Deriving the Euler Equation I want to derive the Euler Equation for the following:

$$max \sum\limits_{t=0}^{T} = \beta^{t}U(C_t)$$

$$s.t. C_t+K_{t+1} \leq f(K_t) , t=0,1,2,...,T-1$$
$$-K_{T+1} \leq 0$$

I'm a bit confused about why the F.O.C. have that:

$$\frac{d\mathcal{L}}{dK_{t+1}}=-\lambda_t+\lambda_{t+1}f'(k_{t+1})$$

and how we combine the F.O.C to yield the Euler equation:

$$U'(C_t)= \beta U'(C_{t+1})f'(k_{t+1})$$

I assume the F.O.C w.r.t. $K_{t+1}$ is such because of the inclusion of the intensive form of the production function but I am not exactly sure how and I really want to understand this completely.  I also need to make sure I understand how we are using the FOC to produce the Euler Equation.
Can anyone provide a bit of clarity?

The question is quite straightforward, and you do not need the first
step that you have. You have (for some reason) a different multiplier
here for each time period. That is not the case- you simply have a
no-ponzi scheme condition, aka a transversality constraint. The problem
is expressed as $max\,\sum\beta^{t}U(c_{t})$ st $c_{t}+k_{t+1}\leq f(k_{t})$.
The stream of utility to the agent is $U=\sum\beta^{t}U(c_{t})=U(c_{0})+....\beta^{t}U(c_{t})+\beta^{t+1}U(c_{t+1})+...+\beta^{T}U(c_{T}).$

Substitute in $f(k_{t})-k_{t+1}$ for $c_{t}$. Then, we derive wrt
$k_{t+1}$ and set equal to 0:$\frac{\partial U}{\partial k_{t+1}}=\beta^{t}U'(c_{t})-\beta^{t+1}U'(c_{t+1})f'(k_{t+1})=0\Rightarrow U'(c_{t})=\beta U'(c_{t+1})f'(k_{t+1}).$

You do not need the other equation you referred to.


Financial markets predictions How do economists/investors predict supply and demand in financial markets such as indices, currencies, etc?

There’s a lot of techniques used, but it is extremely rare that forecasts are on “supply and demand”, rather prices. To go from supply and demand to prices (which is what investors are interested in), you need supply and demand functions, which nobody can really estimate well.

I will first comment on supply and demand analysis for the bond market (the market I am familiar with), then have a few quick comments on price forecasts.

One does see attempts to match up the supply of government bonds (based on projected deficits) to what end investors want to buy, but such analysis is only of a niche interest. For example, governmental debt agencies try to gauge what maturity structure of issuance will match investor needs.

Meanwhile, on the demand side, people look at the effects of regulatory changes, and how much demand that will create for long-term bonds. The UK gilt curve was greatly affected by pension demand for long-term gilts.

There’s a wide range of techniques used in practice to forecast prices (rates in fixed income). If one looks at the debate about market efficiency, most do not work very well (bulk of investors underperform the market average). Some analysts use methods like story-telling or technical analysis, which are generally looked down on by academic studies. However, the bulk of the analysis is fair value analysis.

For example, if I a wanted to gauge fair value for a 2-year bond, I am just going to look at the outlook for central bank policy. This is because the fair value of a bond is the expected cost of financing over the life of the bond (the short rate), which is set by the central bank. If one is interested in corporate bonds, one estimates the probability of default, to get an idea what spread is fair. Equity investors attempt to forecast future dividends and earnings. All of these analyses can be done in a wide variety of ways.


Can I make this assumption about the utility function? My question concerns a household's utility function U(c,l). We're told that the partial derivatives dU/dc and dU/dl > 0, and the second order partial derivatives are negative. Now, can I assume anything about the sign of the partial derivatives Ucl (or Ulc)?

No.

In general, based on the information you have, there are no results you can infer for the partial derivatives $U_{cl}$ / $U_{lc}$. In this context they could be anything.

As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.


Banks and Stock Market in Germany Why does Germany have such a small stock market, but very large powerful banks?

I don't know if your statements are true. By what margin is the German stock market small? And the only bank that did matter on the international level (Deutsche Bank) is far from being large and powerful.

That being said, your notion might have to do with the fact how the German economy has been organized since WWII. The Rhine capitalism used to be, among others, characterized by strong institutional links between banks and industry. Whereas firms would finance on the stock market in the U.S., they received credits from banks in Germany who, in turn, often held stocks of these firms. This construct was called "Deutschland AG", but has been deteriorating since the early 2000s at the latest.


Solow model, time and steady state Suppose we have a Solow model:

$$
Y(t)=C(t)+I(t)
$$

$$
I(t)=sY(t)
$$

$$
\dot K=I(t)-δK(t)
$$

With a given Cobb-Douglas:

$$
Y(t)=Z(t)K^aL^{1-a}
$$

$$
y(t)=Y(t)/L(t)
$$

$$
k(t)=K(t)/L(t)
$$

$$
y=Zk^a
$$

We also know these:

$$
L=L(t), \dot L/L=n
$$

$$
Z=Z(t), \dot Z/Z=g
$$

Therefore we have:

$$
L(t)=L(0)e^{nt}
$$

$$
Z(t)=Z(0)e^{gt}
$$

We get to this point:

$$
\dot k=sZk^a-(n+δ)k
$$

In the steady state:

$$
k=k^*
$$

$$
k^*=\left(\frac{sZ}{n+δ}\right)^{1/(1-a)}
$$

The question has to do with the time dimension in Z. Time enters the final expression if we substitute Z with what we found above, but I'm not sure what to make of this.

$$
k^*=\left[\frac{sZ(0)e^{gt}}{n+δ}\right]^{1/(1-a)}
$$

Surely there's something this neophyte is mixing up. I thought that in a steady state capital per worker remains the same.

In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and Martin book, Chapter 1.


Rationale for discounting I am interested in understanding the rationale behind discounting in the context of government policy appraisal.

As far as I understand it, there are three main components to the discount values normally used:

Estimate of catastrophic risk.
GDP growth.
Time preference value of money.

Adjustment of future costs/revenues to the first two of these make sense to me, but the third seems to include a "normative judgement" that I want to check I understand.

By adjusting for the time preference value of money in policy appraisal we are explicitly saying that we value the utility of present people more than the utility of future people? Not just because an event may lead to the destruction of society (1), or because the economy will grow and hence costs/benefits will be smaller in the future (2), but because future benefits mean less to current people now.

Any confirmation or challenge to my understanding would be appreciated.

As Kitsune rightly points out the usual reason for the inclusion of the time preferences for money into the discount formula (or Ramsey fomula) is that we observe this kind of behaviour in people. Given that many economists take a positive approach towards the study of economics, that is a logical stance.

That being said the last word is not spoken on this, and discounting and its use in benefit cost analyses for especially long-term policies is an active area of research. The Stern review on the economics of climate change sparked intense debate on whether we should include a pure rate of time preference and if so, what its size should be. On the one had you had economists like Stern who argued for a very small rate (0.01 % if memory serves), only to account for catastrophic risk, on the other hand there were economists like Nordhaus who argued for a rate close to 3%, based on observed choices by people in real life and the interest rates. Wikepedia covers the debate here

Martin Weitzman did a study among 2160 economists what the real discount rate should be  for climate change (that is pure rate of time preference plus discounting because of GDP growth) called Gamma discounting and published in the AER. He found a range from -3 to 27%, which may give you an idea of the disagreement among economists on the issue of discounting.

Drupp, Freeman, Groom and Nesje did a more detailed follow-up in 2015, called discounting disentangled. Their range for the full discount rate is lower, from 0 to 10%, and they also asked explicitly what the pure rate of time preference should be. The mean recommended value for the latter by their 200 surveyed experts was 1.10%, but even that rate had a range from 0 to 8%.

Interestingly, the person who first came up with the Ramsey formula, Frank Ramsey, in a mathematical theory of saving, argued for a rate of time preference for societies of 0, calling a positive rate "ethically indefensible", although he does use a positive rate later on in the paper.


What are some good resources to read on Development Economics from a neoliberal or Austrian perspective? I have tried to look for development economics from a neoclassical (Chicago school) or Austrian School approach. However, I could not find any references for this topic. Is it possible that this school had nothing to say on development economics or development studies? But that seems implausible with the existence of the Washington Consensus.
Thank you all.

The reason why you cant find any sources on this is that you seem to be confused about what development economics and those different schools of economics are. I will first address the confusion and then give you references for some of the sources you requested.
Confusion about different terms:
First, development economics is a positive not normative field. Following the
Debraj Ray and Clive Bell  Development Economics, the development economics is defined as follows:

a subject that studies growth, inequality, poverty and institutions in the developing world

