How dominant assurance contract fix free rider problem? Dominant assurance contract is a proposed solution to free rider problem in crowdfunding. It pays more money to payers (for example 105%) in case of failure of campaign. Is it make paying a dominant strategy? I think not.

Suppose a city need a road that make 100\$ benefit for each person. It costs 8000\$ and entrepreneur offers 5\$ in case of failure for every 80\$.

normal strategy (paying 80\$):

in case of failure: 5\$ benefit
in case of success: 20\$ benefit

free rider strategy:

in case of failure: 0\$ benefit
in case of success: 100\$ benefit

That shows a free rider can risk on the success probability (that helping does not affect it so much) and make a higher profit. Where I am wrong? Is there any additional hypothesis that dominant assurance need to work?

It was established in the comments that the source of this claim is Tabarrok: The private provision of public goods via dominant assurance
contracts via Wikipedia.

Tabarrok writes

  assume that a plain is flooding and that a dike must be built to hold back the river. It
  is known with certainty that the flood will be averted if the dike reaches 6ft.
  If the dike doesn’t reach 6ft then the river overflows and the builder’s efforts
  are wasted. Assume that the agent’s choice is binary, he can help to build the
  dike or not.  [...]
  note that the game really consists of two distinct elements [...] Consider column one; the problem in this column is not free
  riding. [...] The problem in column one is that the agent doesn’t want to waste his effort
  in building a dike which won’t hold the river. Each agent lacks “assurance”
  that the others will contribute in the event that he contributes.
  The problem in column 2 is quite different. In column two each agent
  assumes that the public good will be provided regardless of his actions. Each
  agent, therefore, wishes to “free ride” on the others.  [...]
  The assurance problem can be overcome via an assurance contract.

[emphasis by me]

So assurance contracts are not claimed to overcome the free riding problem.


Limit of utility function in Ramsey-Cass-Koopmans model I have given the utility function for the Ramsey-Cass-Koopmans model, as follows:
$$ u(c_t)=\frac{c^{1-\theta}-1}{1-\theta} $$
The claim is that as $\theta \rightarrow 1$, then the funciton is logarithmic, that is
$\lim_{\theta\to 1} u(c) = ln(c)$
I was told to use L'hôpital's rule, but I can't figure it out.
How can I prove this?
Thank you

The answer of @tdm shows you how it is done using De L'Hospital's rule. If you want to avoid this rule you can write $u(c)=\frac{c^{1-\theta}-1}{1-\theta}$ and note that for $\theta\ne 1$ this implies $u(1)=0$. Differentiating with respect to $c$ then gives you $u'(c)=c^{-\theta}$ and therefore $\lim_{\theta \to 1}u'(c)=c^{-1}$. Then integrate again to get $\lim_{\theta \to 1}u(c)=\ln(c)+K$ and comparing for $c=1$ gives you $K=0$, so $\lim_{\theta \to 1}u(c)=\ln(c)$.


Why is the Marginal Utility of losses deminishing in Prospect Theory? This is Kahneman's value-plot on prospect theory:

QUESTION: Why is the Marginal Utility of losses deminishing?

CONTEXT:

I fully understand that the Marginal Utility of gains deminishes: 100 dollar gains has less utility to me when I am rich than when I am poor.
But to me it is different with losses: If I can choose to lose 100 dollar for sure or a 50% chance to lose 200 dollar I would definately not go for the bet! Why would I risk losing even more? After all, like most people, I am risk-averse. My reference point would be that I already lost 100 euro's.
Furthermore: If all I own should be 200 dollar I might regret losing 100 dollar very much, but I will regret losing a second 100 dollar even more because then I would be left with nothing and I can't buy any food...
In other words: I guess my reference point is always zero and compared to that I will always be risk-averse... but isn't that what most people do or should do?

UPDATE: This article shows that the Value-curve can also be shaped differently, for example under time-pressure: https://www.sciencedirect.com/science/article/pii/S0749597815000722

The value function used in Kahneman's prospect theory (which your plot shows) is supposed to capture empirically observed behavior of people's attitudes towards gains/losses as well as to risks in those domains.

In the domain for gains, people are usually risk averse. However, in the domain for losses, people do tend to take larger risks. A nice illustration is the observation that when a gambler is losing money (relative to the amount he came to the casino with), he tends to wager on higher stakes (larger risks), perhaps in an attempt to "win back" the losses.

So the convex curvature in the domain of losses is meant to capture this type of behavior.

In fact, prospect theory's treatment of people's risk attitude is even more nuanced than the above. In addition to the S-shaped value function you showed, it also has a component that allows for non-linear probability weighting. Together, the value function and non-linear probability weighting generate the following four-fold classification of risk attitudes:
\begin{array}{c|cc}
&\text{gains}&\text{losses}\\\hline
\text{low probability}&\text{risk loving}&\text{risk averse}\\
\text{high probability}&\text{risk averse}&\text{risk loving}
\end{array}

See Kahneman and Tversky (1992) for more detail.


Would a Weather and Climate Change class be helpful to peruse a career in environmental economics I’m an undergraduate student and want to become an environmental economist. Would it be beneficial to take a class on weather and climate change or should I stick with economics courses geared towards the environment?

It wouldn't be detrimental by any means, but I would regard it as non-essential.

non-essential, because you'll likely use quite a simplified (as opposed to simple) framework for thinking about interactions between the economy and the environment. See for example Golosov et al. (2011) https://www.nber.org/papers/w17348

and non-essential because the larger models with more complex interactions between the economy and the climate/weather systems, would normally be conducted alongside some climate scientists. The division of labour here is arguably preferable, given that climate scientists and economists bring their own respective strengths to the table. I believe this to be the case in some of the integrated assessment models (IAMs) such as RICE/DICE etc.

But overall, if you believe it to be beneficial to your cause; I don't think it would be detrimental. Certainly it would enrich any modelling you'd consider doing.


Is a lack of any barriers to entry a necessary condition for ensuring pareto optimality is reached? I understand that a competitive market is necessary for a market to equilibriate to a pareto optimal state, and that things like economies of scale, network effects, government restrictions, and government imposed costs can prevent reaching pareto optimality. But do other barriers like startup capital also prevent reaching pareto optimality? I would think the answer is no, since something like sunk costs can be amortized over all products sold by a market actor. Is this correct, or am I missing something?

Let us quickly define a competitive market. A competitive market satisfies the following:

1) All firms sell an identical product

2) All firms and buyers are price takers (have no control over price)

3) Buyers and firms have complete information about the market

4) The market is characterized by free entry (no barriers to entry)

If this is the market structure, we can have the existence of pareto optimality, so I am assuming this is the competitive market to which you are referring in your question. What happens when we try to relax assumption 4?

We can show this using your example of startup capital:

Suppose we have a competitive market and the amount of startup capital needed to enter this market is $A$. When deciding to enter this market the firm would subtract $A$ from the long term profit possibility $\Pi$ (because it would be another cost). By definition $A\geq 0$. Because the situation where $A=0$ is trivial (because that means there is no startup capital needed), we know that $A>0$. Suppose at a point in time there is opportunity for profit in this market. As long as $\Pi\geq 0$ firms will decide to enter the market, but if $\Pi <0$, a firm would decide not to enter the market. However, note that firms who are already in the market are earning positive profit. This is a contradiction because if positive profits persist, we do not have a competitive market.

As you noted in your question, a competitive market is a necessary condition for the existence of pareto optimality. If the nonexistence of barriers to entry is a necessary condition for a competitive market, nonexistence of barriers to entry is also a necessary condition for pareto optimality.


Econometrics - Why does adding a dummy varaible improve my Ramsey RESET test result? As I understand, the Ramsey RESET test (although called ovtest on Stata), is not actually a general test for omitted variable bias. Rather, it is a test for misspecification. Specifically, if the model is properly specified, "no nonlinear functions of the independent variables should be significant when added to the estimated equation". So now I'm confused because after estimating three models, I get these results.

So according to this, the null of no omitted variables (or no misspecification) will be rejected for the first two but not the last (at 5% sig. level). Yet, the translog is essentially the log-log with higher powers of the independent variable, so I'm confused as to what to conclude from this. I'm inclined to say that the dummy variable was an important omitted variable, but then again RESET is not a general test for OVB.

Perhaps also worth mentioning is that I am assuming I can conduct the ovtest as a test for endogeneity to check the OLS assumptions (because this is the only relevant test we were taught that isn't the 2SLS/Wu-Hausman).

As I understand, the Ramsey RESET test ... is not actually a general test for omitted variable bias. Rather, it is a test for misspecification. Specifically, if the model is properly specified, "no nonlinear functions of the independent variables should be significant when added to the estimated equation"

This is completely correct, the idea of the RESET test is that if you have following some model:

$$ y = \beta_0 + \beta_1 x_1 + ... \beta_k x_k + e_i$$

The RESET test will test for correct functional specification by estimating auxiliary regression:

$$ y = \beta_0 + \beta_1 x_1 + ... \beta_k x_k + \gamma_1 \hat{y}^2 + \gamma_2 \hat{y}^3 +  e_i$$

and then testing with an $F$ test whether $\gamma_1$ or $\gamma_2$ are statistically significant. Hence, what the test actually does under the hood is to check whether the squared predicted values and cubed$^1$ predicted values can still explain some variation in $y$ after accounting for all the independent variables.

This can only be the case if the original model did not already explained $y$ fully as the original model is still included in the RESET test and hence the RESET test tells you there is some functional miss-specification. The intuition is that if non-linear combinations of the explanatory variables (through $\hat{y}^2$ and $\hat{y}^3$) have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated some non-linear function.

However, the RESET test does not tell you what kind of miss-specification it is. It can be an omitted variable, it can simply be the case that data are highly nonlinear, it can be there is some multiplicative relationship between the variables so log-log regression where everything is log linearized would be more appropriate and so on.

