner's article. The model is based on an assumption of homogenous income levels across the city. However, a more realistic assumption would be to assume the presence of different income groups, which theoretically results in different bid-rent curves. How might I modify the model given in the article to reflect this?

One way to expand the model is to allow for different income classes as stated in assumption (1) below. Let the model be completely standard with all the usual assumption of the monocentric city model as laid out in Brueckner (1987) the paper you cite.
I will only present the basic idea of the expanded model. Hence I do not give proof of existence of equilibria, proof of uniqueness, nor a comparative static analysis or numerical procedures to solve for equilibrium given values of exogenous variables.
I will allow myself to assume Cobb-Douglas preferences with residents utility given as
$$U(c,h) = A\left(\frac{c}{1-\alpha}\right)^{1-\alpha}\left(\frac{h}{\alpha}\right)^\alpha,$$
where $c$ is composite good and $h$ is square meters of housing.
It is well known that the Marshall demands are $c^\star=(1-\alpha)I/p_c$ and $h^\star=\alpha I/p_h$ with $I$ being income. Plugged into utility function results in the value function
$$V(p_c,p_h,I) = \frac{AI}{p_h^\alpha p_c^{1-\alpha}} = \frac{I}{p_h^\alpha},$$
having assumed that $p_c=1$ and $A=1$. The value function tells us that the max price a consumer can pay for a unit of housing $h$ if she is to attain the utility level $u$ is
$$\frac{I}{p_h^\alpha} = u \Leftrightarrow p(I,u) = \left(\frac{I}{u}\right)^\frac{1}{\alpha},$$
found by inverting the value function to express price as a function of income and utility (which offcourse is the well known bid-function).
The next step is to find the bid-lot size which is the amount of housing consumed by a resident attaining the utility level $u$ having income $I$. To derive this I will simply use that the Marshall demand $h^\star_M(p,I)$ with bid-function $p(I,u)$ substituted for price is the bid-lot size $h_b(u,I)$ [same holds offcourse for the Hicks demand]. Hence the bid lot size is given as
$$h^\star_M(p(I,u),I) = h_b(I,u) = \frac{\alpha I}{p(I,u)} = \alpha I^{1-1/\alpha}u^{1/\alpha},$$
which is decreasing in income because when consumer gets higher income and utility attained is held constant the price offered for housing is increase and hence less housing consumed. Importantly this is not a description of equilibrium behavior, what is being said is that if the consumer has higher income she can offer a higher price for a unit of housing while retaining utility level $u$ but that will require her to substitute away from housing over to consumption of $c$.
I now add the linear geometry and linear transport costs of standard monocentric city expositions. Hence the city expands from center at $d=0$ to potentially infinity $d \in [0,\infty)$ and transport costs are $t(d) = t d$ such that income net transport is $I(d) = w-td$.
I now introduce the extending assumption that
Assumption (1): There are two income classes $j=\{1,2\}$ with incomes exogenously given $w_2>w_1$ and size of population income segments exogenously given as $N_1$ and $N_2$.
and for ease of exposition I will let housing be exogenously
Assumption (2): At location $d$ the housing is given as $H(d)$,
the second assumption is not so restrictive as it may appear (but that is easier to illustrate later).
Given the wages I have income net transport for the two classes $I_1(d) = w_1 - td$ and $I_2(d)=w_2 - td$ which are plugged into bid function $p(u,I)$ and bid lot size $h_m(u,I)$ to get the main result that
$$(3)\ \ \ p_j(u_j,d) = \left(\frac{w_j - td}{u_j}\right)^{1/\alpha} \ \ \ \ h_j^m(u_j,d) =\alpha (w_j - td)^{1-1/\alpha}u_j^{1/\alpha},$$
which is really not new in the sense that it is completely like the model with homogenous residents as in the Brueckner (1987) model just adding Cobb-Douglas preferences.
To illustrate the use of these functions select parameter values for $(w_1,w_2,\alpha,t)$. Assume existence of equilibrium with equilibrum utilities $u^\star_1$ and $u_2^\star$ and select values for these ($u_2^\star > u_1^\star$ rich get higher utlity in equilibrium). Also select a value for the alternative use of land $r_a$ referred to as the land rent (not to be confused by the rent paid for land by producers of housing if that sector was endogenized in the model).
In the figure below the bid function $p_j(u_j,d)$ of the high and low income are illustrated. Residents will live where they bid the highest value for housing. The city expands to where the bid-rents of either income class is higher than the land rent.

Two observations are made:
Observation (1): The low income resident will live in the center of the city (this requires housing to be normal good)
Observation (2): There are perfect income stratification with low income in one area and high income in another area
To close the model the housing market must be in equilibrium. Let $[0,\hat d]$ be the locations where the low income reside. Then for $d \in [0,\hat d]$ housing demand is given as $h_{j}^m(u_{j},d)$ per person and units of housing is $H(d)$ so population density at $d$ must be $H(d)/h_{j}^m(u_{j},d)$ and integrating over $[0,\hat d]$ this must be equal to $N_1$ the number of low income residents I have exogenously stipulated to live in the city. So equilibrium conditions includes that
$$N_1 = \int_0^{\hat d} \frac{H(t)}{h_{1}^m(u_{1},t)} dt$$
and
$$N_2 = \int_{\hat d}^{\bar d} \frac{H(t)}{h_{2}^m(u_{2},t)} dt.$$
As is hopefully apparent from the figure above it is the case that: Given two utilities $(u_1,u_2)$ the bid functions determine $\hat d$ and $\bar d$ - where $\bar d$ is the value of $d$ where $p_2(u_2,d) = r_a$ - and given these values the integrals can be calculated to get $N_1(u_1,u_2)$ and $N_2(u_1,u_2)$. When the equilibrium is unique these equations
$$N_1 = N_1(u_1,u_2) \ \ \ N_2=N_2(u_1,u_2),$$
will be invertible such that equilibrium utilities can be found from the exogenously given $N_1$ and $N_2$.
Proof of existence and uniqueness of equilibrium for a model with $K$ income classes and standard restrictions on the utility functions (not assuming Cobb Douglas) are given in
John Hartwick (1976) Comparative Statics of a Residential Economy with
Several Classes Journal of Economic Theory 13, 396-413.
which also include comparative statics of the model.
Proof of existence and uniqueness of equilibrium for a model is also given in
Fujita (1989) Urban economic theory: Land use and city size
which also offers an algorithm for solving for the equilibrium.
See also
Puga and Duranton (2015) Urban Land Use
for discussion of main results of this and other extensions and empirical relevance of the models (but not detailed exposition).
Note on housing supply: Instead of assuming housing to be exogenously given I could have followed Brueckner (1989) and assumed land to exogenously given $L(d)$ and then housing be produced by constant returns to scale production under perfect competition using capital and land with the capital supply being infinitely price elastic and having price $r_K$ in which case housing supplied is a function of $d$ through $p(d)$. So housing supply is a function of the price $H(p)$ but at any distance $d$ the price residents are willing to give is set by the bid function which is a function of $d$ hence either way we get $H(d)$.
Here is some really fast written R-code implementing the solution algorithm in Fujita (1989):
library(rootSolve)