An important takeaway is that this definition implies that development economics (as well as a wast majority of economic fields) is a positive field. It is concerned with what is not what ought to be.
On the other hand both Chicago School and Austrian Schools nowadays do not exist as a separate positive fields but they are almost purely normative schools that do interact with positive economics through study of political economy (i.e. the field of economics that, among other things, examines what are implications of different ideologies on economic policy. See the ‘Political Economy’ by Peter Groenewegen or this investopedia entry).
Historically in their respective time periods these schools were associated also with some positive economics. An example of this would be the problem of socialist calculation debate where members of Asutrian school such as Hayek were heavily involved.
However, in present day those positive propositions of Austrian or Chicago school that panned out were incorporated into mainstream neoclassical theory and the schools for most part do not pursue any positive research projects.
Next in you question you seem to equate Neoclassical economics with Chicago school which is simply not correct. You should not confuse Neoclassical economics with Chicago school as Neoclassical economics is a set of tools rather than doctrine.
For example, Sargent is considered to be a representative of 'Chicago school' because of his ideological beliefs but his whole work is Neoclassical. At the same time, Stiglitz who is ideologically as left leaning and anti-free market as reasonably possible within the mainstream of the profession (we are talking about a person who praised Hugo Chavez) is also purely Neoclasical economist and would probably get a stroke if you would try to associate him with Chicago school or Austrian school both of which he criticizes heavily. Next, a firmly New Keynesian economist such as Gregory Mankiw is ideologically pure libertarian/classical liberal clearly ideologically influenced by Milton Friedman and Hayek, yet his whole work could not be further from Neoclassical economics.
When it comes to 'Neo-liberal economics' there is not even such thing within economics. The New Palgrave Dictionary of Economics does not even has entry for 'Neo-liberal economics' only for Neo-liberalism as an ideology, and certainly in political economics you might study how Neo-liberal ideology can lead to different economic policies but there really is not a 'Neo-liberal economics' as a proper sub-field or even school of economic thought.
Hence a big problem with your question is that you keep mixing positive approaches and sets of tools in economics such as the Neoclassical economics with a normative largely ideological and political economic perspectives that Austrian or Chicago school in present day consist of.
Resources:
Now finally after getting all  the above out of a way, if you want to get some resources on Neoclassical perspectives on development economics then any textbook will do. A large part of development economics consist of purely neoclassical models. A good source is a Handbook of development economics by Rodrick & Rosenzweig (2009). A large number of models presented in the book can be safely classified as purely neoclassical models (albeit definitely not all). An excellent example would be chapters in the handbook that focus on economic growth that are for most part purely neoclassical.
Furthermore, you can have a look at sources cited in the handbook to get to see even more Neoclassical models and perspectives on development economics although the handbook is quite exhaustive when it comes to the current research and thinking on economic development.
If you want to get Austrian or Chicago perspective on economic development then you really have to study political economy and not positive fields such as development economics.
For example, you can get some idea about the perspectives of Chicago school on economic development by reading Milton Firedman's Capitalism and Freedom, where he shares among other things also opinions on why Hong Kong or Japan successfully developed.
For an Austrian perspective you can read some of the works of Benjamin Powell who identifies as Austrian economist (although his academic work seems to be more or less neoclassical) who shares his thinking on political economy and development in books such as Out of Poverty: Sweatshops in the Global Economy (Cambridge University Press, 2014) or Making Poor Nations Rich: Entrepreneurship and the Process of Development (Stanford University Press, 2008).


Multivariate linear regression: how to test for whether the slopes are the same? If I regress wages on education and the dummy variable gender using a linear conditional expectation function (wage = a + b(education) + c(gender)), how can I test that the slope b is the same for both males and females? i.e. the returns to education are the same?

Intuitively, I was thinking that one could conduct simple linear regressions of wage on education for males and females separately and then do a two-sample hypothesis test for whether the slope estimates are significantly different. Would this introduce issues with the omitted variable gender? Thank you.

Estimate the model
$$W_i=a_1+a_2E_i+a_3G_i+a_4E_iG_i+\epsilon_i,$$
where $W,E,G$ denote wage, education, and an indicator for gender (e.g. 1 if male, 0 if female), respectively. The term $EG$ is an interaction term obtained by multiplying the gender dummy by education. A simple t-test of the hypothesis $H_0: a_4=0$ will tell you if the effects of additional education on wage are significantly different for different genders.


Graduate level texts/notes that cover the Short Run and Long Run Costs of firms' production In undergrad econ we learned that it would cost a firm more money to increase its production in short term than in long term.

MWG does not seem to cover this topic. I need a Graduate level texts/notes/papers that cover the Short Run and Long Run Costs of firms' production (or supply curve).

Textbooks:

Mas-Colell, Whinston, and Green (1995). Microeconomic Theory. Chapter 5
Geoffrey Reny. Advanced Microeconomic Theory, 3rd edition. Chapter 3
Kreps (1990). A Course in Microeconomic Theory. Chapter 7
Varian (1992). Microeconomic Analysis, 3rd edition. Chapter 5

Lecture notes: I think undergraduate notes are more useful on this respect. For instance:

Chapter 6 to Chapter 8 here.
Set 6 and 7 here (go to downloads to get slides).

In my experience, graduate notes deal with deeper issues. For instance this, this, or this one dig much deeper, skipping SR/LR cost functions altogether.

In any case, here Google is your friend. The trick (imo) is to add "pdf" to the search, to get lecture slides. For example:

http://google.com/#q=microeconomics+cost+function+short+run+long+run+pdf


Value representation theorem for countable set How can we prove the following:  if a relation $R$ is a weak order on a set $X$ and $X_\sim$ is finite, then there exist a function $v:X\to \mathbb{N}$, which is a value representation of preference relation $R$.

Here I know the construction of the function $v$ when the range is the set of real numbers (as $X_\sim$ is countable). But,I am confused  when the range of $v$ is the set of natural numbers.

Could anyone please help me?

I use the usual notation $(\succeq,\succ,\sim)$ instead of $R$. I assume that $X_{\sim}$ in your question is the set of equivalence classes of $\sim$ on $X$.

Suppose that the cardinality of $X_{\sim}$ is equal to $n \geq 1$. There exist $x_1\cdots,x_n$ such that $x_1 \prec \cdots \prec x_n$. Define $v(x_1)=1,v(x_2)=2,\cdots,v (x_n)=n$.

Since $\sim$ has $n$ equivalence classes, for any $y \notin \{x_1,\cdots,x_n\}$ there exists a unique $x \in \{x_1,\cdots,x_n\}$ such that $y \sim x$. Define $v (y)=v(x)$.

The function $v$ defined on $X$ takes values in $\mathbb{N}$. I leave it to you to check that $v$ is a utility representation of $\succeq $.


Is there objective evidence that the US economy is "booming"? I realize that it's the political pundits that are saying it, but they never seem to provide evidence that the  economy is "booming".  The dictionary definition of "booming" is having a period of great prosperity or rapid economic growth.

A fairly standard measure to use would be Gross Domestic Product (GDP). Over the past 30 years, US GDP has been growing at approximately 2.5%.

According to trading economics

  The US economy expanded an annualized 2.3 percent on quarter in the first quarter of 2018, below 2.9 percent in the previous period but beating market expectations of 2 percent

Using this measure, calling it a boom is a bit much.

Another way to cut the data is by comparing it to the ten-year trend of about 1.5%. Growing 80bps above trend looks a bit better.

So, to answer your question, depending on how you cut the data, it could be considered booming but I wouldn't.


What empirical evidence is there of distortions when capital gains taxes are introduced or raised? I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0. My question is what observed evidence is there of the distortions or "side-effects" that have been created do to the introduction of or the increase in capital gains taxes? Is there also empirical evidence of the opposite, i.e. an increase in investments (and any other "desirable" side-effects) due to the decrease or elimination of the capital gains tax?

Correcting Misconceptions in the Question:
Before providing an answer it is worth noting the premise in your question is simply incorrect. You state:

I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0.

But this is not rationale for the famous 0 capital income tax Chamley-Judd result. Virtually all taxes create distortions, and any modern government, due the amount that modern governments spend, has to resort to distortionary taxation at some point. However, the famous Chamley-Judd result showed that (Chamley, 1986; Judd, 1985) the burden of capital income tax is in long run shifted to labor, so there is no point to those distortions and one should just levy those taxes on labor incomes instead (although more recent papers showed that the result does not always hold and that only some of the burden of capital income taxes gets shifted not all of it - e.g. see Straub & Werning 2014 or Diamond & Saez 2011).
So the 0 capital income tax result had nothing to do with distortions directly. Indeed taxing labor causes non-trivial amount of distortions as well and in spite of that those economists recommended taxing it directly.
Answer:
Now to your main question on whether empirically capital taxation causes distortion, the answer is clearly yes. There are numerous studies showing that, and I think it is a fair to say that the question is not if there are distortions but how much and how economically significant they are.
The deadweight loss and distortions caused by capital income taxes will vary from country to country and from time period to time period, and it is beyond scope of a single answer to cover every country on a planet and every time period but here are examples of some studies that found capital taxes create distortions:
Devereux MP (2004) Measuring taxes on income from capital. In: Sørensen PB (ed) Measuring the tax burden on capital and labour. MIT Press, Cambridge
Becker J, Fuest C (2004) A backward-looking measure of the effective marginal tax burden on investment. CESifo Working Paper No. 1342
Goolsbee, A. (1998). Taxes, organizational form, and the deadweight loss of the corporate income tax. Journal of Public Economics, 69(1), 143-152.
Shan, H. (2011). The effect of capital gains taxation on home sales: Evidence from the Taxpayer Relief Act of 1997. Journal of Public Economics, 95(1-2), 177-188.
These are just few examples, but in public economics there is no question that capital taxes create some distortions (generally the only non-distortionary taxes are Pigouvian taxes and lump-sum taxes plus some special cases), so the question is not if but how much and answer to that will differ in time and place. Moreover, this has nothing to do with the zero capital tax result even pro-capital income tax economists would not deny taxing capital is distortionary.
For example, Goolsbee (1998) founds that corporate taxes (which are economically taxes primarily on capital incomes) in the  US between years  1900–1939 caused distortions that resulted in deadweight loss of 5–10% of their revenue. Again that number will vary by place and time.
The positive effects of elimination of capital income tax on economy are elimination of the above mentioned distortions. Hence, if capital taxes caused in the US between 1900-1939 deadweight loss of 5-10% of revenue, then eliminating them would mean that this loss would not occur in the first place.


Penance strategy game theory I am having some difficulty in understanding the following use of the penance strategy as a strategy to support compliance in the repeated game presented in Asheim et al. (2006). I understand penance in the following way:

  Begin in the cooperative phase where both states choose compliance; if
  State A then defects, the other State B immediately switches to defect
  for all subsequent periods until and including the first period that
  State A reverts to compliance. After that period, State B also reverts
  to compliance

Now, in their work Asheim et al. state the following:

  We assume that compliance in the global agreement is sustained by way
  of the "Penance" strategy. This strategy specifies that a
  participating country plays cooperate except if another participating
  country has been the sole deviator from Penance in the previous
  period, in which case defect is played.