This being said it is fully possible that the miss-specification was the fact that you omitted a dummy variable. Dummies actually help to solve certain kind of non linearity. Having a dummy variable allows regression to have not just 1 intercept but two separate intercepts. For example, if you are examining the regression of wages on education and dummy is female, it allows for all females in the sample to have different intercept than Males (see the picture below I took from Wikipedia). In essence the dummies allow you to control for a non-linearity in data as you can imagine there being two data clusters one around the line with female dummy set to 1 and one around the line where female dummy is set to 0 (i.e. the male line). In fact data where you need to control for some qualitative state like female and male can sometimes even look in a way look like there is quadratic or cubic relationship if you plot them on scatter-plot.

Consequently adding the dummy might have genuinely solved the problem. An important caveat is that a lot of other things can get wrong in statistics so there could be other problems to see that one would have to examine the whole estimation. For example, there might be some outliers in your data that RESET test confuses for nonlinearity. This being said it is entirely possible the dummy variable solved the problem.

Also It is possible to also include higher orders of $\hat{y}$ but often it is often argued that quadratic and cubic are sufficient in most applications (see Wooldridge Introductory Econometrics: A Modern Approach, Fifth Edition).


Why does optimal allocation call for unlimited distribution of information? In his seminal paper Arrow (1962) states that information should be distributed without limit if an optimal allocation is to be achieved.
Quote (p. 614-615):

  The cost of transmitting a given body of information is frequently
  very low. If it were zero, then optimal allocation would obviously
  call for unlimited distribution of the information without cost. (...)
  The owner of the information should not extract the economic value
  which is there, if optimal allocation is to be achieved; but he is a
  monopolist, to some small extent and will seek to take advantage of
  this fact.

Quote (p. 616-617):

  In the first place, any information obtained, say a new method of
  production, should, from the welfare point of view, be available free
  of charge (apart from the cost of transmitting information). This
  insures optimal utilization of the information but of course provides
  no incentive for investment in research. In an ideal socialist
  economy, the reward for invention would be completely separated from
  any charge to the users of the information. In a free enterprise
  economy, inventive activity is supported by using the invention
  to create property rights; precisely to the extent that it is
  successful, there is an underutilization of the information.

Arrow makes the point that creating a market for information is difficult given the peculiar nature of the commodity information. But if information did not have those special attributes: Couldn't information be "produced" by an organization and sold without welfare loss? Why do property rights lead to suboptimal allocation in the case of information?

Reference:

Kenneth Arrow, 1962.
"Economic Welfare and the Allocation of Resources for Invention," NBER Chapters, in: The Rate and Direction of Inventive Activity: Economic and Social Factors, pages 609-626
National Bureau of Economic Research, Inc.
url: https://ideas.repec.org/h/nbr/nberch/2144.html

Suppose you have a product that you can distribute for constant marginal cost $c$. For every $v\geq0$ assume there are some consumers who value the good at $v$. The net welfare created when someone consumes the good is their value minus the cost of production.

Thus, if we want to maximise the total social surplus (net of costs), we should give the good to every consumer for whom $v-c\geq0$. Since a consumer will choose to buy the good if the price, $p$, is below their willingness to pay (i.e., if $v-p\geq0$), setting $p=c$ ensures the efficient outcome where exactly the consumers with value greater than marginal cost consume the good.

Now suppose we have an information good that can be distributed digitally for zero marginal cost ($c=0$). The above reasoning implies that the socially efficient price is $p=0$! Intuitively, giving you a copy of the good does not cost society anything (because information can be infinitely digitally duplicated), so even if you only get a tiny benefit from consuming it, that benefit will produce a (small but positive) net increase the total social surplus. But the only way to make sure that people who get very small (but positive) value from the good choose to consume it is to give them the good for free.

Arrow's broader point is that this produces a problem: if the price of an information good (e.g. a movie) is zero then firms have no profit incentive to produce movies at all, and then nobody gets any surplus. Eeek! The way society solves this problem is to say "if you make a movie, you get copyright, which means you have the monopoly right to sell that movie". The profits that come from being a monopolist over the movie are the rewards to the firm for having made it in the first place. But, of course, the monopolist movie studio will set $p\gg0$, so some consumers will not buy it even though they could be served as zero cost and this will increase welfare.

If you know your Econ 101 then this can be understood through the lens of some textbook models. Remember that in a basic textbook model of a market, the social surplus is the area below the demand curve but above the marginal cost curve. Social surplus is maximised when $p=MC$, which, for information goods, means $p=0$.

now think of a texbook monopoly model (where the monopoly comes because you own the copyright or patent): the monopolist will set $p>MC$ and this produces a deadweight loss, which is the inefficiency Arrow refers to.


How to check whether the production function below is neoclassical or not Say I'm given the production function of the form

\begin{equation*}
Y= F(K,L,A) = (\alpha K^{\epsilon} + (1 - \alpha)Y^{\epsilon})^{\frac{1}{\epsilon}}
\end{equation*}

Where $\epsilon, \alpha \in (0,1)$.

Now I need to check three things:

That marginal products are positive and they are decreasing
Inada conditions are satisfied
Essentiality
Linear homogeneity

The conditions 2 and 4 imply the condition 3 so no need to check for that. When taking derivatives with respect to $K$ I see that

\begin{equation*}
F_K = \alpha  K^{\epsilon -1}(\alpha K^{\epsilon} + (1 - \alpha)Y^{\epsilon})^{\frac{1 - \epsilon}{\epsilon}} >0
\end{equation*}

Which is positive. Now I need to show that $F_{KK}< 0$. See:

\begin{equation*}
F_{KK} = \alpha (\epsilon -1) K^{\epsilon -2}(\alpha K^{\epsilon} + (1 - \alpha)Y^{\epsilon})^{\frac{1 - \epsilon}{\epsilon}} + \alpha^2 K^{2\epsilon -2} \frac{1- \epsilon}{\epsilon} (\alpha K^{\epsilon} + (1- \alpha) L^{\epsilon})^{\frac{1}{\epsilon}-2}
\end{equation*}

However I couldn't find a way to decide on the sign of this expression. Any help would be appreciated. Thanks.

C.E.S production functions are indeed tiresome to differentiate, and it is very easy to make a mistake. But hey, this (algebraic manipulation) is part of the job, and one needs to practice moving around symbols , creating common factors, etc.

But there is another way to show that $F_{KK} <0$ in totally abstract terms.

First, prove 4. (linear homogeneity), it is easy.

Second, exploit Euler's Homogeneous Function Theorem, and decompose $Y$ using the first derivatives and the variables (this is a very well known result).

Third, re-arrange the previous to obtain an alternative abstract expression of the first derivative of say, capital.

Fourth, differentiate what you got in step Three, to obtain an expression for $F_{KK}$.

All the above are done using abstract notation $F_K, F_{KK}$,etc and nowhere the actual C.E.S functional form. After step Four, you will need to calculate the cross-partial, which is easy, even for the C.E.S. function.


Where is the split in the wealth gap? I've been curious where the split in the wealth gap has been over time, and if it was possible to graph it.

Put another way, at what net worth is the number of those worth more shrinking in proportion to the population and the number of those worth less growing in proportion to the population?

I would love to be able to put a number on this and also see a graph of it over time. If I'm misunderstanding anything in this, feel free to inform me I'm not an economics major.

Turns out I was looking for median net worth in the US over time. For some reason, I thought it was more complicated than that.


What does the sun product of price times quantity divided by average price I have a very simple question, what is the meaning, if any, of this?

I.e. Economically what does the following calculation mean?

$$
\left(\frac{(Q1\cdot P1)+(Q2\cdot P2)+(Q3\cdot P3)}{Q1 + Q2 + Q3}\right)
$$

Consider the values being in a table like the one shown below:

$$
\begin{array}{c|lcr}
n & \text{Factory A Generation of good x} & \text{Factory B Generation of good x} & \text{Market price for good x} \\
\hline
1 & Q1 & Q1 & P1 \\
2 & Q2 & Q2 & P2 \\
3 & Q3 & Q3 & P3
\end{array}
$$

Thank you!

Note that $Q_i P_i$ is the total revenue from good $i$. So dividing total revenue from all goods by the amount produced of all of them should give us something like "average revenue per item produced"?


Did the idea of “making money” originate in the United States? I am currently reading Atlas Shrugged by Ayn Rand and I came across an interesting paragraph related to Economics and American History.

  If you ask me to name the proudest distinction of Americans, I would choose—because it contains all the others—the fact that they were the people who created the phrase ‘to make money.’ No other language or nation had ever used these words before; men had always thought of wealth as a static quantity—to be seized, begged, inherited, shared, looted or obtained as a favor. Americans were the first to understand that wealth has to be created. The words ‘to make money’ hold the essence of human morality.

Is this a true piece of history? Did we create the idea of "making money"?

No, the idea of wealth as something to be created did not originate in the United States.  It was part of the mercantilist approach to national economic policy that was widely adopted in Europe in the 16th to 18th centuries.

Mercantilism involved a range of policies, many of which were designed to increase the wealth of one country at the expense of others and probably reduced wealth overall, eg restrictions on international trade, and warfare to gain control over sources of goods.  However, it also involved policies that were not at the expense of other countries, including promoting industry via research or subsidies, maximising the use of domestic resources, and (see under France in the link above) decreasing internal barriers to trade, reducing internal tariffs, and improving infrastructure.


GDP Calculation and treatment of Personal Savings While calculating GDP of nation, how personal savings which is not invested in economy or has become part mandatory reserve ratio of government is treated? Similarly do we include import and export of financial asset in Net Exchange(Export - Import)

The GDP, as you implied, is represented as follows

  Y = C + G + I + nX

Where Y is the GDP, C and G are private and government consumption, and I is the invest in fixed assets

This formula represents the demand for GDP

In a closed market I = S since all savings, for a certain amount of real interest r are allocated to investments in projects that would return r or higher

S then is a supply factor for the demand for I

In an open market, local savings can be invested abroad and foreign savings can be invested locally, via financial assets

nX are net exports of goods and services, these can be financial or other. tourism is, in a way, an export / import service


What could happen if a country changes their currency rate constantly? Introduction

Imagine this random country. "Yolistan", it is a new country that appeared after a war/revolution/independence/whatever.

The currency it's the "Yolo"

Yolistan needs to import resources to start again the industry, for some time they are not able to produce enough raw materials so they decide to change their currency rate to 1:2 (Yolo/Dollar) to favor Imports.