a <- 0.35
t <- 200
w <- c(6000,12000)
N <- c(500,500)
J <- 2
x <- rep(1,J+1)

u <- rep(170,J)

# Function to make bid-max lot size for agent of type j
# with income and utility (w_j,u_j).
# Bid-max lot size is a function of distance to center d
# given type specifics (w_j,u_j).

make_g <- function(u_j,w_j,t,a)
    {
        g <- function(d)
            {
                nominator <- a*u_j^(1/a)
                denominator <- (w_j-t*d)^((1-a)/a)*.6*pi*d
                bid_lot <- nominator/denominator
                # return should be 1/bid_lot
                # but denominator can be 0 so do not calculate it
                # and use it as denominator instead
                denominator <- ifelse((w_j-t*d)>0,denominator,0)
                return(denominator/nominator)
            }
        return(g)
    }

# Solve the integral of g with respect to r from x_L = x_j to x_U = x_j+1
# such that the integral equals N_j

# Initiate with j=1 so x_l is set to 0 and u_j is some guess
j <- 1
x_L <- x[j]
u_j <- u[j]

    # Get exogenous income and population size for relevant segment
    # of population.
    w_j <- w[j]
    N_j <- N[j]

    g <- make_g(u_j,w_j,t,a)
    v <- function(x_U) {return(N_j-integrate(g,x_L,x_U)$value)}
    x[j+1] <- uniroot(v,c(x_L+0.0001,500))$root

# Initiate j=2
j <- 2

    # Find u_2 given I_1,I_2 at x and u_1
    u[j] <- u[j-1]*(w[j] - t*x[j])/(w[j-1]-t*x[j])

x_L <- x[j]
u_j <- u[j]

    # Get exogenous income and population size for relevant segment
    # of population.
    w_j <- w[j]
    N_j <- N[j]

    g <- make_g(u_j,w_j,t,a)
    v <- function(x_U) {return(N_j-integrate(g,x_L,x_U)$value)}
    x[j+1] <- uniroot(v,c(x_L+0.0001,500))$root

# Calculate price at x[j+1]
j <- 3
p_edge <- ((w[j-1] - t*x[j])/u[j-1])^(1/a)
x
u
p_edge

p_max <- (w[1]/u[1])^(1/a)+10
plot(x[1]:(x[J+1]+1),type="n",ylim=c(0,p_max),ylab="housing price",xlab="distance to city center")
colors <- c("blue","red")
for (j in 1:J)
    {
        x_L <- x[j]
        x_U <- x[j+1]

        d <- seq(0,x[J+1],length.out=100)
        u_j <- u[j]
        p <- ((w[j] - t*d)/u_j)^(1/a)
        points(d,p,type="l",col=colors[j])
    }

abline(v=x[2],col="grey")
abline(h=p_edge,col="black")
legend(12,20000,c("bid function (low income)","bid function (high income)","sorting threshold",
"land rent"),
col=c("blue","red","grey","black"),cex=0.7,bty="n",pch=24)
#savePlot("monocentric.jpg",type="jpg")


What was the "invariable measure of value" problem and how did Sraffa solve it? Fonseca (HET Website) writes:

Only much later did Piero Sraffa (1960) finally solve the "invariable measure of value" problem and re-ignited interest in Ricardo's theory.

What was the "invariable measure of value" problem and how did Sraffa solve it?

The working paper by  Kurose & Yoshihara (2014) addresses both of your questions.
What was the "invariable measure of value" problem:
According to the Kurose & Yoshihara:

An invariable measure of value can be defined as a measure that is invariable
with respect to changes in both income distribution and technique (Ricardo, 1951A,
chap. 1). The advantage of the invariable measure of value, if it exists, is that we can
distinguish between the variations which belong to the commodity itself and those
which are occasioned by a variation in the medium by which values or prices are
expressed, when relative prices change (Ricardo, 1951A, p. 48).

The invariable measure of value problem is the problem of finding such measure. Ricardo considered gold/silver money as good although flawed approximation. The authors provide more detailed overview of the idea at the start of chapter 2.
How did Sraffa 'solve it':
I put the '' around solve it as whether he indeed solved the issue is somewhat up to the debate (see the paper sec. 2.3), but the Sraffa's solution was as as follows (Kurose & Yoshihara, 2014):

... Sraffa (1960) revived the concern about the invariable measure of value, which had fallen into oblivion since the so-called Marginal Revolution.
Unlike Ricardo, he divided the problem of identifying an invariable measure of value
into two parts: the first is to search for a measure of value that is invariable with
respect to changes in technique, left aside the change in income distribution, and
the other is to search for a measure of value that is invariable with respect to the
change in income distribution, left aside the change in technique. Sraffa concentrated
on the latter by constructing a special, composite commodity termed the standard
commodity. He also demonstrated an interesting relationship with respect to income distribution if the standard commodity is adopted as the numéraire: the linear
relationship of income distribution.

