Why is EV<CV if the good is inferior? Through the Slutsky equation I know that if the good is inferior the marshallian demand function is steeper than the hicksian demand but I cannot understand why the Compensating variation is higher than the equivalent variation. Is this because the increase in utility shifts the hicksian to the left instead of shifting it to the right (in a graph with prices as y and demand as x)?

I believe Marshallian demands are less steep than Hicksian demands because we reverse the y and x axis in economics. Thus a larger derivative of x with respect to p will be less steep since p is on the vertical and x is on the horizontal.

(image from here)


Why is discriminating monopoly better than competitive markets for producers? In intermediate microeconomics by Hal Varian, It is said that discriminating monopoly is better than competitive markets for producers. Why so?

You're looking for an answer while it's clear you are very early in your studies and will get to this stuff in due time.

With that said, here is some loose intuition about why a monopoly is better than competition for a single producer:

In a competitive market, every single consumer and every single producer has the same amount of "bargaining clout".  In particular, no single producer can set prices however it wants, because there is always another producer waiting in the wings to offer a better price.  The market-clearing price and quantity can be viewed as an equitable compromise between relatively equal powers (producers and consumers), and is "fair" in that sense.  There are nuances to this that you will learn, but in economic terms producers do not earn "profit" in a competitive market.  They break even.

By contrast, in a monopoly market, the single producer has (nearly) all of the "bargaining clout" because they control the means of production.  Therefore, they have tremendous freedom to set the price or quantity in such a way that they earn a positive "profit" because there is no other supplier who is able to undercut them on price.  This profit for the producer comes at the expense of the benefits consumers would otherwise enjoy in a competitive market (e.g., other things purchased with the price savings).

Again, these are kind of hand-wavy statements, but illustrate the general idea.


How does IS-LM model reduced investor confidence? I am doing an essay on how a sovereign debt rating downgrade to junk status affects the economy. I wanted to use a theoretical framework to think about it, and seeing as an increase in the interest rate is involved, I decided to use the IS-LM model. Please tell me if the following is correct or not. I think it's a bit unconventional to shift the LM curve when money demands shifts, but the reason why I do that is because output was not the driving force of the shift in money demand. As I understand it, if output increases, that increases money demand, which is a movement along the LM curve. However, in this case, investor confidence affects the interest rate directly, without an initial effect on output. That is why I have shifted the LM curve upwards.

I presented the model as follows. Investor confidence drops with the news of a rating downgrade, so financial investors sell their bonds, which increases the demand for money, and decreases the prices of bonds. Because the prices of bonds decrease, interest rates on those bonds increase. Also, new bonds will have higher interest rates because they are seen as more risky. This is shown graphically in the money market model below as the first increase in money demand from $D_{M_1}$ to $D_{M_2}$. (I find it a bit confusing to think about foreign investors demanding more money (like M1), because I think they would want to sell the rand in exchange for their own currency, but I think this is true for the domestic economy.)

Because the money market is the basis of the LM relation, this shifts the LM curve upwards as the interest rate increases (shown as the increase from $LM_1$ to $LM_2$). Higher interest rates lead to lower real investment (i.e. investment in capital, land, labour and enterprise), government spending and consumption. This is equivalent to a movement upwards along the IS curve. The IS relation shows equilibrium in the goods market, which is given by $Y=C+I+G$. $I(i)$ is a (decreasing) function of the interest rate, and the interest rate increasing is what causes real investment to drop. It is a negative relationship because higher interest rates make it harder to borrow money to finance investments. In our situation, government spending decreases because the cost of credit for them increases, but rising Treasury bond interest rates also raises other interest rates throughout the economy due to competition, which then reduces private investment. The result is a decrease in the growth rate of output. The drop in output actually dampens the increase in the demand for money, because income decreases. So, the hike in interest rates is attenuated as output growth slows down. This can be reflected in the money market as a downwards shift in the demand for money curve from $D_{M_2}$ to $D_{M_3}$.

Another thing I want to check is if it is fine to think of there being only one interest rate in the graph on the left, due to the way central bank interest rate and other financial instrument interest rates rise due to arbitrage. The interest rate on Treasury bills is obviously what is affected, so is it fine to think of the demand for 'money' (is that demand for all the money in the entire economy?) as interacting with that interest rate?

I presume that reduced investor confidence is a fairly standard problem to model, yet I'm not so sure about how I've done it.

I studied maths not economics so I won't comment on your graphs or theoretic models. In my experience as a rates trader, some economic models have their place but their inherent assumptions do not capture the practical market dynamics. I'll comment on the parts of your question I disagree with. Hopefully you appreciate a devil's advocate.

Firstly, a "a sovereign debt ratings downgrade" by a ratings agency has little value in the world today as a binary event. Most financial practitioners have made up their mind about the credit quality of issuers through independent or public research before ratings agencies update their databases. Ironically, one possibility is that this may indeed be a classic "sell the rumour, buy the fact" scenario, whereby reasonably large speculators sell the bonds preemptively, profit from general declining markets before the ratings downgrade and then use the liquidity and market wide focus generated by the event to exit their positions, and in fact the bonds may actually increase in value slightly due to this. This is classic Jessie Livermore style of trading (Reminiscences of a Stock Operator). There will doubtless be managed funds whose rules preclude investing in junk status bonds so demand for the bonds overall will likely decline overall although I expect more over a period than specifically because of that binary event.

The notion of "bonds" is a very broad term. If an issuer is downgraded the impact may be greater for various different maturities. The UK in the wake of the financial crisis was under considerable pressure due to its reliance on the financial sector and its government bonds underwent considerable re pricing, but the riskier, longer dated bonds suffered considerably worse that the much less riskier, shorted dated bonds, so in that scenario money market bill rates fell but longer bond rates rose.

When I think about the currency wide issue of money, if you have a reasonably large amount of sovereign debt held by investors that is sold those investors then have only a set of options available to them with respect to that money:

reinvest in some other asset denominated in the same currency, corporate bonds or equities.
sell the currency and buy assets in another currency.
deposit the cash with banks or on repo for some form of collateral.

The first of these is just an asset reallocation and weakens sovereign bond prices relative to other assets in the economy. Your argument about government financing is true and the GDP output would be expected to decline, further as confidence also becomes a factor.
The second would weaken the spot FX rate and the cross-currency basis swap levels. Indirectly this threatens to increase inflation since import prices will be affected by the currency movement. However, a weaker currency might tempt outside speculators at some equilibrium point. The third is difficult; no asset manager will deposit funds unprotected with a bank (particularly a domestic one) and the alternative of receiving back sovereign bond collateral is unappealing, so the repo rates will probably be increased in line with the rates on the bonds themselves, although this is another form of interest rate that does have significant impact and is not necessarily the same the bond rates. Indeed there will also be curvature to this as well.

As an example in 2017 the German government bond curve had 0-1Y yields at -70bps, and a repo rate often at similar levels. But the 20y-30y bond yields were at +100bps whilst the repo rate might be at -100bps. This may sound like arbitrage is possible, but it actually isn't - there is no scenario you can gain without running risk, and those different risks were considered and deemed to be worth the values potentially attainable by those who dared.

Finally I will point out to you the transmission mechanism. A central bank presides over its own currency and attempts to influence its economy by changing its deposit rate. However, a central bank is inherently connected to the government. I do not consider it wise to treat the two as separate entities. As a national bank, whose fortunes are intimately tied to the government, you would probably be better off buying and holding the sovereign debt if the short dated bonds offered considerably more interest than the central bank deposit rate, since from their point of view a government failure or a central bank failure might be one and the same. If you consider the holdings of Italian sovereign debt by Italian banks you will recognise this strategy as a kind of moral hazard. So the transmission mechanism is likely to remain in place for the short dated bonds and be ultimately tied to the deposit rate.

I appreciate there is no general conclusion to this thread, and some ideas are not necessarily coherent or consistent. I just found it an interesting question and thought I would explore my own thoughts in raw form. Possibly you find them useful.


Net present value of studying? Suppose I attend college with 10k tuition fee per year meanwhile I have a part time job for 5k annually. After 3 years, I get a job for 45 years and earn 40k each year. What is my net present value in term of interate rate x?

I did try to form a formula like this, is it correct?

I did try to form a formula like this, is it correct?

No. You need to include the NPV of all years. Right now you are only considering two of those years (and it is not clear which year you refer to in the 2nd term).

Also, it is unclear from the problem whether tuition is paid in the beginning of each year or at the end thereof.

The discounting factor in the 2nd term is wrong. The discounting factor for a cash flow at the end of year $n$ is $$1/(1+x)^n$$.


What's the difference between public debt, government debt and national debt? When it comes to non-private debt, I hear these three terms the most, but I am not sure if they're interchangeable or discernible. Also, are they recorded in the balance of payments? I'm guessing if they are, they'd be recorded in the transfers part of the current account and the capital and financial accounts.

I understand that this is a duplicate question, but my previous question had no contributions other than my own, so I figure it would be sensible to repeat it.

With all these types of debt, there are various descriptions rather than standard definitions, and sometimes the same thing can be labelled with different names or the same name used for different things.  In addition, they can be measure gross or net, and there will be questions of how money owed by one part of the public sector to another is treated.

National debt can mean the debt of the national or central government.  This is how it was used for example with the debt the British government built up during the Napoleonic wars, which became the risk-free asset basis of 19th century banking
Government debt can mean the combined debt of different layers of government, so combines national, municipal and other levels
Public debt may include government debt and debt of other parts of the public sector, such as public corporations

None of these explicitly appear in the balance of payments statistics, especially as most public debt is owed to the domestic private sector.

What does appear is in the international investment position, where all debts (public or private) owed to foreign residents or governments appear as liabilities.  Payments of interest to foreigners count as debits in the primary income part of the current account, while borrowing or repayments of principal are flows in the financial account.