The First Question is,

Where could this country acquire dollars at such rate or how they do it?

Yolistan(yolos) wants to buy Iron from the US(dollars) do they pay is Yolos or they have to buy dollars first?

After some time of importing goods, now they are able to supply themselves so they change the Yolo to 2:1 (Yolo/Dollar) to favor Exports

The Second Question,

"What happens to their economy and currency if that country changes the rate constantly to favor Exports/Imports?

Thanks for the interest in this question.

At the outset, I'll point out that it would be neither healthy nor realistic for Yolistan's regulator or central bank to arbitrarily modify exchange rates. Issuing decrees about what the new exchange rate is does not guarantee that foreign entities will be willing to perform transactions at that rate.

  Where could this country acquire dollars at such rate or how they do
  it?

Yolistan could acquire dollars through exports, revenues from providing goods and services to tourists visiting Yolistan, money transfers from Yolistanis living abroad, persuade a country to lend money, and/or selling bonds (which is a form of debt).

  Yolistan(yolos) wants to buy Iron from the US(dollars) do they pay is
  Yolos or they have to buy dollars first?

Yolistan needs to buy dollars first, as it is unlikely that the US would want to have reserves in a new and uncertain currency such as Yolos.

  What happens to their economy and currency if that country changes the
  rate constantly to favor Exports/Imports?

Markets would not consider the Yolo a reliable currency (see the
discussions regarding the volatility of bitcoin). Markets expect a
currency to be stable.
Frequent and abrupt manipulations of the Yolo would discourage other
countries (and foreign companies) from signing trade agreements with(and invest in) Yolistan because of (1) concerns about unfair competition, and (2) excessive uncertainty about purchase power, rendering investments
and obligations too risky to commit with.

This answer is not exhaustive.


A question of deriving an industry supply curve from cobb-douglass production function I have a question about a proper derivation of industry supply curve from cobb-Douglas.

The production function is given as:

\begin{equation}
q_i = 2x_1^{1/2}x_2^{1/2}
\end{equation}

the price of $x_1$ input is $w_1=1$ and $x_2$ is $w_2=4$

These are the steps I used in trying to derive the supply curve:

I used the well-known fact that technical rate of substitution must be equal to factor prices so:

\begin{equation}
\frac{\text{MP}_{x_1}}{\text{MP}_{x_2}} = \frac{w_1}{w_2}
\end{equation}

So after substituting prices and the marginal products which we get from taking the partial derivatives of $q_i$ we get:

\begin{equation}
\frac{x_1}{x_2} = \frac{1}{4} \text{ or } x_1 = 4x_2
\end{equation}

Now I substituted this back into the cobb-douglas to get:

\begin{equation}
q_i = 4x_2 \text{ or } \frac{q_i}{4} = x_2
\end{equation}

Now I just used general profit equation:

\begin{equation}
\pi_i= pq_i - w_1x_1 -w_2 x_2
\end{equation}

I substituted the solution for both $x_1 = 4x_2$ and also that $x_2 = q_i/4$ and I got:

\begin{equation}
\pi_i= pq_i - 2q_i
\end{equation}

So my thinking was that by solving for optimal $q_i*$ and multiplying it by $n$ - the number of firms I could get the supply curve. But since the profit function is linear the $q_i$ drops from the derivation.

Please, could someone give me advice on how to proceed?

Edit: Also I forgot to mention that it is assumed that industry is perfectly competitive.

Your method and numerical calculations are correct.

Then, it is a well-known fact that Cobb-Douglas (i.e exponential with constant returns to scale) production functions with price-taking behavior lead to linear cost functions, i.e. to constant marginal cost and equal to average cost, and linear profit functions, which means that profits in the long run will be either zero or infinite: If price is above $2$ ($=MC=AC$) each firm would want to produce an infinite amount of output. if it is equal to $2$, firms that produce will make zero profits. If it is below $2$ no firm will produce and profits will again be zero.

At market level, the assumption of perfect competition (costless entry especially) in turn implies that the long-run (no fixed costs) market supply curve will be horizontal at $P=MC=AC$, and the level of output will be determined by the demand curve and consumer preferences. The output per firm and hence the number of firms remains undetermined here.


What does it mean when someone says “2 trillion wiped off markets”? After a major global event, sometimes the markets move and headlines are written saying something like “2 trillion wiped off markets”. But what does this actually mean? Does it mean the aggregate value of companies on the exchanges has been reduced?

After Brexit similar headlines were generated, has the market recovered its value yet?

That's connected to the way currency or prices fluctuate, particularly in the stock exchange. It comes from the approach that value (e.g. of a firm) equals the sum of all the stocks at current price. That's often called the "Market Cap(italization)".

Think of a firm A with 10.000 stocks. If from one day to the other the stock of company A goes from 100€ to 50€, then its market capitalization/market value goes from 1M€ (=10.000 stocks * 100€) to 500k€. In this conditions you could say that half a milion was wiped off the market or from this company.

That answers directly to your question, but keep in mind that there are many other ways of calculating the value of a firm.


How to find the variance co-variance matrix for 4 assets and market returns using the CAPM? I understand the process on excel:
calculate betas, calculate covariance with $Cov(R_{i,t},R_{j,t}) = \beta_{i,m} \cdot \beta_{j,m} \cdot \sigma^2_m$

where m is the market return and i and j are assets.

I'm confused as to what the assumptions are in order to be able to do this?

Can I do this since I have market data 4 US stocks and the S&P500 meaning that I'm only considering systematic risk?

Please could someone give me the assumptions in order to use the CAPM relation to find a variance covariance matrix?

Sharpe says you can always do a linear approximation to the returns of a family of assets and use that approximation to calculate a covariance matrix. For such an approximation to have economic content, one typically assumes that 1) residuals are uncorrelated with factors and 2) each asset's residuals are uncorrelated with that of any other. The second one has more bite and in particular, is likely not to hold in a one factor model.

It is worth mentioning that in your equation above, the diagonal entries are not exactly what you write. Instead it is:
$Var(R_{i,t}) = \beta_{i,m}^2 \cdot \sigma^2_m +\sigma^2_\epsilon $
where $\sigma^2_\epsilon $ is the variance of the residual of the linear approximation equation.


Will a shift in the average total cost curve mean that the marginal cost curve will also shift up? I understand the basic concept of marginal cost and average total cost. I am currently learning about costs curve for an oligopoly

I read online that the marginal cost is not determined by fixed costs, as the derivative of them is zero.

Based on this, is it possible for the average total cost curve to shift upwards without the marginal cost curve also shifting upwards? For example, when the fixed costs increase (rent goes up for example), would this cause a shift in the average cost curve upwards? And would marginal cost not increase?

No, a shift in the average total cost curve does not necessarily mean that the marginal cost curve will also shift up.

Your intuition is correct. Here is a simple numerical example:

Consider a shift from the total cost function $C_1(q)=q$ to $C_2(q)=1+q$.

Then the average cost functions shifts upward from $AC_1(q)=1$ to $AC_2(q)=1/q+1$.

But the marginal cost function which was $MC_1(q)=1$ remains unchanged at $MC_2(q)=1$.


Monopolistic and Bertrand (Nash) Competition Can we view the monopolistic competition equilibrium (a la Dixit-Stigliz) as the limit case of a Bertrand competition with an infinite number of firms providing differentiated products, where the resulting equilibrium is a Nash equilibrium in prices?

I am looking for references on this topic. Thanks.

The following paper compares the efficiency of the Bertrand and Cournot game in the case of product differentiation. However, their utility function is more general than the Dixit/Stiglitz case. You also do not have an infinite number of firms competing in each differentiated product but just one firm per differentiated product (the number of differentiated products however may be infinite).

Vives, Xavier. "On the efficiency of Bertrand and Cournot equilibria with product differentation." Journal of Economic Theory 36.1 (1985): 166-175.


Anonymity in social welfare function Currently reading David Adler's text on distributional weights in benefit-cost analysis (https://scholarship.law.duke.edu/faculty_scholarship/3110/), I cannot manage to fully grasp the definition of the anonymity axiom.

The isoelastic/Atkinson social welfare function (SWF) allows for an inequality aversion parameter; greater weight is put on changes affecting worse-off individuals. Adler states that this function satisfies the anonymity/impartiality axiom, defined here as "meaning indifference between any given utility vectora and all rearrangements of component numbers". I understand this, broadly, as meaning that the levels of utility, and not the persons enjoying those levels, are what counts.

I can't manage to reconcile the fact that the "anonymity/impartial SWFs focus only on the patern of well-being, and not on the identities of the people who end up at particular well-being levels" (page 267 of the text) with the use of an inequality aversion parameter, which, by definition, treats individuals differently. Would this not mean focusing "on the identities of the people who end up at particular well-being levels"?

What am I missing ?

Any links towards readings on this subject would be much appreciated as well.

Many thanks.

Anonymous/impartial SWFs focus only on the pattern of well-being,
  and not the identities of the people who end up at particular well-being levels.

Identities here simply means names. When applying an impartial SWF, one only looks at the profile of utilities, not who those utilities are associated with. Consider the following two cases:
\begin{array}{ccc}
&U_{\text{President}}&U_{\text{CEO}}&U_{\text{Farmer}} \\\hline
\text{Case 1}& 100 & 80 & 10\\
\text{Case 2}& 10 & 100 & 80
\end{array}
The patterns of the distribution of well-being/utility in the two cases are essentially the same, and yet they differ in who is better/worse off. An impartial SWF would generate the same outcome in both cases, since it only focuses on the pattern (i.e. the utility numbers) of the well-being distribution and not on who (i.e. the identity) has which level of utility. In particular, the fact that the President is the happiest person in Case 1 but the most miserable in Case 2 has no impact whatsoever on the outcome of the impartial SWF.


Understanding Pareto Optimal I am looking at an exchange economy where I have two types of goods and n consumers. Half of the consumers have a utility function given by $U(x)= 5\ln{x} +m $ and the other half of the consumers have a $U(x) = 3\ln{x} + m$. All of the consumers have been given an initial endowment of $20$ of good $x$ and $10$ of good $m$.