Again you will find more detailed overview of Sraffa's solution in the paper as well as generalization of said solution by the authors.


Which is the origin of the "economic" approach to production? Economic theory states that under competitive markets, economic profits are zero (even though accounting profits are not zero). What is the origin of this analysis? Who was the first author to write about this? Who was the first to put this into mathematics? I don't know how to search for this. I know Alfred Marshall is an important economist, who wrote a widely used book, but I do not have access to it. Is it him?

Any references are most welcomed.

The book from Alfred Marshall you are talking about is "Principles of Economics"

Principles of Economics PDF

As much I know the idea for Economic Surplus (Consumer Surplus and Producer Surplus) originated from Alfred Marshall, he is considered the person who "put" mathematics in the Economics theory.


Why is the oil future price negative? I don't understand why the price of oil futures went negative today.

News say it is because sellers cannot sell it. But, why don't they just hoard it until the coronavirus crisis is off?
Why don't speculators jump in and bring up the demand again to positive?
Alternatively, I would like to get paid for receiving a barrel of oil (I can keep it in my back garden) and then sell it in the future.

I'm answering from my memory of Reddit comments like blitzkrieg9
 and AdmiralAkbar1. Don't hesitate to edit this post to link to them, but I can't remember where exactly I saw them. And I haven't just copied and pasted – I've rewritten out typos.

  Alternatively, I would like to get paid for receiving a barrel of oil (I can keep it in my back garden) and then sell it in the future.

Unprocessed crude oil is full of toxic and nasty stuff, and if not properly contained, many dangerous gasses like benzene or hydrogen sulfide will just evaporate out of it. You definitely don't want to be anywhere near an open swimming pool full of crude oil.

You can't really do anything with it at home, and if you dump it, you go to prison. You get nasty byproducts that you'll have trouble selling in the tiny quantities you produce, but which are illegal to dump. But hey, you might get a gallon or two of usable gas out of it!

  But, why don't they just hoard it until the coronavirus crisis is off?

Crucially, this is the price for crude oil which still has to be refined into usable products, and cannot easily be disposed of. The negative price reflects the costs associated with storing the stuff indefinitely.

And you can't turn off producing wells like a tap. It takes time to shut them down without ruining them, so that they can be turned on again. Look how long it took to get Kuwait up and running again after the first Gulf War, when the retreating Iraqis destroyed wells.

It's simple to shut in natural gas by turning the valve, as natural gas has a natural pressure. Oil does not have natural pressure. Once oil's shut in, it's not easily "turned back on". It's part of what causes wild price swings as supply can't move as fast as demand.

The leases for oil and gas in America have "shut-in clauses" where after being shut in for a certain period of time (usually one year or so, but they vary) the lease holder has to go back to the mineral owner, to re-lease. Again with gas, operations can go and turn on a gas well for a month, then turn it back off just to get enough production time to avoid the shut-in clauses. You can't do this with oil.  Even old stripper wells need about $18-20 to pay to keeping them operating.

Even if you can stop extracting oil, the oil producing countries are often in a sort of Mexican stand-off. If you produce too much, oil drops and you lose money. If you produce less, the other countries may produce more and you lose money.

  Why don't speculators jump in and bring up the demand again to positive?

A futures contract is simply a contract to buy something in the future, like next month or next year. Some futures contracts "settle to cash". This means we determine the price when I bought the contract and the actual real-world price on expiration date, then one of us pays the other person the price difference.

But oil and most commodity contracts settle for the actual product! If you don't close out your soybean contract, your exchange calls you asking what you'd like to do with your soybeans that are waiting for you in Kansas City! Same with oil. Tankers are showing up in Houston with millions of gallons of oil and somebody needs to take delivery. Nobody can, because all the storage facilities are already full. So people are literally paying you over $30 a barrel for you to possess oil!

You're referring to a specific contract for specific delivery to an oil pipeline in Oklahoma on Apr 21 2020. That is the crux of the issue. The supply vs demand kerfuffle is very localized at a major crude oil pipeline junction in middle America. This is unprecedented. But it is a LOCALIZED phenomenon.

What's happening now is what happened in Trading Places when the real report comes out: people realized that they overpaid for their futures and are selling to try and recoup some of their losses. There's less demand for oil right now, there's less hope of more demand for oil in May, and there's little storage space left. Low demand + high supply = low prices. A few days ago, the settlement price for a barrel of oil for futures contracts with May settlement dates was locked at ~\$18 a barrel. People expect it to be way lower than \$18 next month. So they're trying to sell off their futures contracts. It's going negative because they're literally paying you to take it off their hands if you have room for it. Although they know it'll be a loss, they figure it's less of a loss than holding onto worthless oil that they can't store.


Trembling hand perfection and weakly dominated strategies It is well known that players cannot use weakly dominated strategies in a trembling hand perfect equilibrium. My question, however, is a little different: does iterated deletion of weakly dominated strategies preserve trembling hand perfect equilibria? In other words, if $\sigma$ is a trembling hand perfect equilibrium of a finite game $G$, is $\sigma$ also a trembling hand perfect equilibrium of the game $G'$ obtained by iteratively deleting weakly dominated strategies from $G$?

Apologies in advance if the answer is obvious!