Is comparative advantage undermined by a welfare state? When liberalising trade with developing economies like China, states like the UK seek to benefit from the comparative advantage of cheaper imports.

Cheap imports undercut some native industries (eg. steel making, ship building, pottery) causing (at least in the short term) joblessness and associated negative social outcomes.

The thinking (IIUC) is that those displaced from the marketplace by the imports should move to higher-value-add industries, but in many cases this does not happen. For example towns like Stoke on Trent in the UK whose native industry was decimated (at least in part) by overseas competition, are characterised by chronic joblessness and associated social problems.

In a welfare state like the UK, the cost of those social problems are then borne by the state.

To what extent does this undermine the comparative advantage seen by the UK from cheap imports?

The basic result about the effect of trade on income distribution is Stolper-Samuelson in the Heckscher-Ohlin model, which says that trade liberalization is associated with a rise in the prices of the factors of production that are used intensively in the production of goods the country has a comparative advantage in, and vice versa. If you take "skilled" and "unskilled" labor to be two different factors of production, for instance, Stolper-Samuelson says that a country with abundant skilled labor lifting trade barriers should lead to an increase in income inequality. A welfare state committed to keeping relative incomes of workers within a certain range may need to raise distortionary taxes to expand various income transfer programs to "unskilled" workers in this case, which could lead to inefficiency for various reasons. (Even if the taxes used to finance the income transfer are lump-sum, as long as the benefits phase out this leads to high effective marginal tax rates on the lower end of the income scale, i.e a disincentive to work and to acquire new skills.) In this sense, it's indeed true that a generous welfare state can reduce the efficiency gains from free trade.

The rest of the answer is a rant about the concept of "strategic market dumping" that was brought up in the comments, so if you're not interested you can skip that part.

Things like "strategic market dumping" are ill-defined concepts that are usually used to justify tariff increases without having much backing in theory. If "dumping" is expected to go on for a finite period of time and the associated good can be stored at no cost, for instance, then in a perfect foresight world there's an arbitrage bound on the price of the good today which forces it to grow at a rate less than or equal to the real interest rate available in the broader economy. Adding storage costs does not change the basic result very much. In that case, "dumping" is not a good strategy to begin with - if you try to do this, people will buy up the goods you're selling at low prices, hold them until you reverse your policy, and then flood the market with goods they have stored up in their inventories once you stop the "dumping", leaving you worse off. To justify a policy of industry protection, you have to argue that the government can see the reversal of the dumping policy coming and private agents can't - in this case, the real return on an investment in producing and storing oil would be higher than the market real interest rate is saying it will be, so the market outcome will be suboptimal. It's easy to justify any government intervention by saying that the government is clairvoyant and private agents are not, however.

What about labor market (or, more generally, factor market) frictions? Isn't the point of "dumping" to eliminate your competitors through "artificially low" prices which force them to go out of business? Well, here you're making the mistake of assuming that the government knows about labor market frictions and the private sector doesn't, therefore the government should do something about the situation. In real life, the private sector is well aware of the search frictions in the labor market, the difficulty of firing workers in some countries, etc. This means that if they expect the "dumping" strategy to be reversed in the future, they will choose to borrow money or raise capital to stay in business. They will downsize in response to the "dumping" policy, but not as much as you think they would, and certainly not as much to make a government "protection" of the industry a good policy.

Does this mean markets can never allocate resources poorly? Of course not - if the country doing the dumping manages to fool private agents into thinking their "dumping" policy was permanent while it was temporary, then the market allocation will be suboptimal. However, in hindsight every price and every allocation is wrong. We certainly don't expect market allocations to be optimal ex post, so it's not surprising to see cases where it seems like the market allocation was poor and led to many workers being stuck in jobs where wages fell due to international price changes of goods. In fact, this would happen in a stochastic world where markets were functioning perfectly and giving you an ex ante Pareto optimal allocation of resources. Even a social planner with access to all information available at a given date and the mental capacity to solve the associated economic calculation problem would act in ways that sometimes seem foolish ex post. This doesn't mean the social planner is not doing his job properly, it just means there are things about the future that even a social planner can't know with perfection. The argument is no different for market allocations - allocations that are poor ex post do not justify government intervention ex ante.


Why do Latin American countries fare among the worst at managing inflation Does it have to do with cultural factors? Ex: they were colonies of Spain and Portugal who themselves are not that great at managing their economies relative to other western European countries.

Perhaps you should look up the following collection of studies,

Dornbusch and Edwards (ed.), The Macroeconomics of Populism in Latin America(1991)

It is freely downloadable from its official site. Here is the opening paragraph of the Editors' piece

  "Latin America’s economic history seems to repeat itself endlessly, following irregular and dramatic cycles. This sense of circularity is
  particularly striking with respect to the use of populist
  macroeconomic policies for distributive purposes. Again and again, and
  in country after country, policymakers have embraced economic programs
  that rely heavily on the use of expansive fiscal and credit policies
  and overvalued currency to accelerate growth and redistribute income.
  In implementing these policies, there has usually been no concern for
  the existence of fiscal and foreign exchange constraints. After a
  short period of economic growth and recovery, bottlenecks develop
  provoking unsustainable macroeconomic pressures that, at the end,
  result in the plummeting of real wages and severe balance of payment
  difficulties. The final outcome of these experiments has generally
  been galloping inflation, crisis, and the collapse of the economic
  system. In the aftermath of these experiments there is no other
  alternative left but to implement, typically with the help of the
  International Monetary Fund (IMF), a drastically restrictive and
  costly stabilization program. The self-destructive feature of populism
  is particularly apparent from the stark decline in per capita income
  and real wages in the final days of these experiences."


relation between the demand function and the marginal utility function Can one say that the demand curve equals the marginal utility function? I have one course where this is stated.

However, the demand curve dan be derived from combining the utility function with the budget curve, while varying the price of the one good of interest and plotting the corresponding quantities found from the equilibria in the goods space and the budget curve.

While I do understand this method, I don't understand how the demand curve can be seen as the marginal utility function using this method (or any unknown other).

Please enlighten me :).

This is true for quasilinear utility function. Suppose
$$U = d + V(x),
$$
where $d$ is the numeraire. The budget constraint is given by $p_x x + d = I$. The utility is then
$$U = I - p_x x + V(x).$$
Maximizing this, we have the following FOC: $p_x = V'(x)$. That is, for a given $x$, my willingness to pay is $V'(x)$, the marginal utility.


How does 'building land' fit into economic modeling? According to the ONS National Accounts 2018 & RICS land valuations and ...

Land that is built on has a value that makes it half of the UK's wealth.
The rest of the land has a value less than 3% of UK wealth.
Building land is created when planning permission is granted.

Another question asked "Why is Land not considered as a fixed capital?".

The top answer said ....

  Physical capital is reproducible, and does have a finite lifetime.

  Hence, almost all land is not physical capital.

  There are corner cases, where land is reproducible, and may appear and
  then disappear again in less than a handful of human lifetimes: in
  those very rare cases, that particular land behaves more like physical
  capital rather than how land generally behaves, and so could be
  classed as such.

Building land can appear and disappear(e.g. post war prefab sites which have reverted to open space) and planning schemes can change building land into something else.

How would this very large asset, building land, fit into economic models?

I can't see how it would fit into a Cobb_Douglas function.

Land would fit into production functions or many economic models the same/similar way as other factors of production.

An example of general Cobb-Douglas function with land would be:

$$Y(K,L,T) = A K^{\alpha}L^{\beta}T^{1-\alpha-\beta}$$

Where $K$ is capital, $L$ labor, and $T$ land. Into most economic models land can be integrated rather easily as the example above shows. The land improvements necessary to clear land for building can be in principle modeled as capital accumulation with depreciation but for some general production functions there is not that much difference made between the inputs.


Markov decision processes, contractions and value iteration I am reviewing Markov decision processes (MDP) and there is something I am missing with respect to the contraction argument. I am pretty sure it is a silly mistake somewhere (maybe computational), but anyways, I cannot figure it out. Here it goes.

Consider a simple MDP with two states and two actions defined as follows.

$$ r(s,a) = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix},$$

$$ P(s,s',1) = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix},$$

$$ P(s,s',2) = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix},$$

$$ \beta \in (0,1). $$

Now suppose we start with two guesses for the value function

$$ V_1 (s) = \begin{pmatrix} 100 \\ 0 \end{pmatrix}, $$

and

$$ V_2 (s) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$

If we iterate on these approximate value functions using the Bellman operator we get

$$ T(V_1) = \begin{pmatrix} \max_a \begin{cases} 1 +  100\beta, \qquad \text{ if }  a = 1, \\
                                                 1 +  50\beta,  \qquad \text{ if }  a = 2.                                               \end{cases}\\
                             \max_a \begin{cases} 1 +  100\beta, \qquad \text{ if }  a = 1, \\
                                                 1 +  50\beta,  \qquad \text{ if }  a = 2.                                               \end{cases} \end{pmatrix} =
\begin{pmatrix} 1 + \beta 100 \\ 1+ \beta 100 \end{pmatrix}$$

and

$$ T(V_2) = \begin{pmatrix} \max_a \begin{cases} 1 +  0\beta, \qquad \text{ if }  a = 1, \\
                                                 1 +  0.5\beta,  \qquad \text{ if }  a = 2.                                               \end{cases}\\
                             \max_a \begin{cases} 1 +  0\beta, \qquad \text{ if }  a = 1, \\
                                                 1 +  0.5\beta,  \qquad \text{ if }  a = 2.                                               \end{cases} \end{pmatrix} =
\begin{pmatrix} 1 + \beta 0.5 \\ 1+ \beta 0.5 \end{pmatrix}$$

But then for $\beta$ close enough to $1$ and taking for instance the Manhattan norm, we have

$$ d(V_1(s),V_2(s)) \approx 101,$$

and

$$ d(T(V_1(s)),T(V_2(s))) \approx 199.$$

Now that sounds weird to me because I thought $T$ was supposed to be a contraction mapping. Where did I screw up? Is there a mistake in my computation? I am forgetting to apply an important hypothesis? Or am I misunderstanding something about contraction mappings?