I need to find what is the maximum amount of good $x$ that the first type of consumers can get at a Pareto Optimal allocation under the constraint $m>0$ for all consumers

To solve this, I began with creating the following equation:
\begin{align*}
\frac{\frac{\partial U_1}{\partial x_1}}{\frac{\partial U_1}{\partial m_1}} &= \frac{\frac{\partial U_2}{\partial x_2}}{\frac{\partial U_2}{\partial m_2}}\\
\frac{\frac{5}{x_1}}{1} &= \frac{\frac{3}{x_2}}{1}\\
\frac{5}{x_1} &= \frac{3}{x_2}\\
\frac{5}{x_1} &= \frac{3}{20-x_1}
\end{align*}

However, this approach does not give me the correct solution, nor does it include the initial endowment of $m$. Can someone please help me understand what I am missing?

I see no problem in $m$ being absent from the equation, as it enters additively in the utility function. The partial utility obtained from this good is represented in the 1 that you get in the derivative. This is, the marginal utility obtained from an extra unit of $m$ is independent of the amount the individual consumes of $m$. Additionally, as both type of consumers value the $m$ good equally, the distribution of $m$ is then irrelevant for the optimality of any allocation. Your solution tells you that the relation between the level of consumption of $x_1$ and $x_2$ is that of 5 to 3.

Then, what you need to do is to allocate the aggregate resources among all consumers such that the proportion holds. If two two type of consumers have in total 40 units of $x$, then the allocation is that of 25 for type one consumers, and 15 for type two consumers.


Natural real interest rate and output gap I am a bit confused about the concept of natural real interest rate. I've read that it is the level of real interest rate consistent with the output being at its potential or natural level and with the inflation being static. Is that true?

Furthermore, I do not understand how natural real interest rate is able to affect output and natural level of output (then, the output gap) in the usual New Keynesian framework. I mean the dynamic is curve is defined as
\begin{gather}
x_t=E_t\{x_{t+1}\}-\frac{1}{\sigma}\{i_t-E_t\{\pi_{t+1}\}-r^n_{t}\}
\end{gather}
where $x_t=y_t-y^n_t$, i.e. the output gap and $r^n_{t}$ is the natural real interest rate. ($\frac{1}{\sigma}$ comes from a CRRA household utility with elasticity $\sigma$).

Thus it seems that an increase in the natural real interest rate leads to an increase in current output gap, but how is this true? Does it mean that natural level of output increases more than output?

Regarding you first question, the answer is 'yes', at least in your model. For if the real interest rate, $r_t=i_t-E_t\pi_{t+1}$, is equal to its natural level $r^n_t$, then in your model $x_t=E_tx_{t+1}$ and $x_t=0$ for all $t$ is a solution which is consistent with the dynamic equation. (Note that other solutions are consistent with the dynamic equation in the case $r_t=r^n_t$ for all $t$, such as $x_t=1$ for all $t$.)

According to my experience, the natural level of real interest rate is usually defined as the level of real interest consistent with output being at its 'natural', 'long-run', 'equilibrium' or 'potential' level, given stable inflation, absence of shocks to demand, etc. I.e., natural real interest rate, $r^n_t$, is such that whenever $r^t=r^n_t$ and certain stability/regularity conditions are fulfilled, then $x_t=0$. Read a FRBSF Economic Letter for more information.

We may also view the natural level of real interest rate as the real interest rate at which the output gap $x_t$ is following its trend $E_tx_{t+1}$, i.e. $x_t=E_tx_{t+1}$. This definition is similar to the one given by Knut Wicksell.

Regarding your two last questions, I may say the following. In your model, a ceteris paribus increase in $r^n_t$ increases $x_t$ by $\frac{1}{\sigma}$. (Just take the partial derivative with respect to
 $r^n_t$ on both sides in the dynamic equation, holding everything else constant.) This means that either output, $y_t$, is increasing, or the natural level of output, $y^n_t$, is increasing. However, if such ceteris paribus changes cannot be made, then taking the partial derivative holding all else constant is not meaningful. Indeed, in new keynesian models, the natural real interest rate is often a function of $y^n_t$, and in such cases we cannot increase the natural real interest rate without affecting the natural level of output. One such case is when $r^n_t=\rho+\sigma(E_ty^n_{t+1}-y^n_t).$

More concrete examples may be provided, I will see if I can find some.


Why are people rich? Different people become rich for different reasons, such as:

Developing/selling a great product
Investing
Winning the lottery
Inheriting family money

Some of these seem more fair than others, which leads me to wonder: Are there any studies that attempt to measure what percentage of people get rich in different ways?

You have some incorrect ideas about wealth. First of all, inheriting money does not concentrate wealth, it disperses it. Usually the recipients of inheritances, both organizations and wayward children spend the money the receive so the inheritance is dispersed completely.

Most people who are wealthy got that way not by earning a lot of money, but by saving the money they did earn. How much a person saves and invests is a much bigger indicator of wealth than income. A plumber that saves and invests will eventually become much richer than a doctor who spends all his money.

Also, most people who wealthy are not manufacturers who "make a great product," they are ordinary professionals and businessmen: doctors, lawyers, business executives, electricians, plumbers, and builders. You can even find many blue collar workers who have become rich by saving and investing: policeman, firemen, teachers. Where I live there are two teachers, a husband and wife couple, who started out as ordinary high school teachers and now they are quite wealthy and own their own private school. There are probably more millionaire garbage men, than there are millionaire manufacturers. Owning a bunch of garbage trucks is enough to make you a millionaire. Now think how many garbage trucks there are.

Here is an article about typical characteristics of wealthy people:

https://www.nytimes.com/books/first/s/stanley-millionaire.html


Question on General Equilibrium: how to write offer curves? QUESTION:
Consider simple two-person, two-good economy in which agents’ utility functions are given by
$U_1(x_{11}, x_{21}) = min\{x_{11}, x_{21}\} $, and  $U_2(x_{12}, x_{22}) = min\{4x_{12}, x_{22}\} $.
Endowments are w1 =(30,0) and w2 =(0,20).
If neither agents can have negative consumption of either good, what is Walrasian equilibrium?
SOLUTION:
With Leontiev preferences, the indifference curves of both agents are right-angles. For agent 1 their vertices lie on the line $x_{11} = x_{21} $, whereas, for agent 2, they lie on the line $x_{22} = 4x_{12}$. These lines are respective offer curves of both agents when both prices are strictly positive. For the case when one of the prices is zero, we have following offer curves:
————
$OC_1(P_1, P_2)=$
if $(P_1, P_2) \in $ {0} $\times R_+ $ then $OC_1(P_1, P_2)= \{(x_{11}, w_{21}): x_{11} > w_{21} \}$,
if $(P_1, P_2) \in $  $R_+ \times $ {0} then $OC_1(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} > w_{11} \}$,
$OC_2(P_1, P_2)=$
if $(P_1, P_2) \in $ {0} $\times R_+ $ then $OC_2(P_1, P_2)= \{(x_{11}, w_{21}) : x_{11} > (1/4) w_{21} \} $,
if $(P_1, P_2) \in $  $R_+ \times $ {0} then $OC_2(P_1, P_2)= \{(w_{11}, x_{21}) : x_{21} > 4 w_{11} \} $,
my question is how to write this offer curve? I don't ask to solve this question. I dont understand the point where the offer curves are written. Especially I find to be difficult to write the OC2 for the second consumer. I mean writing OC_2 is especially difficult for me. If you show how to write offer curves, I will really glad. Thank you.

$EO_2BO_1$ is 2's offer curve and $EO_1A$ is 1's offer curve. Set of competitive equilibrium allocations is given by the intersection of the two offer curves that is the set of allocations on the line segment $O_1B$.


Are there any real world examples of potential Pareto improvements? Professor George Fallis in his book "Housing Economics" writes when discussing the notion of a Pareto improvement
(Page 123)

In the real world, almost all policies recommended by economists are potential Pareto improvements, not actual Pareto improvements.

My Question
Are there any real world examples (from the past and/or present) of policy recommendations by economists which are potential Pareto improvements? If so, how?

By "potential Pareto improvement", what Fallis means is a "Kaldor-Hicks improvement" or more simply an overall increase in wealth.

The argument by Kaldor (1939) and Hicks (1939) was that if there is an overall increase in wealth, then we could potentially compensate the losers, so that there is potentially a Pareto improvement. Therefore, a "potential Pareto or Kaldor-Hicks improvement" (i.e. an increase in wealth) is "just like" an "actual Pareto improvement".†

So, a free-trade agreement is a "potential Pareto" or "Kaldor-Hicks" improvement so long as it boosts our country's GDP. However, it is probably not an "actual Pareto improvement" because there will likely be at least one person in our country who suffers a loss due to the free-trade agreement.

And so, as you can tell and as Fallis says, most policy recommendations by economists are "potential Pareto" or "Kaldor-Hicks" improvements, but not "actual Pareto improvements".

†Note that this argument by Kaldor and Hicks is controversial, but is also one that most economists have gleefully latched on to, precisely because it is nearly impossible to ever have an actual Pareto improvement. Further reading: Feldman, 1998, "Kaldor-Hicks compensation".


Mathematics of the income and substitution effects I have recently been learning about the concept of utility and the indifference curve. I am having some problems understanding the effects on consumption of two goods $X$ and $Y$ of a change in the price of $X$. I understand what the substitution effect generally means. It refers to a substitution of the good $Y$ for the good $X$ that has now become more expensive. Graphically, it is represented by a shift along the indifference curve from point $e$ to point $e^{'}$.

The scenario given in the diagram is as follows: Income $M$ = $ {$100}$ while original prices are at $p_x$ = $ {$10}$ and $p_y$ = $ {$10}$ The three downward-sloping lines seen in the diagram are the so-called "budgetary constraints". The original budgetary constraint line that is tangent to point $e$ has equation $Y = \frac{M}{p_y} - (\frac{p_x}{p_y})X$. However, I am unsure how the two other budgetary constraint lines are derived. The effect here is a doubling of the price of $X$ from $ {$10}$ to $ {$20}$. Hence, I would think the straightforward effect is that the the gradient of the budgetary constraint line changes from $-1$ to $-2$ but that is not what is directly observed. In fact, there is also a change in the vertical intercept of the budgetary constraint line.

