Solving Constrained Optimization Problem with Two-Period Model of Human Capital I'm trying to solve a constrained optimization problem in human capital model. The objective function is
$\max_{c_1,c_2,\nu} U = u(c_1) + \beta u(c_2)$,
subjected to
$c_1 = w +(1-\nu)\theta_1 h_1^a$ and $c_2 =  \theta_2 h_2^a$.
The list below summarizes the variables used:

$c_t$ is the consumption in period $t$. Note that $u(c) = log(c)$.
$\theta_t$ is the wage rate in period $t$.
$\nu$ is time spent in the first period to accumulate human capital. $\nu$ is normalized to be between [0,1], and $(1-\nu)$ is time spent working in first period.
$h_t$ is the human capital in period $t$. Note that $h_2 = h_1(1+\nu)$.
$a$ is the innate ability. Together, $\theta_t h_t^a$ represents income in period $t$.
$w$ is the initial wealth.

So, given $(w,a,h_1)$, individuals choose optimal $\nu$ in the first period that determines consumption in both first and second period. $\theta$ is exogenous variable. Now solving this optimization problem using the method of Lagrange equation:
$L = u(c_1) + \beta u(c_2) - \lambda_1(c_1 - w -(1-\nu)\theta_1 h_1^a) - \lambda_2(c_2 - \theta_2 h_2^a)$.
Solving for $\dfrac{\partial L}{\partial c_1} = \dfrac{\partial L}{\partial c_2} = 0$ gives the following two equation:
$c_1 = \dfrac{1}{\lambda_1}$ and $c_2 = \dfrac{\beta}{\lambda_2}$.
Now solving for $\dfrac{\partial L}{\partial \nu} = 0$:
$\dfrac{\partial L}{\partial \nu} = -\lambda_1 \theta_1 h_1^a + \lambda_2 \theta_2 a h_1^a(1+\nu)^{a-1}$ = 0.
Substituting $\lambda_1$ and $\lambda_2$, we get:
$\dfrac{\theta_1 h_1^a}{c_1} = \dfrac{\theta_2 a \beta h_1^a(1+\nu)^{a-1}}{c_2}$.
Substituting the equality constraints and solving for $\nu$, we get:
$\dfrac{\theta_1 h_1^a}{w+(1-\nu)\theta_1 h_1^a} = \dfrac{\theta_2 a \beta h_1^a(1+\nu)^{a-1}}{\theta_2 h_1^a (1+\nu)^a}$
$\dfrac{\theta_1 h_1^a}{w+(1-\nu)\theta_1 h_1^a} = \dfrac{a \beta}{1+\nu}$
$\nu = \dfrac{a \beta w + a \beta \theta_1 h_1^a - \theta_1 h_1^a}{(\theta_1 h_1^a)(1+a \beta)}$.
What I do not understand is why $\theta_2$ does not play a role in determining the optimal $\nu$. Logically, individuals invest in human capital in the first period, by foregoing possible income in the first period, to earn more income in the second period. However, even if $\lim_{\theta_2 \to 0}$, this solution will still recommend individuals to invest in human capital in the first period, by exactly $\dfrac{a \beta w + a \beta \theta_1 h_1^a - \theta_1 h_1^a}{(\theta_1 h_1^a)(1+a \beta)}$ much.

The problem is that you are ignoring the division $ \frac {0} {0} $, which is in $ \frac {\partial L} {\partial v} $. Before looking at the solution, and seeing that indeed when $ \theta_ {2} = 0 \ \Rightarrow \ v ^ {*} = 0 $, I want to note that the first constraint $ c_ {1} = w - (1 -v) \theta_ {1} h_ {1} ^ {a1} $ can be more realistic and logical. It is easy to see that $ \frac {\partial c_ {1}} {\partial v}> 0 $ in the restriction, which implies that by saving to invest in human capital, your income increases in the first period. This implies that there is no trade-off between investing human capital and consumption (if the interest rate were less than one it would reduce the problem a little, but it would still be inherently wrong). And second $ c_ {1} = w - (1-v) \theta_ {1} h_ {1} ^ {a1}$ does not allow saving wealth, only income. A more reasonable restriction would be this $ c_ {1} = (1-v) (w - \theta_ {1} h_ {1} ^ {a1}) $
\
I'm going to ignore the second observation and go on to answer your question, I just thought it relevant to point it out. We are going to work with this restriction $ c_ {1} = w + (1-v) \theta_ {1} h_ {1} ^ {a1} $. This does not affect $ \frac {\partial L} {\partial c_ {1}}, \frac {\partial L} {\partial c_ {2}} $, but it does $ \frac {\partial L} {\partial v} $. The third first order condition would be:
\begin{align}
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} h_ {1} ^ a - - \theta_ {2} h_ {1} ^ a) (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0 \\
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} - \theta_ {2} (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0
\end{align}
We derive the euler equation by introducing the constraints at $ \frac {\partial L} {\partial v} = 0 $:
\begin{align}
\frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} = \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a} }
\end{align}
It seems that you can safely say that $ \frac {\theta_ {2}} {\theta_ {2}} = 1 \ \forall \theta_ {2} $, but this is not true when $ \theta_ {2} = 0 $, but this is not the case because when this occurs $ \frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v $. This is made clear by multiplying $ \frac {\partial L} {\partial v} = 0 $ by $ \theta_ {2} $:
\begin{align}
\frac {\partial L} {\partial v} = \frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = \frac {\theta_ {1} \theta_ {2}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v
\end{align}
Therefore the solution of the euler's equation for $ v $ is valid if only if $ \theta_ {2} \neq 0 $. What is the optimal value of $ v $ if $ \theta_ {2} = 0 $? Since we have three unknowns and $ v $ only appears in 2 of them, we cannot derive a solution by substitution. You have to see what the effect of v is on the utility function. To achieve this, we obtain the optimal values of $ c_ {1} $ and $ c_ {2} $ by solving for the euler equation (without substituting the constraints and we obtain the following:
\begin{align}
c_ {1} ^ {*} = \frac {\theta_ {1}} {(1 + v) a} \\
c_ {2} ^ {*} = \frac {\theta_ {2}} {\theta_ {1}} (w (1 + v) ^ {1-a} - (1 + v) ^ {- a} h_ {1} ^ {a} \theta_ {1} = 0
\end{align}
But this implies that $ c_ {2} ^ {*} $ is a corner solution, therefore the constraint $ c_{2} = \theta_ {2} h^ {a} $ is not binding, so which is not valid. I'm not going to go into much detail why, but you can learn more in Chapter 18 of Simon and Blume's  mathematics for economists  book. The idea is that if  $c_ {2} $ is 0 the constraint of this variable cannot be fulfilled, it remains as an inequality and is irrelevant to the optimization problem. So it becomes an optimization problem in 2 veritable $ v, c_ {1} $ but how $ \frac {\partial c_ {2} ^ {*}} {\partial v} <0 $. So the optimal level of $ v $ is 0? Without any restriction on the value of $ v $ the solution is $ v = - \infty $ !!!. With restriction its optimal value is equal to 0. Note: this is only true if you make a modification to the utility function, otherwise the optimization problem is not defined. See explanation in "edit".
Edit
I made some corrections to what I had previously written. On that $ v $ does not depend on $ \theta_ {2} $, it does; only in a similar way to what happens when you use quasilinear utility functions. In these types of functions, the good that appears linearly in the utility function does not depend on income if you derive first-order conditions. The problem with this is that the first order conditions are only valid, in this case, when the quantities consumed of both goods are positive. This comes in any intermediate microeconomics book.
It turns out this is general. The first order conditions of an optimization problem in which it is not explicitly specified that the values of the endogenous variables must be greater than or equal to 0. This implies that there will also be parameter values that make the optimal solution let 0 be the endogenous variables. In this case, $ \theta_ {2} $.
The problem is that when $ \theta_ {2} $ equals 0, the objective function is undefined $ \theta_ {2} = 0 \ \Rightarrow \ c_ {2} = 0 \ \Rightarrow \ log (c_ {2 }) = - \infty $, so the value of $ theta_ {2} = 0 $ is not possible. But with slight modifications like changing in the utility function $ log (c_ {2}) $ for $ log (c_ {2} +1) $, the problem is defined and by substituting the constraint of $ c_ {2} $ in the utility function, it becomes independent of $ c_ {2} $ and the optimization problem becomes 2 variables and since $ v $ is only a cost, the optimal solution is $ 0 $ with restriction and $ - \infty $ Without restrictions. So when $ \theta_{2} = 0 \ \Rightarrow \ v^{*} = 0$, and when $ \theta_ {2} \neq
 0$ the optimal value of $v$ is a constant that comes out of the problem of optimization. So there is a dependency, but it breaks when $ \theta_ {2}> 0 $.
Now why is there no dependency when $ \theta_ {2}> 0 $ ?. As @Bertrand commented, this could be changed if there were disutility of accumulation of human capital in the utility function. But it is not the only way, you can also allow to save wealth to invest in human capital, this will create the expected dependence for $\theta_{2}>0$. I think that change is very reasonable; why the agent would be not allowed to save wealth to invest in human capital?
Regarding what they told you about what happens when $ w = 0 $, again you will see that this puts more restrictions on the parameters, in fact it has to be true that $ a \beta = 1 $, otherwise the optimization problem is not definite.


What does the adjective "structural" mean in "structural reforms" or "structural effects"? Economists frequently speak of structural reforms and structural effects.

For example, Manmohan Singh (1992):

  The economic crisis was used as an opportunity to implement basic structural reforms in the areas of industrial policy, trade policy, tax policy, public sector management and the financial system.

Donald Walker (1984):

  money plays a central role as a means of payment, and its presence in the economy has structural effects.

What purpose does the adjective structural serve in the above sentences? What difference would it have made if the word were deleted or replaced by non-structural?

Are there mere reforms and effects that differ from structural reforms and structural effects? Are there non-structural reforms and non-structural effects?

Is the word structural just a fairly vague term with a connotation similar to fundamental or important?

A structural reform is a reform that literally tries to change the fabric or structure of an economy. A structural reform has to significantly change the regulatory or institutional framework in which economy operates.