A trembling hand perfect (THP) equilibrium may not survive iterated deletion of weakly dominated strategies, as is shown by the following example. Let game $G$ be

\begin{array}{|c|c|c|c|}\hline
&A&B&C\\\hline
A&0,0&0,0&0,-2\\\hline
B&0,0&1,1&-1,-2\\\hline
C&-2,0&-2,-1&-2,-2\\\hline
\end{array}

There are two pure strategy NEs: $(A,A)$ and $(B,B)$, with the former involving iteratively weakly dominated strategies. And yet $(A,A)$ is a THP equilibrium. To see this, consider the following sequence of totally mixed strategies
\begin{equation}
\sigma_i^k=\left(1-\epsilon^{-2k}-\epsilon^{-k},\epsilon^{-2k},\epsilon^{-k}\right),\qquad i\in\{1,2\} \text{ and }\epsilon>2,
\end{equation}
which converges to $(1,0,0)=A$ as $k\to\infty$. It is easy to verify that, for any $k$, the best response of player $j\ne i$ to $\sigma_i^k$ is $A$:
\begin{align}
u_j(A,\sigma_i^k)&=0 \\
u_j(B,\sigma_i^k)&=\epsilon^{-2k}-\epsilon^{-k}<0 \\
u_j(C,\sigma_i^k)&=-2<0.
\end{align}
Therefore, $(A,A)$ is THP.

However, the game $G'$ that survives iterated deletion of weakly dominated strategies does not contain the strategy $A$ for either player. Hence, the THP equilibrium $(A,A)$ of the original game $G$ is not preserved in game $G'$.


State Level Unemployment Data I thought that the St Louis FED was the ultimative resource for macro data, but even they don't appear to have a time series of state level unemployment numbers for the United States.

Who does?

The Bureau of Labor Statistics Local Area Unemployment series of data provides monthly figures at the state level from 1976-2014 downloadable as Excel, CSV, or images.

I looked into the following Series:

LASST100000000000003
LASST100000000000004
LASST100000000000005
LASST100000000000006

LAUST100000000000003
LAUST100000000000004
LAUST100000000000005
LAUST100000000000006

Where LAU meant not seasonally adjusted, and LAS was seasonal, with the 3,4,5,6 standing for labor force, employment, unemployment, and unemployment rate, respectively, and the 10000000000000 (I believe) indicating the region I chose (in this example, the State of Delaware).

You can use the One Screen Search Tool to find whatever you need.


Does Bangladesh and India's economy face similar economic obstacles? I had a question : Does Bangladesh and India's economy face similar economic obstacles (Poverty, education, unemployment etc) or are they different in their own terms ? I'd be extremely grateful if anyone could answer this question as I really need it for my EPQ. Many thanks.

A good starting point is to look at the World Development Indicators (WDI) of the World Bank. You may compare Bangladesh and India's economy over hundred indicators.

For instance, you will find that the percentage of the population living below the national poverty lines was around 31.5% in Bangladesh (in 2010) versus 21.9% in India (in 2011). The 2014 annual report documents that the unemployment rate was quite low in both countries between 2008 and 2012, around 4-5%.


How to take the derivative and show that MPL in the Solow model is equal to real wage w I am taking macro course this Fall and my calculus is quite rusty. So in the lecture notes they derive the following:
$$MPL = dY/dL = d(ALf(k))/dL = Af(k) + ALf'(k)  (-K)/(L^2  A) = A(f(k) - kf'(k)) = w$$
Specifically, I don't quite understand how they got $Af(k) + ALf'(k) (-K)/(L^2  A)$ from $d(ALf(k))/dL?$
Honestly, I am stuck at this point and guess will be doing a lot of differentiation like this during the entire course.

They used a product rule, quotient rule and chain rule, I won't be discussing the rules themselves  as such topics belong to Mathematics.SE, but the reason why you might not realize this outright is the particular notation in a growth theory, which would be on topic here so I will focus on that.
In Solow growth model with labor augmenting technological change $f(k)$ is the production function expressed per 'effective worker' so:
$$f(k) = F \left(\frac{K}{AL},1 \right)$$
so you might as well rewrite the problem as:
$$d\left(AL F \left(\frac{K}{AL},1 \right) \right)/dL=A F \left(\frac{K}{AL},1 \right) +AL F '\left(\frac{K}{AL},1 \right) (−AK)/(LA)^2$$
Now we can cancel some terms and use the fact that $k =K/AL$:
$$A F \left(\frac{K}{AL},1 \right) + F' \left(\frac{K}{AL},1 \right) (−K)/(AL)=A F \left(\frac{K}{AL},1 \right)  -A k F' \left(\frac{K}{AL},1 \right)  $$
Now finally go replace the $ F \left(\frac{K}{AL},1 \right)$ back with $f(k)$ so you get:
$$A F \left(\frac{K}{AL},1 \right) - Ak F' \left(\frac{K}{AL},1 \right) =A f(k)  -Ak f'(k) = A(f(k)  -k f'(k)) $$
So it is all about paying attention to what different definition of the arguments used here are. If you can't follow the differentiation itself that is question for Mathematics.SE or just further self study (Essential Mathematics for Economic Analysis by Sydsaeter covers all you need for this sort of problem).


Optimal price function: application of calculus of variations The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. Despite the problem being framed in the finance literature, the mathematical problem associated with it is far more general and is based on calculus of variations. Let me briefly introduce the problem:

Let $V \sim N(0, \sigma^2_V)$ denote the random asset value, $U\sim N(0, \sigma^2_U)$ the random liquidity trader's demand, $X$ the order by the strategic trader, and $Y=X+U$ the total order flow
The strategic trader observes the realization of the asset value $V=v$, the exogenous liquidity trades $U=u$, and submits an order $x(u,v)$ that maximizes his trading profits: $\pi(x,u,v)=x(v-P(y))$
The market maker observes total order flow $Y=y$ and sets the trading price equal to the conditional expected value of the asset, i.e. $P(y)=E[V|y]$.
A market equilibrium is characterized by a trading strategy $x(u,v)$ and a pricing rule $P(y)$ such that: i) Given the pricing rule, for every $(u,v)$, the trading strategy $x(u,v)$ maximizes the informed trader's profit $\pi$ and ii) given the trading strategy, the pricing rule $P(y)$ sets the price equal to the expected value of $V$ conditional on $Y=y$.

