How does price-elasticity of supply affect the incidence of tax on consumer and producer From what I've read only price-elasticity of demand (PED) (not price-elasticity of supply (PES)) affects the burden of tax on the consumer/producer. However, my graphs suggest PES does affect it. Why is this??

(The red lines represent the supply curves of firms with elastic PES and the blue lines are for inelastic PES. The S1 lines show supply before tax and S2 shows supply post tax. I have used a demand curve with unitary PED.)

According to the graph the proportion of tax paid by the consumer on goods with price elastic supply (P1P2AB out of P3P2AC)is significantly greater than the proportion paid on goods with inelastic PES (P1P2DE out of P3P2DF. Why is this?

You are right: both the price-elasticity of demand and the price-elasticity of supply affect the respective proportions for producers and consumers of the burden of a per unit tax imposed on sales by producers.

The proportions depend on the relative slopes, which reflect price-elasticities, of the demand and supply curves. Considering the elastic supply case in your diagram, the slope of the demand curve between points AE is much steeper than that of the supply curve between CE.  Hence E is much closer, in the vertical (price) dimension, to C than to A. Thus the post-tax price to consumers (p2 corresponding to A) is much further from the pre-tax price (p1 corresponding to E) than is the post-tax price to producers (p3 corresponding to C),and so most of the burden falls on consumers.  For the inelastic supply case the position is reversed.  A detailed explanation may be found in this NTRC paper (see section III).


Utility Function Implies Consumption of Not All Goods Suppose we have a utility function with three inputs, $j, k,$ and $s$ described by $$u(j,k,s) = A\ln(k^\alpha + \beta j^\alpha) + B\ln(s).$$ The price of $j, k,s$ are $p_j, p_k, p_s$, respectively, and we cannot spend more than amount $Y.$

We are asked to find conditions where the optimized result is to consume only two out of the three inputs. In other words, we wish to find conditions where one input is not consumed at all.

Is there a systematic way to approach this type of problem? I tried taking the marginal utilities with respect to each good, and finding a case where exactly one of the goods was negative for all $j, k, s \geq 0$, but I don't think that's the right way to go.

There might be a systematic way (e.g. some variation of the gradient method), but it is probably not trivial (see here), unlike this problem.

By checking $U(j,k,0)$, $U(j,0,s)$ and $U(0,k,s)$, you will find that one of them can actually never occur when utility is maximized. Then you can move on to examine the two goods where 0 consumption can occur, and compare their marginal rate of substitution to their price ratio. Whenever these are unequal in optimum, it signals a corner solution.


Calculate Elasticity of demand in this example Suppose that house prices increase by 10%, and the total quantity of homes purchased
decreases by 8%. What is the elasticity of demand of housing? Interpret your answer in the
context of a 1% change in home prices?

Is the answer just -0.8? Can someone explain this please, my professor really sucks.

Yes, the answer is -0.8.

Price Elasticity of Demand is defined as:
$$e_{\langle p \rangle} = \frac{\mathrm{d} Q/Q}{\mathrm{d} P/P}$$

You're given :
$$\mathrm{d} Q/Q = -0.08$$
and
$$\mathrm{d} P/P = 0.1$$

ergo:
$$-0.08/0.1 = -0.8$$

I'm reasonably certain that by "Interpret your answer," your professor means that s/he wants you to explain what that number means, in layman's terms.

You can always check the Wikipedia entry.


Show that the dividend price ratio is a ARMA(p, q) process Let the log dividend growth evolve according to $\Delta d_{t+1} = \epsilon_{d, t+1}$ where $\epsilon_{d, t+1}$ is just white noise. Let the log returns be $r_{t+1} = x_t + y_t + \epsilon_{r, t+1}$ where $x_t = b_x x_{t-1} + \delta_{x, t}$ and $y_t = b_y y_{t-1} + \delta_{y, t}$ and $\epsilon_{r, t+1}$, $\delta_{x, t}$, and $\delta_{y, t}$ are all white noise. Solve for the dividend price ratio, $d_t - p_t$ and show that it is an $ARMA(p, q)$ process, find $p$ and $q$.

What I did was to start off with the famous Campbell-Shiller decomposition: $d_t - p_t  = E_t \sum_{j=1}^{\infty} \rho^{j-1}(r_{t+j} - \Delta d_{t+j})$ and one can easily show that $E_t(r_{t+j}) = b_x^{j-1} x_t + b_y^{j-1} y_t$ and $E_t(\Delta d_{t+j}) = 0$ and so we get

$$d_t-p_t = \frac{x_t}{1-\rho b_x} + \frac{y_t}{1-\rho b_y} $$

I'm now stuck with trying to show that this is an ARMA(p, q) process. What I've tried is substituting in $x_t$ and $y_t$ to get:

$$d_t - p_t = b_x \frac{x_{t-1}}{1-\rho b_x} + b_y \frac{y_{t-1}}{1-\rho b_y}+ \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y}$$

But I am stuck on manipulating the RHS to form a $d_{t-1} - p_{t-1}$, specifically, I know that $d_{t-1}-p_{t-1} = \frac{x_{t-1}}{1-\rho b_x} + \frac{y_{t-1}}{1-\rho b_y}$ but the coefficients $b_x$ and $b_y$ on the RHS makes things difficult.