For example if a country has inflexible labor market laws and they change them to flexible labor laws that actually changes the structure of economy, the optimal way of how people are hired and fired, the optimal way of how goods and services are produced, workers trained and so on. Often structural reform is used as synonym for supply side reform - which is not entirely correct but not without merit as a supply side reforms are inherently structural in nature as they focus on improving regulations in way that allow for higher production (although some demand side policies can be considered structural as well).

Opposite of structural reform is not non-structural reform but just an reform that will not make much difference to the fabric of the economy. For example, if you have inflexible labor market laws and you just reform your laws to keep them inflexible but maybe modernize them in some way or do some changes that fundamentally do not change how economy operates. Also an reform that would be only temporary, for example for some crisis situation, would not be structural.

A structural effect would again be some effect that has impact on the fabric of the economy. For example R&D can cause structural change in economy. Thanks to the internet now we have all the ride/house sharing services that fundamentally change how taxi or hotel markets operate.


intertemporal utility function usage : calculating consumption I have encountered this a lot in my exams and can not seem to understand how to use these functions here is an easy exemple :

A consumer who will only live 2 periods receives 1000€ in the first period and 5000€ in the second period, if the interest is at 0% how much will he consume in both periods knowing that his utility function is :

  U(C¹,C²) = 3×C¹×C²

Where C¹ is the consommation of first period and C² of second.

How do I make use of this utility function to calculate the consumption in both periods  ?

This is the two-period budget constraint:

  C1 + C2/(1+r) = Y1 + Y2/(1+r)

Derivation is straightforward. On the LHS, you have the present value of consumption (considered during period 1), and on the RHS you have the present value of income.

Intuitively, think about 1/(1+r) on the LHS as the price ratio between Good 1 and Good 2. Now you can solve the u-max problem as per usual.


Are there economic models that predict (only) a medium/long-term negative effect of protectionism? Going by some ECB models/estimates from last year, protectionism would result in negative consequences pretty fast, e.g.

For illustration, I will use the results of simulations carried out by ECB staff using both the ECB's global model and the IMF’s multi-country model. As with all models, the uncertainties involved mean precise estimates from these scenarios should be treated with caution, but they are useful to explain the different channels at work.
To illustrate the potential effects of rising protectionism, I do not want to dwell on the specifics of the tariffs currently being discussed. This would miss the bigger picture. I rather want to consider a hypothetical scenario where the United States raises tariffs on all imports of goods by 10 percentage points, and its trading partners impose the equivalent on US exports.
According to our model simulations, such a scenario would have significant adverse effects on the global economy, including, and in particular, on the economy that raises tariffs in the first place. Specifically, real economic activity in the United States could be up to 2½% lower than in the baseline in the first year alone. [Gives some reasons for this.]
In other words, the overall scenario is clearly a net negative for the world economy as a whole. According to ECB staff simulations, world trade in goods could fall by up to 3% already in the first year after the change in tariffs and world GDP by up to 1%. Euro area GDP would also decline, but by less than in the US.

They don't make any longer term estimates for the effects of such tariffs, just mention that the EU would have been worse of by 1/5 without the economic integration since the 1950s.
Frankly it looks doubtful that US economy is showing any of that (short-term) trouble.
Are there any theoretical models of protectionism that predict short-term gain but medium or long-term loss for a country?
(Well, there's one BIS paper that advances a sort of short-term masking effect for the current US economy by advanced shipments/stockpiling:

It’s paradoxical that the United States is starting to put obstacles in the road at a time when its
economy is firing on all cylinders. Even looking through the short-term boost to GDP in the latest quarter [citing data from July 2018]
– ironically enough, partly owing to companies bringing forward cross-border shipments to skirt tariffs –
its short-term prospects are promising. But in the long term, protectionism will bring not gain, but only
pain. Not just for the United States, but for us all.

But this looks to me fairly anecdotal...)

This is not exactly what I was looking for, but seems "good enough":

I found a recent empirical paper by IMF staff (Furceri et al., 2018) applying Jorda’s local projection method to a panel of data of 151 countries over 1963-2014. They found/predict negative effects from the first year after the measures are introduced, but the effect size on output looks pretty small and given the confidence band it can well still go up for some years after the measures are introduced.

They also have bunch robustness checks (alternative models) which mostly preserve/confirm that conclusion

The "drop early years (<1980)" seems to produce a different enough result on output though (in that robustness graph), but I can't find any commentary on that particular check in the paper. I guess it's because it still falls within the 90% confidence region of the base model. The paper does point out in its introduction that older data had more protectionism:

  Our panel of annual data is long if unbalanced,
  covering 1963 through 2014; more recent data is of greater relevance, but older data
  contains more protectionism. Since little protectionism remains in rich countries, we use a
  broad span of 151 countries, including 34 advanced and 117 developing countries.

As an US addendum, regarding the Trump economy in particular, Kenny LJ answered another (much broader) question by citing/meniotining Fajgelbaum et al. (March, 2019)

  We analyze the impacts of the 2018 trade war on the U.S. economy. We estimate import demand and export supply elasticities using changes in U.S. and retaliatory tariffs over time. Imports from targeted countries declined 31.5% within products, while targeted U.S. exports fell 11.0%. We find complete pass-through of U.S. tariffs to variety-level import prices. Using a general equilibrium framework that matches these elasticities, we compute the aggregate and regional impacts. Annual consumer and producer losses from higher costs of imports were \$68.8 billion (0.37% of GDP). After accounting for higher tariff revenue and gains to domestic producers from higher prices, the aggregate welfare loss was \$7.8 billion (0.04% of GDP). U.S. tariffs favored sectors located in politically competitive counties, but retaliatory tariffs offset the benefits to these counties. We find that tradeable-sector workers in heavily Republican counties were the most negatively affected by the trade war.

So noticeable effect on trade already, but very small on GDP, percentage-wise. Given the IMF/Fuceri model, we don't really expect a large impact on GDP anyway.


Can a government or central bank work to reduce inflation without hiking interest rates and causing the currency to appreciate? Since trying to lower inflation will have some undesirable consequences, such as currency appreciation, which hurts exports, should governments and central banks keep hands off on inflation?
But exports would become uncompetitive anyway when inflation is high (as prices increase).
In a high inflation scenario, what should government or central bank do to protect exports (in an export dependent economy)?

Yes, but only if you also introduce capital controls. You should check out the Mundell–Fleming trilemma:

  The Impossible trinity (also known as the Trilemma) is a trilemma in
  international economics which states that it is impossible to have all
  three of the following at the same time:

  A fixed foreign exchange rate
  Free capital movement (absence of capital controls)
  An independent monetary policy

Wikipedia Impossible Trinity


Convention / standard methodology for calculating real prices? Is there an established professional standard used by economic / financial institutions for including or excluding i) the starting year/period, and ii) the end year/period when calculating real prices (i.e. when multiplying nominal values by a multiplier)?

Theoretically, there are four (mutually exclusive) possibilities

Include start period inflation, include end year inflation
Include start period inflation, exclude end year inflation
Exclude start period inflation, include end year inflation
Exclude start period inflation, exclude end year inflation

Example

Converting $100 from 2008 into 2010 dollars.

Suppose inflation was 2% in 2008, 3% in 2009, and 4% in 2010.

Applying the four methods gives four answers, like so

Method 1

Multiplying by the inflation in all of 2008, 2009, and 2010 (i.e. including start year and end year)

$100 * (1 + 0.02) * (1 + 0.03) * (1 + 0.04) = 109.26

Method 2

Multiplying by the inflation in all of 2008 and 2009 (not including end year)

$100 * (1 + 0.02) * (1 + 0.03) = 105.06

Method 3

Multiplying the 2008 price by the inflation in 2009 and 2010 (not including start year)

$100 * (1 + 0.03) * (1 + 0.04) = 107.12

Method 4

Multiplying by only the years in between (i.e. not including the start year nor end year)
i.e.

$100 * (1 + 0.03) = 103

Would all professionals calculate the real value the same way and, if so, which way?

This is not an issue of "established practice", but of methodological and conceptual consistency.

In theoretical models, for "real" magnitudes we write, say $W_R = W_N/P$ where $P$ is the "price index" and we're done.

But in reality, in order to obtain an actual value for this price index (and hence also of inflation), we must specify a base period, against which we will evaluate the price index for the other periods (past and future). This in practice means that we set the value of the price index at the base period equal to $1$ (or $100$).

In the OP's example, the base period is 2010, since they want to express a 2008 amount in "2010 dollars". Let's see then:

The "start period inflation" is an indication of how much prices increased in 2008 compared to 2007. This cannot be relevant to the task at hand. We are examining the relation between 2008 and 2010, the year 2007 is out of the picture.

So conclusion 1: "Start period inflation" is NEVER included in the calculations.

This immediately eliminates the first two possibilities stated by the OP, as being wrong.

Then the issue is, do we include "end-period inflation"? In our example, end period inflation is how much prices increased in 2010 compared to 2009. This is certainly relevant to the task at hand, because we are examining the evolution from 2008 to 2010, and 2009 lies inside.

So conclusion 2: "end period inflation" is ALWAYS included in the calculations.

Combined the two conclusions, tell us that "possibility 3" is what should and is being used.

And it is also evidently logical, because, in order to have the same purchasing power in 2010 as in 2008, we want to cover for the prices increases that happened in 2009 compared to 2008, and the price increases that happened in 2010 compared to 2009.


Gravity Equation Interpretation I have a question about the gravity equation. In the Feenstra textbook, on page 144, it is stated that

  In the monopolistic competition model [...] the countries are completely specialized in different product varieties. Trade in these product varieties is referred to as intraindustry trade...

On page 145, it is stated that

  Then it follows that a good produced in any country is sent to all other countries in proportion to the purchasing country's GDP.