The value iteration operator is a contraction with respect to the supremum norm. Your example probably provides a counterexample for the statement that it's a contraction with respect to the Manhattan norm.


what is potencial GDP? The Wikipedia defines potencial GDP as "the total capacity of a nation to produce goods and services". If this definition is correct than the potential GDP can only occur when the unemployment rate is 0, all the land is exploited for the purposes of production (agriculture, factories, roads so on.) and the infrastructure is not outdated.

Speaking primitively and according to the definition provided above, if country A still has some part of it covered by forests (i.e. not factories and roads) and part of the population (say 4 percent) are out of work, then potential output can not be reached i.e. output gap is negative. This hypothetical example describes basically all countries present in the real world.

However, OECD indicates (the link below) that number of countries (Belgium, Denmark, Czech Republic just to name a few) had their real GDPs above the potential. In non of the cases the unemployment rate is O or the land is fully exploited.

https://stats.oecd.org/index.aspx?queryid=51655

Are my assumptions wrong? How should I look to the potencial GDP?

That quote from Wikipedia page is not correct. Actually both OECD and also commonly in economic research we define potential GDP/output as:

  Potential gross domestic product (GDP) is defined in the OECD’s Economic Outlook publication as the level of output that an economy can produce at a constant inflation rate. Although an economy can temporarily produce more than its potential level of output, that comes at the cost of rising inflation. Potential output depends on the capital stock, the potential labour force (which depends on demographic factors and on participation rates), the non-accelerating inflation rate of unemployment (NAIRU), and the level of labour efficiency.

Also note what’s written in Wikipedia was probably misunderstanding of the term full employment in economics, as potential GDP is often defined as a GDP at full employment of all resources. In economics full employment does not mean that unemployment rate is necessary 0, or that all resources are used.

In economics full employment is an economic situation in which all available factors of production (labor, capital land etc) are being used in the most efficient way possible.

Unemployment at full employment will still be positive number because even in case when all resources are used in the most efficient way possible some people won’t have work, some capital will just lay unused and some land will just lay unexploited.

Also, always when you look at data you should check the data’s own metadata/explanation for what’s measured not Wikipedia otherwise you won’t really understand what the statistics is showing you.


Is undervaluing environmental impact a subsidy? This is in relation to a recent question on skeptics which to me seems to broil down to whether you accept that form of accounting, i.e. consider undervaluing environmental impact a hidden subsidy.

So, is it this a common or uncommon practice in economics research? Or is it somewhere in between, e.g. does it vary with the school of thought?

Yes, undervaluing environmental impact is a subsidy, within the context of the IMF report that you are referring to.

Here's what that report says in its abstract (my emphasis):

  it focuses on the broad notion of post-tax energy subsidies, which arise when
  consumer prices are below supply costs plus a tax to reflect environmental damage and an additional tax applied to all consumption goods to raise government revenues.

It goes on further on pages 10-11 to define some aspects of the subsidy in more detail.

In particular:

  post-tax consumer subsidies represent the amount by which the cost borne by the consumer falls short of the total economic cost of consumption. This excess cost (or subsidy) is either covered by governments in the form of budgetary support or foregone revenues or passed to the society in the form of environmental damage.

and

  Producer subsidies exist when producers receive either direct or indirect support that increases profitability above what it otherwise would be (that is, the support is not passed forward in the form of lower consumer prices). This support can take many forms, including receiving a price for the output above the supply cost, paying a price for inputs below supply costs, receiving preferential tax treatment, or receiving a direct transfer from the budget

This makes it clear that within the context of that report, undervaluing environmental impact is a subsidy. And that is entirely consistent with how "subsidy" is used within the field of environmental economics generally.

See, for example: The Economic Cost of Global Fuel Subsidies, Davis (2014):

  Fuel  subsidies  are  different  from  subsidies in  most  other  markets  because  of  the  substantial  external  costs  of  driving.

And Energy subsidies: How large are they and how can they be reformed?, Clements et al 2014:

  The subsidy estimates capture both those that are explicitly recorded in the budget and those that are implicit and off budget ... Tax subsidies
  arise when energy taxation, which includes a corrective component for externalities and a component for revenue consideration, is below the efficient level. The efficient taxation of energy requires corrective taxes to capture negative environmental and other externalities owing to energy use.

And Fossil fuel subsidies in the Pacific island context: Analysis of the case of Kiribati, Peltovuori (2017)

  Subsidy in this report is defined according to the definition of World Trade Organisation's (WTO) Agreement on Countervailing Measures (ASCM) (Uruguay Round Agreements, 1994). This definition has been recommended to G-20 by the Global Subsidies Initiative (The Global Subsidies Initiative, 2010). The methodology for quantifying fuel subsidies in this study is the price gap approach. In the price gap approach, a cost recovery price for fuels is first estimated based on the world free on board (FOB) market prices. All additional monetary and non-monetary costs such as transport, storage, distribution costs, and corrective taxes to account for externalities are then added on top of the FOB price. This cost recovery price is then compared to the actual price. The gap between the two prices is the amount of subsidy per price unit.

These pollution costs are included within the broader definition of subsidy as used by the IMF, because these negative production externalities are costs incurred by production, that the producer itself does not directly bear. Thus, those costs are not reflected in prices, leading to market failure and an unfair competitive advantage, of the same kind and market impact as if they'd received a cash subsidy on their directly-incurred costs.

The "Economics Help" site explains the issue

The deadweight loss from coal pollution is the red striped triangle: this comes about because coal producers do not have to pay for their pollution, leading to a competitive advantage, and an artificially high quantity sold Q1 at price P1.


Why do firms adjust their fixed costs in response to a change in price in a perfectly competitive market? When firms make a profit in a perfectly competitive market, new firms enter the market and drive the price down. My textbook says that the existing firms will then adjust inputs that are fixed in the short term, such as capital, in order to operate with a lower average cost:

This is in order to maximize profit, but I don't understand why the short-run average cost curve would change at all. In my opinion, the minimum point of the long-run average cost curve, where there are constant returns to scale, is the profit-maximizing point in the long run. In order to substantiate this claim, I came up with a short-run total cost function

$$TC=0.001Q^3-0.09Q^2+2.95Q+20$$

and derived the average total cost

$$ATC=0.001Q^2-0.09Q+2.95+20/Q$$

and marginal cost

$$MC=0.003Q^2-0.18Q+2.95$$

Assume that the price is variable. Then profit

$$\Pi=Q(MC-ATC)=0.002Q^3-0.09Q^2-20$$

can be thought of as a function that relies only on marginal cost and average cost, because at the profit-maximizing point, $MC=MR=P$. But as $Q\to \infty$, $\Pi \to \infty$ because $\Pi$ is a cubic equation with a positive coefficient on the $Q^3$ term. This means that the higher the price is above the minimum point of the short-run average total cost curve, the higher profit will be. Thus, profit is maximized when the lowest possible short-run average total cost curve is selected, in any perfectly competitive market condition, at least for cubic total cost curves (and I assume that the cubic function is the most applicable for a total cost curve). The lowest ATC curve is obviously the one where there are constant returns to scale.

You can see the difference between marginal cost and average total cost getting larger and larger here:

I think you're right - entry should not shift the SRAC in a normal model of a market. The textbook pages that you include read very oddly to me. (It is possible that this is because the context being described is somehow very specific, so I may be unfair in the criticism that follows).

A potential entrant compares price to min of ATC and sees that there is a profit to be made. It enters and produces the quantity such that it achieves min ATC. What does this do at the level of the market? At the clearing price before entry, there is now excess supply equal to the output of the new firm. Somehow ... and introductory models of perfect competition and price formation are usually silent on exactly how this happens ... the price must fall for there not to be unsold goods in the market.

There are lots of unsaid things in the above story. What do other firms do in response to the entry? How exactly are prices formed. One natural answer is  Bertrand price competition: the new firm undercuts the clearing price; all customers switch to it; it supplies the whole market ... upon which other firms react by undercutting it ... until undercutting no longer makes sense because each firm is at the min of the ATC. Again, there is lots here that is unsaid (are fixed costs sunk or not? what determines capacity constraints ...)

But in a way, the root of the problem with the story told in your textbook is the upward sloping aggregate supply curve. How is this compatible with a micro foundation of perfect competition? If each firm can produce according to the ATC function given, then the aggregate supply curve is actually flat. Under the assumptions of perfect competition, you can enter without constraint. So it is always possible and profitable to produce at the min ATC. So the aggregate supply curve is not upward sloping. I think the text book is trying to bring together two models that don't really fit coherently, and you have picked up the inconsistency involved.

So here's what I think about your question and the answer you have posed:
1. you're right & the book's wrong
2. don't take economic models too literally as representations of reality.


How can two countries of roughly the same size, population and GDP have such big differences in minimum wage? We could answer this question theoretically, or give a real-world example of two such countries.

Is there anything more interesting to say than: that's just something decided by the government? Could geographic location play a role? Is an equal GDP an indication of equal minimum wage, or isn't it?

The minimum wage is usually the outcome of a bargaining process between the government, businesses (via lobby), and the trade unions. As such, a key driver of the final outcome depends on the balance of power. In countries like France and Germany, with strong unions in the former, and wide collective bargaining on the latter, the minimum wage is relatively high, compared for instance with Poland. Meanwhile, in Poland the minimum wage is roughly two times that of Chile, both roughly equally sized in terms of GDP per capita. How this reflects differences in power is not easy to tell.