EDIT: I've done a bit more working out and here is what I've done:

\begin{align*}
d_t - p_t & = \frac{x_t}{1-\rho b_x}  + \frac{y_t}{1-\rho b_y} \\
& = \frac{b_x x_{t-1} + \delta_{x, t}}{1-\rho b_x} + \frac{b_y y_{t-1} + \delta_{y, t}}{1-\rho b_y} \\
& = b_x \frac{x_{t-1}}{1-\rho b_x} + b_y \frac{y_{t-1}}{1-\rho b_y} +  \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\
& = b_x\left(d_{t-1} - p_{t-1} - \frac{y_{t-1}}{1-\rho b_y} \right) + b_y\left(d_{t-1} - p_{t-1} - \frac{x_{t-1}}{1-\rho b_x} \right)+  \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\
& = (b_x + b_y)(d_{t-1} - p_{t-1})-b_x\left(\frac{b_y y_{t-2} + \delta_{y, t-1}}{1-\rho b_y} \right)-b_y\left(\frac{b_x x_{t-2} + \delta_{x, t-1}}{1-\rho b_x} \right) +  \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\
& = (b_x + b_y)(d_{t-1} - p_{t-1}) - b_xb_y\left(\frac{x_{t-2}}{1-\rho b_x} + \frac{y_{t-2}}{1-\rho b_y} \right) - b_x \frac{\delta_{y, t-1}}{1-\rho b_y}- b_y \frac{\delta_{x, t-1}}{1-\rho b_x}+  \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y}  \\
& = (b_x + b_y)(d_{t-1} - p_{t-1}) - b_xb_y(d_{t-2} - p_{t-2}) - b_x \frac{\delta_{y, t-1}}{1-\rho b_y}- b_y \frac{\delta_{x, t-1}}{1-\rho b_x}+  \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y}
\end{align*}

So it seems like there are two lags for $d_t-p_t$ but I am not sure how to manipulate the moving average terms...

Hi: I don't know where you got the expression for $d_{t-1} - p_{t-1}$ but assuming that's true, then I think you are basically there. Take your final expression and  consider the two lagged white noise terms first. They can be added together to make one lagged white noise term because I'm pretty certain that the sum of two white noise processes is still white noise. Similarly, the two non lagged white noise terms can be added to make one non-lagged white noise term. So, you end up with an MA(1) so the whole expression is ARMA(2,1). Granted the MA(1) part will have some complicated constant coefficients on both the lagged and non-lagged term but they are not terribly important. Also, you can always normalize so that the non-lagged coefficient of the MA(1) piece is 1.0.  If you or anyone thinks this reasoning is flawed, let me know because I don't see the flaw in it and am all ears. Thanks.

$\boldsymbol{EDIT}$

Elbarto: I've dealt with ARIMA models quite a bit over the last 5-6 years so this was bothering me so I kept at it. It turns out ( after many readings that may have been in my sub-conscious initially ) that the 4 last terms of your original expression do represent an MA(1) process. The mistake I made is that my argument was el-wrongo. The following is the argument that's used in the literature-texts for showing what process any ARIMA type "process" reduces to.

1) Construct the autocorrelations at each lag of the process.

2) If that  calculation results in a functional form that behaves in the same way (in terms of its shape) as some well known ARIMA model, then the original unknown process is equivalent to that well known process.

Unfortunately, we were unable to derive the process was using your methodology which is more direct and, to me, more intuitive.  The 1) and 2) argument above stems from the fact that any stationary and invertible ARIMA model has a 1 to 1 mapping between the model parameters and the model  autocorrelations. Therefore, if you know the autocorrelation shape, you know the process. I'm not absolutely sure if the stationary part of the theorem is even needed and, in fact, I don't even know if it's an actual theorem. I just think of it as a theorem but it is true.  The best analogy to this argument would be knowing the impulse response in the DSP framework. Still, for the intuition,  I like your approach better.

So, to make a long story short,  using this approach,  take those 4 terms as some unknown  ARIMA process and calculate its autocorrelation. Since that process clearly has an non-zero autocorrelation at lag 1  and zero after that, it is an MA(1) process.  This is a specific case of the generic argument used in the literature  but it's slightly unsatisfying given how your method seems quite reasonable also. My thinking is that the  one to one mapping  theorem must be equivalent to what you (and I later) were trying to do.

Finally, if you want to be bothered ( I didn't want to be ), you could calculate the autocorrelation of that 4 term process. It will be some function of the three parameters. Then set that function equal to $\frac{\theta}{(1+\theta^2)}$ and  hopefully ( it looked ugly to me  and I'd probably make a mistake somewhere even if I could be bothered) solve for $\theta$. That implies that the four term  process is  an MA(1) with parameter $\theta$. Anyway, the usefulness of calculating it is that you would then have $\theta$ as some function of of the other parameters: $b_{x}$, $b_{y}$ and $\rho$.

With that in hand, it may become more clear as to what the algebraic argument is that would show the same thing ??? We must be thinking about it in some incorrect way that I can't see (a scaling-normalization issue perhaps ?).

I'm reasonably satisfied now so, if you can't be bothered, I understand. All the best and thanks for neat question. I learned something from you and also during my googling journey.