Now the authors show that this problem is equivalent to the following optimization problem $\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]$, that is, the problem can be solved by the calculus of variations. They state the answer to be $P(y)=\frac{\sigma_V}{\sigma_U}y$ which is true, once we "guess and verify" the solution. However, I want to find the solution from scratch, that is, without "guess and verify".

My problems arise as the function over which the functional of $P$ is defined is itself an optimal value. We know that the insider maximizes $x(v-P(x+u))$ or equivalently $(y-u)(v-P(y))$. From the first-order condition, we can calculate the optimal demand $y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$ which results in an optimal profit of $\pi(y^*,u,v)=\frac{(v-P(y^*))^2}{P'(y^*)}$.

Therefore, the market maker's optimization problem can be rewritten as $$\min_{P(\cdot)} E [ \max_x (v-P(x+u))x]= \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \pi(y^*,u,v) f_V(v) f_U(u) du dv = \int_{ \infty}^{\infty} \int_{-\infty}^{\infty} \frac{(v-P(y^*))^2}{P'(y^*)} f_V(v) f_U(u) du dv  $$ with $f_V$ and $f_U$ being the density functions of $V$ and $U$ respectively.

From the fact that $\pi^*$ is the maximum of the informed trader's objective function, we can use the envelope theorem to simplify: i) $\frac{d \pi(y^*)}{d u}=\frac{\partial \pi(y,u,v)}{\partial u}|_{y=y^*}=-(v-P(y^*))$ and  ii) $\frac{d \pi(y^*)}{d v}=(y^*-u)$.

This is where I am stuck. My problem is the fact that the argument $y^*(u,v)$ of the function we are searching for $P(\cdot)$ is itself the solution of a maximization problem. Especially, I do not know how to calculate the integral with respect to $v$ and $u$.

Any comments are highly appreciated. Thank you!

Literature:

Rochet and Vila 1994, "Insider Trading wihtout Normality", The Review of Economic Studites, Vol. 61, No. 1, pp. 131-152
Kyle 1985, "Continuous Auctions and Insider Trading", Econometrica, Vol. 52, No. 6, pp 1315-1335

Not really an answer, but too long for comment.

The $P$ in your
$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$
expression from the insider's problem and the $P$ in the expression
$$
\min_{P(\cdot)} \cdots
$$
from the market maker's problem should not be the same. That is not the definition of rational expectation equilibrium in this context.

The insider takes the pricing rule as given, per definition of rational expectation equilibrium. This means insider has optimal demand
$$
y(u,v, P_1(\cdot))^*= \frac{v-P_1(y^*)}{P_1'(y^*)} +u
$$ for any pricing rule $P_1(\cdot)$. Then market maker minimizes
$$
E[ ( y(u,v, P_1(\cdot))^* - u) \cdot ( v - P_2(y(u,v, P_1(\cdot))^*) )  ]
$$ over $P_2$---taking $P_1$ as fixed. It's a simultaneous game.
If the minimizing $P_2$ coincides with $P_1$, then $P= P_1 = P_2$ is an equilibrium pricing rule. Doing this in the one-period Kyle setting recovers Kyle's lambda, whereas your calculation would not.

The optimization problem you're solving is therefore not the correct one.

The insider optimizes with respect to a conjectured pricing rule $P_1$. In equilibrium, the market maker confirms the insider's conjecture, making the latter's expectation rational.

Your calculation assumes that the market maker acts after the insider. This is not the case in Kyle and I don't believe it's the case in Rochet and Vila either.


Regression discontinuity questions I am considering a regression discontinuity design (RD) where the "treatment" has a definite sorting rule (below the threshold, you are not fined - above the threshold, you are fined). The outcome I am considering is how much you are fined the FOLLOWING year. There are 4 options: 1) remain without a fine, 2) go from no fine to fined, 3) fined less than you were the previous year, or 4) fined more than you were the previous. While I have listed these as categorical, I have the continuous amount of the fine as well.

Here are my questions:
1.) I have never seen a "longitudinal" RD design where the treatment is at a time period before the outcome. Are there any methodological issues with this approach?
2.) Assuming this approach is valid, how best should I model the outcome given the multiple possibilities?

Thanks in advance for any advice!

The fact that your outcome is measured at a later point in time than the treatment is not a problem, and not unusual either - this is true to some extent in all RD studies (if this were not the case, there couldn't be causation).

As for how to analyze your outcome, I suggest a couple of different specifications. First, you should use the continuous fine measure as an outcome. Second, you could use an indicator variable equal to 1 if they got a fine, and equal to zero if they didn't. You might be able to do similar dummy regressions for the more/less cases too, but that's a little more complicated. If you want to be a little more advanced, it would certainly be interesting to take the RD framework to a tobit model. Shouldn't be too complicated, I expect, but I haven't thought that much about it.


Bartik Instrument Intuition I have a question regarding the Bartik Instrument.

I understand that this instrument is a particularly important tool that is used in labor economics. From my understanding, this instrument attempts to isolate demand shocks from supply shocks.

Consider the following thought experiment:

Say that we have an equilibrium quantity determined both the labor demand and labor supply. Call it total labor employed in period t
  in region i.
  We can express it as:$$L_{it}=\sum_{j}L_{ijt}$$
  where the RHS is the summation over all industries hiring labor in this region.