To formalize this, consider a multicountry framework where $i,j=1...C$
  denotes countries and $k=1,...N$
  denotes products. Let $y_{k}^{i}$
  denote country $i's$
  production of good $k.$
  Since prices are the same across all countries, we normalize them to unity, so $y_{i}^{k}$
 actually measures the value of production. The total GDP in each country is measured by $$Y^{i}=\sum_{k=1}^{N}y_{k}^{i}$$
  and world GDP is $$Y^{W}=\sum_{i=1}^{C}y^{i}.$$

Here is where I am confused. I was under the impression that each country produces a single variety of the differentiated good. In that case, the total GDP in each country should just be $$Y^{i}=y_{k}^{i}$$
  where $k$
  is specific to $i$.
  In other words, I understand the statement as each country producing just one variety of the same good. I believe this is an incorrect interpretation for two reasons:

If that were the case, the number of goods should actually be $C$,
not $N$.
(It could be that $C=N$
but the author does not specify this)
The GDP
by the product would be $y_{k}^{i}$
but by the expenditure approach would be $Y^{i}=\sum_{k=1}^{N}y_{k}^{i}$.

Confronted with the likely possibility that my interpretation is fallacious, I don't know what the alternative interpretation is. What else could the statement “countries are completely specialized in different product” varieties mean?

My understanding is that the gravity equation can be derived from different settings.

You can assume that each country

1/ produces only one product. This is knwon as the "Armington assumption" as in Armington (1969) and allows a simple derivation of the gravity equation as in Anderson and Van Wincoop (2003).

2/ is specialized in different products (or varieties of a product, given that each variety of a good counts as a distinct product). This is the case worked out by Feenstra, where products are differentiated by their country of origin. So, each country produces many distinct products.

Then the question is to know if the number of products is fixed, as in Armington (1969), or endogeneous and varies due to free entry, as in the monopolistic competition models.

References

ANDERSON, J. E., AND E. VAN WINCOOP (2003): “Gravity with Gravitas: A Solution to the Border Puzzle,” The American Economic Review, 93(1), 170–192.
ARMINGTON, P. S. (1969): “A Theory of Demand for Products Distinguished by Place of Production,” International Monetary Fund Staff Papers, 16, 159–178.


Weighted voting: Vote of representative weighted by number of election votes I was wondering if there is any literature on voting systems where voters elect representatives, and in the representatives' assembly the vote of each representative is weighted by the number of votes she was elected with.

E.g.: A special but easy example would be the US supreme court. The 'voters' here are the US senators, their 'representatives' are the supreme court judges. Instead of each judge having an equal vote their vote could have a weight equal to the number of votes they were confirmed with. This would make a difference, as consensus candidates seem to be confirmed by more votes - at least in the current system. Changing the supreme court's voting mechanism could of course affect confirmations, but this may be a way to incentivize consensus.

In case of general elections and parliamentary representatives such a weighted voting system would incentivize representatives to try to represent all the voters in their district, not just the majority. (Assuming representatives care about their voting power, but one could always tie their salary to the number of votes they were elected by.)

The idea is simple, so it has probably been raised before. Please refer me to some literature.

You ask an excellent question, as it has been partly but not fully discussed in the scientific literature.

Your exact proposal (weighting representatives by the amount of votes they got in the election) has been discussed on a blog post (in french) by Jean-François Laslier see here.

In practice, representatives are often elected on a geographical basis, and their weight is determined by the local area they represent (typically: its population). There is a whole literature on that subject. Balinski and Young [BY] focus on how to design (integer) voting weights that are as close as possible to proportional to the populations. A consequent literature has questioned the desirability of the idea of proportionality, a representative paper in this area is Barbera and Jackson [BJ].

Besides, your proposal seems to be one feature (among other) of the liquid democracy system advocated by democracy activists. In that system, each one can delegate or vote on each decision, and delegates' weight equals the number of received delegations. The system has been experimented by the German Pirate party. It has been studied recently in computer science. One issue is that some influent members may get too many votes [GKMP].

[BY]: Balinski and Young. Fair Representation. 1982

[BJ]: Barbera and Jackson. On the Weights of Nations: Assigning Voting
Weights in a Heterogeneous Union. 2006

[GKMP]: Gölz, P., Kahng, A., Mackenzie, S., & Procaccia, A. D.. The Fluid Mechanics of Liquid Democracy. 2018


Which currency is preferred when countries with different currencies purchase imports/sell exports? For example, when a company in the United States sells goods to a company or distributor in China, does the U.S. company accept payment in yuan (which it then exchanges for dollars on the currency market) or dollars?

Likewise, when a Chinese company sells goods to an American company, does the Chinese company accept only yuan or will dollars suffice (even if they have to be sold to purchase yuan)?

Are some currencies (for example, the dollar, British pound, or euro) preferred in lieu of weaker currencies when two parties are exporting/importing on the international market?

There isn't a typical answer.

The vendor may want paying in whatever currency represents the highest proportion of their costs, as this reduces their currency risk.

But there may also be restrictions on what currency they can trade in; how much of some particular currency they're allowed to hold; there may be capital controls in place; some currencies might be easier to hedge than others; trade in some currencies might attract that country's tax authorities, if significant movements have to be registered; ... and so on.

They might want to do business across various buyers in a basket of currencies, as a way to minimise currency risk through diversification.

They may prefer to do business in currencies with a higher liquidity (USD, GBP, EUR, CHF, AUD, JPY, CAD).

There are lots of variable factors in play. And it's a negotiation between vendor and buyer; the above is written as if the vendor holds all the power, but similar issues apply to the buyer too. For example, the buyer may want to pay in whatever currency represents the highest proportion of their income.


What statistical concepts are used in introductory econometrics? I am an undergraduate taking an upper-year introductory course in econometrics. I have taken calculus but I am not familiar with mathematical statistics.

What are the basic concepts from statistics that I can read up on in preparation for the course? I am familiar with the idea of distributions but my knowledge is very limited.

I'll list the statistics-related topics from my basic econometrics course.

Expected value, mean, variance and covariance as well as conditional expectation and conditional variance.
Basic probability and probability density functions (I'd suggest getting somewhat familiar at least with Normal, Student's T, and F distribution and maybe Bernoulli, Chi-Square and Poisson) and what random variables mean.
The central limit theorem (closely related with PDFs).
Correlation. Also understanding why correlation does not imply causation is useful, but you'll probably discuss this in class.
Methods of regression/estimation. You'll probably be introduced to Ordinary Least Squares early on and a basic understanding of what regression models are for might be useful although probably not assumed.
Statistical hypothesis testing. This will probably be a main subject in any applied econometrics course, so you might take a sneak peek on the basics.

You might have to understand some matrix and equation algebra depending on how theoretical the lecturer makes the course, but these were probably included in your calculus course.


Demand for utility function given prices and budget I have to find the consumer's demand at prices $\textbf{p} = (1,1,1)$
and budget $m=4$ when the utility function is given by
$$u(x_1,x_2,x_3) = 3 \sqrt{x_1+x_2+2x_3}$$ or when transformed
$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3 $$

I know the answer is $\textbf{x} = (0,0,4)$

What I don't understand: For perfect substitutes the demand is given by $x_1^* = \frac{m}{p_1}$ so the demand should have been
   $\textbf{x} = (4,4,4)$. But why is $x_1^*,x_2^* = 0$?

For the same utility function I want to find the demand at prices
$p=(2,3,p_3) >>0$ and budget $m>0$.

The answer is  \begin{equation} \textbf{x} = \left(0,0,
    \frac{m}{p_3} \right) \quad \text{when} \quad p_3<4 \end{equation}
\begin{equation} \textbf{x} = \left(\frac{m}{2},0,0 \right) \quad
    \text{when} \quad p_3>4 \end{equation} \begin{equation} \textbf{x} =
    \left(\frac{\alpha m}{2}, 0, \frac{(1-\alpha)m}{4} \right) \quad
    \text{when} \quad p_3=4 \end{equation}
where $\alpha \in [0,1]$ because the consumer is indifferent between the bundles

What I don't understand: Why is the demand different depending on what $p_3$ is and how do I know when the demand is different? Personally, I would've said the demand is $\textbf{x} = \left( \frac{m}{2}, \ \frac{m}{3}, \ \frac{m}{p_3} \right)$

Regarding the first question, @Oliv suggestion in the comments to formally write down the consumer's problem, budget constraint, Kunh-Tucker multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:

Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function

$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3$$

will take its maximum value if we purchase only good $x_3$? For any tiny amount $\Delta x_3$ of $x_3$ that you don't purchase, you lose $2\Delta x_3$ in utility terms, while you gain (in utility terms) only $\Delta x_1 + \Delta x_2 = \Delta x_3$ for any combination of $\Delta x_1$ and $\Delta x_2$. Since $2\Delta x_3 > \Delta x_3$ it will be suboptimal to purchase anything else than $x_3$. That was the "plain logic" part, and I hope it answers the "what I don't understand..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.


What does national debt really mean? and why is it bad to pay it off? I can't get my head around the concept.
This question didn't help much either.
Could you please explain what it really means? and why is it bad to pay it off and have zero national debt?
Thanks.

You borrow \$100 from John, at 3% interest per annum. You are indebted to John, to the tune of \$100.
The Wakanda government borrows \$100M from the public, at 3% interest per annum. The Wakanda government is indebted to the public, to the tune of \$100M.

Conceptually there is no difference between (1) and (2).

But in practice, (1) is more likely to be informal. Maybe the agreement between you and John is not even written down and is simply verbal. Or maybe it is casually scrawled on a scrap of paper.

Whereas (2) is more likely to be formal. Indeed, officially, governments don't call it "borrowing". Instead, they say that they "issue bonds". Anyone in the "public" (including Joe on the street or the Chinese government) is free to buy these bonds. If say the bonds are one-year bonds that pay 3% interest per annum, then the government promises to repay the bond-holders $103M in a year's time.

  why is it bad to pay it off and have zero national debt?

I don't think any economist believes it is "bad to pay it off and have zero national debt". Instead, as with most questions in economics, the answer is "It depends." You can have too much debt or too little debt. Having zero national debt is neither always ideal nor always bad.

But in practice (i.e. in the real world), many governments tend to err on the side of having too much debt.


Bernoulli transformation in " A Close Form Solution to the Ramsey model" by Smith (2006) I am trying to understand the way Smith arrived from (1) and (2) to (6) and (8).

Could anyone give me a hint? Thank you!