$\boldsymbol{Last~~ Comment}$

Elbarto: Just one last thing. Keep in mind the same autocorrleation argument could have been used for the full expression that includes the AR(2) term. I have no idea how difficult that is but it's not nearly as easy as the MA(1) because now the ACF dies out slowly or oscillates depending on the roots. In fact, I bet it would be a lot more work than you did to show the AR(2) directly so not really worth pursuing.


Good book on capital flows? Are there any good textbooks/books for capital flows?

I have found some from searching online, but they are a bit older:
International Capital Flows by Feldstein in 1999
Capital Flows and Crises by Eichengreen in 2002

I am wondering if there is anything out there that is more recent?

Here are some recent books:

Taming Capital Flows: Capital Account Management in an Era of Globalization (2015)
Capital Flows and Exchange Rate Management (2013)
Managing Capital Flows: The Search for a Framework (2010)
Capital rising: how capital flows are changing business systems all over the world (2010)


Given a Utility function, U(x,y), why is multiplying U(x,y) * x not a monotonic transformation? If $x$, is a commodity and it is s.t. $x \geq 0$ (it is always nonnegative) how will multiplying it times a utility function $U(x,y)$ NOT yield a simple monotonic transformation?

Because $x$ is not a constant, and therefore multiplying $U(x,y)$ by $x$ does not necessarily preserve the ordering between bundles.

For instance, consider the function $U(x,y)=x+y$, and the bundles $(x_0,y_0)=(2,0)$ and $(x_1,y_1)=(1,2)$. We have
\begin{equation*}
U(x_0,y_0)=2 < U(x_1,y_1)=3
\end{equation*}
but
\begin{equation*}
x_0 U(x_0,y_0) = 4 > x_1 U(x_1,y_1) = 3
\end{equation*}
Hence multiplying $U$ by $x$ changes the preferences, therefore is not a monotonic transformation.

As a side note, multiplying the utility function by a constant provides a monotonic transformation if the constant is positive, and not only nonnegative as you assumed (if the constant is zero the orderings are obviously not preserved).


Ttest and heteroscedasticity problem in no intercept model I was running a t test over two regression betas with the assumption of equal variance. I know that if the condition of homoscedasticity do not hold then there are chances to have type 2 error but what if in my regression I do not have the intercept? (The models without errors do not require the error mean to be zero).
I am very confused, I may have mixing two different concepts so, please tell me the associated readings to get it more clear.

You are generally better off asking statistics questions at cross validated, simply because there are more statisticians there. Also, you will get a better answer if you supply some more detail of what you want to test exactly, and what program you use if you want hints on where to read up on these things

I assume that you want to test whether two coefficients that you found in a linear regression are equal or different from another, where your regression model has no intercept. That is an example of a linear restriction. These kind of restrictions are generally tested with F-tests. You can read up on them a.o. here (which implements them in R).

How to implement them depends on your stats program.


Research at the county-level: Macro or Micro? Is research at the county-level considered applied micro or applied micro?

The standard undergrad definitions ("micro" = "individual", "macro" = "nation") are of no help with units of measurement that lie in between.

Any help is appreciated.

I would consider county-level research macroeconomics.

Microeconomics focuses on individuals and firms acting as decision makers. I think, looking at county-level aggregates, you are too high level to look at decision makers.


What happens when I leave out empty cells in regression? I'm using Stata 14.1 to do a regression, and I got a matsize too small error. It gave some more output to tell me possible reasons for this problem, and I think this was the reason that applied to me

If you are using factor variables and included an interaction that has
lots of missing cells, either increase matsize or set emptycells drop to
reduce the required matrix size; see help set emptycells.

Yes, I am using factor variables (one for each state in the US and Washington DC), and yes there are a lot of empty cells.

I'm trying to figure out which option is better, drop empty cells or increase the matrix size so it can include all the factor variables.

It has help articles on how to do both of those things: drop empty cells or increase matsize. It's not the "how" but the "why" that I'm trying to figure out. What happens mathematically when I drop empty cells? (I probably won't understand the math, so if there's a dumbed down answer just about whether it's good or bad to leave off empty cells, that would probably be better.)

When you have empty cells (and by empty cells, I assume that you are referring to a missing value of an observation specific variable), then STATA by default drops the entire row. So in a sense, even when you have a single missing value on a variable and you have say 100 variables, you lose the whole row (the entire observation). This is by default. If you want to avoid this, perhaps think about imputing the missing data points.


Trying to understand ceiling in supply and demand on picture I am having one problem, which I do not understand. Let's assume the government applies ceiling price below the equilibrium price. I colored it red in my graphs.

Of course, supply would decrease, there would be a shortage. But why does everyone plot the supplier and consumer surplus as it is in the left picture?

Let's think about it. If the government says the new price is N dollars for something, this means that there are a lot more people who are willing to pay the N price. I just don't understand why we always plot it like those who are willing to pay the most will get it, since new N dollar price is now acceptable for many more people and they would grab for an item as fast as those who are willing to pay the most. What I am trying to say is, if there is a ceiling price for something, it's not affordable only for those who are willing to pay the most but by everyone who is willing to pay above the threshold of ceiling price. But I see no one plotting the model as I did on the right side of the picture. Could you please explain to me why?