My doubt has to do with the use of the word "sole". Specifically, following the penance strategy in a game with 3 or more players, is the punishment the same whether a single player deviates or more than one does?

I see two ways to produce histories with multi-state deviations (not including the degenerate case in which all states defect):

Multiple states defect in different periods.  This possibility is ruled out so long as Equation 3 (the necessary and sufficient condition for unilateral defection at time $t-1$ to be punished by all other states at time $t$) is satisfied - in that case, only one state can ever defect at a time because all other states have credibly committed to Penance.
Multiple states defect in the same period. There are two obvious ways this can happen: 1) regional agreements are in effect and one of the regions is punishing a defection.  Equation 7 gives the conditions required for both regions to revert to Regional Penance in the following period, so this case isn't interesting.

The other way to have multiple states defect in the same period is the interesting case. It implies collusion between a subset of states. This reduces to the Regional Penance case if the subset is equal to half the states.  If it isn't, Footnote 10 explains that such an arrangement is not weakly renegotiation-proof. This makes sense, as collusion implies that (Regional) Penance is not, in fact, subgame perfect.


Interest rate parity: Counter intuitive What is the basis for interest rate parity to hold?
I know, the reason stated is excess returns

Excess returns as such should not be the reason for a currency depreciation. for e.g. when we posit that forward rate converge to the spot rates over time and back by arbitrage argument it is actually because forward prices, if less than spot prices, would result in an increase in the demand for forwards and subsequently increase its price. (talking about long positions here).

Hence, in relation to interest rate parity, excess returns(which is wrongly stated sometimees as arbitrage) should put downward pressure on the interest rates. (you keep investing more and more money in a bank at a given rate obviously it cant keep it up and has to lower the rates, also it now has a lot of reserves so it wants to lend again leading to lowered rates) why it should affect the exchange rate beats me.

To the contrary a high interest rate(say of country A) should put upward pressure on exchange rates(foreign/home); People want to invest more into A's banks so they convert more foreign currency to A and hence the A's currency should appreciate

In your last paragraph you write "high interest rate in Country A". There is no such thing. There is higher interest rate, per some given criterion. Say higher than historical average. In out case, it is "higher interest rate in Country A compared to the interest rate in Country B".

So your last paragraph says that if returns (from nominal interest rates) in Country A are higher than returns (from nominal interest rates) in Country B, that would put upward pressure to the exchange rate (units of currency B per one unit of currency A): people will demand more currency A, etc.

Now a different but equivalent  way to say "higher returns in A than in B" is "excess returns", in the specific context we are talking about. Of course the term "excess returns" is very general and can be used in any situation. Sometimes it may used to mean "abnormal" returns (in some sense and again compared to some base). But in the specific situation we are talking, it has the exact same meaning with "higher returns from nominal interest rates in Country A compared to returns (from nominal interest rates) in Country B".

So moving to your penultimate paragraph you say that "if returns in A are higher than returns in B, that should put downward pressure on the interest rates in country A (because funds flow in), and it should leave the exchange rate unaffected".

Although there is logic to the mid-term effect of an increase in the flow-in of funds, still it appears that your last paragraph already answers the question "why should exchange rate be affected?", stated  in your penultimate paragraph.

ADDEDNUM
Responding to the OP's comment, let's write the Uncovered Interest Rate Parity expression

$$(1+i_{A,t}) = \frac {S^e_{A|B, t+1}}{S_{A|B, t}} (1+i_{B,t})$$
where $S_{A|B}=$ units of currency A per unit of currency B.
This is the claim.  The numerator is the expected future exchange rate

Assume that $i_{A,t} > i_{B,t}$. Standard economic logic (and a few assumptions on ability to invest over borders), says that then investors would want to invest in country A. To do that they will increase demand for currency A. This should lead to appreciation of currency A. Does the UIRP say something like that?

If $$i_{A,t} > i_{B,t} \implies 1+i_{A,t} > 1+i_{B,t}$$
and so the UIRP asserts that we must have

$$\frac {S^e_{A|B, t+1}}{S_{A|B, t}} > 1 \implies S^e_{A|B, t+1} > S_{A|B, t}$$

Given how $S$ was defined the last inequality means that we expect that the currency will depreciate. Indeed, in the future. Because, what is affected in the first place by the reaction of the investors is $S_{A|B, t}$, not $S^e_{A|B, t+1}$. The reaction of the investors will decrease $S_{A|B, t}$ (which reflects the appreciation of currency A), creating expectations of future depreciation, i.e. leading to $S^e_{A|B, t+1} > S_{A|B, t}$, as the UIRP asserts. Why should we expect depreciation of currency A in the future, compared to the present? Because the inflow of funds in country A will eventually pressure $i_{A,t}$ downwards, as the OP remarked, investors will stop loving currency A so much, demand for it will fall, hence depreciation.

The UIRP therefore compacts two effects from a discrepancy in interest rates: a first effect of currency appreciation and a second (expected) effect of currency depreciation in the future.


Firm Concentration and Low Inflation Inflation in the advanced economies since the financial crisis has been surprisingly low. Yet at the same time we’re told that the US economy is losing its dynamism in recent decades. This is reflected in often massive firm concentration and rising markups.

Are the two – persistently low inflation and oligopoly structures – not in contradiction with one another?

Interesting!

A possible way to resolve this contradiction is by taking cost reducing technologies into consideration.

Perhaps (example with made up numbers follows) in 1960 a gallon of milk cost \$1 to produce, package and market. Due to strong competition, the consumer could get this for \$1.02, meaning a markup of only 2%. Today, due to technological advancements, the same gallon of milk's production, packaging and marketing cost could be lower, say \$0.80. However perhaps there is much less competition, and the markup is now 30%. This would make the consumer price of this gallon of milk \$1.04. The price jump from \$1.02 to \$1.04 seems negligible, something most people would call low inflation. (Especially if this took 60 years.)


Walrasian Equilibrium under Leontief Preferences There are two people in the economy, A and B.  Both have the utility function: U=min{x,y}.  A has an initial endowment of (x,y)=(50,100) and B has (x,y)=(100,50).

I know how to draw the Edgeworth box and establish the contract curve, but how do I determine the Walrasian equilibrium ratio of Px/Py?  I'm confused by the fact that I can't differentiate U, nor is there money exchanged to establish a budget constraint.

You do not need money exchanged to establish a budget constraint, most of these standard micro questions work in cashless economies.

Your Px/Py is just the exchange rate between the goods. If it's 3:1, it means that you exchange 3 $y$ for one $x$ - no cash involved.

Not being able to differentiate sucks, but actually requires you to know what you're doing. The point of differentiation is finding a place at which all Pareto improving trade has been exhausted.

Here, that means that, indexing the two players by i:

$$ x_i, y_i \in \arg\max_{x,y} U^i(x, y) s.t. p_x x + p_y y = p_x x^i + p_y y^i \, , \forall i $$

Without first-order conditions, this problem is potentially hard to solve. However, most non-differentiable utility functions give you a hint on how the outcome must look like.

Here, aggregate resources add up: $A=x_1 + x_2 = y_1 + y_2$. Hence, the planner would chose some allocation where x=y for both players, as for any positive $a$ and $\epsilon$ it is inefficient to give one player $(a + \epsilon, a)$ and the other one $(A - a - \epsilon, A - a)$.

However, that leaves a continuum of distributions as potential outcomes: Those where we give players one and two $(a, a)$ and $(A-a, A-a)$ for any $0 \leq a \leq A$.

Which equilibrium we actually get depends on the relative value of the initial endowments. I will leave you with this: What does $p_x/p_y$ necessarily have to be given total endowments and preferences? What does this imply for the unique equilibrium distribution of resources among the players?


Construct utility function for a risk-averse agent I am trying to construct utility function for an agent who can be risk-seeking or risk-averse. We have an agent $i$ who has an ideal point $x$ in a policy space $X = [0,1]$. There is a policy (option) given by $a \in X$. Can we construct the agent $i$'s utility function as a distance between his ideal point and $a$ as $u_i (a) = -(x - a)^\gamma$, where $\gamma > 0$ represents $i$'s attitude towards risk, such as $\gamma > 1$ means $i$ is risk-averse and $\gamma < 1$ means that $i$ is risk-seeking?

If $-|x-a|$ represents the monetary payoff associated with the policy choice $a$, then $u(a) =-\left(|x-a|^\gamma\right)$ is risk averse for $\gamma > 1$, and risk loving for $0<\gamma < 1$.


When should one use Lagrange Multipliers as opposed to calculating the ratio of Marginal Utilities for MRS? As far as I could see in the examples I found, they led to the same result through relatively similar processes.  I have only looked at 2 variable utility functions, so is does the use of Lagrange become evident beyond that?  I.e. U(x,y,z) and if so is that the only real use for Lagrange Multipliers?

Edit:  For clarity, I was wondering about why Lagrange Multipliers are used instead of simply calculating partial derivatives individually to find the MRS.  What MRS-related situations are Lagrange Multipliers used in?  Why are they used in those situations?

Lagrange multipliers not only incorporate constraints on a maximization/minimization problem, but the multiplier itself can have a meaningful interpretation. For example, say we have the problem:

$$\max_{x, m}: U(x, m) = m + \phi(x) \quad s.t. m + px = y$$

Where $m$ is a numeiare good, $p$ is the price of good $x$, and $y$ is income.