  It is well known that the dynamics per capita consumption $c$ and capital $k$ are determined by the following pair of differential equations:
  \begin{eqnarray}
\dot{k} &=& k^\alpha - \delta k - c \tag{1} \\
\frac{{\dot c}}{c} &=& \sigma(\alpha k^{\alpha - 1} - \delta - \rho) \tag{2}
\end{eqnarray}
  A Solution

  The system of equations (1)-(2) is dauntingly nonlinear. To approach a solution, define the capital-output ratio (a Bernoulli transformation) by $z = k^{1-\alpha}$ and the consumption-capital ratio by $x = c/k$. Using these  transformations the system can be rewritten as
  \begin{eqnarray}
\dot{z} &=& (1-\alpha)[1 - (\delta + x)z] \tag{6} \\
\frac{{\dot x}}{x} &=&  \frac{\sigma \alpha - 1}{z} + \delta (1 - \sigma) - \rho\sigma + x \tag{7}
\end{eqnarray}
  The Bernoulli transformation converts the capital accumulation equation [Equation (1)] into the linear, albeit non-autonomous, differential equation in Equation (6). Despite this simplification, the system still does not allow an analytical solution. Equation (7) is still non-linear.

  Suppose, however, that $\sigma = 1/\alpha$. In that case $z$ disappears from Equation (7), and the system becomes recursive: Equation (7) reduces to
  $$
\frac{\dot{x}}{x} = - \frac{\delta(1- \alpha) \rho}{\alpha} + x \tag{8}
$$
  while $z$ evolves according to Equation (6), given the forcing process $x$ determined in Equation (8). This is a simple, autonomous logistic equation.

Citation: Smith, William. (2006). A Closed Form Solution to the Ramsey Model. Contributions to Macroeconomics. 6.

If $z = k^{1-\alpha}$ then

\begin{eqnarray}
\dot{z} &=& (1-\alpha)k^{1-\alpha-1}\dot{k} = (1-\alpha)k^{-\alpha} \dot{k} \\ &=& (1-\alpha) k^{-\alpha} [k^{\alpha} - \delta k - c] \\
&=& (1-\alpha) [1 - \delta k^{1-\alpha} - c k^{-\alpha}\color{blue}{(k/k)}] \\
&=& (1-\alpha) [1 - \delta z - c(k^{1-\alpha})/k] \\
&=& (1-\alpha) [1 - \delta z - x z]
\end{eqnarray}

I will leave $\dot{c}/c$ for you to work out


Calling 118 118 (UK) - A market failure? Is the shocking cost of directory enquiry numbers in the UK a market failure, or the economy working perfectly well?

EDIT: I know that proponents of free markets often like to blame over-regulation or government interference for perceived problems in markets, so I wonder if anyone could point to some regulation that has caused this incredible series of price hikes.

When there was only one telephone service supplier, BT directory enquiries used to be free. If you did not know the number of who you wanted to call, you could not call. Once you got the number from the phone book or directory enquiries, you could make the call and would be charged for it - instant payback for the phone company.

After the telephone market was opened to competition, BT started charging for directory enquiries eventually reaching 40p per call.

After de-regulation of directory enquiries in 2003, hundreds of suppliers have sprung up. The cheapest is currently 40p per minute plus your phone provider's Access Charge. The most expensive is £15.98 per call, plus £7.99 per minute after 60 seconds, plus your phone provider's Access Charge.

Prices for a number of 118 services have risen substantially over the years.

118 118 prices have risen sharply in recent years...

1 July 2015 - £5.74 per call, plus £2.99 per minute after 60 seconds, plus your phone provider's Access Charge.

1 March 2016 - £6.98 per call, plus £3.49 per minute after 60 seconds, plus your phone provider's Access Charge.

1 May 2017 - £8.98 per call, plus £4.49 per minute after 60 seconds, plus your phone provider's Access Charge.

It is no surprise that the most well-known services such as 118118 and 118500 are the most heavily advertised.

It should come as no surprise that the most heavily advertised services are among the most expensive. All that advertising has to be paid for somehow.

The cheapest services have no budget for advertising. In order to advertise, they would have to increase their prices.

What is needed is a third-party to rate and review all of the suppliers and their services and for this information to be freely available and easily accessible.


How to evaluate if an activity is likely to produce a positive ROI? I am trying to establish if it would be worth my while to sell a product in a particular online marketplace but am unsure how to evaluate if it would be financially worth my while.

The marketplace in question allows me to know basic kinds of sales data for any given item in the marketplace such as:

When was the item first offered for sale?
What is the total number of sales for this item?
What is the price of this item?
Who is the seller of this item?
What kind of item is this?

The marketplace also allows me some other high level information such as:

What is the total number of items for sale on the website?
How many user accounts are there on the website?
What are the most popular items for sale on the website?

I would love to know if this information is enough to help me make an educated decision as to if it would make economic sense in order to produce an item to sell in this marketplace and if so how would I go about figuring that out?

Look into what similar/same products this platform and similar platforms are selling for compared to your pricing.

Ask yourself the following:

How does their pricing compare to yours? If you are equal to or above their pricing- too risky. If you are below their sales prices, compare their online shelf life to estimate your projected holding period and carry cost between your purchase and sale time?
Do the other sellers have strong reviews (credibility)? Are you starting out without any reviews, or are you an existing seller with reviews?
What are your profit margins? If your costs go up 10-20% does will the demand allow for an increase your sales price?

Finally, run a 3-5 year cash flow plan to estimate your total sales, costs, and profits. Compare your projections(profits) with the amount of time you anticipate it will take you on a daily, weekly, monthly, annually.

Remember, selling online has its challenges, don't underestimate the amount of time it will take out of your day.

There's this story about a paintball gun dealer who bought their products from China and sold them out of their store, plus online through a major online selling platform. Their online sales grew so large their online platform (partner) went direct to the manufacture in China and made a deal cutting out the reseller, selling it cheaper than the paintball company could. They became a victim of their own success.

Anyway, good luck! :)


Does a fall in the wage rate always reduce hours worked? In the Neoclassical Labor-Leisure model, does a fall in the wage rate always reduce hours worked?

Or does it depend on whether the substitution or income effect dominates?

It is more common to consider an increase in the wage rate: in that case, assuming leisure is a normal good, the income effect should increase leisure (more money available to pay for it, making it more affordable) and the substitution effect should reduce leisure (each hour worked brings greater benefit than before the increase, encouraging work), with a net effect which could go either way

So with a reduction is the wage rate, you can simply reverse this: assuming leisure is a normal good, the income effect should reduce leisure (less money available to pay for it, making it less affordable) and the substitution effect should increase leisure (each hour worked brings less benefit than before the reduction, discouraging work), with a net effect which could go either way

In the improbable situation that leisure were an inferior good, the income and substitution effects would work in the same directions, so a wage rate increase would lead to less leisure and a wage rate reduction would lead to more leisure


Comparing utility functions I'm doing an econ course after not having any math or micro for a few years, now I'm totally missing the basics again.
I'm wondering how to show that utility functions are an equivalent to each other:
3xy+2, (xy)^2, e^xy, lnx + lny
should be equal to xy.
Could someone explain me in words what to do? Do you have to take the derivative or can it also be found in another way?
I've seen this post, but for me the explanation is too mathematical.
How can I tell if 2 different utility functions represent the same preferences?
Kind regards and thanks for your help!

When someone says two things are equivalent, what that means is that they give the same result for some test; they are equivalent with respect to that test. For instance, in Texas Hold 'Em, getting dealt the ace of spades, then the ace of diamonds, is equivalent to getting dealt the ace of diamonds and then the ace of spades, because strength of hands doesn't depend on the order that you received your cards. So to answer the question "How are these functions equivalent?", we have to know what test they're equivalent with respect to.
A utility function is a tool that's used to determine, given two options, which one someone would take. The idea is that we look at what utility each option has, and the person will take the one with the higher utility. For this test, we only look at which is larger, not the absolute size. So if you have two functions $f$ and $g$ that always give the same order, they are equivalent with respect to this test. That is, given any two options $a$ and $b$, if $f(a)<f(b)$, then $g(a)<g(b)$.
Now, if we set $f$ to $(x,y) \rightarrow xy$ and $g$ to $(x,y) \rightarrow (xy)^2$, we can introduce $h: u \rightarrow u^2$. Then $g(x,y) = h(f(x,y)$. In other words, $g$ is $h$ composed with $f$. If we assume that $x$ and $y$ are positive, then $h$ is monotonically increasing, and thus composing it with any function doesn't change ordering; if $n<m$, then $n^2<m^2$ (again, for positive numbers). Any time two functions are such that one can be written as a monotonically increasing function composed with the other, the two functions are equivalent with respect to ordering.
However, it is important to note that utility functions are also used for evaluating a concept called "lotteries". This is where someone chooses between options that involve different probabilities of different outcomes. For instance, someone might have a choice between "1% chance of getting \$1m and 99% of getting nothing" versus "100% chance of getting \$100k". When evaluating lotteries, the functions that you list are not equivalent when applied to the outcomes.


Is there any formal definition of a relationship between resource availability and demand? I do not have any background on economics. So my question might sound too simple and I hope it to be clear enough. I am curious about the following:

Is there any formal definition of a relationship between resource availability and demand? Intuitively I would expect that resource shortage, demand would increase. But this is not expected by the law of demand, assuming that the price of the resource would increase. Am I confusing something here?

Thank you,
best

EDIT 1: @Brythan Thank you for your answer. @EnergyNumbers and @erik thank you for your contributions. They were very useful to understand these concepts deeper. I was reading a little more about this and I wonder if the concept of scarcity is instead better applied here. I reformulate my question:

(1) does supply-induced scarcity implies that demanded is greater than supply?

(2) what is the correlation between supply-induced scarcity and quantity demanded?

EDIT 2: To be more clear, I understand that scarcity does not necessarly refer "to whether a good is in shortage or surplus, nor to whether there are only a few or whether there are many". That is to say that scarcity is not the same as availability. Instead, scarcity "refers to the possible existence of conflict over the possession of a finite thing" (https://mises.org/library/goods-scarce-and-nonscarce). The scarcity of a good (or a resource) happens when supply is not sufficient to meet the demand. "Supply-induced scarcity occurs when environmental degradation, pollution, natural variation or a breakdown in the delivery infrastructure constrains or reduces the total supply or local availability of a specific resource. As the supply of natural resources is reduced, options for pursuing productive livelihood strategies are undermined, potentially creating competition between livelihood groups." (http://www.un.org/en/events/environmentconflictday/pdf/GN_Renewable_Consultation.pdf). That is, supply-induced scarcity is correlated with the resource availability.