Needless is to say that the ideology of the government is also crucial in this respect. Left-wing governments are more keen on raising the minimum wage than right-wing ones.

Additionally, the structure of production and workers' skills varies among countries. This means that some countries could have a lot of workers around the minimum wage, giving more incentives to unions to fight for them, with greater opposition from business. Yet, businesses in a country with few people around the minimum wage are more willing to accept a minimum wage, because it affects less people.

Last but not least, what matters is the real minimum wage. Equally sized countries can have different price levels, because the basket of non-tradable goods might be very different (prices of tradable goods are supposed to equalise in the long run, according to the Law of One Price).


Complicated utility function I am trying to answer this past paper question on microeconomics and a rather complicated utility function question. The question is below as well as my answer.
In my answer, I use the fact that the marginal rate of technical substitution (MRS) is equal to their price ratio. I think that I derive both demand functions for pizza and apple. I am not sure if they are right and I don't know how to answer:
How James responds to an increase in the price of Pizza?
Just looking at the price of pizza (p1) in the demand functions would mean that an increase in p1 would result in a decrease in demand for apples.
I feel like I have done something wrong and that I would have to consider substitution and/or income effects? I'm terribly confused.

The demand functions you have derived are correct.
The Marshall demands are given as
$$ x^\star(p_x,p_y,M) = \sqrt{\frac{M}{p_y}} \\[8pt]
y^\star(p_x,p_y,M)  = \frac{M - p_x \sqrt{\frac{M}{p_y}}}{p_y},$$
where $M$ is income and $p_x$ is pizza price and $p_y$ is the apple price.
To answer the question of how the agent responds to change in pizza price it would be standard to find derivatives and/or elasticities of demand with respect to the price $p_x$ for pizza.
The derivative of pizza demand with respect to own-price is given as
$$ \frac{\partial x^\star(p_x,p_y,M)} {\partial p_x} = 0, $$
which implies that the elasticity is
$$El_{p_x}x^\star(p_x,p_y,M) = \frac{p_x}{x^\star(p_x,p_y,M)}\frac{\partial x^\star(p_x,p_y,M)} {\partial p_x}  = 0,$$
so you can conclude that demand is perfectly inelastic. When the price increase the consumer simply keeps demand constant and substitutes away from apples. This is manifest from the fact that
$$ \frac{\partial y^\star(p_x,p_y,M)} {\partial p_x} = - \sqrt{\frac{M}{p_y^3}}, $$
which is clearly negative and by implication the cross-price elasticity $El_{p_x}y^\star(p_x,p_y,M) <0$.
This is "unusual" if you expected apples and pizza's to be substitutes with the demand of one increasing when the cross-price increase.


Saddle path equilibrium on financial market with rational expectations In his 1978 paper introducing the Tobin tax Tobin states that :

  As a technical matter, we know that a rational expectations
  equilibrium on markets of this kind is a saddle point. That is, there
  is only a singular path that leads from disequilibrium to equilibrium.
  If the markets are not on that path, or if they don't jump to it from
  wherever they are, they can follow any of a number of paths that lead
  away from equilibrium - paths along which, nonetheless, expectations
  are on the average fulfilled.

He seems to consider this result as common knowledge among the economists and gives no reference. To which result or paper is he referring ? Do you have an idea of what he means with "markets of this kind" ?

Suppose you have a dynamic system
$$ x_{t+1} = Ax_{t} $$
with a stationary point (or steady state as used in growth or RBC literature), say, $x^*$, i.e. $x^{*} = Ax^{*}$.

Now, consider the following question. Starting from an initial value $x_0$, how many paths are there leading to the stationary point $x^*$? If there is an unique path going from $x_{0}$ to $x^{*}$, then your model is well behaved in the sense that you can track the vector of variables, $x_t$, along the transition without worrying which path you are actually on. This is the saddle-path stability case that you are referring to. On the other hand, if the answer is not affirmative, it means that you have at least two routes from $x_0$ to $x^{*}$. There is another case: no matter where you start from you eventually end up at the stationary point $x^{*}$, in which case your model is said to be indeterminate.

So you can think of saddle-path stability as a desirable feature that you would like that your model to manifest in order to analyze the problem at hand. For example, standard RBC models all posses this property.

There are some mathematical conditions that ensure the saddle-path stability.

For the details and more, check out the section 7.8 (The q-Theory Investment and Saddle-Path Stability) in Introduction to Modern Economic Growth (Acemoglu, 2009).

Ok, I'll try to explain intuitively for the Ramsey model (but it's not rigorous). Suppose that from the first order conditions, you can conclude that the equilibrium is unique and furthermore you can express consumption as a function of capital stock, that is $c_{t} = g(k_t)$, for some (smooth) function $g$. If that is the case, for a given initial capital stock $k_0$, you know how the economy is going to evolve. That is, if we denote the vector of variables by $x_t = (c_t, k_t)$, you know $\{ x_{t}\}_{t=0}^{\infty}$; because

$t=0: \quad$ $k_0$ is given, and $c_0 = g(k_0)$,

$t=1: \quad$ $k_1 = (1-\delta)k_0 + f(k_0) - c_0$, and $c_1 = g(k_1)$,
(here $f$ is the production function and $\delta$ is depreciation)

$\vdots $

$t=\tau: \quad$ $k_{\tau} = (1-\delta)k_{\tau -1} + f(k_{\tau -1}) - c_{\tau -1}$, and $c_{\tau} = g(k_{\tau})$,

so on ....

Now, suppose the economy loses some of its capital stock for some reason while it was on $x^*=(k^*, c^*)$ and the new capital stock is, say, $k_{\tau}$.

Where will the economy be with this new capital stock?

Well, we have just claimed that there is a corresponding consumption $c_{\tau}=g(k_{\tau})$, and therefore the economy will jump to the point $x_{\tau}=(k_{\tau}, c_{\tau})$ on impact the shock and then move toward back to steady state $x^*$. You can think of $x_{\tau}$ as the red point in the figure.

Finally, a few remarks on the intuition behind the mathematical conditions ensuring saddle-path stability.

1) Note that, in the model we have actually two  variables $c_t$ and $k_t$, but we have assumed  that one of them can be  expressed in terms of the other, so the number of free variables is reduced to one, which is $k_t$ here. Basically, in order to be able to do this, you want to have a smooth curve mapping capital stock to consumption, i.e. $g: \mathcal{R} \to \mathcal{R}$ is a smooth curve (1-dimensional manifold).

2) Once the economy is at $x_{1}$ after the shock, you want it go back to $x^*$ and this is related to the number of eigenvalues that are less than one in absolute value.

For precise statements (in a general environment) see Acemoglu as I mentioned before.

Regarding rational expectations(RE): it is a solution concept and RE alone does not imply saddle-path stability.


Approximately Optimal Mechanism Design vs. Revelation Principle The abstract of the article "Impossibility Results for Truthful Combinatorial Auctions with Submodular Valuations" (Dobzinski, Vondrak, 2016) states

  A long-standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions, with performance guarantees close to those possible without considerations of truthfulness. In this article, we answer this question negatively: the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions.

My question is--  given that the revelation principle says we can WLOG restrict our attention to direct-revelation IC mechanisms in mechanism design, how can there be a loss of revenue/lower approximation ratio by restricting the search of approximately optimal (i.e. revenue maximizing) mechanisms to IC ones?

A key part of the paper is that they are interested in

  computationally efficient truthful mechanisms

The VCG mechanism is a truthful and efficient direct mechanism, but it is not computationally efficient. In many combinatorial auctions, it is impossible to compute the VCG transfers in a reasonable amount of time using current technology.

In the past, economists have largely ignored the problem of computational efficiency in mechanism design. This problem has been mainly addressed by computer scientists. Read for instance the introduction to Milgrom and Segal (2020) Clock Auctions and Radio Spectrum Reallocation. JPE


Mathematical derivation of the Production Possibility Frontier What are the mathematical basics of production possibility frontier? How can I derivate it? Can I have an example for it?

The question is broad, but I believe there is plenty of literature that defines this concept in similarly broad terms.
The following is adapted from the Wikipedia on Pareto Efficiency,
which is the mathematical basis of the Production Possibilities Frontier.

There may be better definitions out there, but this one should probably work in a lot of cases:

The Production Possibilities Frontier, $P(Y)$, may be more formally described as follows. Consider a system with function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where $X$ is a compact space of feasible decisions (including allocations of time and endowment goods) in the metric space $\mathbb{R}^n$, and $Y$ is the feasible set of criterion vectors (say, final goods and services) in $\mathbb{R}^m$, such that $Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}$.

We assume that the preferred directions of criteria values are known so that more of any good in $Y$ is better. A point $y^{\prime\prime} \in \mathbb{R}^m$ strictly dominates another point $y^{\prime} \in \mathbb{R}^m$, written as $y^{\prime\prime} > y^{\prime}$, means that for each element index $i$, $y''_i \geq y'_i$ and there is at least one element $j$ such that $y_j'' > y_j'$. The Pareto frontier is thus written as:

$P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} > y^\prime, y^{\prime\prime} \neq y^\prime \; \} = \emptyset \}. $


Gross interest rate ($R$) v. Net interest rate ($R-1$) I am reading a macro-finance article that deals with asset financing.

While it defines the gross interest rate to be $R$, it defines the net interest rate to be $r\equiv R-1$. In equilibrium, the interest rate on one-period loan is said to be $R=\frac{1}{\beta}$, where $\beta$ is the discount factor applied to the dividends firms obtain over an infinite horizon.