Can someone help to explain to me why the substitution effect leads to consumption at point $e^{'}$ and why the income effect leads to consumption at point $e^{*}$?

The example is taken from the book Economics with Calculus by Michael C. Lovell (Chapter 4, p. 152).

The (Hicks-) substitution effect is by definition the change in consumption of X induced by a change of the relative prices, holding utility fixed. Thus, the original budget line is "rolled along" its indifference curve until it has the same slope (same relative prices) as the actual budget constraint after the price change. This determines $e'$. You can think of the tangent to $e'$ as a hypothetical budget constraint that compensates you for the lost utility resulting from the increasing price of $X$ by adding income until you can just afford a bundle with the same utility as the original bundle $e$. In total, the change from $e$ to $e^*$ is then separated into a change from $e$ to $e'$, (rolling the budget line along the indifference curve: relative price change, holding utility constant = substitution effect) and a change from $e'$ to $e^*$ (parallel shift of budget line: income change, holding relative prices constant = income effect).


Necessary conditions for the existence of a competitive equilibirum I got that in an exchange economy, conditions as preferences being continuous, strictly convex and strongly monotone and $\sum_i \omega_i\gg 0$ are sufficient conditions for the existence of a Walrasian (or competitive) equlibirum.

Are there necessary conditions for the existence of an equilibrium?

If the competitive equilibrium exists, it must lie on the contract curve where two consumers' marginal rate of substitution are the same. Also, this equilibrium lies inside the zone of mutual advantages (area where the indifference curves intersect with the endowment) because trade is voluntary. If consumers’ marginal rates of substitution vary continuously with their consumption levels then a competitive equilibrium exists


What is the name for a market where everyone is basically selling the same thing? Consider the market for retail banking. Everyone is governed by the same interest rates, and it's not like one bank can offer something remarkably different from the others.

Retail pharmacies are the same way. They're all selling the same drugs.

In these markets, competitors have less "room to maneuver" when competing. They have to differentiate based on intangibles like service and amenities. They can't actually change the product.

It's like a 100-yard race. How do you win? You run faster than everyone else...full stop. There's no strategy. There's no way to subtly maneuver yourself into a better position. Everyone is selling the same thing -- speed in a straight line for 10 seconds.

Is there a name for this type of market? Or a name for the type of product?

Such markets are said to be homogeneous goods markets or commodities markets. And the goods traded within them are sometimes known as commodities. From Wikipedia:

  "In economics, a commodity is an economic good or service that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them. Most commodities are raw materials, basic resources, agricultural, or mining products, such as iron ore, sugar, or grains like rice and wheat. Commodities can also be mass-produced unspecialized products such as chemicals and computer memory."


Endogenous Growth in the case of externalities (Learning by doing) so for the Learning by doing model productivity is defined as $$A_t=BK_t$$.

Inserting this into the optimum conditions leads us to the following:$$r_t=\alpha BL^{1-\alpha}$$ and $$w_t=(1-\alpha)B^{1-\alpha} K_t/L$$.

As we can see, the interest rate here is constant and does not change with capital $K_t$. Intuitively, the reasoning is that more capital used in an individual firm reduces marginal product of capital. However, more capital of an individual firm level also leads to a higher aggregated capital stock (if all firms increase their stock) and hence also to higher productivity A, that can compensate for the decreasing marginal product of capital.

However, could someone explain to me why $w$ increases with $K_t$. What does this intuitively mean?

You don’t explain what $w$ even is but I will assume it’s wage. The wage increases with the capital accumulation for several reasons. First, as the capital becomes more abundant the labor inputs become relatively more scarce and hence more valuable. Thus, also the price of labor relatively to capital should increase through higher wages.

Second, in most commonly used production function capital needs to be used with labor for example with standard Cobb-Douglass production function $AK^{\alpha}L^{1-\alpha}$ if you only use capital and no labor output is 0 no matter how much capital you employ. Hence using these kind of production function imply that when the capital stock increases, demand for labor increases as well putting more pressure on wages.

Another way how to think about it more intuitively is to think about labor productivity. It’s well established that wages vary positively with labor productivity and working with more capital makes people more productive hence increasing their wages.


Computing the continuous time survival rate We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the distribution of ages over time.

Let $m(a, t)$ denote the mass of people at age $a$ and point in time $t$. I will start with a discrete time approximation and let $\Delta$ go to zero. At each discrete point in time,

$$ m(a+\Delta, t+\Delta) = (1-P(a, t))m(a, t)$$

where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$. As I'm going to let $\Delta\to 0$, I can approximate $P$ with $(1-\Delta d)$:

$$ m(a+\Delta, t+\Delta) = \Delta d(a,t)m(a, t)$$

Issue Already here I'm having trouble understanding what happened: $\Delta d(a,t)$ denotes the mass of people who died during $\Delta$ at age $a$ and time $t$. Shouldn't this be negatively affecting the mass of people alive? I was expecting something along the lines of

$$ m(a+\Delta, t+\Delta) = (1- \Delta d(a,t))m(a, t)$$

I think the step

  "...where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$..."

is the problem.

In continuous time I guess we have

$$\dot m(a,t) = -d(a,t)m(a,t) \implies m(a,t) = m_0\exp \{-d(a,t)t\}$$

We then have, discretizing,

$$\frac {m(a,t+\Delta) - m(a,t)}{m(a,t)} = \frac {\exp \{-d(a,t)(t+\Delta)\}-\exp \{-d(a,t)t\}}{\exp \{-d(a,t)t\}}$$

$$=\exp \{-d(a,t)\Delta\}-1 \approx -d(a,t)$$

$$\implies d(a,t) \approx 1-\exp \{-d(a,t)\Delta\}$$

So it should be $P(a, t) = 1-\exp(-d(a,t)\Delta)$. Now $a$ changes exactly in the same way as $t$ does (at $t+\Delta$ those of age $a$ at $t$ will be of age $a+\Delta$). So $m$ changes in both arguments together.

$$m(a+\Delta, t+\Delta) = (1-P(a, t))m(a, t) = \exp(-d(a,t)\Delta m(a, t)$$

$$\approx [1-d(a,t)\Delta]m(a, t)$$

I understand the left hand side as "the people that were of age $a$ at time $t$, and are still alive at time $t+\Delta$, where they have become of age $a+\Delta$. Is this what you are after?


Relationship between expected utility and independence axiom Jonathan Levin in "Choice under Uncertainty" wrote in Theorem 1 " A complete and transitive preference relation on a set of lotteries P satisfies continuity and independence if and only if it admits as expected utility representation". So it seems that if we have an expected utility function, preferences will definitely be independent and continuous, and vice-versa. Am I correct? But I am unsure that if preferences don't have expected utility form, will they necessarily not be continuous and independent?

i.e. if preferences are inconsistent with expected utility, does that necessarily imply that the independence axiom is violated?

If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom  of independence.

For example in prospect theory, consumers display loss aversion, which  means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected utility form, but the utility function is still continuous. This preferences violate the independence axiom.

Similarly, you could have preferences that do not admit the expected utility form, and satisfy the independence axiom, but not continuity.


Does it make sense to 'deflate' a sectoral price index in a regression analysis? For instance, if one is running a regression with deflated prices for a given year and one of the independent variables is a price index for a given sector, does it make any sense to 'deflate' this variable or somehow extract the content that might be related to the general economy (assuming I am deflating with a more agregated variable, like a consumer price index)? One might think of demand estimation or supply estimation, if that helps.

My first thought was that I shouldn't bother, but now I'm thinking of running an auxiliary regression with the agreggated price index and the sectoral price index as a dependent variable and save the residuals -- does that make sense?

Edit: Added some more intuition, as follows:

Think of an equation of a residual demand such as $q_t$$=$$\beta p_t$+$\epsilon_t$, where $q_t$ are the quantities, $p_t$ is the price and $\epsilon_t$ is an estochastic error. But since $Cov(p_t,\epsilon_t)≠0$ -- the price $p_i$ is inherently endogenous in a demand function because of the existence of another equation defining price and quantities, the supply equation --, one should estimate demand adequately through instrumental variables estimation. Eligible instruments for demand are supply shifters, such as the company cost. But one could also use some sort price index of a sector as an instrument, a sector that sells inputs to the company out of which you are estimating the residual demand. The thing is that this price index might be related to the price index that you are using for deflating your variables. In that situation, how do you extract these fluctuations related to the general economy out of your price index being used as an instrument?

In that situation, how do you extract these fluctuations related to
  the general economy out of your price index being used as an
  instrument?

Regress the sectorial price index on the general price index, and use as instrument in the demand equation the "residuals plus estimated constant" from the previous regression.

These residuals will be by construction orthogonal to the general price index which indeed could be also a demand shifter, something that would invalidate the sectorial price index as an instrument.

As is the case with any time-series data, one should also deal with possible issues of non-stationarity, co-integration etc.


Effect in the Ramsey model of a decrease in population growth In the basic Ramsey model with technological growth, assuming that the economy is in a steady state, how would a sudden decrease in the population growth rate $n$ impact the steady states value of consumption and capital?

I obtain the following dynamics for the model:

$\dot{k} = f(k) - c - (\delta + n + g)k$

$\frac{\dot{c}}{c} = \frac{1}{\theta}[f'(k) - \delta - \rho -\theta g] $

where $\delta$ is the depreciation rate, $g$ the rate of technical progress, $\theta$ the risk aversion factor, $\rho$ the discout factor, $k$ the capital per effective worker, $f(k)$ the production per effective worker, $c$ the consumption per effective worker and $ f(k) = k^\alpha $

The steady state values, ($\dot{k}=0, \dot{c}=0$):

$\hat{c} = \hat{k}^\alpha - (\delta + n + g)\hat{k}$

$\hat{k} = \left( \frac{\delta + \rho + \theta g}{\alpha} \right)^{\frac{1}{1-\alpha}}$

I believe that there would be a sudden increase in the steady state consumption that would persist in the long run and no change in the capital. Is it right?

Let $\left(\hat{c},\hat{k}\right)$ and $\left(\tilde{c},\tilde{k}\right)$ be the old and new steady-state levels of $c$ and $k$ respectively.