Now concerning the questions (1) and (2), it seems reasonable to say that in circumstances of supply-induced scarcity, when a decrease on the availability of a resource happens, because supply is not sufficient to meet the demand, the quantity demanded is higher than in circumstances closer to equilibrium. Does this make sense to you?

Now returning to my original question:

Is there any formal definition of a relationship between resource availability and demand?

I do not know how to answer it. But taking into account the conclusions above it seems plausible to say that there is a relationship between resource availability and quantity of demand (instead of demand): in circumstances of supply-induced scarcity the quantity of demand may increase.

What do you think?

I'm going to take a stab at providing the information that I think that you want, but I won't be answering the question that you asked. If this doesn't help you, consider changing the question as I don't think that we are necessarily understanding it.

TL;DR:  Resource availability and demand are mostly unrelated.  If anything, low resource availability will reduce demand (in the long term, not the short term).

Resource availability has no direct effect on the demand curve.  The demand curve reflects how people value the resource.

Availability determines the supply curve, which measures the cost of production.  I.e. at a specific quantity supplied, the supply curve will give a specific price.  That is often a marginal price.  That is to say, it is the price at which the last unit can be produced.

In general, in a market economy, the quantity supplied and the quantity demanded are the same.  And that value occurs when the demand and supply curves intersect.

I've been using a term, quantity demanded, with which you may not be familiar.  It may be that what I call quantity demanded is what you are thinking about when you say demand.  That happens sometimes.  Demand in economics is generally a graph of how the quantity demanded will change based on price.

In general, if a resource is constrained, this moves the supply curve (and may change its shape).  A constraint will move the supply curve to the left.  Since supply curves generally slope up and demand curves slope down (the higher the price, the more people will find a way to supply; the higher the price, less people will demand at that price), moving supply to the left will reduce the quantity supplied/demanded.

Meanwhile, there will be a bunch of people who would like to buy the resource but not at that price.  These also may be the people you think of as "increased demand."  From an economic perspective, this unmet demand is not measured beyond that.  It is the portion of the demand curve to the right of the quantity demanded.  That area will increase if the demand curve stays the same while the quantity demanded shifts left.  Most economists would not refer to this as increased demand, but you may be.

Now all that's the short term reaction.  In the long term, lack of a resource may cause demand for the resource to drop.

Consider gasoline.  Let's assume that it becomes constrained.  Initially, in the short term, my only real options are to pay more or drive less.  But in the long term, I can get a different car.  My new car may run on natural gas or electricity.  But if I do that, then the gasoline demand curve changes.  Not by a lot, as I'm only one person.  But if many other people do the same thing, collectively we shift the demand curve.  It would be reasonable then to say that lack of availability reduced demand in the long term.


How does stock market drop affect the economy if it doesn't affect the corporations? Sorry if it's a basic question.

Suppose corporation ABC went through public offering many years ago. ABC retained 30% of the shares, and 70% are being traded. The very initial offering, ABC made money. But thereafter, all the share trading and the associated money doesn't go to ABC, rather between the share holders.

How does a drop in share price due to trader fears (or an action ABC took) affect ABC itself? According to How can a company run out of cash when its stock prices is dropping? it does not affect the corporation.

  Stocks usually don't affect the immediate operation of a company, since they are traded on secondary markets (stock exchanges) among stock owners.

So how does it affect the overall economy?

Companies don’t issue stocks once and for all. It’s one of their sources of financing. Even if a majority of stocks is traded on secondary market companies still routinely issue new stocks when they are in need of cash.
Stock owners may feel less wealthy if the stock values fall and hence they might spend less or postpone their consumption. People who were just to cash out, for example due to retirement, might decide to postpone that and lower their consumption until the market recovers. This is negative demand shock and it’s bad for the economy.
There is also reverse relationship. A stock price is basically derived from the discounted expected value of future profits that the company will earn. If the expected future profits are lower due to some economy wide problems stock market will pick on that. The causation there is exactly opposite - it’s not stock market that affects the economy it’s economy that affects the stock market. Often when policy makers talk about improving the stock market they actually talk about policies that affect companies profitability not about policies that target sellers and buyers on secondary market.

The above list is not exhaustive but it should give you some idea why there might be some relationship between how the economy and how the stock market is doing.


Equilibria in non regular economies We know that in regular economies general equilibrium theory predicts a finite and odd number of equilibria, using the properties of the excess demand function and the index theorem.

How about the non-regular economies? From my understanding, these are the economies in which at least one price vector equilibirum generates a singular matrix of price effects. Geometrically, this can be interpreted as the excess demand function having a zero slope at one of the equilibria as shown in the picture below

Can we say something more about the equilibria, in case they are a finite number? Do they need to be even, odd or there is no restriction?

@Afreelunch gave you the right intuition. Non-regular economies will exhibit an infinite amount of equilibrium points. Intuitively since there is no local uniqueness, the continuum can be expressed as an interval on the real line so there is an infinite number of such points.


$x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$, then the preference is linear? $x,y,a$ are vectors in $\mathbb R^n$

We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero.

We want to prove (or disprove by counterexample) that:

Suppose $x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$,

Then the preference is linear.

One definition of linear preference is that $x\sim y$ implies $x+a\sim y+a$ for any $x,y,a$.

It is not true. Let us consider $\mathbb{R}^2$ so bundles are $x = (x_1,x_2)$.

Consider the preference:

(i) If $x_1 \leq 0$, preferences are lexicographic, i.e.
$$
x \succ y \Leftrightarrow
\begin{cases}
x_1 > y_1 \\
\text{ or } \\
x_1 = y_1 \text{ and } x_2 > y_2
\end{cases}
$$
(ii) If $x_1 \geq 0$, $u(x_1,x_2)=x_1+x_2$.

Notice that no indifference occurs on $\mathbb{R}_{<0} \times \mathbb{R}$, and the required condition holds on $\mathbb{R}_{\geq 0}\times \mathbb{R}$ since it is linear there.

However, $(0,4) \sim (2,2)$, choose $a = (-1,-1)$, we get
$$(2-a,2-a) = (1,1) \sim (0,2) \succ (-1,3) = (0-a,4-a)
$$
hence the second condition fails.


Problem with interpretation of exchange rate I found an article in which one of explanatory variable is "neer". From this article "The rise (decline) in this variable points to the appreciation
(depreciation) in excess of the appreciation of the Swiss franc."

Further we have "The positive sign of the coefficient means that the appreciation of domestic currencies vis-à-vis those of their trading partners relative (rise in neer) leads to higher price levels ceteris paribus."

So, I have several questions. Is neer local currency/swiss franc? How will look like the interpretation when we have swiss franc/local currency? Will in this situation the negative sign of the coefficient be?

NEER stands for nominal effective exchange rate and not the Swiss local currency. Essentially, this takes into account how the local currency moves with respect to its trading partners.

Example: if Switzerland trades only with the UK, an appreciation of the Swiss franc compared to USD would not make a difference to exporters/importers in Switzerland if GBP also appreciate against the USD by the same amount. In this case, NEER would remain the same.

Without knowing which article you're referring to (which "coefficient"), I can't say much about your other questions. In general, however, an appreciation in the currency, holding other currencies constant, will lead to an increase in NEER.


Elasticity of demand I'm reading  a paper from Pindyck (1988) where $$P_t=X_t-\gamma Q_t.$$ $P_t$ is the price,  $X_t$ is the demand state variable, $Q_t$ production and $\gamma\geq0$ is apparently the ''(constant) demand elasticity''. I'm not quite sure how to interpret this number (percentage change of what wrt what?)
According to wikipedia, I could consider
\begin{align*}
\text{Elasticity}&=\frac{\partial X_t}{\partial P_t}\frac{P_t}{X_t}=\frac{X_t-\gamma Q_t}{X_t}=1-\gamma\frac{Q_t}{X_t}\\
\text{Elasticity}&=\frac{\partial P_t}{\partial X_t}\frac{X_t}{P_t}=\frac{X_t}{X_t-\gamma Q_t}\\
\text{Elasticity}&=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}=-\frac{1}{\gamma}\frac{X_t-\gamma Q_t}{Q_t}=-\frac{1}{\gamma}\frac{X_t}{Q_t}+1 \\
\text{Elasticity}&=\frac{\partial P_t}{\partial Q_t}\frac{Q_t}{P_t}=\frac{-\gamma Q_t}{X_t-\gamma Q_t}=\frac{X_t}{\gamma Q_t-X_t}+1
\end{align*}
None of these expressions is really constant

When people talk about 'elasticity of demand' without further qualification they normally mean 'price elasticity of demand':
$$\text{E}_p=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$$
However, note your calculations are not entirely correct as in this case the elasticity should be:
$$\text{E}_p=-\frac{1}{\gamma}\frac{P_t}{(X_t-P_t)/ \gamma} = - \frac{P_t}{X_t-P_t}$$
However, the paper never claims $\gamma$ is elasticity and that the demand has constant elasticity in this case. Based on discussion in the comments which mentions that the claim was actually not made by the paper but by some student I assume it was a mistake (confusing slope of a demand with elasticity is actually surprisingly common among students).


Question related to conjoint analysis I am studying for an exam in marketing, and there's the following sample question:

The main reasons that people choose products in category X is: 1. Brand 2. Price. Where the importance ranking is 50% Brand and 50% Price. Brand A’s utility is 50 and Brand B’s utility is 30. What is the maximal pricing among the following of Brand B’s product so that it will be preferred by the market if the pricing of A is $80?

a: $54.9

b: $59.9

c: $64.9

d: $69.9

The correct answer is (b). How do they get to this number? My only idea of how to solve this would be to multiply $80 by 30/50 but that's not even an option. Thank you.