Now, the problem is as follows: the changes in total labor hired in each industry is a result of both supply and demand shocks. What the Bartik Instrument does is that it constructs local labor demand shocks in the following manner:$$\tilde{L_{it}}=\sum_{j}\omega_{jt}L_{ijt-1}$$
 where the LHS is region $i's$
  predicted employment. The summation is basically a weighted average using weights which corresponding to growth rates in national level employment in industry $j$
  times the labor force employed in industry j
  by region $i$
  at time $t$.
  In a sense, these are changes that are unrelated to local labor supply shocks. The Bartik instrument is then calculated as $\frac{\tilde{L_{it}}-L_{it-1}}{L_{it-1}}$

This is where I am lost. Once I construct this 'instrument', what would be my first stage? Do I need a first stage anymore? My intuition tells me yes. What I mean is that is this already the predicted value that we obtain after a first stage? Let me phrase my question in a more intuitive manner:$$L=f(L^{d},L^{s})$$

As a result,$$dL=f_{L^d}dL^{d}+f_{L^S}dL^{s}$$

Now, in a stochastic environment:$$dL=f_{L^D}dL^{d}+f_{L^S}dL^{s}+v
 =f_{L^D}dL^{d}+\epsilon$$
 where I assume that $$cov(dL^{d},\epsilon)=0$$
  or that demand shocks and supply shocks are unrelated. In the first stage then, is the RHS the constructed Bartik instrument? In that case, I would regress the total observed change in labor on the Bartik instrument and obtain $\hat{dL}$
 . Or is it the case that the constructed Bartik instrument by itself serves as $\hat{dL}$
 ?

Thanks a lot!

I think the "first stage" would be $L_{it}$ on $\tilde{L_{it}}$. In the Peri paper above, the Bartik instrument is actually just included directly as $\tilde{L_{it}}$ as a control variable because it is an exogenous regressor in that form. If you are running labor supply elasticity regressions (and thus want to see the effect of $L_{it}$ itself on labor supply), if you can argue that the Bartik instrument is in fact exogenous, you can use it as an instrument for $L_{it}$. But, putting it directly in, as you suggested, would amount to something very similar (i.e., the Reduced Form rather than the Structural Eq.).


causal time series analysis economics I want to analyse the relationship between the level of globalization,and the level of income inequality between two specific countries. however, I'm quite lost as to which method is good to use (in spss). I'm certain I have to perform a time series analysis, but can someone advise which one is good to use? Kind regards

If you run a vector autoregression you could also follow up by testing for Granger causality. I don't know whether this is implemented directly in SPSS, but once you have an estimated model it is easy to calculate the statistic as it is basically just an F-test. In R you can do it directly via for example the package vars.


Payoff from an option contract In period 1 the consumer of type $\theta$ selects an option contract consisting of an up-front fee, $B>0$, and exercise price, $\bar{R}$. The consumer pays $B$ at the end of the first period. In period 2, he realises his valuation, $\theta$, distributed on $[0,1]$ by CDF $G(\theta)$ with density $g>0$. His expected payoff, from choosing contract $(B, \bar{R})$, should thus be $-B+\int_{\bar{R}}^{1}(\theta-\bar{R})g(\theta)d(\theta).$ However, in the paper I am told that it is instead: $-B+\int_{\bar{R}}^{1}(1-G(\theta))d\theta.$ Are these two expressions equivalent? What does it mean to integrate a CDF in this fashion? Thank you.

When integrals look different than what pops into your head, often the reason is integration by parts. For your example note that
$$\int_R^1 (\theta -R) g(\theta) d \theta + \int_R^1 G(\theta) d \theta = (1-R) - 0,$$
where the right-hand side is equivalent to $\int^1_R 1 d\theta$.
Hence, the two expressions you consider are equivalent.
It's of the form
$$\int u(x) v'(x) dx + \int u'(x) v(x) dx = \int u(x) v(x) dx.$$


Cost Minimization of $f(x) = min(x_1,x_2) + x_3$ The following production function is given,
$f(X) = min\{x_1,x_2\} + x_3$
There is a solution here https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-minx-1-x-2, which is similar to this function. However, it does not include an additive $x_3$.
My initial solution attempt
Redefine the production function, such that,
$f(X) = x_1 + x_3$ if $x_2 < x_1$ and $f(x) = x_2 + x_3$ otherwise. This gives the following Lagrangian;
$L = p_1 x_1 + p_2 x_2 + p_3 x_3 - \lambda_1(x_1 +x_3 - Q) - \lambda_2(x_2 + x_3 - Q)$
With each $\lambda_i > 0$ both constraints are binding, such that,
$Q=x_1+x_3$ and $Q=x_2 + x_3$; and then the constraints reduces to
$x_1 - x_2 = 0$ implying that $x_1 = x_2$. However, this contradicts the initial conditions that I made on the relationship between $x_1$ and $x_2$ and, thereby, the arising production functions.
I'm caught in a bit of confusion here. How should I proceed with this problem?

While not said explicitly in the question I am guessing from the Langrangian function you set up that the problem you intend to solve is
$$\min_{x_1,x_2,x_3} p_1x_1 + p_2x_2 + p_3x_3 \\[8pt]
h(x_1,x_2,x_3) = \min\{x_1,x_2\} + x_3\geq z$$
This assignment combines perfect complements in production $f(x_1,x_2) = \min \{x_1,x_2\}$ with perfect substitues $g(y_1,y_2) = y_1 + y_2$.
One approach is sometimes to solve problems in two stages. Intuitively, with $y_1:=\min\{x_1,x_2\}$ firm will produce $y_1$ at the unit cost $p_{y_1} = p_1+p_2$ for whatever level of $y_1$ is produced it can never be cost minimizing to use more of $x_1$ than $x_2$ or other way around. Hence $x_1=x_2 = \min\{x_1,x_2\} = y_1$ and the seperate cost for each unit of $y_1$ is therefore $p_1+p_2$.
Next, solve cost minimizing with production
$$g(y_1,y_2) = y_1 + y_2 = y_1 + x_3 = z$$
with input prices $p_{y_1}$ and $p_{3}$. Since factors $y_1$ and $x_3$ are equally productive (constant equal marginal productivity) production will happen with cheapest factor so cost must be $C(z,p) =\min\{p_{y_1},p_3\}z = \min\{p_1+p_2,p_3\}z$.
I avoid using Lagrangian multiplier method because the production function is non-differentiable.