$\textbf{My Question}:$ How do you interpret the difference between $R$ and $r$? Is it conventional to define net interest in macro-finance literature in this way?

It is often conventional. R is the amount of the loan plus the interest and r is just the interest rate. So R=1+r.

Suppose you borrow 100€ at a 10% interest rate. The interest you pay is 10€, since 10% of 100 is 10. That is where r comes in, with an interest rate of r=10%=0.1.

On the other hand, R is the rate implied by that interest rate r plus the original amount, so it is relevant for the total you pay back. Before adding interest payments, you must pay back 100% of the original loan or 1x the original loan, so R=1+r=110%=1.1.

So, in our example, with R you pay back 100×1.1= 110€, while with r we have a payment of 100×0.1=10€.


Calculating elasticity between terms in a regression equation Given the following regression: $ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$ I am asked to calculate the elasticity of wages with respect to age. Is the following attempt valid?

$$\frac{\partial ln(w)}{{\frac{\partial age}{age}}}=\frac{\partial ln(w)}{\partial age}age=(\beta_2+2\beta_3age)(age)$$

Is this consistent with the definition of an elasticity as percent change over percentage change? Thank you.

No that would not be correct definition of elasticity. First, mathematically in multivariate function elasticity is defined as follows:

$$ EL_x =\frac{ f_x '(x,y)}{f(x,y)}x$$ or in your case it would be: $$ \frac{ \partial \ln [w(age,Y,T,Mar)]}{\partial age} \frac{age}{\\ln [w(age,Y,T,Mar)]}$$

However, even if you would plug in the expressions in this formula you would get an elasticity of a log of wage with respect to age - I am not sure if thats what you actually want. You could exponentiate the original function but that would be messy as hell.

One way how to estimate the elasticity directly from the regression would be using log-log form for variables of interest. In your case (assuming there are no people with 0 age in your sample):

$$ln(w_i)=\beta_1+\beta_2 \ln (age) +\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

In this specification $\beta_2$ will already give you an elasticity. This specification also already controls for possible non-linear effect of an age so no quadratic term is necessary. This being said logs are only rarely used for age, but you could estimate this specification as an additional robustness check and still keep the main specification with quadratic term.

Alternatively, already in your original equation which is in log-level form:

$$ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

all the beta coefficients are already semi-elasticities (as also pointed out by @chan1142). So if you would only care about semi-elasticity you will already get your result from your original model as in your case semi elasticity of wages with respect to age is $\beta_2+2\beta_3age_i$.


Basic question on calculating real GDP My book says the way we calculate real GDP in a given year is by adding the quantities of all goods and services produced in that year multiplied by their prices in some base year. There are actually two things I can't quite get:

How do we account for goods and services that exist now but weren't
around in the base year? I mean, especially today, with things such
as the market for apps, startups, and the increasing number of new
products that get released on an almost daily basis, wouldn't the
real GPD calculated that way become dated really fast?
This might turn out to be a really silly question, but how do we
take into account quantities of services that are provided in a
given year, such as lawyer services, marketing services etc.? All
examples I've seen of calculating real GDP were of the sort
"consider an economy that produces only apples an orange...".

Thanks very much in advance.

First, note that GDP is not the quantities of all goods and services produced in that year multiplied by their prices in some base year, but the quantities of all final goods and services produced in that year multiplied by their prices. Intermediate goods and services are not included.

1) GDP is about value created in the economy. So, if there are new services or products, we should simply include them. For the price, we take the price in that year and deflate it using a factor that accounts for inflation. Note also that new services and products typically come at the expense of other products produced. People that would otherwise be, say, working at a factory, are now programming apps.

2) As you would with oranges and apples. Lawyers typically charge a price per hour. So, it would be number of hours times the price. For more general services, like, say, the police force, you would just calculate the costs of the police force for a given year.


Is this a realistic solution to world poverty? The idea: By convincing the U.S. that ending world poverty would actually return so much money back to the U.S. that it would more than compensate for the cost of providing the resources needed to end poverty.

Example: The influence of terrorist groups in the Middle East could be dramatically decreased if we hire the locals there to be soldiers, and this could be far cheaper than risking the lives of our own soldiers and more importantly, more effective because we could afford more of them.

Anyway, the proposal is here:
http://genexie.com/the-most-realistic-solution-poverty-yet/2015/03/31/

I am afraid the solution to world poverty will not be two pages long. Some flaws:

You cannot simply give poor people in other countries money, their government controls their financial system. Even if the government agreed and you transferred millions of dollars it would not make more housing and consumer goods available at their location, they would just have credit. They could use these dollars to import the desired things but this may cause a lot of additional costs and again the government might interfere with large scale imports by controlling the distribution.

On hiring soldiers in Syria: This was an idea early on, but how can you be sure that they are doing what you pay them to do? Also there might not be a sufficient number of soldiers for hire at the desired location.

On giving poor people in the US unlimited access to basic goods. I like this idea, but to my knowledge no country in the world has implemented such a scheme. Also access to some goods does not eliminate crime, look at all the white collar crime that wealthy people commit or consider that they also struggle with drug problems.

Humans are complex creatures who will always compete with their environment.


Choosing Data for CES Production Function So I am currently trying to write a paper estimating the Constant Elasticity of Substitution Production Function of the USA. I am using the simple version with two inputs capital and labour. Since the estimation is at the aggregate level I thought of using the perpetual inventory method to create the data for aggregate capital, but I wasnt sure what kind of data I need for aggregate labour. Maybe the number of employed people, work force, in the USA? I'd be glad if someone could help me out.

The Penn Wolrd Tables have the data you need.


How does Egypt prevent people from arbitraging their currency? Apparently Egypt uses a fixed exchange rate so there is a substantial difference between the government rate for the Egyptian pound and the price on the street. What prevents someone from just buying pounds on the street using dollars and then depositing the pounds in a bank and converting them back to dollars at the official rate, thus allowing the person to instantly make a large profit?

I assume that there is some mechanism to prevent this. Is there?

Used to be capital controls. You could only buy dollars (or any other foreign currency) through official channels at the official favorable rates by providing documentation displaying need for said currency (e.g. Travel or import documents).


Difference-in-Differences setup for policy evalution I have a panel data set of housing prices from different neighborhoods in a city - and wish to test whether or not a policy has had an impact on the housing prices. The policy (treatment) is simply a sign signaling whether it is a good or bad neighborhood (based on parameters which are already known).
Initially I figured I could mirror the paper by Card and Krueger (1994) with a difference-in-differences estimation however what complicates my issue is that the policy enables neighborhoods to go from good-bad and bad-good over a span of 10 years. So they could potentially get a treatment multiple times.
Any suggestions into papers which has studied policies in a similar fashion or at methods which could be applied is greatly appreciated.

I think a DiD will not work here as the parallel trend assumption probably doesn't hold in your case.
However, given that the treatment is based on a number of parameters that are (I assume) observed, you could potentially use some type of discontinuity design to estimate the effect of the policy around the threshold values of these parameters.
To be more precise, your treatment neighbourhoods would be the the neighbourhoods that have the sign but are very close to loosing it, while the control neighbourhoods would be the neighbourhoods that are just below the threshold for obtaining the sign. Then if prices increases are a continuous function of all other covariates, the difference in treatment vs control would measure the effect of the signs for neighbourhoods around the threshold.
Of course there are potential endogeneity issues that you'll need to take into account as you can expect neighbourhoods to do some effort to get above the threshold.


Does the 2nd welfare theorem involve government intervention? How is the initial redistribution of resources carried out?

The second welfare theorem does not necessarily involve a government as such. It does not matter for the theorem who redistributes the resources. Nevertheless, in practice this will most likely often involve the government.


A term for utility functions based on the max operator What is a standard term for utility functions of the type:

$$ u(x_1,\dots,x_m) = \max(\frac{x_1}{w_1},\dots,\frac{x_m}{w_m}) $$

where $x_i$ is the amount of commodity type $i$, and $w_i$ is a constant weight?

This is similar to Leontief utilities, only with max instead of min.

What term describes the type of goods with such a utility function? Initially I thought they were called "perfect substitutes", but now I see that this term is used for linear utility functions.

The Rawlsian welfare function, which takes the form of the min of all agent's utility, is often called the maximin function, because it maximizes minimum utility. In the spirit of that nomenclature, your function is a maximax utility function. And, apparently, this function does see some use. It is sometimes called the optimistic decision criteria, because a person evaluating projects based on their best outcome uses a similar evaluation function.

  The maximax looks at the best that could happen under each action and
  then chooses the action with the largest value. They assume that they
  will get the most possible and then they take the action with the best
  best case scenario. The maximum of the maximums or the "best of the
  best". This is the lotto player; they see large payoffs and ignore the
  probabilities.

Jones (2002), Decision Theory Notes


Destruction in Exchange Economies I am reading about exchange economies and ran across something that is counter intuitive.

How is it that in an exchange economy with no production that a person can make himself/herself better off by destroying a portion of his or her endowment? If there is no production how, in the long run, is a person better off destroying some of the available resource if that resource can't be replaced etc.?

Can anyone illustrate this in a simple manner to help with the intuition?

I know that if there could be some way that destroying a bit of a resource can drive up prices such that the remaining quantity is now more valuable than the original quantity at original prices. Is that what is happening here? If so, why and how?

Consider an exchange economy with two goods $x$ and $y$ and two agents $A$ and $B$. The utility functions are
$$
U_A(x_A,y_A) = x_A \cdot y_A, \hskip 20pt U_B(x_B,y_B) = \min(x_B,y_B).
$$
Let the initial endowments be
$$
\omega_A = (\omega_A^x,\omega_A^y) = (z,0), \hskip 20pt \omega_B = (\omega_B^x,\omega_B^y) =  (0,4),
$$
where $0 \leq z \leq 8$ is left as a parameter. In equilibrium
$$
x_A = \frac{z}{2}, \hskip 20pt y_A = \left(4 - \frac{z}{2}\right), \hskip 20pt x_B = \frac{z}{2}, \hskip 20pt  y_B = \frac{z}{2}.
$$
(A detailed calculation showing this is at the bottom of this answer.)