This is due to not enough supply. So yes you are correct, more people can afford it now, but because the price is less, there is not enough supply for everyone, even if they can afford more than it cost.

It seems you are not taking into account dead weight loss.
Here is a link to help you understand it and use it.
https://corporatefinanceinstitute.com/resources/knowledge/economics/deadweight-loss/

An inefficiency occurs since at the price ceiling quantity supplied the marginal benefit exceeds the marginal cost. This inefficiency is equal to the deadweight welfare loss.

When it comes to rent ceilings, certain things almost always happen that make the people willing to pay more in order to get the product vs people willing to pay only the rent ceiling price.

If a price ceiling is set, then there must be a way to assign who gets the low supply of the product. Of course, since there is a legal limit on the price, the price can't simply be raised. There are several ways this is done without raising the price:

Black Market: For those lucky enough to get some of the short supply, they are often better off selling what they have obtained to the demanders that will get more benefit out of it. In some cities there have been ceilings put on the apartment rent. While the demand for apartments increases, the rent remains the same. When some renters are ready to move, they sublease their apartment instead of ending their contract. If they were renting for \$500, but someone who  is  willing  to  pay  \$1000, then the subleaser can continue paying \$500 and  pocket the extra \$500 he gets from the subleasee.


Effect of today's decision in the US to no longer mandate contraception coverage Today the Trump administration in the USA announced they are removing the mandate that company provided health insurance include free contraception coverage. My understanding is that providing contraception for free actually reduces the overall healthcare cost in the US, since the price of pregnancy and births is extremely high compared to inexpensive pills. Therefore, from a strictly economics standpoint, I would expect that companies that elect to not cover contraception might pay higher premiums than if they were to choose policies that included coverage. Is this the case? Prior to today did religious organizations and private companies that chose not to cover contraception actually pay higher premiums, or did insurance companies just raise everyone's premiums to compensate? (Or is my original premise flawed that premiums should be higher when contraception is not covered?)

There are different arguments here, depending on the points of view of the government and the insurance providers. I'm trying to answer to the general question on the title.

Insurance providers aim at minimizing their costs with healthcare and the probability of health issues. Free birth control not only prevents unwanted pregnancy, but might also have other health benefits. This prevention would save insurance providers from the costs of unwanted pregnancy, which are not only the birth but also psychological costs. I am not aware of any studies that would compare both situations. Insurance providers could benefit from this measure if they know that women will buy the birth control anyways, so the insurer free rides on the benefits of the pill.

The government should try to maximize the "society's profit" or well being, coming from healthcare, but also from other aspects like the fulfillment of values. The logic is to balance the "harm" inflicted by this policy to some groups (say, some religious citizens could say they feel bad for living with these policies) and the losses in profit inflicted to insurance providers or employers, with the potential benefits to society. The book Freakonomics makes an interesting correlation with approving abortion and reducing crime in New York, which could be an argument to give it for free (social benefit of less crime). Note that the argument was later refuted in The Economist, but might still be used by some people who only read the book.

Who could benefit from this change in policy, besides what is mentioned above (my own opinion, discussable)

Pharma industry, with people choosing more expensive pills when they are not covered by the insurer
Religious groups and anyone who disagrees with the concept of free birth control
President Trump, by gaining more approval from these groups
(Far fetched) Hospitals and doctors who could increase their services


Reference request about econometric analysis of Environmental Kuznets curve I could not find from where to learn the application of Econometrics on data analysis relating to environmental Kuznets curve. Are there any books or articles which talk about the above?

There is a very good review article by Richard Carson in the review of environmental economics and policy, which (perhaps unfortunately) shows that much of the empirical evidence for an environmental Kuznet's curve is weak at best.


Principal Agent Question Private Benefit If an agent can take action which impacts social welfare in a noisy way, the private net benefit of the action is $P(x,t)=x-s(\frac{x^2}{2} + t)$, $s$ is $(0,1)$
How can I determine the privately optimal t?

If we try to directly optimize with respect to $t$, given this system (two unknwowns, one equation), we get a nonsensical result.

$$\frac{\partial P}{\partial t}: -s = 0$$
Which may not be true, and I'm assuming you are taking $s$ as given, not something to choose. You can only optimize with respect to $x$.

$$\frac{\partial P}{\partial x} = 1 - sx = 0 \implies \boxed{x^* = \frac{1}{s}}$$

What does this tell you about the optimal value of $t$ if anything?


What are good mathematics books to learn decision theory? I am looking for a set of mathematics books to understand proofs in decision theory. Examples would be proofs of utility representations and social choice (im-)possibility results. I found many textbooks on both decision theory and social choice, some of which contain proofs which contain references to mathematical results.

However, I could not find a collection of mathematical results commonly used in these fields. Usual "Mathematics for Economists" textbooks fall short. What is the standard reference/textbook researchers use to find the results they employ?

I do not know about social choice, but for utility representations I think the two most cited books are "Convex analysis" by Rockafellar and "Infinite Dimensional Analysis: A Hitchhiker's Guide" by Alliprantis and Border. They contain most (if not all) results on convex analysis and functional analysis used by economists.