Now you could just rewrite the constraint as

$$m = y - px$$

and substitute that into the original maximization problem. But you can't always do this easily. So you can also solve with a Lagrangian.

$$\mathcal{L} = m + \phi{x} - \lambda(m + px - y)$$

Solving for $\lambda$ will give you the shadow price of income, or the marginal price of income. That is, it represents the rate of increase in maximized utility as income is increased. You also need to find $\lambda$ to get a nice full Marshallian demand, or to relate indirect utility to Marshallian demand in Roy's identity.

Even for problems that aren't utility maximization, $\lambda$ is still representing the rate of change of the quantity being optimized as a function of the constraint variable.

Here's a useful example with the firm maximization problem and bribing problem.


How to interpret "scale efficiency" in DEA? I am reading the book Benchmarking with DEA, SFA, and R by Bogetoft and Otto. In Section 4.8 they discuss the concept of scale efficiency, and I am having trouble interpreting this concept.

The usual output of Data Envelopment Analysis (DEA) is a collection of efficiency scores. If we consider the input efficiency scores, then these scores represent the fraction of a firm's current costs which should, in principle, be able to be used to produce a firm's current output. Thus when a firm is judged as inefficient, a reasonable reaction might be to reduce costs and focus on making internal practices more efficient.

When investigating scale efficiency, one compares two different DEA outputs: one using the assumption of constant returns to scale, and the other using varying returns to scale. This expresses whether a firm is operating at it's "optimal size." If it is not, then using further comparisons of DEA outputs (using increasing or decreasing returns to scale) it is possible to see whether the firm is "too large" or "too small."

While the concept of scale efficiency makes sense to me intuitively, I don't see how the practical response to a scale efficiency analysis is any different from the response to the usual DEA efficiency analysis. For example, suppose we do a DEA input efficiency analysis and find that a firm is inefficient. Then the practical response is to reduce the budget and focus on making internal practices efficient. Suppose now instead that a firm is found to be scale inefficient. The practical response again seems to be to reduce costs and focus on making internal practices more efficient... but what exactly are the proposed gains? With the original DEA analysis we had some estimate of how much we could reduce costs and keep the same level of production. But with a scale efficiency analysis, how much production should expect to keep up given the reduction in the size? Should we expect a proportional reduction in production as well? If so, what is the advantage of downsizing?

Finally, there is the somewhat more confusing case (whose interpretation may help me to make sense of all this) where a firm is judged to be maximally input efficient (i.e. is assigned a score of 1 by the DEA) but is not scale efficient. What is the practical response to this case?

Cross posted at Cross Validated: https://stats.stackexchange.com/questions/144554/how-to-interpret-scale-efficiency-in-dea

First of all, let me tell you that you're doing great. I remember that, being a non-parametric method, it was more difficult for me to understand even the simplest things of DEA.

Now, your interpretation of DEA is correct. When a firm is considered inefficient, it is because it could have gotten the same output but with a lower cost if it had used the best practices. Nonetheless, there is another way of looking at it (the output oriented way): if the firm adopts better practices the output will be increased with the same costs. Keep that in mind.

To explain the scale efficiency, i'm gonna build a very simple case. Imagine a firm that manufactures some product (y) and uses only labor as an input (x) and both the product price and the wage are $1 (for easier calculation). This firm has 4 plants: a, b, c, and d. Table 1, shows the number of workers, production, profit and productivity of the 4 plants.

Table 1

It is clear, that plant a is the least productive of them, because every worker produces only 1 unit. Plant b produces the same as a with less workers. c, on the other hand, has the same number of workers as a, but with more production and it is the most productive plant(2.5 units per worker). d is the biggest plant producing 29 units.

Now, let's calculate both the Constant Returns to Scale (CRS) efficiency and the Variable Returns to Scale (VRS) efficiency and the scale efficiency:

Table 2

Now, let's plot what it looks like:

Figure 1

Where every point represents a plant, the black dotted line is the CRS efficiency frontier and the blue line is the VRS efficiency frontier. This blue line implies that b, c, and d are as efficient as they can be. That is, they already have the best practices. The only difference in their productivity is due to economies of scale. Now, if you were to decide the future of a, the least efficient plant, what can you do? you have two options:

Fire some people keeping production constant: move to b.
Increase production keeping all the workers: move to c.

You should be aware by now that the smart move is to increase the production without getting rid of anyone (move to c). So far, this is the usual DEA analysis.

But now, let's forget about a. Let's see what we can do to increase the profitability of b. b is already in the efficiency frontier of VRS DEA. This means that b is already using the best practices. Nonetheless, because of economies of scale it is impossible to achieve the same productivity. So, the smart move, would be to invest \$5 hiring 5 new workers: move to c in Figure 1. Note that b is not adopting efficiency measures this time: it's just increasing the scale, while maintaining the best practices it already has. This investment would be profitable indeed: we hire 5 new workers and production goes from 10 up to 25, which represents an increase in profits of \$$10=\$15-\$5$. Put in other words, for every \$1 we got \$2 return. Not bad!

But wait a second... We already have 5 extra workers in d! What would happen if we just move workers from d to b? Now the three firms would have \$15 profit, leaving a total profit of \$50 without investing a single penny! The profit increased by 47.05% just by adjusting scales.

Now, what role did the scale efficiency measure of DEA played here? Take a look back to Table 2 and divide b's productivity by b's scale efficiency $2/0.8=2.5$ which is c's productivity. Repeat this procedure with d: $1.93/0.77333=2.5$. So you can say that b is achieving 80% of the optimal scale efficiency, while d achieves 77.33% of it.

I have to make a remark: note that both b's and d's scale efficiency have a positive sign, but one has to increase its scale while the other has to reduce it. In this sense, the analysis is kind of blind, but you have other tests to know whether it's increasing returns to scale or decreasing returns to scale.


Clarification on the impact of a change in the Bank Rate (UK) I'm currently revising for my AS economics exam and I am going over Monetary Policy. In a handout, I've read the following phrases which I am a little confused about...

"A cut in the base rate, assuming it reduces market interest rates across the economy..."

This quote implies that the Bank Rate only changes the Central Bank's Interest Rate not the IR for commercial banks; so therefore, how are IRs determined in commercial banks? Do they have to be within a certain range of the Bank Rate? (Otherwise what is the point of influencing the Bank Rate if commercial banks are not forced to comply?).

"An increase in the base rate will not only reduce post-debt repayment income..."

This implies that mortgages etc. vary depending on the IR rate rather than being a fixed amount independent of changes in the IR (which is what I thought it was).

Interest rates in commercial banks are determined by market forces - supply and demand.

Their demand will be influenced by their customers' expectations - which will be influenced by the Central Bank interest rate. Their supply will be influenced by the Central Bank (the Bank of England) interest rate.

Some UK mortgages have fixed interest rates for a particular term - for example, the first three or five years of the mortgage. After that, they revert to the standard variable rate, which loosely tracks the Central Bank interest rate plus a few percentage points.

Some UK mortgages are trackers - that is to say, they are designed so that they track the Central Bank interest rate plus a fixed offset, and they change very quickly when Central Bank rates change: if the Central Bank rate changes by x%, then within one day to one month, the tracker interest rates also change by x%.


In Supply and Demand is Quantity the Amount of Inventory of the Amount of Units Sold? I am new to economics in school and we are going over supply and demand.  This may sound like a very simple question, but I just need one thing clarified.  When quantity is mentioned in the graphs is that referring to our total inventory or just the number of units sold?  Thanks.

Demand and supply are functions indicating the price at which a certain quantity would be traded. They represent the intentions of producers and consumers to buy or sell a given quantity at a certain price.
Reciprocally, you can consider that these curves show you, for a given price, how much the producers would sell and how much the consumers would buy.

Now in reality, for the portion of the demand curve which is higher than the supply curve (ie. Consumers would buy the quantity at a higher price than producers would sell it), the market clears the quantity in question at whatever price lies in-between. The portion of the graph where the demand curve is below the supply curve can contain some inventory: that's exactly what the order book of a stock exchange is, with bid (buy) and ask (sell) offers.


What can we know for sure among ambiguous impulse responses in RBC? I'm trying to understand the section on responses to productivity disturbances in Understanding Real Business Cycles by Plossor.
It seems to me ambiguity of representative agent's response in consumption, saving and leisure rooted in two reasons:

Wealth effect is ambiguous. As far as I know, wealth effect means a temporary shock or a permanent change make the choice set determined by production techonology and budget bigger or smaller. The curvature of indifference curve is to some degree arbitrary, so the reallocation of consumption, saving, working, leisure could be arbitrary.
The relative strength of wealth effect against substitution effect is also arbitrary.

Here's my attempt: Suppose that, a higher value of (unexpected) temporary technological shock hits the economy at time $t$.
$$\sigma_{high}F(K_t, N_t)=C_t+(K_{t+1}-(1-\delta)K_t)$$
Substition effect indicates more labour at $t$, but wealth effect makes this indetermined. $\{K_{t+1+i}\}_{i\in \mathbb N}$ is higher in each period for sure. But it's unclear for $\{C_{t+i}\}_{i\in \mathbb N}$ and $\{N_{t+i}\}_{i\in \mathbb N}$. It could be the case both wind up with higher value for all $i$, or only one of them wind up with higher value for all $i$.
It seems to me the qualitive result for a permanent increase of productivity is the same.

Yes and no.

Temporary Shock

It is correct, that a temporary increase in TFP has unclear effects on leisure:

The **income effect* will increase consumption and leisure
The **substitution effect* will decrease leisure.

On total, effects onto leisure are unclear. But, unless you have very specific preferences, consumption should increase. What are the cases in which it doesn't? Let $l$ denote leisure, $1-N$:

Violation of local nonsatiation: Households do not prefer more consumption at their current $(C,l)$ locus, so an income effect will not make them consume more (shut down income effect)
Strong and negative elasticity of substitution between $(C, N)$: People that work more, prefer more consumption. Hence, if substitution effect increases $N$, it also decreases $C$ (putting in $C$ into the substitution effect)

1. is generally not accepted, both empirically and theoretically (remember, we're here not talking about individual preferences, but under aggregation). 2. is more disputed, but elasticities of substitution found typically range between $0$ and small positive amounts, not strongly negative as required here.