Derive the Hicks demand function for $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$ Consider the utility function $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$ and the budget line $p_1x_1+p_2x_2 = m$. Then I have to find the Hicks demand function.

I know that to do this I have to solve the following expenditure minimization problem
$$
\min_{x_1,x_2} p_1x_1+p_2x_2
$$
subject to
$$
x_1^{1/2}x_2^{1/3} = U
$$
for some utility level $U$. I have already found the Marshallian demand functions to be $x_1^* = \frac{3m}{5p_1}$ and $x^* = \frac{2m}{5p_2}$.

I thought maybe we could use the well-known Cobb-Douglas demand functions but I don't think we can as $1/2 + 1/3 = 5/6 \neq 1$.

I then tried to use Lagrange and got that

By Lagrange we have that
$$
L = p_1x_1+p_2x_2 - \lambda \left( x_1^{1/2}x_2^{1/3}-U \right)
$$
Thus the three FOC will be
\begin{align}
\frac{\partial L}{\partial x_1}: p_1- \lambda \left( \frac{1}{2} x_1^{-1/2}x_2^{1/3} \right)
\\
\frac{\partial L}{\partial x_2}: p_2 - \lambda \left( \frac{1}{3} x_1^{1/2}x_2^{-2/3} \right)
\\
\frac{\partial L}{\partial \lambda}: x_1^{1/2}x_2^{1/3} = U
\end{align}
Dividng the first equation with the second equation one gets
$$
\frac{3x_2}{2x_1} = \frac{p_1}{p_2}
$$
Thus
$$
x_1 = \frac{3p_2x_2}{2p_1}
$$
Inserting this into $U(x_1,x_2)$ we have that
$$
\left(\frac{3p_2x_2}{2p_1} \right)^{1/2} x_2^{1/3}
$$
which gives me after some few calculations that
$$
x_1^* = \frac{3p_2}{2p_1} \left( \frac{U\sqrt{2}\sqrt{p_1}}{ \sqrt{3} \sqrt{p_2}} \right)^{6/5}
$$
$$
x_2^* = \left( \frac{ U \sqrt{2} \sqrt{p_1}}{ \sqrt{3} \sqrt{p_2} } \right)^{1/(5/6)}
$$
but this doesn't seem right. Is there an easier solution to this kind of problem?

Let $f(U)=U^{6/5}$. This is a positive monotone transformation of $U$ on $\mathbb{R}_0^+$. So the preferences represented by $U$ are also represented by $V(x_1,x_2):=f(U(x_1,x_2))=x_1^{3/5}x_2^{2/5}$. The utility function $V$ has Cobb-Douglas form and you can use the formula for the Hicksian demand for Cobb-Douglas utilities: $$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}V,\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}V.$$ Substituting for $V$ you get $$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}U^{6/5},\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}U^{6/5},$$ or $$x_1^*=\left[\left(\frac{3p_2}{2p_1}\right)U^3\right]^{2/5},\quad x_2^*=\left[\left(\frac{2p_1}{3p_2}\right)U^2\right]^{3/5}.$$


Printing money to put into a Superannuation or UBI fund It is fiscal policy to keep inflation at between 1-2% (in New Zealand at least)
Various tools are used to keep inflation in this range.

Let's say the average inflation rate is 1.25%

What would happen if a small country (this probably would not work if the US tried it) printed* enough money to push inflation to 2%. Then put that money into a global index fund (and gold as a hedge).

This fund would be earmarked for superannuation, Universal Basic Income (UBI),  'a rainy day', quantative easing during a depression, etc.

In say 50 years the fund could become quite substantial. It could then be used to offset (all?)taxes with the fund's dividends.

What is the obvious flaw in this plan of printing money to finance a long term government fund?

*Not literally print, but you know what I mean.

Here is an interesting article on the link between helicopter money (“printing money”) and UBI:
Helicopter Money and Basic Income: Friends or Foes?
First, helicopter money (or funding a SWF) is arguably a more straightforward way than QE (asset purchases) to increase inflation to the target:

Instead of QE, the ECB could start a helicopter money scheme by giving
200 euros per adult citizen for one year – no strings attached, no
taxes involved, simply courtesy of the ECB’s (digital) printing
presses. This would involve about one-third of the money printing of
that under QE and yet would be more likely to fulfil the ECB’s
objective. … The ECB’s temporary scheme would allow some time for EU
policymakers to create the institutional and fiscal infrastructure for
such a eurodividend to be functional.

Instead of paying a UBI directly to citizens, or fostering economic activity by buying government debt like in QE, the central bank could also use newly created money to set up SWF to make consistent UBI payments over time.
The UCL IIPP is proposing the wider use of SWF:
Public wealth funds: Supporting economic recovery and sustainable growth
November 2020

Previous pandemics have, according to some studies, depressed growth
for decades (Jordà, Singh and Taylor 2020). One explanation of this is
that state-backed loans given out during the pandemic build a
corporate debt mountain that stifles many companies’ growth for years
to come. ... Under such conditions, public sector equity investments
or debt-for-equity swaps become an attractive option. By taking stakes
in threatened, strategically important firms, governments can not only
maintain jobs and provide a financial cushion to allow for firms to
recover and invest, but also ensure the taxpayer earns some reward
when the economy does emerge on the other side.

While you would likely want to use public leverage to impact investment towards public policy objectives you would arguably not want to end up with the government actively managing the economy.
The most neutral investment would be for the government to invest in land given that who owns land does not affect supply, which is fixed, or in practical terms governed by public planning laws.


Comparing currency prices If I know that one USD buys 0.90 euros or 1.42 Australian dollars, can I reliably conclude that one Australian dollar buys 0.63 euros? (0.90 / 1.42)

That seems to make sense. But would this be true for every case? For example, if you were to collect historic currency data for all currencies with respect to a single base currency, could you then accurately calculate all relative exchange rates, or are there are other factors to consider?