USA seigniorage revenue I came to know about the term seigniorage meaning the difference between the value of money and the cost to produce it. I also read somewhere that many countries use seigniorage as a means of revenue inorder to reduce the requirement of or add to the revenue from taxation. I want to know:

How much revenue does a country make by printing money? (Information on any country will be appreciated)
What are the specific laws (in the country mentioned in the answer to Q1 ) that govern the specifics of printing money (like amount, date, period etc.)?

N.B : Information on any country(or countries) will be appreciated.

In former times, seignorage was a de facto fee charged by a government authorized mint for converting precious (or semi-precious) metals into de facto trademarked hard currency.  Typically a mint would charge about a ten percent fee for this service.  Anyone in possession of the hard currency could "redeem" the currency by melting it down back into the original metals, and thereby recover the remainder of the currency's value.

In countries that do not rely on hard currency for most daily transactions, seignorage occurs in two stages:

A government borrows purchasing power, and issues a coin or note in exchange.  The government promises to repay the purchasing power upon return of the coin or note.  An accountant would say that this is the "cash flow" part of the seignorage transaction.
The government defaults on the promise to repay the purchasing power.  This can happen explicitly (like when the U.S. refused to repay silver certificates in silver, or when the U.S. "closed the gold window") or implicitly via inflation of price levels stated in government notes.  An accountant would say that this is the "revenue" part of the seignorage transaction.

According to the U.S. Federal Reserve, about 1.5 trillion (1.5 million million) U.S. dollars were in circulation in 2017.  This amount has been increasing at the rate of about 90 billion (90,000 million) U.S. dollars per year.

So the rate of the U.S.' seignorage incremental borrowing per year is about 90 billion dollars per year.

The official U.S. inflation rate has been in the range of 0 to 3 percent per year during the last few years.  If the average is 1.6 percent per year, then the amount of seignorage revenue (i.e. de facto default on the purchasing power of U.S. dollar notes in circulation) is about 25 billion (25,000 million) U.S. dollars per year.

In addition, the U.S. Federal Reserve purchased about 4 trillion (4 million million) U.S. dollars worth of mortgages and other securities during and shortly after the 2008 financial crisis.  It "increased the monetary base" when it purchased these securities.  In other words, it used seignorage to provide the electronic money used to purchase these securities.  At an average inflation rate of 1.6 percent per year, this is about 65 billion (65,000 million) U.S. dollars of seignorage revenue per year.

The U.S. Federal Reserve uses its net income from (after being laundered from seignorage revenues into net interest on securities and capital gains on securities sales) to make dividend payments to the United States Treasury.  From 2010 through 2013, these dividend payments were about 75 billion (75,000 million) U.S. dollars per year.


Prove relation between expected utility and utility of the mean Assume $U(x)$ is a concave, monotonically increasing function. If we know that

$$U\left[E(p)\right] > U\left[E(q)\right]$$

where $p$ and $q$ are probability distributions, how can we prove that

$$E(U_p) > E(U_q)$$

The conclusion you want to prove is not always true. I will give you a graphical and mathematical answer.

Graphical analysis

Consider the example in the image below:

There is one lottery, with expected payoffs $z_1$ and $z_2$, and average $E(z)$. The expected value of this lottery in terms of utility is $U[E(z)]$.

Now, assume a second lottery, with higher $z_1$ and smaller $z_2$, such that the expected value is the same, i.e. $E(z)$. The expected value of this lottery is the same, $U[E(z)]$. Yet, because of lower variance in the second lottery, the point E in the second lottery is closer to D. (In the extreme case of no uncertainty, D and E coincide). In other words, $E[u(z)]$ in lottery one is lower than $E[u(z)]$ in lottery two, even if they pay the same on average.

The contradiction of your statement follows immediately. Take the second lottery just introduced, and decrease $z_1$ by a small $\epsilon$. Now, the expected income of the lottery is marginally lower than $E(z)$, which is the opposite relation to the one in your question. There clearly is a small $\epsilon$ such that the relation between $E[u(z)]$ in the two lotteries remains as before.

The proof follows the same logic for the case of risk loving individuals (where uncertainty is preferred to certainty and the utility function is convex).

Actually, the only case where your statement does hold is for risk neutrality. This can trivially be seen in the graph too.

Mathematical analysis

First, we know that $\dfrac{d U(x)}{d x}>0$. Then, it is true that:

$$ U(x_p + \pi)=E[U(x_p)] $$

$$ U(x_q + \phi)=E[U(x_q)] $$

where the term inside the parenthesis in the left-hand side are the certainty equivalent. The sign of $\pi$ and $\phi$ depend on the nature of the utility function. There are three cases:

Risk aversion

If utility is concave (i.e. agent is risk averse), $\pi$ and $\rho$ are positive. Therefore:

$$ U(x_p) < U(x_p + \pi) = E[U(x_p)] $$

$$ U(x_q) < U(x_q + \phi) = E[U(x_q)] $$

Assuming that $U(x_p) > U(x_q)$:

$$E[U(x_p)] = U(x_p + \pi) > U(x_p) > U(x_q) $$

However, $E[U(x_q)] > U(x_q)$. It is evident that nothing more can be said regarding the relation between $E[U(x_p)]$ and $E[U(x_q)]$. The result you want to prove does not necessarily hold in this case.