So, to conclude, temporarily, $C$ responds positively to a positive TFP shock.

Permanent Shock

Yes, a temporary shock could end up however, given how we calibrate it. But here's the gist:

We calibrate models to data.

One standard calibration goal are the Kaldor facts, among which we will find:

Despite an strong rise in productivity during the 20th century, working hours per capita have remained constant.

This means that in calibrating an RBC model, we will chose specifications that give us exactly this result. A simple way of doing so is having an additive separate utility function of King-Rebelo-Plosser type (Note that Prescott in his original paper claimed that only log-log preferences would reproduce this independence, which is incorrect).

Temporary Shock revisited

To conclude, we calibrate the model such that an income and substitution effect on leisure cancel out exactly. The income effect of a permanent shock is bigger than the income effect of a temporary shock. Hence, if it holds that permanent increases in TFP do not affect working hours, it must be the case that temporary increases in TFP do positively affect working hours, since the substitution effect is the same, but the income effect is smaller.


Why did Federal Reserve buy already-issued corporate bonds, ETFs, and agency commercial mortgage-backed securities? I screen-shot WSJ on Sep 18 2020.

I'm assuming that Fed Primary Market Corporate Credit Facility didn't buy ALL newly issued corporate bonds?

Please see question in title. Why didn't Federal Reserve just bloat or maximize its Primary Market Corporate Credit Facility? Why use its Secondary Market Corporate Credit Facility and Open Market Account?

Federal Reserve | Published July 10, 2020 | Full text published under the following link (with info-graphic):
https://www.clevelandfed.org/en/newsroom-and-events/infographics-library/smccf.aspx
FULL TEXT

The Fed’s Secondary Market Corporate Credit Facility, explained

America needs a bridge—a bridge that will get households, communities, and businesses over the unanticipated challenges created by the COVID-19 shutdown. In helping to build that bridge, the Federal Reserve, with authorization from Congress, has created and revived a number of rare lending programs, each providing targeted assistance to the needs of those impacted.

The challenge: Investors buy corporate bonds, which are essentially IOUs that companies issue when companies need money. While on the investors’ books, the bonds generate income as issuing companies pay interest and, after some period of time, the original amount financed. In the event an investor wants to sell off some of its bonds before they are paid off by the borrower, they typically can do so because many other investors are willing to buy corporate bonds. Current conditions have made investors less willing to buy such bonds, leaving those that want or need to sell the bonds without buyers. At the same time, the pandemic-related shutdown has caused lost revenues for many companies, some of which invest in corporate bonds, and that means more investors may want to sell off investments such as corporate bonds to turn them into cash they can use to pay their bills.

The response: The Fed is buying corporate bond-related assets so that investors that want or need to sell them off are better able to do so and get the cash they need.

The details: This work is intended to stimulate the marketplace in which corporate bonds are bought and sold after they have been issued. Corporate bonds are first issued by large businesses to raise funds. Investors buy the bonds, lending businesses the cash they need, and expect to be paid interest in addition to the original amount borrowed. In the secondary market, those initial investors sell the corporate bonds to other investors. The Fed’s purchases are intended to allow investors to buy and sell corporate bond investments in the secondary market as needed. Should losses occur, the US Treasury has invested $25 billion to cover losses arising from this lending.

The goal? To help ensure a reliable market for investors that need cash in order to pay off debt and sustain themselves until economic conditions normalize.

The Fed is committed to transparency and makes periodic reports and updates related to this lending available.

For more information: https://www.newyorkfed.org/markets/secondary-market-corporate-credit-facility

Federal Reserve | Published July 10, 2020


Key points for a speech on Capitalism I was given an English task in which the teacher has asked us to do a presentation/speech on Capitalism. The thing is he didn't specify which aspects (historical, economical and the political) we should be focusing on as the concept is rather vague. Thus, I was wondering if anyone here could suggest some points that the presentation should contain (we'll be using slides), and some references (movies, books and videos) so as to enlighten the class more on the concept with simple ideas. Thanks in advance.

PS: I don't study history nor economics nor politics, so the presentation will be just for general culture.

Search for 'Milton Friedman' on youtube, he was a nobel prize winning economist, probably the best advocate of capitalism, especially his Q&A's with college students are very entertaining and also very accessible.


When Does a Country's External Debt Become a Problem? Here in the UK, the external debt is currently over 300% of GDP. To me this looks very worrying, using my basic financial logic, it seems like this debt will only go up (because of interest combined with the fact we don't have enough money to pay it off). The fact that it's over 3 times the value of the British economy worries me even more, like there's no end in sight...

I understand a country can remain in debt forever (so long as they keep paying their lenders) but how long can a country like the UK keep going like this? Public debt is 94% of GDP so that's just about manageable, but what happens if the external debt keeps rising? or the GDP begins to fall? Can the UK maintain a level of debt that is 300% of GDP forever? Basically, what would it take to turn this debt into a total financial collapse? I've heard of something similar happening in Argentina in the late 90s and they had a catastrophic recession, could that happen in the UK anytime soon?

The position is both worse and better than you describe

If you look at Section 7 and Figure 6 of  https://www.ons.gov.uk/economy/nationalaccounts/balanceofpayments/bulletins/balanceofpayments/quarter1jantomar2018 you will find that UK liabilities (public and private) abroad were estimated to be £10,894.5 billion at the end of March 2018, about 5.30 times UK GDP in the 12 months to March

But at the same time UK assets abroad were estimated to be £10,632.2 billion, about 5.18 times UK GDP

So the net position here was a liability of £262.3 billion, about 0.13 times UK GDP (a minor rounding issue here), i.e. 13% of annual GDP.  In fact the net position is the difference between two enormous numbers largely resulting from financial activity in the City of London, both of which are estimated with errors, so large revisions to the estimated net position is common

A small UK net international liability represents the position for many years and so in that sense empirically seems to have been sustainable.  For Greece this net figure became large and unsustainable, contributing to its crisis, and not helped by the Eurozone restricting the possible solutions.  The answer to your question "When Does a Country's External Debt Become a Problem?" seems to be "when people start to worry that they will not be able to get their money back and so stop lending more money".  Turkey may be facing this issue at the moment.  But I doubt there is a fixed figure you can identify


Causal effect of population growth on GDP per capita increase I'm wondering if there is a logic for the causal influence of population growth on the increase in GDP per capita.

I guess,

increase in population as a factor of production might move aggregate supply curve to the right, lowering the price.
increase in population as consumers might move the aggregate demand curve to the right, increasing the price.

In both cases, the total amount may increase. But I don't understand how these reasons combine to explain the causal effect (if any!). So, why they are so worried about low birth rate?

Birth rate affects not only total population but also the age structure of the population.  If the birth rate over many years is below the level at which the population replaces itself, the result will be a falling population with an increasing proportion of elderly and often economically inactive people.  With a falling proportion of economically active people, GDP per capita is likely to be less than it would have been with a higher birth rate.

This effect can to some extent be offset by policies to encourage older people to continue working for longer, such as raising the age at which state pensions become payable.


Why the stock market went up when the currency being weakened? From Bloomberg Market Concept (BMC), in 2013, Japan is traditionally a heavy exporter, especially automobiles. When the yen being weakened, the Japanese automobile being more attractive because it seems to be cheaper to the target market.BMC also documented that "The automobile won more order as a result, so the stock market went up."
Why the stock market went up, is it because investors optimistically value these firms because of better performance? And the result seems to be triggered by the reduction in the value of the yen compared to USD, so the stock market went up, but whether the Japanese stock market still went up if the stock price being calculated in USD?

Quoting from the question

When the yen being weakened [...] "The automobile won more order as a result..."

and more orders made the automobile sector more profitable, hence the higher stock price.


For a certain good, if there is a one-time fixed cost for a consumer to switch to a different firm, then this good is? For a certain good, if there is a one-time fixed cost for a consumer to switch to a different firm in a later time, then we say this good is ____ (or has property ___). Examples include

Consumer: House renter
Firm: Landlord
Good: Stays in a house
One-time Cost: Moving cost

Consumer: Mobile phone user
Firm: Wireless carrier
Good: Uses of phone over a carrier network
One-time Cost: New account activation fee

The good has a switching cost / is subject to a switching cost. Rarely used: it is a Diamond good. (Referring to the Diamond 1971 paper.)


How does the treasury determine the rates of the bonds they sell? My understanding is that price and yields are determined via supply and demand. How are the interest rates determined when the bonds are auctioned off?

The question appears to refer to what is normally called the coupon rate (but the US Treasury does call the interest rate in its documentation). This is different than the yield, which is determined by a 1:1 function of price.
Issuance conventions vary across markets. I will just use the U.S. Treasury as an example. Firstly, this only applies to new bonds/bills, since reopenings already have a coupon, and bills are discount instruments (no coupon). I will ignore inflation-indexed bonds.
For the auctions inquestion, competitive bidders specify yields to 3 decimal places. Then:

Single-price auctions. The interest rate we establish produces the price closest to, but not above, par when evaluated at the yield of awards to suc- cessful competitive bidders.

That is, the coupon rate is set to get a price closest to par from below, using 1/8% increments.
Quote from: section 356.17 oflink to regulations
Treasury auction information page.


Retaliating against a foreign country's import tariffs When Trump imposes tariffs on imports from China, he is hurting American consumers who won't be able to take advantage of cheap Chinese goods. If China was to retaliate by imposing tariffs on imports from the US, they would be hurting Chinese consumers in the same way. So, is there is any need for a country to retaliate against another country's import tariffs?