The utility of agent $A$ is
$$
U_A\left(\frac{z}{2},4 - \frac{z}{2}\right) = \frac{z}{2} \cdot \left(4 - \frac{z}{2}\right).
$$
The maximum of this is at $z = 4$. If the initial endowment of agent $A$ is larger than 4 he has incentive to destroy part of it, because assuming a competitive equilibrium outcome this will improve his utility.

As @Foobar pointed out in the comments further utility could be gained by not destroying but simply withholding part of the initial endowment. There are two arguments to be made for destruction:

Agent $B$ sees the supply and refuses the monopolistic outcome, insisting on the competitive price ratio.
One could divide agent $A$ into several 'smaller' agents. Agents who would have the same utility functions and equal shares of the initial endowment. In this case withholding part of the endowment is a cartel move. It is not individually rational. One could argue that destroying the 'surplus' is a commitment necessary for coordination.

In equilibrium total demand equals total supply in both markets.
\begin{eqnarray*}
x_A + x_B & = & \omega_A^x + \omega_B^x \\
\\
y_A + y_B & = & \omega_A^y + \omega_B^y.
\end{eqnarray*}
Our next step is determining demand. Here is the information we gather from the utility functions:

Let $y$ the numeraire good, i.e. let $p_y=1$. Let us denote the price of $x$ by simply $p$. The value of the initial endowment of agent $A$ is $z \cdot p$. Using the Cobb-Douglas property
$$
x_A = \frac{z\cdot p}{2\cdot p} = \frac{z}{2}, \hskip 20pt y_A = \frac{z\cdot p}{2}.
$$
Using this we have
\begin{eqnarray*}
x_A + x_B & = & \omega_A^x + \omega_B^x \\
\\
\frac{z}{2} + x_B & = & z + 0 \\
\\
x_B & = & \frac{z}{2}.
\end{eqnarray*}
It follows from agent $B$'s utility function that in equilibrium
$x_B = y_B$. Using this we have
\begin{eqnarray*}
y_A + y_B & = & \omega_A^y + \omega_B^y \\
\\
y_A + x_B & = & \omega_A^y + \omega_B^y \\
\\
y_A + \frac{z}{2} & = & 4 \\
\\
y_A & = & 4 - \frac{z}{2}.
\end{eqnarray*}


Are low interest rates dangerous? Interest rates keep dropping. In my country in northern Europe we now have negative interest rate - banks have to pay the central bank for lending it their money.

I have two questions:

Why are interest rates lower than any time ever before?
Are low interest rates dangerous? The low interest rates will make it easier to borrow money for projects that would otherwise not have been possible. Irresponsible lending has proven dangerous in the past.

Are the low interest rates a way that the central banks are "artificially resuscitating" a poor economy which may finally lead to a crash?

Adding that we have never seen this situation before I ask if this situation with extremely low interest rates is dangerous?

One aspect of the question is: Are these low interest rates expected to stay low or are they considered an extreme value in a cyclic variation?

If by dangerous it is meant they add possible downwards volatility of assets, that could easily be argued for as:

$ PV = D / (1+r)^t$

Where D denotes any future income. This is especially the case if the reason for the low rates is very easy monetary policy, as such policy cannot be maintained and thus the rate is not equilibrium rate and must rise, causing a crash in asset values when that happens.

Central banks determinate the interest rates in the short run, their policies are targeted to lower rates, thus a low rate. The other reason is that of supply and demand. There is not much demand for credit (no profitable ideas, no more room for consumer credit) and on the other hand the supply of credit is significant due to central bank action. High supply and low demand equals a low rate.


A growth model with regime switch I have a very general question. I am reading this paper :

http://www.webmeets.com/files/papers/eaere/2015/177/Discounting-HelsinkiBlind.pdf

There is a catastrophic event probability and after the catastrophic event the consumption level is reduced to zero. However, authors are making a steady state analysis prior to the catastrophic event. The catastrophic event probability is $h\left(X\right)$ (let s say that it follows a Poisson process.) $X$ stands for the pollution for example.

Let $T$ be the event occurrence time and denote $F\left(t\right)=Pr\left\{ T\leq t\right\}$  and $f\left(t\right)=F^{'}\left(t\right)$ as the corresponding probability distribution and density functions respectively.

$$h\left(S\left(t\right)\right)\Delta=\frac{f\left(t\right)\Delta}{1-F\left(t\right)}=-\frac{d\left[ln\left(1-F\left(t\right)\right)\right]}{dt}$$

where $\Delta$ is an infinitesimal time interval. The term $h\left(S\left(t\right)\right)\Delta$ specifies the conditional probability that an abrupt event will occur between $\left[t,t+\Delta\right]$.

My question is that : Due to this specification, the probability distribution function will be equal to 1 when $t$ tends to $\infty$. Then, in this case, for sure that catastrophic event will occur at the long run.

So, how is it possible to discuss about a steady state with catastrophic events ? At some moment in time, the economy will change regime with the catastrophic event and this steady state will not be a "permanent" one. How can it be possible to justify this ?

Continuing on the comments exchange, the conclusion that the overall effect is to change the discount factor, is similar to the one reached in the original Overlapping Generations Model of Blanchard, where individuals are faced with a "probability of death" (surely a "catastrophic event" I believe).

See the Blanchard & Fischer book "Lectures in Macroeconomics", p. 117. The authors note that the result has been originally obtained by Cass and Yaari (1967) "Individual Saving, Aggregate Capital Accumulation, and Efficient Growth", a paper that is part of the book Essays on The Theory of of Optimal Economic Growth


What are the economic benefits of SLAPP in file sharing/piracy? Spin-off from Piracy/File sharing - Why aren't songs, movies or ebooks given for free (+ads) like TV?
What are the economic benefits of SLAPP in or out of file sharing/piracy?
There's a comment in above link that says 'The relevance is that the recording industry is not obviously losing the piracy battle. The basis of the OP's question is that they are' (I'm OP, btw ^-^).
If that's the case, why did the RIAA heavily prosecute people for what some (yay, weasel words!) people may consider outrageous fines for outrageous expenses (not to mention opportunity cost)? Examples are Joel Tenenbaum (4.5m USD at one point) and Jammie Thomas-Rasset (1.92m USD at one point).

From Wiki (emphasis mine):

A strategic lawsuit against public participation (SLAPP) is a lawsuit
that is intended to censor, intimidate, and silence critics by
burdening them with the cost of a legal defense until they abandon
their criticism or opposition.
The typical SLAPP plaintiff does not normally expect to win the
lawsuit. The plaintiff's goals are accomplished if the defendant
succumbs to fear, intimidation, mounting legal costs or simple
exhaustion and abandons the criticism. In some cases, repeated
frivolous litigation against a defendant may raise the cost of
directors and officers liability insurance for that party, interfering
with an organization's ability to operate.

I'm assuming that there are some economic benefits to the government, society, the RIAA/the plaintiffs, someone or something by having pursued those individuals so persistently. But then again if it was so beneficial, then I expect we'd hear about a lot more than just these 2 SLAPP cases. (Or, what, everyone else really decided to settle? Idk. I kinda think it's more like RIAA/MPAA decreased its litigious tendencies.)
Perhaps in cases besides file sharing, this might have some benefits, but when it comes to current copyright law, file sharing, piracy, a digital world, etc, I'm drawing blanks particularly due to the 'Internet principle of the Hydra: You know, you can stomp one person, but there’s going to be seven more of us.'
Chairman and CEO of the Motion Picture Association of America (MPAA) Glickman 'concedes that piracy will never be stopped, but states that they will try to make it as difficult and tedious as possible.'
So, what exactly is the MPAA/RIAA getting out of SLAPP's? If there is no direct answer to be given, an alternative could be: What are the benefits of SLAPP's in general (then reader e.g. me infers analogous benefits)?

Background

Since the marginal cost of distribution for a creative work such as a song is now essentially zero, the efficient thing to do would be to provide all songs to listeners at a price of zero (i.e. to allow piracy). However, this neglects the fact that producing creative works involves a fixed up-front cost (e.g. studio recording time, or simply the artist's time composing the work) that cannot be covered if the price is zero. Moreover, it is efficient for producers of 'good' songs (i.e. those that create value in excess of the production cost) to incur the fixed cost, and for producers of bad songs not to do so.

The way that society mostly addresses this issue is through the grant of intellectual property (specifically, copyright) protection. The trade-off made is that artists are allowed to act as monopolist suppliers (with all of the attendant inefficiencies) because the resulting profits will give them an incentive to produce in the first place. The fact that a seller's rewards are tied to their monopoly profits means that the most popular artists should, in principle, have the biggest incentive to create.

Efficiency effect of piracy

If one buys the idea that we need IP protection to induce the production of creative work then tolerating widespread piracy is a bad idea because it takes away the profit that is supposed to provide artists with the incentive to create. Whilst each individual instance of piracy might be efficient, the overall effect of reducing the number of works produced might not. It is then necessary to ensure that enough people do not pirate enough of the time to keep the rewards from production at a sufficient level.

People should be expected to pirate if the benefits from doing do (free consumption of creative works) exceed the costs (determined by likelihood of being prosecuted and the penalty when prosecuted). An obvious way to stop people from pirating is to increase this cost by prosecuting more people—even if you don't really care that much about winning. Indeed, the main objective is not to win the case and get what, to the RIAA is pocket change; rather, the objective is to scare people enough that they start buying rather than pirating music. [n.b. The RIAA would have to win at least some of the cases, otherwise the threat would be completely empty. As Densep notes in a comment, this raises the issue of whether we can really call these cases SLAPPs or not].