There can be market imperfections if information is not passed on immediately - historically that definitely happened all the time before telecommunication was well established.

Furthermore, if they have sufficient control of the in- and outflows countries can fix exchange rates for their currency at rates different from those of the free market. That happened in many socialist countries. The free market exchange rate for the Eastern and Western German currencies was substantially different from what the official exchange rate that was used by Eastern German banks. After unification of the countries again an exchange rate that was different from the market rate was used for the monetary union.

In Cuba a convertible peso was introduced in 1994 with a fixed exchange rate with the US dollar internally. It cannot be traded outside of Cuba.

The currency crises in Mexico and Asia in the 90s, and in Argentina in 2001-02 are associated with attempts to fix exchange rates that were in conflict with the market rates.


Is it possible to peg a cryptocurrency to a stock or bond index? What if the creators of a new cryptocurrency gave out new units of the cryptocurrency in exchange for shares of a stock or bond index fund, held these shares in reserves, and allowed owners of the cryptocurrency to redeem them back for shares in the index fund? Wouldn't this effectively peg the cryptocurrency to such a fund?
Wouldn't such a system offer a lot of stability, desperately needed in today's most popular cryptocurrencies?
What kind of issues would the creators or the consumers of the cryptocurrency run into with such a system?

There are already cryptocurrencies that are backed 1:1 by dollar, see Tether and USDC, or even backed with gold, see Paxos Gold, and yes all of that are redeemable. there is no problem with it so far.
There may be some trust problem, are the issuer really back the cryptocurrency with real asset? (this can solved easily by external auditing though) At 2018 there are some issues that tether aren't pegged 1:1 with USD, the result is Tether value are decreasing to around 0.9 USD. Weirdly enough, even without an external audit(they never got an audit since that scandal), people still believe in tether and the price of tether jumped out to 1 USD Again.


Interaction between productivity growth and interest rates Isabel Schnabel, Member of the Executive Board of the European Central Bank point at the interview out "The real interest rate (...) emerges from the economy’s growth potential in the long run.“ https://www.ecb.europa.eu/press/inter/date/2020/html/ecb.in200211~d2b4afb5d3.en.html

Another article also touches upon this question "It is well-known that productivity is a long-term determinant of return on capital and thereby of interest rates, which explains why there is a positive correlation between these two indicators" https://www.clevelandfed.org/newsroom-and-events/publications/economic-commentary/2017-economic-commentaries/ec-201720-productivity-real-interest-rates-long-run.aspx

However, for me it seems counter-intuitive. If productivity is low i.e. growth is week, than the risk to lend is high, meaning the lenders would demand higher premium for the risk -> IR would be higher on average. Can you please come up with an explanation on why slower productivity growth would result in lower interest rates?

Starting at the end, the simple answer is that intreset rates are decided as a counter measure for slow growth, which may derive from productivity slump. Meaning interest policy is a result, not a cause

On a wider level, the interntion of Central bankers is that there is a productivity rate, internally set in the economy set it's pace of growth. Imagine the added value of internet applications, as opposed to the world before that, as an example.

Their goal, is to set interest rate to match the proper appetite for risk, to encourage that productivity growth: take loans to pay for internet services, as per the example

The main parameter lacking in your equation, is that monitary policy cannot be the main economic policy to encourage productivity. It's the government in general: investinh in education, lowering barries for good healthy markets, provide a supporting legal system for business disputes etc.

Monitary policy can only be effective to a certain extent, in a way, like a wide (regressive) tax deduction


A timeline on the the key achievements of Economics and its evolution Something I've never seen in Economics but have in Maths and Chemistry and really enjoyed is that when I took it we got to the see the evolution of the science (and yes Economics is a [social] science ) showing us the very building blocks and the story of how it evolved into its modern counterparts.

Like with maths: Something like the basis 1) Indian numbers, $+$, $-$, $x$, $\rightarrow$; 2) fractions, radical numbers, square roots; 3) complex numbers.

Something like that it doesn't come to mind well but I quite enjoyed that. It really set the tone.

I don't think I can really do this question justice. A lot of us probably have different things that come to mind when we think of the history of economics. I tend to take an academic interpretation of your question.

A potpourri of random (mostly microecon, but maybe it'd be better to sort them) events that come to mind:

1838 - Supply and Demand:
Antoine Augustin Cournot publishes a book about the supply and demand model (Alfred Marshall makes it famous later).

1870s - General Equilbirum Theory:
William Stanley Jevons, Carl Menger, and French economist Leon Walras develop around the same time the idea of marginal utility, and Walras in particular creates his work in 1874, "Elements of Pure Economics"

1915(?) - Slutsky Decomposition:
Eugen Slutsky relates Marshallian and Hicksian demand to separate income and substitution effects. In 1927, he did something with statistics and business cycles that I don't quite understand, but I think that was important too.

1930 - Intertemporal Consumption:
Irving Fisher lays out early work on discounted saving, though arguably,  Modigliani & Brumberg (1954), Albert Ando, and Milton Friedman (1957) and the work on the life cycle-model is more noteworthy.

1944 - Game Theory:
John von Neumann and Oskar Morgenstern use Brouwer's fixed-point theorem to create the basis for game theory.

1947 - Von-Neumann Morgenstern Utility
John von Neumann and Oskar Morgenstern create their four axioms surrounding rationality with probabilistic utility outcomes.

1951 - Arrow's Impossibility Theorem:
Joseph Kenneth Arrow publishes his book, "Social Choice and Individual Values"

1973 - Black-Scholes Pricing Model:

1977(?) - Mechanism Design:
Leonid Hurwicz, Eric Maskin, and Roger Myerson create the foundation for "reverse game theory".

As you can see, there are lots of gaps. I still don't know a lot about economic history--just that it is very large.

Other topics to add:  Behavioral economics, experimental economics, information economics