You make the correct observation that $\tilde{k}=\hat{k}$ and $\tilde{c}>\hat{c}$.

However, starting from the steady-state $\left(\hat{c},\hat{k}\right)$, and experiencing an unexpected negative shock to the population growth rate $n$, it is easy to see from the equations determining equilibrium dynamics that the economy does not instantaneously transition to the new steady-state $\left(\tilde{c},\tilde{k}\right)$.

It would be a good exercise for you to work out what the qualitative features of the transition path from $\left(\hat{c},\hat{k}\right)$ to $\left(\tilde{c},\tilde{k}\right)$ look like after the population growth shock.


How technology will help economists to do better predictions? Would technologies as AI, BI and Big Data help the economists to make better predictions and test models to solve problems like "What level of fiscal stimulus is better for a especific country?"

Is it something for the future or its already been used?

I'm an economist by training who also is a programmer and works with a lot of data scientists, so I've some insight into this area.  One of my projects at the Urban Institute is trying to bridge the gap between these fields, and part of our work, supported by the Sloan Foundation, is publicly available here.

To some degree the answer is yes, but there are huge obstacles as well.  Some are surmountable with work, like differences in terminology, programming platforms (knowledge of SAS or Stata versus R or Python) and conventions (do others in your field understand and/or accept the results of machine learning models?).  Others are much more difficult to surmount, such as the size of data available.

Essentially, the fields that make extensive use of AI and machine learning have huge amounts of data.  They can afford to, say, set aside a million observations for training the model and then test the model on the other nine million.  Economists, and social scientists in general, however, are often lucky if they have 100 observations.  Many macroeconomic variables are annual, or quarterly.  Some are monthly, which means if you can get 30 years of observations then you've got 360 time periods!  Data-rich by social science standards, but not even a rounding error in many data science models.

That said, computing power and sophisticated data collection are making it more and more feasible to approach social science questions with big data.  Social media is a major area that social scientists are just beginning to reach into.  However, some areas of economics will likely remain closed off to data science methods due to the nature of the data.  Your fiscal stimulus example might very well be one - the relevant data just isn't produced that fast.

So the answer to your question is yes, with reservations and a hope for the future.


Interpretation of price indices How can cpi 3.41 and Wpi 3.54 be interpreted? I know that cpi gives the inflation rate but what is it that we understand from these figures?

In itself, a price index for single period does not tell you much, as any index is, by definition, without dimension. Assuming that 3.41 is the annual change in the CPI (= Consumer Price Index), this would imply an inflation rate of 3.41%. To be specific, it implies that the total price of a representative commodity basket changed by 3.41%. The WPI (= Wholesale Price Index) covers different commodities and hence yields a different inflation rate. Which one to prefer depends on what you are interested in. Consumer Prices are more relevant to private households, but businesses care about wholesale prices.


Where do stocks get their value from? I'm aware that some companies give out dividends to their shareholders, but are there companies that don't? And if they don't, what incentivizes people to buy stocks at the IPO? Is it simply the belief that other people will value the stocks?

Yes, there are companies that DON'T pay dividends to their shareholders. Some of these well-known companies are:

Amazon.com Inc (AMZN)
Facebook Inc. (FB)
Netflix Inc. (NFLX)

The main incentive shareholders get by buying at IPO is CAPITAL APPRECIATION.
98% of the time, the company will only be listed on the stock exchange if it has strong financial fundamentals. Thus, a case in which no one buys into that company as soon as it goes public is very rare.
Share prices change due to the changes in the supply and demand of that company's share. Hence, when the company's IPO is launched, a lot of retail as well as institutional investors buy into that stock and increase the demand for it, while the supply stays relatively stable.
This increases the price of the company's share in the short run. However, if the company has strong financial fundamentals its price would continue to grow steadily. But if a company has weak financial fundamentals, the company's stock price may even drop below its IPO price after a few days of its IPO's launch.
Below is the price of Facebook. Inc since its IPO launch. As the company had strong fundamentals it grew into the social media giant it is today.

However, a company like Groupon had a minute increase in its price just for a few days after its IPO launch due to the rapid increase in demand as discussed above. However, because the company did not have strong financial fundamentals, its price gradually started to lower.

So what incentivizes people to purchase a company's share AT IPO?
The very high probability of the stock to jump above its IPO launch price for a few days.
After which, its price is determined by the company's VALUES/ FINANCIAL FUNDAMENTALS
Cheers!


How to determine demand without the price of a good? If I have the utility function of an individual that exhibits perfect substitutes, think (U=ax+by), and I'm given the price of good x and the budget M, then how am I supposed to determine the demand for good Y if I don't have the price of good Y?

I know for Cobb-Douglas style utility functions the optimal choice of good Y is given by Y=BM/price of good Y

Consider that, if two goods are perfect substitutes, their marginal utilities are constant with respect to each other. Hence with two goods, you will consume only one or the other*.

If $\frac{a}{p_x} > \frac{b}{p_y}$, then $x^*(M) = \frac{M}{p_x}; y^* = 0 $

otherwise if $\frac{a}{p_x} < \frac{b}{p_y}$, then $x^* = 0; y^*(M) = \frac{M}{p_y} $

*(assume their prices are not the inverse of their marginal utilities, in which case you are indifferent).


Bizarre max {} notation My professor gave us the following problem, but it uses a notation/syntax I haven't seen before $(q_{luke} = max \{ \dots \})$. She hasn't explained it or introduced it to us prior to this.

What does this mean? I assume it must be somehow translatable into $y=mx+b$ form given the context of the problem, but I'm not sure how. Thanks!

max {a, b, c} means the highest of these elements.

For example, if "b" is higher than a and c, then max {a, b, c} = b


In quantitative tightening do banks have no choice in refusing to buy bonds from Central Banks (mainly Federal Reserve) In light of the recent unease about interest rate rise, I've been taking a growing interest in how our economy actually works, something I've been trying to avoid.
So this forum is a way for me to clarify my (mis)understanding, so for some of you, I'm sorry for stating the bloody obvious.

Banks stopped borrowing money when people or corporations or even other banks defaulted on their loans. Banks even stopped lending to each other. As a result the economy wasn't growing. So to avoid another recession (or encourage spending in the current recession), the Feds decide to experiment with QE.

With their unlimited power, they created new money into being, and bought these debts, even bad debts (bonds) and equities/assets from these banks.

My focus is on the bonds that the Feds bought, and this is where my understanding starts to get fuzzy and wtfs/minute grows exponentially....

So these banks sold their bonds to the feds, and with this new loan they received, they can start lending back to people or corporations. But this also means they are paying interest rates to the Feds, but hopefully they are making good investments on the new loan (but what if they default on this new loan, rinse and repeat???)?
Because Feds have been introducing these magical money for years, bond prices were up, and thus interest rates decrease...

Quantitative tightening...
But cheap money is unsustainable and cannot go on forever (I'm still unclear on this but I will research more on this later).
So now the Feds are thinking of selling back the bonds (and presumably equities) they bought from the banks that sold it to them. Now, bonds price will plummet but interest rates will rise...

My question for now is:

Will the Feds now pay the banks interest rates on money that THEY magically created or are they exempt from this?

Do banks really really have to buy these bonds bank? Then interest rates just won't rise and the Feds are stock with bad debt until maturity, and it's essentially paid for, more or less.. What am I not understanding?

The question as it stands now contains points that are debatable. So I will just attempt to answer a similar question; the question could be revised in response.

There are a number of other questions on QE that offer some background.

Difference between QE and regular monetary operations
Limits to QE

My revised version of the question is: “In quantitative tightening, the Federal Reserve sells bonds on its balance sheet. Can banks refuse to buy these bonds?

The answer to that question is yes, but it does not matter. Note that my description here is highly simplified, technical details are being skated over.

Let’s assume that Betty is an investment dealer, but not a bank. She has a bank account at Acme Bank. The Federal Reserve has an auction to sell bonds. Acme Bank refuses to participate for some reason or other, but Betty buys \$1 million in bonds from the Federal Reserve.

Betty transfers the money on deposit at Acme Bank to the Federal Reserve. What then happens.

Acme Bank reduces Betty’s deposit balance (at Acme) by \$1 million.
Acme Bank transfers the funds to the Federal Reserve by losing \$1 million of reserves it had on deposit at the Federal Reserve. This eliminates \$1 million of excess reserves in the system.

These operations are neutral for Acme itself. It loses an asset (the reserves), but also loses a liability (Betty’s deposit).

That is, bank reserves are just deposits by banks at the Federal Reserve. A central bank is called a central bank for a reason - most banks have deposits there. It’s just that in American banking jargon, these deposits are called “reserves”. Other countries (like Canada) have abolished resserve requirements, and these deposits are just called “settlement balances.”

Reserves are destroyed by anyone transferring money to the Federal Reserve. The only way a bank could stop it is by not transferring the money, which would be a breach of some law or another.

The only way that quantitative tightening could be stopped is if nobody was willing to buy Treasury bonds at any price. Unless the Treasury was on the verge of default, we normally do not see such behaviour.


Macroeconomics - Fiscal/Monetary Policy and the Price level I was studying for my macroeconomics test by solving test banks and I came upon this question which has been really bugging me.

To decrease the price level

A.the Fed could buy bonds and the government could increase the
corporate profit tax.

B.A decrease in income tax and an increase in corporate profit
tax

C.the Fed could sell bonds and the government could lower the
corporate profit tax.

D.An increase in government spending and a decrease in the price
of raw material

Answer : C

I don't understand why the answer is C. If I take it answer by answer:

A) Expansionary monetary policy + contractionary fiscal policy = unknown result (Aggregate demand curve could shift to the left or to the right depending on which policy was more effective).

B) Either contractionary or expansionary fiscal policy depending on the values of increases/decreases. Employee income surpasses corporate profits income in most cases so I would consider this expansionary fiscal policy which raises the price level.

C) Contractionary monetary policy + expansionary fiscal policy. Why would they choose this as the right answer? It's as ambiguous as the others, the shift in the AD curve is unknown.