If you're using a simple standard model of trade, like the Ricardian model, trade is good and all tariffs, price floors, price ceilings, etc are bad. Retaliatory tariffs will still be problematic for the country imposing them. I think your reasoning is fine in this regard.

However, trade occurs over many many years. Responding to the initiation of new tariffs by imposing temporary tariffs will reduce the short-term welfare for everyone, including the tariff initiator.  The country which began this "trade war" will then have lower welfare from both sets of tariffs.  Hopefully, the initiating country will remove their tariffs and return to normal.

The retaliating country has some short-run cost, but if it leads to a final low-tariff state, retaliation may be worth it in the long run.

Imagine the game is a repeated version of the following:

\begin{array}
\\ & Free Trade & Tariff
\\ Free Trade&(5,5)&(3,4)
\\ Tariff & (4,3)&(2,2)
\end{array}

Where for some reason a party has ended on Tariff, perhaps because they have a "trembling hand" with regard to policy.  Note this is different then the prisoner's dilemma because the payoff does not increase as a result of imposing tariffs.


Why don't we transact with securities rather than fiat money? I don't know a whole lot about economics besides for an intro course I took once so bear with me please...

I'm a software developer working with cryptocurrencies, and I got to thinking that it would be technically possible to tokenize securities (ideally giving legal ownership of that security to the token holder) and use them for transactions within an economy instead of currencies like the USD. One token would represent a certain percentage share in a holding company or a mutual fund or maybe even a government bond.

After looking into this, there seems to be some large legal barriers preventing this sort of thing from happening, and it would be more of a feat in that regard than on the technical side of things (I think), but, more theoretically, it got me thinking about what the effects of replacing fiat with securities might be.

One thing I've heard is that a mildly positive inflation rate is desirable in a stable economy because it increases the opportunity cost of holding money thus stimulating more spending. A deflationary means of exchange might reduce spending and cause bad things? I'm also aware that using fiat allows the Federal Reserve some control over the economy via monetary policy, which is definitely needed in this boom and bust situation we're in now.

On the other hand, securities actually have what all those gold-standard nutjobs want a currency to have, actual intrinsic value! It allows for anybody taking part in an economy to necessarily own part of it. It could be a kind of free-market socialism where the means of production is actually owned by the people (redistribution of wealth still needed of course). Furthermore, In today's world, the very rich tend to have proportionally more of their wealth tied up in investments compared to the working class, further widening the divide. If securities were currency, investing wouldn't be a choice but a requirement.

My main question is, why isn't this the way it works today? Certainly it would be possible for just a regular US dollar to be legally backed by equity or debt with no crypto hacking involved, so why haven't we done so?

You've just reinvented bearer bonds. From that page:

  Bearer bonds are bonds that are owned by whoever is holding them, rather than having registered owners like most other securities. Like most other bonds, they have a stated maturity date and interest rate, but coupons representing interest payments are generally physically attached to the security and must be submitted to the company for payment.

It's been done. Like cryptocurrencies, the one class of transactions they turned out to be well-suited for was the class of illegal transactions; money laundering, payment for criminal services, prohibited capital movements, etc.

And having a fiat currency which is simply backed by a country's economy, has succeeded against all other kinds of currency as the most viable.


Why in the quantitative equation: $MV=PY$, $V$ and $Y$ can be taken as fixed? To  equation is
\begin{align}
MV=PY
\end{align}
where $V=\frac{1}{k}$.
Why $V$ and $Y$ can be taken as fixed or constant?
Why can $V=\frac{1}{k}$ too?
Thanks in advance

There are multiple answers to this question.

In any model you can always make a thought experiment where you hold certain variables fixed.  So one answer, although not very satisfying one, is that you can view it as a thought experiment. For example, in physics distance traveled equals velocity times time or $D=tv$ and you can always make a thought experiment where you choose 1 or 2 variables that will be fixed and see how others behave.

If you are asking why it would make an economic/intuitive sense to take $Y$, $V$ fixed then:

Regarding $Y$: From economic theory we know that $Y$ in a long run depends on productive capacity of an economy that is independent of $M$, $P$ or $V$ (you can learn more about that in any 101 Macro/econ book such as Mankiw Macroeconomics or Mankiw Principles of Economics). So you can consider it to be exogenously given and holding it constant is just performing a thought experiment where you assume you have economy that is not growing so you can see how other variables behave.

Regarding $V$, in the past (before 2008), empirically speaking velocity of money was very stable (see the Fed graph below). As you can see between 60s and late 2000s it was always hovering around 1.9 so on empirical grounds many economists considered $V$ stable, although it is worth while pointing that recently it significantly declined. Nonetheless, given that it was historically very stable this might be just transitional, or it is possible there was a structural break and now velocity will remain approximately constant at some lower level. It is also possible that it was wrong to assume it is constant in the long run based on previous empirical observations.


If aggregate demand and supply do not have a specific value, what is the point of the formula? I have been learning about aggregate supply and demand. I have just read this answer which states:

"Aggregate supply is a relationship of price level and output. It is a function, or a curve, or a table. It is not a single value. If we know a particular price level, then we can determine the level of output that would correspond with that. The GDP for 2006 is determined by plugging in the price level of 2006 to the AS curve for 2006, and seeing what output is produced at that price level."

I don't understand, I have learnt the formula AD = C + I + G + (X - M). Wouldn't that give a specific numerical value?

This should have been a comment, but I have less that 50 reputation and I am new to this site as are you.

First, I think the point is clear and concise, but you may be misinterpreting it slightly.

Think of demand or supply like a pendulum swing that is being pulled by two forces.

For supply this is prices and output, prices can be pulled by output and output can be pulled by prices.

However, you're right, you can get specific values, you can use the formula.

The formula is particularly useful for averages, the prices and output over time.


A case where the solution to Expenditure Minimisation Problem is not a solution to Utility Maximistion Problem I am looking for a preference relation which satisfies the above property.

If the utility is continuous and locally insatiable then the Hicksian demand equals the Walrasian demand.

So you'll need to look for a utility function which violates atleast one of these.

Lets try a simple violation of non-satiation $$
u(x,y) = \left\{
        \begin{array}{ll}
            x+y & \quad x+y \leq 1 \\
            1 & \quad \text{ otherwise}
        \end{array}
    \right.
$$

With $p=(1,1)$ and $w=2$ Then $x(p,w)=\{(x,y): 1\leq x+y\leq 2\}$. This is becuse the consumer's problem is $$\max_{x,y} u(x,y)$$ $$\text{st }x+y\leq 2$$

$u(x(p,w))=1$ so apply the EMP to this utility $$\min x+y$$ $$\text{st } x+y\geq 1$$
This gives $h(p,1)=\{(x,y): x+y= 1\}$. Hence, $h(p,v(p,w))\neq x(p,w)$.

I think a violation of non-satiation is much easier to construct a counter example with. If we have a 'thick' indifference curve then the UMP will choose the entire curve but the EMP will choose at most the lower envelope, as why spend more to get the same utility?

Conversely, we can use continuity to find an example of where $h(p,v)$ is not a subset of $x(p,e(p,v))$. Take $p$ and $w$ as above and

$$
u(x,y) = \left\{
        \begin{array}{ll}
            x+y+1 & \quad x+y \geq 2,\ x\geq y \\
            x+y & \quad \text{ otherwise}
        \end{array}
    \right.
$$

The EMP with $v=2$ is

$$\min x+y$$
$$\text{st }u(x,y)\geq 2$$

$h(p,2)=\{(x,y): x+y=2\}$. Now look at the UMP at $p,w$

$$\max u(x,y)$$

$$x+y\leq 2$$

$x(p,w)=\{(x,y):\ x+y=2, x\geq y\}$. Hence, $x(p,e(p,v))$ is a proper subset of $h(p,v)$.

I found it alot easier to come up with a counterexample without continuity (our $u$ is locally non-satiated). Perhaps with continuity you get $h\subset x$ and local non-satiation you get the reverse? Ill leave that to you.


Mundlak Approach when panel data include time fixed effects What would happen when, starting from a panel data model with group fixed effects and time fixed effects I apply the Mundlak approach?

This is the model:

$ y_{i,t} = c + \beta x_{i,t} + \alpha_i + \delta_2 B2_t + \delta_3 B3_t +....\delta_T BT_t +\epsilon_{i,t} $

where $B_t$ are time dummies (time fixed effects).

Should I also include the sample averages for each $B_t$ so that the equation becomes:

$ y_{i,t} = c + \beta x_{i,t} + \gamma \bar{x}_{i} + \delta_2 B2_t + \delta_3 B3_t +....\delta_T BT_t +  \tilde{\delta}_2 \bar{B2} + \tilde{\delta}_3 \bar{B3} +....+\tilde{\delta}_T \bar{BT} +\nu_{i,t}$

wher $\bar{\bullet}$ denotes the sample average computed over each panel for variable $\bullet$.

If the panel data set is balanced, the averages will cause perfect collinearity and will all be dropped. So you lose nothing.

If it is unbalanced, the averages of the time dummies will explain some aspects of the structure of the data set. For example, let T=3 so we have two dummies. B2bar(i) is 1/T(i) if observed at t=2, and 0 otherwise. Similarly, B3bar(i) depends on whether t=3 is observed and on T(i). So, I think the individual averages of the time dummies control for heterogeneity due to the period effects and data structure heterogeneity. I personally believe that it is safe to include those Btbar(i) terms. But there may be more fundamental issues involved in this unbalanced case as it relates to modeling selectivity.


Microeconomics: Calculating and Comparing MRS for Various Utility Functions I am having a problem determining "which utility function is not like the rest." The following utility functions are:

$$U_1(x,y)=x^{2/3}y^{1/3}$$

$$U2(x,y)=3x^2*y+2$$

$$U_3(x,y)=\frac{1}{3}\ln (x)+\frac{2}{3}\ln( y)$$

$$U_4(x,y)= 4\ln(x)+2\ln(y)-12$$

So my general thought in approaching this question was to obtain the MRS for each of the functions. After doing this, however, I found U1 and U2 to have an MRS of -2y/x. However, I am having a much bigger problem differentiating U3 and U4. My intuition is that U4 would be equal to U1 and U2 because we value x twice as much as we value y. however, can anyone help out with the math?