Risk acceptance

The proof is exactly the same, but reversing the signs.

Risk neutrality

$\pi$ and $\phi$ are zero.

$$ U(x_p) = E[U(x_p)] $$

$$ U(x_q) = E[U(x_q)] $$

Assuming $U(x_p) > U(x_q)$ implies that:

$$ E[U(x_p)] > E[U(x_q)] $$

Thus, in this case the conclusion you want to prove is true.


Translog Preferences What are translog preferences? The wikipedia article only clears up that it stands for transcendental logarithmic preferences, and that they are a generalization of Cobb-Douglas preferences.

Do they have special features that make it more appealing? I have never seen these being used in Macroeconomics.

The translog function can be used not only in preferences but also in production and cost functions. I am not very familiar with its implications in consumer theory, but from the production point of view, i've seen it widely used.
The Translog Function doesn't impose additivity and homogeneity, and hence Constant Elasticity of Substitution. This is interesting because it doesn't require a "smooth" substitution among the inputs (in production analysis). I guess that in consumer theory the interpretation would be similar.
So basically, the translog function is less restrictive than a cobb-douglas. If you impose some restrictions while calculating the translog function parameters, you get a cobb-douglas function. That is why it is a "generalization". In other words, the cobb-douglas is a specific case of the Translog function imposing additivity and homogeneity (i.e. imposing constant elasticity of substitution).

Edit: i added more info to reply your comment.
I think that the other reply is more complete than mine. But i'm just going to add something i consider useful for you to have a broader understanding. I assume that you're familiar with indifference curves. I refer you to this site (from where i took the graphs), in case you're not.
An indifference curve is just a mapping of all the combinations of two (or more) goods that give you the same utility, or "make you happy at the same level".
First, see this indifference curve:

Fig 1: source
This setting is known as "complements". Because as you can see, adding one thousand units of good x (that is moving to the right), without adding good y (that is not move upwards) does not make you happier: you move along the indifference curve. Think of this as the left shoe and the right shoe. It is useless to have a thousand additional left shoes without adding a right shoe because they are perfect complements.
Now, look at this one:

Fig 2: source
This one is called "substitutes". It's the opposite case to the complements. You can think of this as beef and chicken. You can cook using only beef, or you can substitute and cook using only chicken. But you can cook with certain combination too, say 150 grams of beef and 100 grams of chicken beccause they are perfect substitutes (Sorry, i couldn't come up with a better example but this one makes the point).
Now, this extreme cases make easier to imagine all the settings that are "in the middle". That is, two types of good that are not perfect complements neither perfect substitutes. Think of food and drinks. They can't be perfect substitutes because you can't have lots of food without drinks. The're not perfect complements either because the mix of food and drinks is not fixed. For this setting the cobb-douglas could be a nice approximation as can be seen in the next figure:

Fig 3: source
Now, the Cobb-Douglas utility function doesn't solve everything, as it imposes certain constraints by construction. For example, the line that goes from the origin through all the curves (the expansion path) is 45° and straight by construction: it cannot be changed. This means that as you get richer (even infintely rich), your preferences over this goods remains constant. The formal name is homotheticity or homothetic preferences. This is empirically false, as it has been shown that the richer you are, you use a smaller share of your income for food. With Cobb-Douglas preferences, this cannot happen. Translog preferences relaxes this assumption.
In the next figure, you have a utility map relaxing the homotheticity assumption:

Fig 4: source
Think of this graph as good y being food and good x being entertainment. As you get richer (or farther from the origin), you will destin more of your income to entertainment.
Finally, i will talk about the Elasticity of Substitution which is known as $\sigma$ (sigma) which can be imagined as being the curvature of the indifference curve. In the Fig 1, the perfect complements $\sigma = 0$: No curvature. In the perfect substitutes, $\sigma = infinity$: straigt line. In Cobb-Douglas, $\sigma = 1$: a slight curvature. Nonetheless, as you get richer (distant to the origin), this Elasticity of substitution remain constant in the three settings. Even in the non homothetic preferences seen in Fig 4, the elasticity of substitution remains constant. These are the **Constant Elasticity of Substitution (CES)**preferences. But what if you allow the curve to have different shapes as you get richer? Look at Fig 5:

source
In this example, the indifference curves get less elastic every time. Hence, they are not CES preferences. The advantage of the Translog preferences is that, since you don't impose neither CES neither homotheticity, you can test this hypothesis with observed data. You can see that the Translog utility function is much less restrictive than the Cobb-Douglas preferences.
As a final remark, i shall say that it can be the case that you do not reject the hypothesis of homotheticity, CES and $\sigma = 1$ in a dataset of observed behavior. This would leave you in a Cobb-Douglas preferences setting. So, by using Translog you're not necessarily ruling out Cobb-Douglas.