If the result is that artists earn more money and respond by producing more creative works then this can indeed be efficient.

Some tangential remarks

The last qualifier (if the result is that artists earn more money and respond by producing more creative works then this can indeed be efficient) is non-trivial. There is some debate about how elastic the supply of creative works really is to the rate of piracy and whether this effect is big enough to offset the efficiency gains that come from consuming creative goods at marginal cost (i.e. zero).

My sense is also that there is a consensus among economists that the existing degree of copyright protection (typically something like the entire life of the author plus an additional seventy years) is far more generous than the level necessary to induce efficient levels of creative goods production.

Lastly, there are other ways of providing an incentive to create. For example, instead of granting an intellectual monopoly, the state can insist that all creative works are freely available and instead reward artists with a lump-sum prize. This avoids the monopoly distortion, but has a couple of drawbacks: (i) the prize would have to be funded somehow (e.g. from taxation, which is itself distortionary); (ii) it would be necessary to find a way to ensure that prizes are only given to reward the right kinds of music. The advantage of the intellectual property approach is that the market automatically rewards popular songs more.


What effect does changing the retirement age have on employment? I had an interesting casual discussion with a colleague the other day.

A lot of western economies at least, such as New Zealand, have some form of retirement pension. It New Zealand it's called Superannuation. For a single pensioner, the amount received is 40% of the average wage.

I mentioned that as we get an aging population, the cost to the tax payer of paying this pension will increase. Currently about 12% of the budget is used to pay Superannuation.

One of the common solutions put forward is to raise the age that people are eligible to receive the pension.

However, my colleague pointed out that raising the retirement age will likely keep older people in the workforce longer, and pointed out that India in fact lowered the retirement age, as to generate jobs for young people.

So this poses an interesting problem - on one hand, increasing the retirement age saves the government and taxpayer money. On the other hand, it possibly takes jobs from young people and stiffles their career.

Is there any research on the effect of retirement ages and their effect on employment?

"(...) increasing the retirement age (...) possibly takes jobs from
  young people and stifles their career."

Since this is often surfaces as "the conventional wisdom" (not necessarily in economics circles), let's present a simple argument, that shows that there are bound to be distinct effects for people already in the labor force, compared to the people that have not yet started their working period in life.

To bring in the surface the core effect, assume a constant population, and a constant demand for workers (that does not mean necessarily constant output of course, since productivity and efficiency may rise).

Assume that in this society, people start working as they enter their 21st year, and retire when the enter their 61th year (40 years of work). An old person should retire today -but he doesn't: he will stay on for another year (say, because the government raised the mandatory retirement age).

Then, the young person that was about to enter the labor force in order to replace him, will have to stay out of it for one more year. He will now enter the labor force when entering his 22nd year and he will exit when he will enter his 62nd year. But this means that this person will work again for 40 years: Raising the retirement age, does not mean raising the numbers of years worked for the new/future workers. What happens is that their working life shifts as regards its beginning and end on their personal age-time, but not as regards its lifetime duration.

Does this "stifle their careers"? One could reasonably argue that starting one year later gives you one more year to accumulate human capital through education and training -and so when you enter the labor force, you will enter it better prepared than if the change in retirement age had not happened.

Casually observing western societies, this is what appears to be happening: people tend to acquire more and more education, and so also start working later in life. At the individual (or mass media) level, this may be negatively experienced/pictured as "high unemployment for young people" and "difficulty to get a job that makes the young people accumulating more education in their attempt to find a job". I presented above an alternative perspective on the matter.

But as regards those that were already workers when the change happened, and so have entered on the labor force in the beginning of their 21st year, will now work in total 41 years. If we envision a gradual rise in professional ranks and income, the change in retirement age will mean that they may have to stay one more year "where they were", and in that sense, increasing the retirement age may appear to "stifle their careers". But there is also Arrow's "Learning by doing": human capital increase due to on-the-job experience (a verified phenomenon even in a perfectly static environments). So this one more year will not be a total waste in terms of increases in income earning capacity -and it may pay off more for the younger workers, than for the older ones (since the former will gain it earlier in their working life). So the "career-stifling" argument gets a little weaker, even for the class of workers-already.

Obviously this is a very abstract approach to the matter... for example, what happens to societies where population is rising? How many people can a society afford to invest in -create human capital for young non-workers- at any given time?
...but I believe it reveals one aspect of the matter that we tend to overlook.


GARP and SARP assumed mononticity? Monotonicity means the decision maker prefer more goods than less. It is not mentioned in textbook that SARP and GARP preasumed monotonicity, implicitly.
GARP: if $a$ is indirectly revealed preferred to $b$, then $b$ is not directly revealed strictly preferred to $a$.
SARP: if $a$ is indirectly revealed preferred to $b$, and $a\neq b$, then $b$ is not indirectly revealed preferred to $a$.
Because $aRb$ (directly revealed) means $ap_a\geq bp_a$ and $aR_sb$ (directly revealed strictly) means $ap_a>bp_a$, does that implies monotonicty if we assume the price is positive?

Strict monotonicity of the (internal) utility function is not assumed but is implied by GARP thanks to Afriat's theorem

GARP is equivalent to:

  There exists a continuous, strictly increasing, and concave utility function
  which rationalizes the data

The equivalence does not hold for SARP. SARP implies GARP, but not vice-versa.

Recall the actual definitions:

Directly revealed weak preference; $x^a \ge^R x^b \; \text{if} \;p^ax^b \le w^a$
Indirectly revealed strict preference:
$$ x^a >^I x^b\; \text{if } \; \exists x^{i_1}, x^{i_2}, \ldots, x^{i_m}
\; \text{such that} \; x^a \ge^R x^{i_1}, x^{i_1} \ge^R x^{i_2}, \ldots , x^{i_m} \ge^R x^b \\ \text{with at least one of these inequalities strict}$$
GARP: if $x^a \ge^R x^b$, then not $x^b >^I x^a$
an (internal) utility function $u:\mathbb{R}^n_+\to \mathbb{R}$ rationalizes the data $\{x^i, p^i, w^i\}_{i=1}^n$ if

$$ \forall x \in \mathbb{R}^n_+, \; p^ix \le w^i\; \text{implies}\; u(x^i) \ge u(x).$$

So you can argue just "by inspection" that GARP is a "sham" set up just so we get a strict (internal) utility that rationalizes the data. (Why else ask for one of those inequalities to be strict? It's actually a key to the proof.) Apparently, the GARP axiom was formulated by Afriat himself, although he called it "cyclical consistency", so... draw your own conclusions how he may have come up with GARP.

The selling point of Afriat's result, as I understand it, is that you can check the "rational expectation" that the agent is utility maximizing (making optimal choices) by merely looking at a sample of its choices (i.e. testing this sample for GARP); furthermore, if the test is positive, you can also derive an utility function that is "as good as" the one that presumably generated the choices. There's an issue that the method is not robust to measurement error though.

    The revealed preference principle of Samuelson … elaborated by Houthakker … easily gives a condition for the rejection of the hypothesis of existence. But the principle has been absent by which the hypothesis can be accepted or rejected on the basis of observed choice of the consumer, supposed to be finite in number; and, in the case of acceptance, a general method is needed for the actual construction of a utility function which will realize the hypothesis for the data. (Afriat, 1967, 68)

    Afriat’s approach … was truly constructive, offering an explicit algorithm to calculate a utility function consistent with the finite amount of data, whereas the other arguments were just existence proofs. This makes Afriat’s approach much more suitable as a basis of empirical analysis. Afriat’s approach was so novel that most researchers at the time did not recognize its value. (Varian, 2006, 101)

  [...] Houthakker had proven that if demand functions satisfied SARP there would always exist a rationalizing utility function, but provided no practical way to find it, while GARP techniques provided a way to construct such a utility function.


How does inflation in income go 'hand in hand with inflation in prices'? Source: p 594, Economics, 3 Ed, 2014, by N G Mankiw, M P Taylor

  Yet further thought reveals a fallacy in this answer. When prices rise, buyers of goods and services pay
  more for what they buy. At the same time, however, sellers of goods and services get more for what they sell.
  Because most people earn their incomes by selling their services, such as their labour, inflation in incomes goes
  hand in hand with inflation in prices. Thus, inflation does not in itself reduce people‘s real purchasing power.

Is the bolded (independent) clause too presumptuous? I doubt its truth. Even if hand in hand means any positive correlation weaker than 100%, the bolded can be false. For example, despite being in view of inflation in prices, a greedy employer can simply refuse to increase wages. Then no inflation in incomes exists, and so inflation DOES reduce people‘s real purchasing power?

In the situation you describe people's real purchasing power would indeed be reduced. However such a situation is not likely. If the employer was a monopolist, how come employees could afford more than the bare necessities in the first place? (See the works of Carl Marx.)

In real life it is more likely that if an employer does not raise wages when businesses are doing well another employer will make a competitive wage offer. Thus inflation will result in the rise of wages.

Mind that Mankiw's book is not a mathematical treatise, his statements are not precise. The book is merely a collection of stories aimed at students who are just familiarizing themselves with economic phenomena.


Can lying be disincentivized without serious curtailments of free speech? In the new age of Alternate Facts, one might argue that many actors personally benefit hugely from lying, at the expense of other sectors of society.
Economically speaking you might argue that this is a form of moral hazard -- the actors bear relatively small risks to themselves, receive benefits and may impose great costs or risks to others.