The rule for differentiating a natural logarithm is

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

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

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

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

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


What data/factors/attributes do the professionals monitor in a top-down analysis on the macroeconomics? I am learning to do top-down analysis on a number of sectors in the US equity markets. I have never done it before and I would appreciate it if some professionals can share their experiences with me.
Specifically, I would like to know the list of attributes/data/factors I should be measuring and monitoring.
Most of the websites I can find are vague about this subject. They mostly generally list a few vague things such as GDP, inflation, treasury bond yields, etc. However, I would like to know the more specific data used. Take treasury bond yield for example. I have learnt from elsewhere that people are monitoring the 2Y yield vs 10Y yield for indication of an economical reversal. I can't find any website that gives this kind of working examples on a top-down approach.
Can you tell me what other specific measures/data the industry's professionals normally would look at in a top-down sector-wide analysis?

I am experienced with fixed income macro analysis. The short answer is that what you look at depends on what you want to know.

In rates analysis, the key driver for bond yields (hence prices) is the expected path for the overnight rate. The central bank sets the overnight rate. So you need to be interested in what the central bank is looking at.

The Bank of Canada is a fairly typical developed central bank, and has very good primers on their decision-making process. They have some pages describing indicators they look at here: link to BoC page. You could draw analogies to indicators in other countries.

(With regards to the 2/10 slope, that is indication of the expected path of short rates. The usual story is that if the slope is negative - 10-year yield less than the 2-year - rate cuts are expected, which normally happens in a recession. There is a huge literature on the “yield curve” for forecasting; should be easy to find something with a web search.)


Specifying a Diff-In-Diff with multiple treated groups over multiple years Suppose I have a group of 100 counties. Further suppose that the state decides to allocate a lump sum amount of money to some counties each year to boost spending on education in that county (until, say, 50 of the 100 counties have received money). Suppose my goal is to use a dif-in-dif approach to analyze the effect of the spending increase.

How would I specify this DID regression? The basic idea is that, for each year, I want to compare educational outcomes in counties that receive increased funding with those that do not. I want to do this so that I eventually get to evaluate the effect for all 50 counties that receive this educational funding boost.

My guess is something like this: I need to specify a typical DID regression with year and county FE s.t. for any given year, a county that receives funding is in the treated group and any county without the funding boost is in the control group. However, the effect of that funding boost means that a county treated in year $t$ ought not be included in the control group in year $t+1$, despite not receiving a funding boost in $t+1$.

Am I over complicating this? It doesn't seem to me like specifying a normal DID with year and county FE will quite work.

Firstly, the diff-in-diff already is a fixed effects estimation. If you run a regression with FE and your time variant terms ($S_i*T_t$ and $T_t$) as in the plain and simple diff-in-diff setup, as such:

$$ Y_{it}=B_0+B_1T_t+B_2S_i+B_3S_i*T_t+e_{it} (1)$$

You should obtain similar results. Secondly, your specification -- since it will be on a multiple treatment setup -- should consider if your treated groups in different periods of time are disjoint, that is, if a unit of observation might receive the treatment multiple times. Considering that adding a fixed effects in time might be convenient for your estimation, as you were considering, I would work with the following specification if the the treated groups are disjoint:

$$ Y_{it}=\alpha_0+\alpha_1S_i*T_t+\alpha_2S2_i*T2_t+v_t+u_i+e_{it} (2)$$

And if they are not disjoint, I would consider a interaction between the treatment effects:

$$ Y_{it}=\beta_0+\beta_1S_i*T_t+\beta_2S2_i*T2_t+\beta_3(S_i*T_t)*(S2_i*T2_t)+v_t+u_i+e_{it} (3)$$

You should also consider testing the common trend hypothesis, since you still need an adequate control group.


Bank Runs During the 1930s Why were there numerous banks run right after the stock market crash of 1929? Why would a bank fail because the stock market crashes?

The bank runs were not caused by the crash; they were both caused by the same phenomena.  In the run-up to the crash banks around the United States sent requests to city banks and the reserve banks for gold.  In part due to the poor information collection system at the time, there was no way for the Fed to distinguish between a heating up economy and a true shortage of gold.

The Fed interpreted the requests for gold as a heating up economy and so contracted the money supply by 3%.  This exacerbated the shortage of gold around the United States.  It quickly propagated through the world financial system because of the shared gold standard.  The market crashed because banks were issuing margin loans at up to 90% of value.  The older system in use, before the Fed, was to create a banking suspension where banks would collect gold, but not disburse gold.  The Fed made that functionally impossible forcing runs.

The thing to remember about stocks is that they are purchased with your spare money.  Given a choice of eating food or buying shares in Apple, people will choose food.  When the money supply contracted, funds didn't exist for businesses to continue operations, to pay employees, so people sold stocks.  The sales forced margin calls, which triggered more sales, which triggered more margin calls and so forth.

The NY Fed did provide liquidity at the beginning of the crisis, but the Board of Governors were still convinced that the contraction was necessary and forced the end of the NY Fed's liquidity provision.

The stock market was the thermometer, not the cause.  The system had become sick.

To be fair to the Fed, it was itself part of a much larger set of global events without an information base to act on.  It was also true for the Bank of England or the main European banks.  The gold standard propagated economic issues in ways that were not understood at the time.

When World War II ended the nations of the world created a variety of treaty organizations such as the United Nations and the OECD to prevent such a sequence of events happening again.  The problem is that restricting those events from happening means that some large business owners miss out on opportunities and so there is pressure to dissolve or weaken the treaty organizations.  In addition, the major nations sometimes endanger the system.  You can see this very directly through the actions of the Trump administration.

The postwar era institutions really do need reform as the threats have changed, but the political will to protect the world doesn't play well in local political processes.


Constant Regressor in GLS Consider the following regression model:

$y_{i1}=\beta_1 +u_{i1}$
$y_{i2}=\beta_{21}+\beta_{22}x_i+u_{i2}$.

If $E(x_i' u_{i1})\neq 0$ and $E(x_i' u_{i2})=0$, will we get consistent estimators for $\beta_1$ and $\beta_{21}$ using GLS?
I think we will not get consistent estimators for both, because the assumption $E(x_i\otimes u_i)=0$ is violated. Please correct me if I am wrong. Thanks in advance!

The equation-wise OLS estimators are consistent under the further normalization that $E(u_{i1})=0$ and $E(u_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x_i' u_{i1}) \ne 0$ is irrelevant because $x_i$ is not a regressor in the first equation.


More than one Bellman Equation I'm attending to my first dynamic optimization course, and what I don't fully graps yet is that sometimes we have to use more than one bellman equation.
How do you realize that? I mean how do you know when your problem solution require more than one Bellman equation?
For example this problem taken from Sargent's Recursive Macroeconomic Theory 2nd edition.
An unemployed worker receives each period a wage offer w drawn from the distribution F(w). The worker has to choose whether to accept the job – and therefore
to work forever – or to search for another offer and collect c in unemployment
compensation. The worker who decides to accept the job must choose the number
of hours to work in each period. The worker chooses a strategy to maximize
$E\Sigma_{t=0}^{\infty}\beta^{t}u(y_t,l_t)$
and $y_t=c$ if the worker is unemployed, and $y_t=w(1-l_t)$ if the worker is
employed and works $(1-l_t)$ with $l_t$ leisure and  $0<l_t<1$
Analyze the worker’s problem. Argue that the optimal strategy has the reservation wage property. Show that the number of hours worked is the same in every
period.
The solution's manual goes like this for the part of stating the Bellman equations:
Let s be the vector of state variables. We choose $s=(w,0)$ where $w$ is the wage offer and $0=E$ if the worker is employed and $0=U$ if the worker is unemployed. Consider first the situation of an employed worker. Bellman’s equation is:
$v(w,E)= max \{u[w(1-l),l]+\beta v(w,E)\}$
and for unemployed worker:
$v(w,U)= max \{v(w,E);u[c,1]+\beta\int v(w',E)DF(w')\}$
So being more concrete.
Why the solutions requires two bellman equations and how do you realize that when reading the problem?
For example my first guess when trying to solve without looking the solutions I wrote:
$v(w,E)= max \{u[w(1-l),l];u[c,1]+\beta\int v(w',E)DF(w')\}$
Why is this different?
Thanks in advance.

(The second equation for the value function of the unemployed should be
$$
v(w,U)= \max \{v(w,E); \,u[c,1]+\beta\int v(w', U) dF(w')\}. \quad (*)
$$
)

...how do you know when your problem solution require more than
one Bellman equation?

Whenever the state space of the problem contains discrete coordinates,
there would be  "multiple" value functions, indexed by the discrete coordinates.
Here the state space is $[0, \infty) \times \{E, U\}$. The second coordinate is discrete. So there are "two" value functions $v(w, E)$ and $v(w, U)$.

How do you realize that?

Any variable that determines the optimization problem faced by the agent is part of the state.
In this particular example, an unemployed agent has the option to switch to being employed. An employed agent has no choice but to stay in the job---"...to work forever".
The choice set, therefore the decision problem, faced by the agent are different depending on his employment status.
This tells you that employment status is part of the state.
(Note that, if the employed worker has the option to leave the job, the decision problem faced by the agent are still different depending on his employment status. In that case you would still have two value functions and they would be "intertwined", so to speak.)

Argue...the optimal strategy has the reservation wage property.