D) Expansionary fiscal policy (shift of the AD curve to the right) + shift of the AS curve to the right (which lowers the price level), again leads to an unknown result for the price level)

Am I approaching these questions badly or are they as ambiguous as I think they are? Also, when monetary policy and fiscal policy oppose each other, which one takes the lead on the shift of the AD curve? I feel like that it would solve a lot of problems to know that.

I believe that tax policies refer to the supply side (AS). Since no information is given about government expenses in a, b and c, I would only shift AD in response to the monetary policy actions in a and c, and the explicit increase in goverment spending in option D.


Papers on International Trade and Environment everyone.

As a requirement to get my bachelor's in economics, I have to write a paper. The theme I picked is something along the lines of "the impact of environmental crimes and regulation on international trade". The problem is I've always worked on topics that are based on microeconomics, never on macroeconomics.

Therefore, I'd like to ask for paper recommendations on international trade, if possible related to environmental issues, as a way for me to develop a solid foundation regarding this macroeconomic topic.

I'll also be consulting Krugman's book, but I feel reading some papers that use theories and models tailored to international trade might be very useful.

I appreciate any help!

Feel free to ask for any clarifications if needed.

Kind regards,
Pedro.

I have to disagree with the previous answer. There is quite a large literature on international trade and the effect of environmental regulations. To start off there is the whole race to the bottom literature and the environmental Kuznet's curve. A good start of trade versus the environment is given in Esty's 2001 Bridging the Trade-Environment Divide (found a.o. here). Copeland and Scott Taylor have a 2004 article on international trade, growth and the environment in the journal of economic literature.

As for trade in natural resources and related trade policies Ruta and Venables have an overview paper here.

For a more general equilibrium model on differences in environmental regulations and how that affects trade you could start at Chichilnisky's 1993 North-South trade and the dynamics of renewable resources (here).


How is sending money out of a country draining its economy? From this article:

  Every year, Mexicans working in the U.S. send at least $20 billion back to Mexico in the form of remittances, placing a huge drain on our economy.

But I don't see what negative effects it could have on the US economy.
Won't the money find their way back to the US, since that is the only place where you are able to trade them for goods and services?

Would the impact be different for a smaller country with a currency not as global as the US dollar?

Whatever money earned in a country that is spent outside that country is taking away economic activity from that country, GDP can no longer be increased with that money.

Sending money to a country with a different currency usually implies that currency is bought. That will increase the value of the bought currency and decrease the value of the currency sold. A lower US dollar will make it easier for US exporters to sell their products abroad, which will lead to foreign importers selling their foreign currency in return for US dollars. That is how the money can come back at some point.

It can also come back if foreigners invest in the US. If the US has a lot of money from remittances flowing out and doesn't export as much as it imports, which has been the case for a long time, foreigners holding US stocks and bonds will be the result. You could argue this could lead to a dangerous dependency at some point, e.g., if you look at how much the US owes to the Chinese. Mexico in comparison is too small an economy to be such a threat and I agree with the comment above that losing a significant number of Mexican workers would be much more problematic to the US economy.

You should also take into account that the positive development effects of remittances in the receiving country should that country to become a more attractive partner for trade with the sending country, especially given the strong relationship that is built by the migrant community.


Are there plausible scenarios for the U.S. dollar to fall out of use as the world's reserve currency? The U.S. has maintained an overall trade deficit for decades, essentially exporting money. For many countries this fund of 'ex-patriated' dollars has been a big tool of trade. Everyone accepts dollars. Convert Samoan clams to dollars to buy Romanian veeblefetzers.

Everyone trusts the dollar. Under what circumstances can that change?

I've seen and felt the contempt of the world toward the U.S. government since The Great Cheeto took office. He's a wild card. I have some investments in U.S. funds. I'm seriously considering pulling them.

I'm making a point of not buying US products if I can find alternatives.

As a tree farmer, I'm looking for alternate suppliers of the pots I use.

If my attitude is common, then US exports fall, the demand for dollars to buy US goods drops, at some point the holders of US debt decide that holding it is a bad idea, and there is a run to get rid of it.

Is this plausible?  Is there a plausible scenario that would cause nations to abandon the U.S. dollar as the reserve/trading currency?

There are many plausible scenarios in which the dollar ceases to be the de-facto currency. They all have a few things in common. For people to stop using dollars for international trade, there would need to be two circumstances:

A reduction in trust in the USD (not just in US politicians)
A good (better than the USD) alternative currency

While political emotions run strong, both in the US and abroad, these emotions do not normally have a large effect on the economy.  There has been no significant change in American business practices, law, monetary policy, or productivity as a result of the change in leadership.  This has been very clear.  You can easily argue that American reputation abroad has been harmed, but that doesn't translate to negative economic repercussions.  Remember, the Soviet Union was using the dollar for international trade during the cold war.  A reduction in trust in the USD would come about if there was significant inflation of the dollar or if the US stopped producing things that people want to buy.  Neither of those appear to be on the horizon.  As the result, there has been no change in the attitude toward the dollar.

At the same time, the best candidate by far for an alternative currency, the Euro, has had significant problems recently.  It doesn't matter if the world thinks very highly of European politicians and dislikes American politicians; they aren't going to switch from dollars to Euros unless they perceive Euros to be a safer and more liquid currency.

Similarly, foreign investors will not withdraw from American investments wholesale unless they perceive a real risk to the investment potential. These types of decisions (on a large scale) are made in a cold, calculating way.  Until something happens that might actually affect the potential of US-based investments, there won't be a large change in demand for them.

Interestingly, when there are shakeups in the US economy (like when treasury securities were downgraded) the result is often the opposite of what you might expect.  There's a general concern about the worldwide economy and then a flight to quality, causing an increase in demand for relatively safe US securities, including the ones that were the apparent cause of the panic.


If a retailer consumes a good he bought from the producer, does that get accounted into the GDP of a nation? Let's say we have a soft-drinks store. Say the store sells Pepsi and Coke. If the retailer drinks a bottle of coke from his own store, does that get counted into the GDP of a nation?

Or is it only counted if someone else apart from the store owner consumes it by paying for it?

How does this work?

Good question, which goes into the heart of the definition of GDP.

The answer is that some of the value of the drink is accounted irrespective of what the seller does with the drink. Recall that GDP accounts for the value added of final goods, value which is "added" at each stage of production (and equivalent to the final price of the good).

As such, when the retailer buys the can of coke from the wholesaler or producer, a portion of the total value has already been recorded in GDP. It is only the value added by the retailer which is not counted if the seller drinks the coke.

Naturally, this is true only if the seller does not sell the coke to herself. Otherwise it makes no difference at all.


elasticity from inverse demand I was wondering of my thinking here was right:
Given $$ e=\frac{dQ}{dp}*\frac{p}{Q}, $$ where
$ e $ is elasticity, $ dQ/dp $ is first derivative of demand function, $p$ is price and $Q$ is quantity.
With this expression for $e$, could you then state that:
$$ \frac{dp}{dQ} = \frac{p}{(e*Q)} $$
where $ dp/dQ $ expresses first derivative of inverse demand.
Is this correct?

I will denote the demand function by $Q(p)$ and the inverse demand function by $P(q)$. Then
$$
\forall q: Q(P(q)) = q
$$
so for any $h > 0$ and $q$ we have
$$
\begin{align*}
p & := P(q) \\
p_h & := P(q+h) \\
q & = Q(p) \\
q_h & := Q(p_h) = q+h
\end{align*}
$$
From the definition of derivatives
$$
\frac{\text{d} P(q)}{\text{d} q} := \lim_{h \to 0} \frac{p_h - p}{q_h - q}.
$$
If this is a non zero real number then indeed
$$
\frac{1}{\lim_{h \to 0} \frac{p_h - p}{q_h - q}} = \lim_{h \to 0} \frac{q_h - q}{p_h - p} = \frac{\text{d} Q(p)}{\text{d} p}.
$$

There are some technical considerations. We did not prove that
$$
\lim_{h \to 0} p_h - p = 0.
$$
This is true if $P$ is continuous.
We also assumed that at $q$ and $q+h$ both functions are single valued. If there is a price $p$ where $Q(p) = 0$ then $\frac{\text{d} P(q)}{\text{d} q}$ does not exist.


Great "constants" of economics Question

What are the great "constants" of economics?

Background

In his Structural Slumps (Chapter 3, p.20), Phelps refers to the great "constants" of economics. He explains that the natural unemployment rate has been treated as one of these great constants. So, what else is a great constant in economics? The natural interest rate? The natural level of output? And, so on? References would be greatly appreciated!

One of the original statements of this kind were Kaldor's Stylized Facts on growth, Kaldor's Facts, the most important of which are that:

Labor and capital receive approximately constant share of output (about 0.6 for labor and 0.4 for capital)
The rate of growth of output per worker is roughly constant (about 2%)
The rate of return on investment is roughly constant (if you look at long enough intervals, i.e. about 3%...)

More recently, the famous growth theorist Paul Romer came up with a renewed set of facts New Kaldor Facts paper. The most salient ones are that:

Market is constantly increasing its dominance as a way to organize economic activity.
The rate of economic growth has been accelerating for millenia.
Measured inputs explain only a small part of income differences across countries.
Human capital is constantly being accumulated, and yet, the human capital premia has not changed over time.

Also, a newly minted constant, as theorized by Minsky:

The nature of financial markets and risk taking implies periodic financial crisis.


When are banks allowed to fail? If the Federal Reserve acts as a lender of last resort and steps in to prevent bank runs, then how do banks still fail?
Do they only step in if the bank is solvent but in emergency need of funds?

First, it is actually not Fed that prevents bank runs. Bank runs in the US are prevented by Federal Deposit Insurance Corporation (FDIC)  which insures deposits that people have at the bank up to \$250000. This is what prevents the bank runs as runs are created when people worry bank will fail and they will loose their deposits. This is because bank never has enough money to let all people withdraw them at the same time as loans are made long term while deposits can be withdrew at a depositors whim and consequently if people worry bank will fail they will all 'run' to their bank in order to make sure they get their money before bank fails and stops paying out deposits.
Second, generally and broadly speaking the literature agrees that there is some rationale for saving banks under two conditions (Hanson, Kashyap & Stein 2011; Osinski Seal & Hoogduin, 2013):

They are systemically important (their collapse would lead to collapse of other institutions).
They are only illiquid not insolvent.