Interest rates, inflation and liquidity effect I am currently reading Alvarez, Lucas, Weber (2001) where the authors argue that changes in money supply is more important for affecting inflation, than changes in the interest rate. However, I am confused about this part:

  To be useful in thinking about the role of interest rates and open market operations
  in the control of inflation, a model of monetary equilibrium needs to deal with
  the fact that most coherent monetary theories do not have anything like a downward
  sloping demand for nominal bonds: With a complete set of financial markets, it is
  just not true that when the government buys bonds, the price of bonds increases. We
  may believe that such a “liquidity effect” occurs in reality (though it is hard to see
  it in the data) and may regard it as a deficiency that so much of monetary theory
  ignores it, but the fact remains that one cannot take take a Sidrauski (1967), Brock
  (1974), or Lucas and Stokey (1987) model off the shelf and use it to think about
  increases in money reducing interest rates.

How come when the government purchases bonds, the price of said bond doesn't increase? The authors argue that with an unsegmented market, this does will not happen (i.e there will be no liquidity effect). However, when markets are segmented there is a liquidity effect. The market is segmented in to two groups, traders (have access to the bond market and goods market) and non-traders (only trade in the goods market).

They do develop a model and I more or less understand the mathematics behind it. They show the phenomenon mentioned above using their model. However, I don't quite understand the intuition behind it. Why is there no liquidity effect when markets are unsegmented? Why don't bond prices rise with government purchases?

The reason why supply effects would not occur is the result of market efficiency assumptions: all financial assets have the same expected returns. Within the bond market, that would imply that the "term premium" is zero.

If the term premium is zero, then:

The value of the 1-period "bond" is the result of the central bank reaction function. (The policy rate.)
The value of a 2-period bond is determined by the 1-period yield, and the unbiased expectation of the 1-period yield, 1-period forward.
(Build up rest of curve similarly.)

They added market segmentation as a way of allowing bonds to trade away from the predicted ('efficient") value, that is, allow a non-zero term premium. Once we allow for a non-zero term premium, supply effects can be seen in predicted bond pricing.

The term premium interacts with supply as a positive term premium will induce bond investors to increase the holdings of bonds relative to the allocation that would happen if the term premium were zero. That is, changes in the term premium is needed to balance the market.


Is it possible to have a signaling game in which all players are both informed (about themselves) and uniformed (about other players)? So far, the models of Signaling Games I have seen involve two players. One player is known as the Sender and has more information. The other player is known as the Receiver and has less information.

But suppose I have two people who each don't have information about the other.  Now they start sending signals and learn about the other.

Are there models of Signaling Games that cover such dynamics where both players are senders and receivers?

A comment on the number of players in signaling games in general.

Signaling games need not have only two players. In the cheap talk literature, there are papers that study signaling games with multiple informed senders and one uninformed receiver (e.g. Krishna & Morgan, 2001), or one informed sender with multiple uninformed receivers (e.g. Farrel and Gibbons, 1989). There are no theoretical reasons to limit the number of players to two.

Is it possible to have a signaling game in which all players are both informed (about themselves) and uniformed (about other players)?

The answer is yes. For example, auctions with pre-play communication, where each bidder is informed about their own valuations of the object but not about the others'. In the communication stage, they use signals to strategically (mis-)communicate their private information, and to infer about other bidders' types from their signals (e.g. Mathews and Postlewaite, 1988). In general, mechanism design is about signaling privately observed types. But the specific model depends on what you want to do after the signaling stage, whether signals are costly, and what kind of outcome you want the model to deliver, etc.


Sum of Homothetic Functions If two utility functions represent homothetic preferences, will their sum also be homothetic?

Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$.

Defn: A function is homothetic if it is a monotonic transformation of a homogenous function.

Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then
$$
\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} = \frac{g'(u(x,y))\frac{\partial u}{\partial x}}{g'(u(x,y))\frac{\partial u}{\partial y}}=\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}
$$
is homogenous of degree zero.

Let

i) $u_1(x,y)=x+y$,

ii) $u_2(x,y)=\log(2x+y)$

Then, $u_1$ and $u_2$ are homothetic functions since they are monotonic transformations of homogenous functions (of degree 1). Let

iii) $u_3(x,y)=x+y+log(2x+y)$. Then

$$
MRS_{u_3}=\frac{\frac{\partial u_3}{\partial x}}{\frac{\partial u_3}{\partial y}}=\frac{1+\frac{2}{2x+y}}{1+\frac{1}{2x+y}}=\frac{2x+y+2}{2x+y+1}
$$
which is not homogenous of degree 0.

Hence, sum of homothetic functions is not necessarily homothetic.


Unsure how to go about computing GDP for the following problem I came across this question where the GDP for the years 1995-1997 is given. Then I was asked to compute the growth rate every year which I'm aware of.The next question was to compute the average which is simple divison.However, the final question asks me to predict the GDP for the year of 2012 using the average rate of GDP between 1995-1997. I understand the tedius way to go about solving this problem is to keep computing the GDP from 1997 all the way upto 2012. Is there a simplar way I can determine the GDP of a particular year, using an average growth rate?

Generic growth formula:

$$\frac{V_t}{V_{t_0}} = (1 + r)^{t-t_0}$$

Solving for $r$ will give you a growth rate in terms of the units of $t$ and $t_0$.  In your example, that's years.

You currently have a growth rate and the value in 1997.  Projected value in 2012 would be

$$V_{2012} = V_{1997} \cdot (1 + r)^{2012-1997} = V_{1997} \cdot (1 + r)^{15}$$

This assumes that $r$ is the decimal (not percentage) average growth rate that you calculated.