Here you have a possible answer (although I am not 100% sure about it):

You said Brand counts 50% and price 50%. We can consider that with higher brand utilities we get higher chance of buying, while with higher prices we get lower chance of buying the product. Following this idea, I made this equation in which both chances (of product A and B) should be equal, since it would lead us to an equilibrium in which it would be the same for the custumer to buy the product A or the product B:

$U_A - P_A = U_B - P_B$

We substitute according to our values:

$50 - 80 = 30 - x$

We simplify:

$-30 = 30 - x$

Again:

$-60 = -x$

And again:

$x = 60$

So 60\$ would be the price of indifference between product A and B. So we should make it a bit lower if we want people to buy product B: 59.9$, the correct answer of your problem.

I hope it helps.


Modelling Pay what you want I have been trying to model a Pay What you Want scheme. Now, for a monopoly it is clear that monopoly will have more profit by setting monopoly pricing assuming all players are selfish. But consider a scenario of 2 sellers one posting a fix price and other a PWYW strategy. In a market of selfish players the PWYW will make huge losses and market would move on to the fix price seller who can now act as a monopoly. So does this imply that there is an incentive for the selfish players to keep PWYW firm in market ? Also, how do I extend this idea to more than 2 firms ?

So does this imply that there is an incentive for the selfish players
  to keep PWYW firm in market ?

No, unfortunately not. In general equilibrium models (as @FooBar already mentioned) buyer's do not think about the influence of their actions on the seller. Therefore, they would just buy at the lowest possible price.

However, if you have a game theoretic model including uncertainty about profits of the PWYW firm and risk-averse consumers who prefer the PWYW firm for some reason you might get Nash-equilibria in which PWYW firms have positive profits.

Another way to model PWYW firms is to include non-monetary benefits from paying. For example a disutility for being seen as a cheapskate or for exploiting the firm or a positive utility from "doing the right thing" (i.e., paying the price that they think is appropriate). The selfish, rational players will pay a positive price and will generate profits for the firm.

I think in reality the impact of non-monetary benefits is much higher than the impact of asymmetric information with risk-aversion.

  Also, how do I extend this idea to more than 2 firms ?

I don't really think it is really interesting to extend this to more than two firms (especially without having solved this for 2 firms), but for both modeling options presented it should be "easy enough".


Financial Economics Textbooks I have an interest in financial economics, and I plan to take the graduate sequence, however I did not take an undergraduate course in that field. I would really appreciate it if someone could recommend a textbook for financial economics. I would prefer the book involved calculus, so I can get the fundamentals of problems (such as asset pricing) I may see in the future.

Two books which have been recommended to me are: "The Economics of Financial Markets" by Roy E. Bailey and "Principles of Financial Economics" by LeRoy and Werner. Are either of these worth reading?

The recommended books are decent. From these two I'd go with Bailey first and if you're comfortable with that, then LeRoy & Werner. The latter requires some background in linear algebra and optimization theory.

If you want to study some econometric applications for financial economics, you might try:

Cuthbertson & Nitzsche: Quantitative Financial Economics: Stocks, Bonds and Foreign Exchange
Campbell, Lo & MacKinlay: The Econometrics of Financial Markets


If I gain, then someone else loses. Correct? On a very small scale, it's certainly true that if I gain, somebody else might lose. If I take away my brother's chocolate, then he will lose it, and will most probably not get anything comparable.

But on greater scale, say, nationally, if one person (e.g. successful start-up founder) makes a fortune, will this generally be bad for the other players? Or can it be beneficial (e.g. if the money is not saved up)? Does it entirely depend on the rich person's spending behavior?

I completely agree with denesp's answer, however I think you can make it even simpler.

  On a very small scale, it's certainly true that if I gain, somebody
  else might lose. If I take away my brother's chocolate, then he will
  lose it, and will most probably not get anything comparable.

OK, let's say I prefer chocolate to wine gums and my brother likes wine gums better than chocolate. Then taking his chocolate away and giving him my wine gums is good for both of us, so we both win and no one loses. So the answer is no.

You could even consider the extreme case in which your brother hates chocolate and you are doing him a favor by taking it. (Works not as well with chocolate, but you might think of recycling.)

In general these "trades" are called Pareto improvements.

But this is only one example, if you are interested in the subject, you might be interested in one of the following basic economic ideas:

Trade between two countries in which one of them is more efficient than the other: Ricardo's comparative advantage example
Your brother likes to give: altruism / warm glow (I really don't like the wiki page here, but was not able to find a decent non-scientific explanation of it.)
Or maybe other other-regarding preferences, for example fairness (your brother has a lot of chocolate and feels better if he gives you some): Theories of Fairness and Reciprocity (On page 3 is a brief "Non-technical summary" which might be interesting.)

As you can see there are many examples for a "win-win" situation and there are many many others, depending on the situation.


What are the economic arguments against (a) variolation (analogue) as a solution for Covid-19? It has been suggested in some parts of the (usually right-wing) press that a solution to Covid-19, especially in economics terms, is to expose the young to the virus sooner, e.g.:

Perednia [...] argued for infecting young people with the virus via a “controlled voluntary infection program.” (The infecting would be followed by quarantine until immunity could be confirmed.) [...]
Perednia quickly got to the crux of the matter, as he sees it. “Either self-isolate and do what we’re doing now—which will bankrupt the next generation,” he said, or, “there is a program that the government operates, which, if you decide it’s appropriate for you, you can become infected voluntarily with the disease.” [...]
Perednia was convinced that the way to adapt his idea to this reality was to make sure that the infecting was done with “the lowest possible dose” of the virus—a concept known as variolation—which, he thought, would cut the death rate among those who chose to take part. “Even without the addition of variolation,” he said in an e-mail, “the whole idea of CVI now appears to be ten times less risky than originally appeared to be the case.” Meanwhile, the C.D.C. continues to recommend social distancing, wearing masks, and washing hands often.

Note that the main argument for CVI/variolation of the young approach to Covid-19 is an economic one, namely that the alternative supposedly is to "bankrupt the next generation". So, how have economists reacted to this type of CVI/variolation proposal?

Although some models would surely be better than mere qualitative [counter]arguments, Tyler Cowen, who teaches economics at George Mason University, put forward some counter-arguments, although his arguments are against a more specific proposal of Robin Hanson (another GMU economist) who proposed creating "Hero Hotels" where [young] people would get "variolated" with Covid-19. Some of Cowen's [counter]arguments (I'm skipping some):

Qualified medical personnel are remarkably scarce right now.  I do not see how it is possible to oversee the variolation of more than a small number of individuals.  [...]

The NBA has an amazing amount of money, on-staff doctors, the ability to afford tests, and so on.  And with hundreds of millions or billions of dollars at stake they still won’t restart a crowdless, TV-only season.  They could indeed run a “Heroes Hotel” for players who got infected from training and play, and yet they won’t.  “Stadium and locker room as Heroes Hotel” is failing the market test.  Similarly, colleges and universities have a lot at stake, but they are not rushing to volunteer their dorms for this purpose, even if it might boost their tuition revenue if it went as planned (which is not my prediction, to be clear).

Does small/marginal amounts of variolation do much good compared to simply a weaker lockdown enforcement for activities that involve the young disproportionately?  Just tell the local police not to crack down on those soccer games out in the park (NB: I am not recommending this, rather it is the more practical version of [Hero Hotels]; both in my view are bad ideas.)

Society can only absorb a small number of very blunt messages from its leaders.  You can’t have the President saying “this is terrible and you all must hide” and “we’re going to expose our young” and expect any kind of coherent response.  People are already confused enough from mixed messages from leaders such as presidents and governors.

(Some of Cowen's later arguments are more into the realm of behavioral economics.)
Cowen is actually somewhat sympathetic to the idea of sorting who is (most) exposed to the risk, but he concludes by saying that

I don’t think Heroes Hotel, and accelerating the rate of deliberate, publicly-intended infection, is the way to a better solution.

It would be more interesting (of course) if these competing ideas/intuitions (Hanson's, Cowen's) were more formally modelled.
I suppose I could add that China did run fairly large centres when they quarantined everyone they thought were infected (though not infected on purpose), the so-called "Fangcang hospitals". There were up to 16 such facilities opened at one point in Wuhan, with about 10,000 people in them at the peak, but reportedly these have all be closed now.
Hanson doesn't seem to make any quantitative estimates how many "hero hotel" rooms would be needed (e.g. per capita) to achieve some concrete effect in reducing the economic downturn, although he does advance a scenario in relative population risk terms as:

For example, in a simulation where old and sick people are selected for isolation, while only the young and healthy are eligible for variolation, there are 40% fewer life years lost, compared to no variolation and random selection for isolation. Each variolation volunteer suffers only an additional 0.20% chance of death to save a random other person from a 6.5% chance.

Alas, he doesn't seem to (even) quantify the (relative) effects of variolation "on top" of non-random isolation of the old (which is a more plausible alternative than random isolation, which Hanson uses as the yardstick.)
I see Hanson has (also) written a reply/rebuttal to Cowen's points. I won't get too much into that here because Hanson e.g. says

If variolation cuts death rates by a factor of 5 (median estimate in this poll) [...]

Where the "poll" is a Twitter poll Hanson ran rather than some medical expert poll (at least). So the arguments seem based on rather dubious/uncertain parameters... which is what I suppose also precludes more serious research/publications on this proposal.


Perfect competition questions I have problems with deciding whether those statements are true or false.

In a short/long run price of equilibrium on a perfectly compettive market might be either lower or higher then average total cost of one company operating on this market.
In a short run total amount of companies in perfectly competitive branch of industry, changes when their economic gains are equal to zero.

Appreciate your help with some further comments about that.

PS. anticipating your questions i am preparing to an exam, that is why i am looking for an answer to those questions.

I'm not going to answer the questions for you, but here is some food for thought, you should be able to conclude yourself.

Ad 1

We know that in the long run equilibrium, there are zero profits: If they'd be positive, firms would enter, if they'd be negative, firms would leave. Can we conclude something about a firm's profits if it has average total costs smaller (larger) than the price?

Ad 2

What induces change in the total amount of companies in such a branch? Is having zero-profits one of these reasons?


What lies behind supply and demand model? This is a very basic question. I can understand that price of goods and stocks change follows the supply and demand law. On the other hand, I find it difficult to understand what happens in the real world.