Some examples:

A politician may willfully lie to the public about intelligence facts concerning a country's weapons of mass destruction and receive increased political capital at the  expense of her country's cost of entering into an unnecessary war.
A populist news agency may willfully may lie about immigration or terrorism facts and receive increased circulation and advertising revenue at the expense of increased civil unrest, or in extreme cases contribute in driving a country into war justified by falsehoods.

Small risks to themselves, but great outsourced costs.

I think we can agree that, in general, lies are a generally 'a bad thing' that it would be wise to disincentivize them in many arenas (politics, science and academia, law, news media).

At the same time free speech seems like 'a good thing' that it would be unwise to curtail or disincentivize.
What, if any, incentives could be introduced to reduce lying without seriously curtailing free speech?

--

Legal systems (which are ideally a forum of free discourse in pursuit of the truth) disincentivize lies by imposing jail-time costs to people who make false statement.

Science (which depends heavily on free discourse) disincentivizes lying firstly by peer-review processes (which ideally would cause a liar to waste their time when submitting falsehoods for publication), and where this fails, by retracting published falsehoods, public shaming and loss of employment.

Yet in politics and media specifically, we rarely see costs incurred for distributing objective falsehoods. Can incentives be introduced in these arenas without seriously limiting the beneficial aspects of free speech to society?

For the purposes of this discussion, its probably good to focus of lying as "willfully making statements that the actor objectively (and provably) knows to be false", and leave the discussion on statements of opinion for another day.

If free speech is narrowly defined to only include true claims, true and false claims can be perfectly distinguished, the harm of false claims perfectly measured, and all claims are either true or false, then false claims are really no different than other forms of pollution. A standard pigovian tax should internalize the externalities and encourage the socially optimal level of truth telling. The social planner just has to tax each lie enough to discourage the social harm it does.

However, I maintain that in practice all of these assumptions are false. All experience seems to show that using the legal system to purge public discourse of lies leads to concentrating enormous power in those who make the lie / truth determination and that they will use that power in a way that is often inconsistent with objective truth And, when there is no objective truth there will still be those who would use the legal system to control public discourse in the name of protecting it.

In addition, the real world's imperfect legal processes are expensive and painful to participate in. That means that many true things that may be confused with lies will not be said, and that will further restrict freedom of speech.


Modelling involving sum of random variables: Simple CDF? This question emerges from a project in microeconomic modeling.

I have $n$ agents receiving noisy i.i.d signals $s$.

In my model, a situation of interest occurs when the average signal across $n$ agents, say $\bar{s} := (1/n) \sum_{i=1}^n s$, passes a certain threshold $t$.

I would like to be able to express the probability that this happens, i.e. $P(\bar{s} > t)$, in an algebraically simple way.

Of course, $P(\bar{s} \geq t) = 1 - P(\bar{s} \leq t)$ so the probability I am interested in is a simple function of the CDF of $\bar{s}$.

My problem is I am unable to think of a distribution for $s$ that would make the CDF of $\bar{s}$ easy to play with and obtain analytical results from.

For example, letting $s \sim N(0,1)$ leaves me with a $P(\bar{s} \leq t)$ depending on the very unhandy error function (I know it's easy to obtain computation results based on $erf(\cdot)$ but I am aiming at analytical results here). Letting $s$ be a Bernoulli or a Uniform leaves me with the CDF of a Binomial or an Irwin-Hall, which aren't much nicer to play with...

Any suggestions as to what I could use for the distribution of $s$?

Bonus requirements:

in another part of my model, I am also interested in $P({s} \geq t)$ itself, so it would help a ton if both $s$ and $\bar{s}$ had simple CDFs.
for reasons I cannot yet lay down myself, I feel like I will be happier further down the line if $s$ has a continuous and somewhat symmetric distribution, but I am happy to hear about discrete and asymmetric solutions as well.

So we have

$S_n \thicksim^{iid} \ ?$

$\bar{s} = \frac{1}{n}\sum{s_n}$

Exponential

Suppose $S_n$ follows the exponential distribution.

$$f(s|\beta) = \frac{1}{\beta} e^{-\frac{s}{\beta}} \quad , \quad 0 \leq s < \infty \quad , \quad \beta > 0$$

Take the simple bivariate case. Say $Z = \frac{S_1 + S_2}{2}$ and $W = S_1$.

So $S_2 = 2Z - W$ and $S_1 = W$. The determinant of the Jacobian: $\frac{\partial s_1}{\partial z} \frac{\partial y}{\partial w} - \frac{\partial x}{\partial w} \frac{\partial y}{\partial z} = -2$. The absolute value is $2$.

Thus, given the join marginal of $S_1 , S_2$

$$f(S_1, S_2) = \frac{1}{\beta ^2} e^{-\frac{S_1 + S_2}{\beta}}$$

we then find the joint marginal of $W, Z$, and integrate out $W$ to find the density function for $\frac{S_1 + S_2}{2}$

$$
\begin{align}
h(W,Z) = & \ f(W, 2Z-W) \cdot 2 \\
= & \ \frac{2}{\beta ^2} e^{-\frac{w+2z-w}{\beta}} \\
= & \ \frac{2}{\beta ^2} e^{-\frac{2z}{\beta}} \\
\end{align}
$$

Integrate with respect to $w$ from $z$ to $0$ gets you the density:

$$f(Z) = \frac{2}{\beta ^2} z e^{-\frac{2z}{\beta}}$$

Then you do a lot of garbage integration by parts (let the buyer beware if I have made an error) to get the cumulative distribution function evaluated from $z$ to $0$:

$$\boxed{F(Z) = -\frac{2z + \beta}{2 \beta} e^{-\frac{2z}{\beta}} + \frac{1}{2}}$$

Substitute $q = -\frac{2z}{2\beta}$ and you can make this look more tolerable:

$$\boxed{F(Z) = \left(q - \frac{1}{2}\right) e^{-2q} + \frac{1}{2}}$$

You may now umm...generalize the case for $n$ agents LOL. It may be more viable to compare this to the CDF of a single exponential distribution, then maybe try the case for $n=3$ and see if you can see a pattern.

Cauchy :P

One thing I looked at for fun online, take a simplified Cauchy distribution:

$$f(x) = \frac{1}{\pi(1+x^2)}$$

By the convolution formula and then a lot of work, the density of $Z = X + Y$ is:

$$\frac{2}{\pi(4+z^2)}$$

You can also verify that if you want the density of $Z = \frac{X + Y}{2}$, it is:

$$f(z) = \frac{1}{\pi(1+z^2)}$$

It's the Cauchy again! Now, the Cauchy has that pesky property where umm...it has no moments. But I'm pretty sure for this Cauchy distribution, the density is symmetric around 0. And technically you did not specify that your signals would have mean and variance, just that they were iid. :)

The cdf of this ends up being:

$$F(z) = \frac{\arctan(z)}{\pi} + \frac{1}{2}$$

So if the average for three variables or four variables also gives the density of the Cauchy, you may feel so bold as to use this.

Final Comments

Uhh, if you are willing discretize the noisy signal (you mentioned Bernoulli), you can try using the Poisson distribution or something. If you are looking into another continuous option you could try Beta. If I have more time I may edit this answer with some investigation into one or the other.


Fama Macbeth and double clustering presents inconsistent results I have a large unbalanced panel data with 460 firms and 1259 days. The model I would like to run is below

$$
Y_{it} = \beta X_{it} + \alpha Z_{t} + \epsilon_{it}
$$

where $Y_{it}$ is stock return, and $Z_{t}$ are Fama French 3 factors, and $X_{it}$ are variables of interest.

I run Fama Macbeth (FM) and double clustering to correct for the standard error, but two models give inconsistent results,i.e., $\beta$ is significant in one model, and not in the other.

I understand that Fama-MacBeth technique
was developed to account for correlation between observations on different firms in the same time point,
not to account for correlation between observations on the same firm in different time points. Traditionally, it should run cross section regression at each time point, then average estimates along time $T$. But in my case, due to the inclusion of $Z_t$, I have to run time series regression first since otherwise, $Z_{t}$ are not identifiable. In this case, does FM actually correct for correlation between observations on the same firm in different time points?

More importantly, does the inconsistent result mean that my results are not robust? Under my case, can I argue one is more appropriate than the other? Can the unbalanced data structure contribute to the inconsistent results?

I'm using fm and cluster2 command on this page Stata command

This is a large comment.

Fama-MacBeth only corrects for cross-sectional correlation in a panel and suffers from the error-in-variables problem (your $Z_t$, also check Chordia, Goyal and Shanken (2015)).

Depending on the exact nature of your dataset's variables the answer to your questions is "it depends"; please check Goyal (2012), Mitchell (2009) and Ibragimov and Mueller (2010).

Goyal (2012) accurately explains the F-M procedure (section 2.5). He states that "whether there is a bias in the traditional Fama–MacBeth approach if expected returns vary with time-varying characteristics is still unexplored.". Moreover, he says that "autocorrelation in returns (negligible at monthly frequency) leads to autocorrelation in risk premium estimates. This is easily accounted for by Newey–
West type corrections to variance formulas".
Mitchell (2009) discusses the SEs in financial panel data and clearly states that we should identify the presence of a firm or time effect to see whether the Fama-MacBeth standard errors are unbiased.
Ibragimov and Mueller (2010) "find that as long as year
coefficient estimators are approximately normal (or scale mixtures
of normals) and independent, the Fama–MacBeth method
results in valid inference even for a short panel that is heterogenous
over time."

You might also find Jagannathan, Skoulakis and Wang. (2014) enlightening to make proper arguments on the appropriateness of the methods if your study is asset pricing related; in corporate finance things are more "flexible".

Last, I am far from an expert in this area, but I think the "pre-made" Stata commands are not exhaustive in dealing with variables with different statistical characteristics (e.g. autocorrelated returns).