The point 1 is considered to be more important than the second so even illiquid institutions might be considered to be worth saving if they are too systemically important.
Now Fed or other government bodies can't be always $100\%$ sure about how systemically important bank is or whether it is illiquid or only insolvent, but central banks around the world typically create and track some measures of systemic importance of various banking institutions. For example, Fed has whole dedicated unit/department just for researching systemic risk.  In addition they will also try to estimate whether bank is insolvent or just illiquid. Lastly, since no research can give you $100\%$ correct answer at the speed Fed usually has to act there is also some discretion involved in the decision process. So partially it also depends on what decision makers at Fed think should be done in each specific case.
However, if some banks are judged to be insolvent and not systemically important  they are routinely allowed fail. In fact as the data from FDIC show the total number of commercial banks that have to register for deposit insurance (basically any 'ordinary' bank) fell by almost 200 just between year 2018-2019.


Notation of a Cobb-Douglas function printed in 1989 I am trying to understand a paper written back in 1989 about long run population growth. It seems like the PDF is a scanned image of the paper.

The notation for the function is on page 11 of the pdf (here):

Is it:

$$ Q_t = A_t*(K_t^\alpha N^t_\beta T_t^\rho) $$

or

$$ Q_t = A_t*(K_t^\alpha N^\beta_t T_t^\rho) $$

Im not sure its a typo or not because it shows up the same way later in the paper on page 14.

*Edit:  I also have no idea what $\hat c$ is in the paper:

I don't see where it is defined.  Only where consumption is compared to it.

Seems to be the second one, so $$ Q_t = A_t*(K_t^\alpha N^\beta_t T_t^\rho). $$

Two clues:

This is the usual specification.
On the top of page 14 it is written that
$$
w_t = \beta Q_t/N_t.
$$
Given a competitive labor market you have
$$
w_t = \frac{\partial Q_t}{\partial N_t}.
$$
If you assume the production function above then you indeed have
$$
\frac{\partial Q_t}{\partial N_t} = \beta Q_t/N_t.
$$


Regarding the Expenditure Function Underlying a Bliss Point I've been looking at expenditure systems and have been really interested in the behaviour of the demand system that underlies bliss points:

Consider the bliss point utility function of the following form:

$$U(x_1,x_2)=-(x_1-\delta_1)^2-(x_2-\delta_2)^2$$

for two dimensions the corresponding hicksian demands are:

$$x_1^c=\delta_1-\left[\frac{\bar{U}}{1+\frac{p_2}{p_1}}\right]^\frac{1}{2}$$
$$x_2^c=\delta_2-\left[\frac{\bar{U}}{1+\frac{p_1}{p_2}}\right]^\frac{1}{2}$$

It follows that the expenditure function is:
$$e(p_1,p_2,\bar{U})=p_1\delta_1-p_1\left[\frac{\bar{U}}{1+\frac{p_2}{p_1}}\right]^\frac{1}{2}+p_2\delta_2-p_2\left[\frac{\bar{U}}{1+\frac{p_1}{p_2}}\right]^\frac{1}{2}$$

Obviously expenditure functions can be much larger. however I'm having a hard time for generating a expenditure function for a number of $n$ goods.

tldr What would the hicksian demands look like for the utility function:
$$U(\mathbf{x})=-\sum_{i=1}^n(x_i-\delta_i)^2$$

Usually the change in variables $X=x-\delta$ allows to write
$$ e(p,u) = \max_{x \geq 0}\{ p'x : U(x-\delta) \geq u \} =  \max_{X \geq -\delta} \{ p'X : U(X) \geq u \} + p'\delta $$
and so $ e(p,u) = E(p,u) + p'\delta.$


Change in the marginal utility of leisure with respect to a change in consumption I am reading a paper that derives a theoretical retirement model. There is a utility function and a budget constraint forming an optimal control problem. The solution to this problem states that
\begin{align}
    U(C(t),0) &= \lambda \quad\text{for }0 \le t \le R,\\
    U(C(t),1) &= \lambda \quad\text{for }R \le t \le T,
\end{align}
where $\lambda$ is the Lagrange multiplier. $U$ is the utility function where the arguments of the function are consumption ($C$) and leisure ($L$). $R$ is the retirement age, and $T$ is the years the agent lives.

The first condition says that the utility is constant during the working years where leisure available during working years is $0$. The second condition says that the utility is constant during the retirement years where leisure available during retirement years is $1$.

The paper then mentions that "This implies that the individual chooses a constant level of consumption for his pre-retirement years, $C_0$, and another (perhaps different) level of consumption, $C_1$, for his post-retirement years. $C_1$ will exceed (be smaller than) $C_0$ if
\begin{equation}
    \frac{\partial^2U}{\partial C\;\partial L}>0\quad(<0)
\end{equation}
The last expression denotes "cross partial derivative of $U$ with respect to $L$ and then $C$".

I am struggling to interpret the inequality. That is, the cross partial derivative says that if the change in the marginal utility of leisure with respect to a change in consumption is positive, then the consumption level during retirement should be larger than that during the working years. Why?

The partial derivative $\frac{\partial^2 U(\cdot)}{\partial C\partial L}$ tells you how a change in leisure affects the marginal utility of consumption. Therefore if it is positive it means that the marginal utility of consumption is higher when $L=1$. In that case, the agent plans her life to be able to consume more when she is retired and leisure is bigger. In contrast, if leisure reduces the marginal utility for consumption, $\frac{\partial^2 U(\cdot)}{\partial C\partial L}<0$, then it is better to consume more while working, than when leisure is big.


Large consumer goods companies not using advertising, and what their reasons are Recently I've learned that Inditex (owner of Zara brand) is the biggest fashion group in the world and it's using almost no advertising at all.

I would like to know more examples of such big companies that use no advertising and why might a large company not invest directly in advertising?

I'd argue that the best example is Jiffy Mix, which spends literally no money on advertising, even in the form of the implicit advertising fees paid to direct marketing affiliates. They just put their product on a shelf and people buy it.

We'd expect to see this no-advertising strategy more often in industries where advertising has a significant impact on the cost of the product, and where consumers are very price-sensitive. As Jiffy is very nearly a commodity good, this holds.

In Zara's case, it's "fast-fashion", a segment of the fashion industry where the goal is to provide popular styles of clothing (so product differentiation, a major goal of advertising, does not apply), at low cost (they're targeting price-sensitive consumers), so this strategy is in some sense not a surprise.


Monotonicity of second price auctions with bidder-specific reserve prices I am having difficulty understanding the problem below:

Given CDFs $F_1, ..., F_n$, prove that the allocation rule of a second-price auction with bidder-specific reserve prices is monotone. The item is rewarded to the bidder with the highest bid that meets his reserve price. The reserve price of bidder $i$ is $r_i = _i^{-1}(t)$ where $_i$ is the virtual valuation function

$$\phi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$$

and $t$ is chosen such that

$$Pr\left[\max_i _i(v_i) \ge t\right] = 1/2.$$
Since the virtual valuations are determined before the auction (if I understand it correctly), how does the reserve prices affect the monotonicity of the allocation rule? The behavior is still the same: if you keep on increasing your bid, you'll have a higher chance of getting the price. How should I go about this?

Some thoughts. If this was a SPA without a reserve price, the monotonicity of the allocation rule is trivial. When the reserve price is uniform across bidders then the allocation rule is clearly weakly monotonic now; since for low bids, increasing them might not change the probability of allocation. The only thing that makes this problem different is that the reserve prices are bidder-specific. So one possibility is that you only need to show that the logic for the previous monotonicity is still valid in this setup. Probably while trying to show that the logic still goes through you will find out if there is some extra argument you will have to make in order to still have the monotonicity.

For example, I might worry that by increasing my bid I might increase my own reserve price, or decrease the reserve price of some other bidder and thus reduce my probability of being allocated the good.


Demand curve for items where consumers typically only buy a single product This question began with rental housing (most people do not rent multiple apartments).

The demand for apartment units in any given period is limited to the number of people searching for units (not all searchers must match with a unit in each period--assuming that most begin searching while their current lease is active).  Supply in the rental market is also fixed in the short term.

I expect that using the normal axes (x: Q, y: P) there would be a kink with a vertical line to the X-axis at the lesser of A) the units available B) the number of searchers.  At a certain price all of the searchers/units are matched. Is this correct?

Is there a different unit to use on the X-axis other than quantity (count) when consumer typically only purchase 1x of an item?

In the simplest case, to draw a two dimensional diagram, you need to assume that apartments are homogeneous.Suppose we go along with this. Then the market demand schedule can be written

$$Q^d(p) = \sum_{j=1}^N I\{w_j\geq p\}$$

where $N$ is the number of persons looking for an apartment, $p$ is the price and $w_j$ is searcher's $j$ "willingness to pay". $I\{\}$ is the indicator function, taking the value $1$ if the condition in the curly brackets is satisfied, and the value $0$ otherwise.

So as long as the price is above its willingness to pay, a searcher won't "demand" an apartment (he won't be willing to rent at that price, that is).

Strictly speaking, this will make the demand schedule a collection of distinct points, rather than a line, since indeed, the quantity axis is in discrete values. But since the price variable is essentially continuous, we can imagine that there exists a price where the number of units demanded equals the number of units supplied.

Also, be careful with the assumption of a vertical supply curve. Ιt does not just imply that the available units are fixed in the short run, it also implies that apartment owners are willing to rent at any price.

This does not appear to be the empirical case. If prices fall, owners are seen to "withdraw" their apartment from the market, waiting perhaps for better times. This means that while we can imagine that eventually, we will have a vertical segment in the supply curve, for lower price we will observe a rising supply curve as usual.

For the rising part we will have analogously, for $M$ owners

$$Q^s(p) = \sum_{j=1}^M I\{s_j\leq p\}$$

where $s_j$ is the minimum acceptable rent by the owner. So an owner too is really "active" in the market and contributes to the supply only when the prevailing price is above that threshold (assumed possibly different for each owner).

A normal situation will be