This is more or less immediate from the equation $(*)$.
The reservation wage $w^*$ is given by
$$
v(w^*,E) = u[c,1]+\beta\int v(w', U) dF(w').
$$
At $w = w^*$, agent is indifferent, $v(w^*, U) = v(w^*, E)$.
You would expect that for $w < w^*$.
$$
v(w,U) = u[c,1]+\beta\int v(w', U) dF(w') > v(w,E).
$$
This is a typical option-exercise problem.
The difference $v(w, U) - v(w,E)$ when $w < w^*$ is the option value. If the offered wage $w$ is too low, the agent would rather keep the option and not exercise it.
(This is an American-type option, which can be exercised anytime. Same "exercise threshold" phenomenon occurs in the continuous-time setting, which is sometimes more convenient. There the threshold is given by the smooth-pasting condition.)
General Comments
The general formulation of the Bellman equation is
$$
V(s) = \max_{c \in \mathcal{C}(s)} \int_{\mathcal{S}} V(s') dF(s'; s, c),
$$
where $\mathcal{C}(s)$ is the choice set faced by the agent at state $s$ and
$s' \mapsto dF(s'; s, c)$ is the Markov transition kernel if agent makes choice $c$ at state $s$.
(For notational simplicity, assume no period utility/discounting/etc. The discussion would not change.)
Therefore, by definition, any variable that determines the optimization problem faced by the agent is part of the state.


Non-axiomatic definition of well-being I first wanted to ask this on philosophy@SE but, on a second thought, decided this might be a better place.

If we defined a general policy's objective being "to maximize well being and minimize suffering", we can say that "suffering" in that sentence is the opposite of well-being. But how do you define "well-being" by using lower level indicators, none of which are regressivly axiomatic and in a way that's so abstract that it's difficult to blame for cultural or any other kind of bias?

I would imagine that, at this point, defining well-being should be a province of economics and not philosophy. Do economists concern themselves with the question of defining well-being at an individual or collective level, by using natural/rationalist terms so that no amount of regression of the origins of any preceding term's definition runs into a proposition that will be considered "axiomatic"?

When you say that a policy's objective is to "maximize well-being", presumably you mean "maximize collective well-being".  And presumably, by "collective" well-being, you mean some sort of aggregate or average of the individual well-beings of all members of society.  So your question breaks into two parts:

What is the right measure of individual well-being?
What is the right way to aggregate individual well-being into a measure of collective well-being?

Question #1 is really a philosophical question (although many of the more philosophically minded economists have thought about it).  Broadly speaking, there are three families of approaches to defining/measuring individual well-being:

Hedonic:   Well-being is an emotional state (e.g. happiness).  Different hedonic approaches differ as to whether well-being is a short-term or instantaneous phenomenon (e.g. pleasure, absence of pain) or a more long-term one (e.g. "overall life satisfaction").
Preference-satisfaction:  According to this approach, well-being is the satisfaction of the individual's preferences.  To the extent that people want to be happy, and to the extent that they are happy when they get what they want, this overlaps the hedonic approach.  But there are situations where they come apart.  Also, different preference-satisfaction approaches differ over whether we care about the individual's actual preferences, or her "ideal" preferences (i.e. those she would have if she was perfectly informed, perfectly rational, etc.).
Perfectionist (or "objective list") approaches.  These approaches propose to measure well-being in terms of some list of objective criteria, such as physical health, education, material consumption, participation in the community, etc.  Whereas the two previous approaches are "subjective" (it is is up to individual to determine her own level of well-being), this approach is "objective" (the individual's own opinion is only one part of the data, and perhaps is entirely irrelevant).   So in an objective list approach, an individual could have a high level of well-being even if she herself thinks she is miserable (and vice versa).

Turning to the second part of the question:  what is the right way to aggregate individual well-being into collective well-being?  This is also a philosophical question (indeed, it is the central question of the entire "welfarist" approach to moral philosophy).  But it has also been explored extensively by economists.  Economists refer to such an aggregator as a social welfare function or social welfare order, and there is a huge literature on the axiomatic characterization of various families of social welfare functions.

For a good introduction to these topics, I suggest the following books:

Handbook of Well-being and Public Policy, edited by Matthew D. Adler and Marc Fleurbaey,  Oxford University Press, 2016.
Handbook of Rational and Social Choice, edited by Paul Anand, Prasanta Pattanaik, and Clemens Puppe, Oxford University Press, 2009.
Handbook of Social Choice and Welfare, volume I, edited by  Kenneth J. Arrow,   Amartya K. Sen and  Kotaro Suzumura,  Elsevier, 2002.
Handbook of Social Choice and Welfare, volume II, edited by  Kenneth J. Arrow,   Amartya K. Sen and  Kotaro Suzumura,  Elsevier,
2011.

Each of these handbooks contains chapters written by different experts on different aspects of these questions.  The first two books are the most "philosophical".  The second two books are more focused on axiomatic analysis of social welfare functions.  The very first book (the Handbook of Well-being and Public Policy) is probably the best place to start.


Multidimensional screening and convexity of the surplus/rent function I'm starting to read the literature of multidimensional screening models for monopolists selling $n$ goods to a continuum of buyers with $m=n$ dimensional types, and Rochet (1987) proves that a mechanism is implementable if and only if the surplus function is convex. Let $s(\theta)$ be the surplus function defined as $\max\{ u(x,\theta) -t(\theta)\}$ where $x \in \mathbb{R}^n$, and $t$ is the tariff function the monopolist announces by the Taxation Principle.

For example, the monopolist solves the problem of maximizing the following functional (assuming 0 cost of producing the good) $$\max J(t) = \int_D t(\theta) \mathrm{d} \theta = \int_D u(x,\theta)-s(\theta) \mathrm{d} \theta $$ where $\theta \in [0,1]^n$ and $t: [0,1]^n \to \mathbb{R}$.

My question is: what is the intuition behind the fact that $s(\theta)$ must be convex? What happens if the function is piecewise linear (and hence trivially convex) but globally concave, i.e. there are kinks along null-measure sets? Can you point me to any resources?

First, it's important to be clear about what Rochet proves. In theorem $1$ of Rochet (1987), he shows that cyclical monotonicity is equivalent to the implementability of a mechanism for general environments (i.e. arbitrary allocation/type spaces). (See the statement of theorem $1$, or a standard text such as Borgers (2015), for the definition of cyclical monotonicity.)

The convexity result (proposition $2$) applies to the case when $u(x,\theta)$ is linear in $\theta$. In this case, we have that

$$ V(\theta) = \max_{\theta^\prime} \, u\left(x\left(\theta^\prime\right),\theta\right) - t\left(\theta^\prime \right) .$$

The convexity of $V$ follows from the fact that it is the maximum of a family of linear functions. I'm not sure there's much more intuition to be given here, but a simple drawing may help. (That is, draw two straight lines that intersect. The maximum over the two straight lines will be convex, while the minimum will be concave.) This may also help.


Travel Cost Method - how to treat people who live at a site We are about to do a survey using the individual travel cost method to value a harbour, located in a fishing society, which is visited by tourists each season. My question is - should we include in the analysis people who live in this society and who, therefore, have zero travel costs to go there? Or should we skip them in the econometric analysis?

Short answer.  It is perfectly correct, and probably simplest, to exclude from the analysis the members of the fishing society, provided  that you present your result as “the tourist value of the harbour” or “the value of the harbour to tourists” (and not as the total value of the harbour).

Long answer. The individual travel cost method involves two stages.  In the first stage, the data collected is used to estimate a trip-generating function, something like:

$$V_i  =  f(TC_i,\pmb{A}_i)$$

where $V_i$ is the number of visits by individual $i$, $TC_i$ is the round-trip travel cost from $i$’s location of origin, and $\pmb{A}_i$ is a vector of characteristics judged to be relevant, eg age, income.  In the second stage, a demand curve for visits to the site is estimated on the assumption (call this the key assumption) that individuals would respond to different hypothetical prices added to their actual travel costs in the same way that, according to the trip-generating function, they respond to different  travel costs.  The site value is then taken to be the consumer surplus, which if access to the site is free is the whole area under the demand curve.

The fact that travel cost for some individuals is zero is not in itself a reason for excluding them from the analysis.  One reason is that their data is relevant in estimating the trip-generating function: a data point with $TC_i = 0$ is still a data point, and to exclude such points would shift the line of best fit.  Also, the second stage methodology means that, other things being equal, individuals with zero travel costs make a disproportionately large contribution to the total consumer surplus (because the sum of a hypothetical price and zero is less than the sum of that price and a positive travel cost, so that the number of visits associated with the former will be greater).

However, there are other reasons for excluding members of the fishing society.  The origin locations of the tourists, it can probably be assumed, are to a large extent randomly spaced.  It seems unlikely however that this is so for the fishing society.  Very likely they live at the harbour because, given their dependence on fishing, they have chosen to do so.  If so, one of the assumptions of the travel cost method is undermined.  Freeman, Herriges & Kling (2014) list as one of the assumptions of the method (for the case of recreation but the point is more general):

  … it is assumed that the individual’s choice of where to live (which is one determinant of the cost of a trip to a recreation site) is independent of preferences for recreation visits.  If people choose residential locations so as to be near preferred recreation sites, then the price of a visit is endogenous.  (p 273)

As a consequence, including the fishing society would tend to result in a biased valuation.

A second reason for excluding members of the fishing society (which would apply even if their travel costs were positive) relates to the key assumption above.  The assumption may be reasonable for a fairly homogeneous group such as tourists, or indeed for the fishing society considered on its own.  But if one puts together two such very different groups, the assumption becomes much more dubious.  Suppose, for example, that the incomes of the tourists are typically much larger than those of members of the fishing society.  In that case the former  might respond to a certain price by visiting less frequently, while for the latter the price might be so large in relation to their incomes that they could not afford even one trip.  Thus a ‘hybrid’ demand curve, based on data for the two groups combined, would be unlikely to give a valid consumer surplus estimate.