If I want to buy a particular stock I can buy it at the daily price and I can sell it at the daily price. I can't choose to sell it at a higher price as I would do if I had the only orange in the world and 1 million people wanted that orange.

Who has the ability to start selling a specific stock at a different price? Who initiate the change in price if the demand goes down? Has anyone a practical simple example for me?

The “law of supply and demand” is a feature of economic models with supply and demand curves. These are usually mainstream equilibrium models. It is somewhat difficult to relate these models to real world behaviour. I am in a theoretical camp that is critical of such models; perhaps someone else can give a better account.

As for your questions about stock prices, you probably would need to look at the mechanics of stock trading. Investors post prices at which they are willing to buy or sell, and then others will transact at those prices. (In some cases, market makers are the ones posting the price.)

Those participants can change their posted prices whenever they want, typically in response to news. The prices can move without a single share trading hands. This can be interpreted as a shift in the supply or demand curve.

If someone has a big order to buy or sell, they can clear out the first set of posted prices, and transact against other orders. This will result in a price change. For example, assume there are orders to buy 100 shares at \$10 and \$5. An order to sell 200 shares would first clear the best bid (\$10) and then hit the next bid (\$5). In this way, big orders can change prices.

A longer version of this example.

There are no existing orders for shares in a company.
Investor A puts in a bid to buy 100 shares for \$5. The best bid (highest bid) is now \$5.
Investor B puts in an order to buy 100 shares for \$10. The best bid is now \$10. The bid price solely as the result of the new order; there was no transaction that moved the price.
Investor C wants to sell, and puts in an order to sell 200 shares. The first 100 shares clears B’s bid at \$10, and the second 100 clears A’s bid at \$5. The prices at which transactions happened was the result of an order clearing out the existing order.


Why is macroeconomics studied through "Shocks"? "Why is modern macroeconomics all about identifying this shock and that shock?"

Coming from a theoretical econometrics background, I have recently delved into empirical macro literature. I am a bit confused about the meaning and significance of the notion of shock. Why we are trying to see the macro-world through the lens of shocks? What is the nature of these shocks that gives them this much explanatory power?

The only explanation I could come up with is that this obsession with shocks is a result of trying to fit macro problems into the framework of Wold decomposition theorem. Any covariance stationary process can be decomposed into a deterministic part and an $MA(\infty)$ part. The deterministic part fits well with the idea of steady-state of the endogenous variables, while the stochastic part gives rise to the idea of seeing the macro variables fluctuation as a result of MA shocks.

What is confusing to me is that it seems macroeconomists are trying to find "real" meaning for this merely-statistical notion of shock. The impulse response function of tax shock, government expenditure shock, etc., are how they see the world.

What am I missing?

Im assuming you're discussing some of the models associated with Real Business Cycle Theory.

These models have a structural underpinnings (meaning that there is some assumptions/empirics with regard to consumer behavior). Starting with the understanding that consumers are also suppliers of labor which face a intertemporal consumption-leisure trade off and Firms are profit maximizing and use capital,labor and technology to produce.Using this setup we can do some standard calculations for what equilibrium and Balanced Growth Paths would be.

Now this as it stands is not a theory of fluctuations. The way Fluctuations are accounted for in this model is VIA exogenous shocks that enter through Technology,A overall change in aggregate preferences and increases in interest rate ect. This enters in the error term of all our equations.

This is the justification for using Impulse response functions, Its to analyse what the implications are for surprise changes in our variables in terms of consumer and producer behavior.

TL;DR:
The main point of this analytical process of shocking is to be cognizant for how economic shocks enter the economy and what the implications are for consumer and producer behavior. This is based on the structure of that is imposed by economic theory.


Dynamic programming, optimal consumption-savings (finite horizon) problem Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$,
the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ at time $t+1$ are determined by a random process in which $w_{t+1}=\alpha w_t$ or $w_{t+1}=\beta w_t$, where $\alpha,\beta$ are postitive constants, each with probability $\dfrac{1}{2}$ and independent of past rates of return on her savings. At time $t$, when she is choosing $c_t$, she knows $w_t$ but she doesn’t know
anything about future returns except the probabilistic law just given; she does know her past consumption decisions, of course, and past rates of return. Considering a log utility, i.e. $u(t)=ln(c_t)$ and assuming that the consumer is impatient, so that he discounts future utility by a factor $b$ each period, where $0 < b < 1$, then the consumer's decision problem can be written as follows:
$$v(c_t)=\max_{c_t \geqslant 0}\sum_{t=0}^{T}b^{t}u(c_t)$$

If the consumer can not borrow, then what is the optimal consumption level for the consumer?

$\underline{Note:}$ The problem is based on David M. Kreps' microeconomic theory book, but it is adjusted to be a finite horizon problem. Kreps, in his book, solves this problem in a fuzzy way for $T=3$, that is not obvious to me. However, due to the fact that I know little of dynamic programming, since I am in the begining of examining this topic, I would appreciate it, if someone could provide a solution in the finite horizon. If the problem is not set in a rigorous way, I would be glad too to see somene making any appropriate change. I beleive that it is a classic problem in the field of economics.

Your value function is as follows:
$$
V_t[w] = \max_{c_t \in[0,w]} \left\{u(c_t) + \frac{1}{2}V_{t+1}[\alpha(w_t - c_t)] + \frac{1}{2}V_{t+1}[\beta(w_t-c_t)] \right\}
$$
with the terminal condition
$$
V_{T}[w_T] = \max_{c_T \in [0,w_T]} u(c_T)
$$

So, we can solve this via backward induction. Clearly, at the final period $T$, since $u$ is monotonic, we consume everything, so $V_T[w_T] = u(w_T) = \ln(w_T)$.

Let us now consider one period previous, so period $T-1$. The value function is:
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{u(c_{T-1}) + \frac{1}{2}V_{T}[\alpha(w_{T-1} - c_{T-1})] + \frac{1}{2}V_{T}[\beta(w_{T-1}-c_{T-1})] \right\}
$$
We already know what $V_T[\cdot]$ is, so substituting,
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{u(c_{T-1}) + \frac{1}{2}u(\alpha(w_{T-1} - c_{T-1})) + \frac{1}{2}u(\beta(w_{T-1}-c_{T-1})) \right\}
$$
considering the case $u(\cdot) = \ln(\cdot)$,
$$
V_{T-1}[w] = \max_{c_{T-1} \in [0,w_{T-1}]} \left\{\ln(c_{T-1}) + \frac{1}{2}\ln(\alpha(w_{T-1} - c_{T-1})) + \frac{1}{2}\ln(\beta(w_{T-1}-c_{T-1})) \right\}
$$
Take first order conditions, and we can get optimal $c_{T-1}^*(w_{T-1})$.

Then we have solved what $V_{T-1}[\cdot]$ is! We can then do the same procedure by considering $V_{T-2}$. Repeat until we get to $V_0$.

EDIT to reflect comment.
You now know that $c^*_{T−1}(w_{−1})=\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}$.
Plugging back into $V_{T-1}$, we have solved for the value of $V_{T-1}$,
\begin{align}
V_{T-1}[w] &= \ln(\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}) + \frac{1}{2}\ln(\alpha(w_{T-1} - \frac{\alpha\beta}{1 + \alpha \beta}w_{T-1})) + \frac{1}{2}\ln(\beta(w_{T-1}-\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1})) \\
&=\ln(\frac{\alpha\beta}{1 + \alpha \beta}w_{T-1}) + \frac{1}{2}\ln(\alpha(\frac{w_{T-1}}{1 + \alpha \beta})) + \frac{1}{2}\ln(\beta(\frac{w_{T-1}}{1 + \alpha \beta}))
\end{align}

Finally, lets now go to $T-2$. The value function is
$$
V_{T-2}[w] = \max_{c_{T-2} \in[0,w_{T-2}]} \left\{\ln(c_{T-2}) + \frac{1}{2}V_{T-1}[\alpha(w_{T-2} - c_{T-2})] + \frac{1}{2}V_{T-1}[\beta(w_{T-2}-c_{T-2})] \right\}
$$
We just solved what $V_{T-1}$ is! Plug it in, and repeat.

This expression will probably blow up quite quickly, so it will be a pain to solve analytically.


The Price and Demand Index in Homothetic Kimball Utility Suppose with Kimball preferences, utility $Q$ from consuming $\left\{q_{\omega}\right\}_{\omega \in \Omega}$ is implicitly given by $$\int_{\omega \in \Omega} Y\left(\frac{q_{\omega}}{Q}\right) d \omega=1$$. The consumer thus maximizes $Q$ subject to both budget constraint $\int_{\omega \in \Omega} p_{\omega} q_{\omega} d \omega=y$ and this implicit function.
By solving this problem, we can get $$\lambda \frac{p_{\omega}}{\delta}=Y^{\prime}\left(\frac{q_{\omega}}{Q}\right)$$ and thus the demand for each variety $$q_{\omega}=Q Y^{\prime-1}(\frac{\lambda p_{\omega}}{\delta})$$, where $\lambda$ is the Lagrange multiplier for the budget constraint and $\frac{1}{{\delta}}=\int_{\omega \in \Omega} Y^{\prime}\left(\frac{q_{\omega}}{Q}\right) \frac{q_{\omega}}{Q} d \omega$.
What I don't understand is that the papers using this preference often set price index $P \equiv \delta /\lambda$ and denote $\delta$ as a demand index. Whereas in CES case, the price index is $1/\lambda$. And I manipulate the FOCs in this Kimball case and find that $Q/\lambda=y$, thus it would make sense if set $P \equiv 1/\lambda$. So why set $P \equiv \delta /\lambda$? Moreover, why is the $\delta$ term a demand index? In particular, given the argument in this answer, it seems that $\delta$ is exactly the implicit function and thus should be 1, which makes me confused.
For some reference, in a recent wp by Baqaee & Farhi ("The Darwinian Returns to Scale") they discuss this issue: "this price index $P$ does not coincide with the ideal price index $P^{Y}$ for the representative consumer: deflating income by $P$ does not yield welfare. In fact, $d \log P=d \log \bar{\delta}+d \log P^{Y}$. Since in general, $\bar{\delta}$ is not a constant, $d \log P \neq d \log P^{Y}$. It is only in the CES two price indices coincide since then $\bar{\delta}=\sigma /(\sigma-1)$ is constant. Going forward, we refer to P as “the” price index without further qualification, despite the fact that it is not the ideal price index.".
Update: By reading Baqaee & Farhi I get the idea that this manipulation of price index offers a tidy expression of relative demand and relative price. However, I still don't fully understand what the role that $1/\delta = Q/\gamma$ plays in this revised demand function, and how as a result the Kimball utility differs from the usual CES case. Most importantly, I guess this is related to what Baqaee & Farhi argues that "Monopolistic competition models with Kimball demand are parsimonious in the sense that competition across varieties is mediated by the price index; every variety competes with the price index. The flexibility of the demand system comes from the fact that it allows different varieties to face different “intensities” of competition, as captured by the different elasticities of the demand curve at different points." This seems to be the essence of the Kimball demand, but I fail to understand how is "competition across varieties is mediated by the price index; every variety competes with the price index" and why this is not the case in the CES?

The problem is the following:
$$
\begin{align*}
\max_{Q, q_\omega} Q \text{ s.t. } &\int Y\left(q_\omega/Q\right) d\omega = 1,\\
&\int p_\omega q_\omega d\omega = y
\end{align*}
$$
The first order conditions give:
$$
1 - \gamma \int Y'\left(\frac{q_\omega}{Q}\right) \frac{q_{\omega}}{Q^2} d \omega = 0,\\
 \gamma Y'\left(\frac{q_\omega}{Q}\right)\frac{1}{Q} - \lambda p_\omega = 0
$$
Then:
$$
Y' \left(\frac{q_\omega}{Q}\right) = \lambda \frac{p_\omega}{\gamma/Q}
$$
So we have that $\delta = \gamma/Q$.

What I don't understand is that the papers using this preference often set price index $P = \delta/\lambda$ and denote $\delta$ as a demand index. Whereas in CES case, the price index is $1/\lambda$.

I think the answer is already given by Baquee and Farhi themselves.
If we define $P = \gamma/(Q \lambda) = \delta/\lambda$ then we can write:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{p_\omega}{P}\right)
$$
Using $Q$ as a quantity index, we have on the left hand side the "normalized" for good $\omega$ and on the right we have a function of the "normalized" price. Under this definition, you can therefore interpret $Y'^{-1}$ as a demand function.
On the other hand, if you would use $P = 1/\lambda$  you obtain from inverting $Y'$ in the first order condition that:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{Q}{\gamma} \frac{p_\omega}{P}\right).
$$
The right hand side now also involves the quantity index $Q$, so this does not take the "form" of a demand function.
As a conclusion, I think the main idea of defining $P$ in this way is to obtain a nice expression and interpretation for the normalized demands as a function of some normalized prices.


Find Change in output from marginal products of labor/capital A firm produces 231 doohickeys with 8.4 units of labour and 22.1 units of capital. the marginal product of labour is 18, the marginal product of capital is 20. Approximately how many doohickeys will it produce with 8.6 units of labour and 22 units of capital?

Is this just as simple as taking the difference between the new and original values and multiplying that by the marginal product?

i.e. (8.6-8.4) x 18 + (22-22.1) x 20 = 1.6

so they will produce approximately 232.6 doohickeys

Or am i supposed to take the integral of the marginal products and do something with those? i tried that but I couldnt get any where

I think your solution is golden.

Taking the integral of the marginal products could only work if the marginal products were constant everywhere. This is not mentioned in the text, and in fact if you assume it to be constant then production given this level of $K$ and $L$ would not be 231.

Another interesting tidbit:
 If you knew the marginal product function of $L$ and also that of $K$ taking the integrals and summing them up would only work if $MPL$ is constant in $K$ and vice-versa.


The equation for the firm's demand for labour I have been stuck on this for a few hours (actually a few days). I can't find much useful information on the web that has made it any clearer.

Suppose that a firm’s production function of output Q is a function of two inputs, labour (L) and capital (K) and can be written Q = 25LK.

Letting the wage rate for labour be w and the price of capital be r, what is the equation for the firm’s demand for labour?

I understand that the answer is L=√(rQ/25w) but I don't know how to get there.

You could set up the profit function as suggested by @Kitsune , but here I take the expenditure minimization approach (which is mathematically equivalent).

Set up the minimization:
$$\min_{L,K} wL + rK \\
s.t. \quad 25LK=Q$$

Using the Lagrange method:
$$\mathcal{L}=(-wL - rK) + \lambda (25LK-Q)$$
Giving us the First-Order Conditions:
$$(1)\quad \frac{\partial\mathcal{L}}{\partial L}=-w + \lambda 25K=0$$
$$(2) \quad \frac{\partial\mathcal{L}}{\partial Q}=-r + \lambda 25L=0$$
$$(3) \quad \frac{\partial\mathcal{L}}{\partial \lambda}=25LK-Q=0$$

Taking (1) and (2):
$$(4) \quad K=\frac{w}{r}L$$
From (3):
$$(5) \quad K=\frac{Q}{25L}$$

From (4) and (5):
$$\frac{w}{r}L=\frac{Q}{25L}$$

Solving for L:
$$L=\sqrt{\frac{rQ}{25w}}$$


How to derive the reaction function from these provided variables? This is the question. I cant seem to reach a concrete solution because of the scarce information.

Firm 1 spends 2 million, plus 33.33% on advertising what firm 2 spends on and firm 2 spends $7 million plus 12.5% of what firm 1 spends. Draw the reaction functions and label them from the provided information.

I want to know the exact procedure on such problems.

The reaction function tells you how a player in a game reacts to the strategy of another player.

As noted by denesp in the commentary, the statements provided in the question are verbal descriptions of reaction functions. The mathematical description (in millions) is given as follows:

Firm 1:
$x_1= 2 + (1/3) x_2$

Firm 2:
$x_2= 7 + (1/8) x_1$

There is not really any procedure to it, besides rewriting the verbal description into a mathematical description.


Finding the % of the total consummation by category using elasticity and MPC Given a total consummation of 1000 and a total revenue of 2000 with

Cat 1 => elasticity of 0.5; MPC of 0.1

Cat 2 => elasticity of 1, MPC of 0.3

How would I go about to calculating the % of the total revenue for each category?

I tried using both these formulas

and tried to figure out the first category by assuming the variation of revenue to be 2000 (2000-0) and the initial consummation to be 0.

which gives me a value of 500 for c2. But then, using the MPC formula, it does not seem to make any sense

Notice that the elasticity can be written as

$$\frac{\Delta C}{\Delta Y}\cdot\frac{Y}{C}$$

So for category $1$ you have $$\frac{\Delta C_1}{\Delta Y}\cdot\frac{Y}{C_1}=0.5$$ and $$MPC_1=\frac{\Delta C_1}{\Delta Y}=0.1$$

This gives you
$$0.1\cdot\frac{Y}{C_1}=0.5$$
which by reordering yields to
$$\frac{C_1}{Y}=\frac{1}5$$

Similarly $$\frac{C_2}{Y}=\frac{3}{10}$$

Finally notice that $C=C_1+C_2=1000$ and $Y=2000$, hence we must have that $\frac{C}{Y}=\frac{C_1+C_2}{Y}=0.5$ which is satisfied since
$$\frac{1}5+\frac{3}{10}=0.5$$

(Since we have that $2C=Y$ and $100\%$ of revenue comes from $C$ is straightforward to see that the $\%$ of revenue associated with each category are twice this numbers)


Fixed-Effects on individual level with explanatory variable on state level I am planing a panel data analysis evaluating a continuous policy intervention. I do have individual level data. I would like to account for individual fixed-effects. However, the explanatory variable does not vary on individual level, but only on state level. The response variable is on individual level. Is that a problem?

If you want to do unit-fixed effects (so eliminating any unobserved variable that varies cross-sectionally but remains the same over time), this should not be a problem, at least assuming that your explanatory variable varies over time (be it a dummy that is 1 in the year the policy was administered and 0 otherwise, or a continuous variable that changes over time according to the intensity of the policy treatment).

It does not matter that the variable is on state level, as long as the policy varies over time. However, if you only have individual observations for the one state that introduced the policy, you will not be able to interpret your results counterfactually. For that you would need similar individuals observed over the same period of time in a state that did not introduce the policy (and apply difference-in-difference or something of the sort).


What are some applications of Real Analysis in Graduate Economics? I am interested as to what areas of masters/PhD coursework that learning the fundamentals of Real Analysis would be beneficial for?

I am aware of its applications in Econometrics proofs and analysis, but what of mathematical economics, microeconomics, and macroeconomics that one would run into during either graduate degree?

Resources to consult or links to relevant papers would be appreciated!

My favorite example of real analysis' application in micro is the proof that lexicographic preference cannot be represented by any utility function. (A succinct version of the proof is given in p.43 of MWG.) It uses the notion of cardinality of sets, in particular, the different cardinalities between the rationals and the reals, which are some of the first things one learns in a real analysis course.

Other materials in real analysis, such as set operations, closedness, openness, continuity, differentiability, boundedness, compactness, etc., find their ways into every nook and cranny of economic theory. Some examples include:

Classical demand theory, which deals mostly with $\mathbb R^n$, the $n$-dimensional commodity space, draws on a lot results from real analysis
Any theoretical work that involves probability, e.g. econometrics as well as risk/uncertainty in micro or macro, would rely on some basic measure theory, an advanced topic in real analysis
Dynamic programming/optimal control, a technique for solving DSGE (and other) models, relies on the contraction mapping theorem
Separating hyperplane theorem is used to prove the existence of general equilibrium
Brouwer's fixed point theorem is used to prove the existence of Nash equilibrium

See some of the book recommendations and related questions on this site (1, 2, 3, 4) for more examples.


Sidrauski Monetary Model and Neutrality In the Sidrausk model, money is in the long run super neutral.
Does this result only apply to the case where money and consumption are separable in the utility function, or also to the case where they are non separable.

The Sidrauski model is equivalent to a model with 2 consumption goods where the second good (money) is produced costlessly by the government. Inflation is equivalent to raising the price of the second good. Money is super-neutral if this does not affect the real allocation (demand for the first good). This only happens when preferences are separable.