I'll leave it up to you whether your original calculations of the growth rate are consistent with this method of calculating the growth rate.  If not, you may want to recalculate it.

The formula that I gave is a generic one.  While you may be applying it to GDP, this will work for the growth rate of any variable.  That's why I used $V$ for value rather than a variable more commonly recognized as representing GDP.


If one always spends the same amount, what's one's price elasticity of demand? Source: p 109, Question 5.9, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw
= Question 5.7, Principles of Microeconomics, 4 Ed, 2008, by N Gregory Mankiw

  9. $\color{green } { \text { Before looking at the price,}   }$ Jessie spends $d$ dollars on something  (abbreviate this S). What is Jessie’s price
  elasticity of demand?

  Given Answer: Jessie's price elasticity of demand is one, because he spends the same amount on S, no
  matter what the price, which means $\color{darkred} { \text { his percentage change in quantity  }   }$ is equal to the percentage
  change in price.

  From p 91: Price elasticity of demand $= \dfrac{ \text{
Percentage change in quantity demanded }} {  \text{ Percentage change in price } } $

The green implies Jessie's ignorance of prices, so % change in price = 0. Yet please explain the answer? Especially the red, because the question implies nothing about the % change in quantity demanded?

Footnote: I slightly generalised the question. Mankiw writes 'gas' instead of  'something'.

The intuition is that if Jessie spends the same amount on $S$ no matter the price $p$, then a % change in price of price will reduce the demand by the same % . The thing that might confuse you is that the formula you are given is an approximation for the true price elasticity of demand given by
\begin{align}
     \eta=\frac{dS\Big/ S}{dp\Big/ p}
 \end{align}
 where $d$ is the derivative operator. The approximation
 \begin{align}
     \frac{\%\Delta S}{\% \Delta p}
 \end{align}
 only works for very small changes in $p$ and $S$.
 To see why the elasticity of demand is $\mid 1 \mid$, let $b$ be the fixed amount spend on $S$. Then the quantity demanded $S$ is given by
\begin{align}
    S=\frac{b}{p}
\end{align}

Assume that there is a small price change from $p$ to $p+\epsilon, \, \epsilon>0$. Then we can define $S_1=\frac{b}{p}$ as the demand before the price change and $S_2=\frac{b}{p+\epsilon}$ as the demand after the price change. Thereby we see that
\begin{align}
    \frac{S_2\Big/ S_1 -1}{\frac{p+\epsilon}{p}-1} = \frac{\frac{b}{p+\epsilon}\Big/\frac{b}{p}-1}{\frac{p+\epsilon}{p}-1}=\frac{\frac{-\epsilon}{p+\epsilon}}{\frac{e}{p}}=-\frac{p}{p+\epsilon}\approx -1
\end{align}
when $\epsilon$ is small enough.


Do governments issue debt at low r to pay debt with high r? I have a question. I know governments do debt rollover from time to time. But, to me it seems logical that governments do this when the interest rate of bonds they have issued in the past is higher than the one they could get now. This way, they pay more expensive debt with cheaper debt. To me this seems like a no-brainer. But is this what they do?

With current QE and low interest rates, are governments doing this? I think they are not. Why?

The phrase governments "do debt rollover from time to time" is an understatement. The entire stock of Treasury bills is rolled over continuously, as well as the coupon debt. For example, a 3-month Treasury bill needs to be rolled over four times a year.

As for the current situation, it is somewhat strange.

Government fiscal agencies (Treasury, Ministry of Finance) have tended to increase the average maturity of their debt issued. (The answer by Facet_Velvet discusses this, but in order to keep this answer self-contained, I will provide the same link: link to OECD report.)
Many (but not all) developed central banks have been buying government bonds, replacing them with short-maturity governmental liabilities (deposits at the central bank).

These two actions tend to cancel each other out. The only defense of the policy that I can offer in this context is that the central bank is the party taking the interest rate risk. (The central bank has nominally independent finances.) As a disclaimer, I think Quantitative Easing as a policy makes no sense, so I cannot come up with a good rationale for this policy mix.

However, the answer to your question is that we cannot think of the central government as being like a household or business. As the currency issuer, it has a very different perspective on finances, and minimising interest costs is actually not an important policy objective.

For example, it is extremely easy for the government to minimise interest costs: it just needs to set interest rates at 0% by legislative fiat. (Some may object that this is impossible; the experience with interest rate pegs during and after World War II show that the policy is feasible.) However, just because we can minimise interest costs does not mean that it is a good idea. In order for monetary policy to work, the government needs to have bonds that set the rate of interest at a positive level, which can then be varied by the central bank to help control the business cycle.

The primary interests of the government in issuing bonds are:

Create a yield curve that act as a benchmark for private sector borrowing. When Australia was running large surpluses, I believe that the Australian banks pleaded for continued government bond issuance to keep the yield curve liquid. (I do not have a reference.)
Have a liquid Treasury bill market that gives risk-averse investors a place to park cash. In particular, central bank foreign exchange reserve managers are not willing to take large amounts of interest rate or credit risk, and so you need a liquid Treasury bill market if you want foreigners to hold your currency as foreign reserve assets.
They need to meet the demands of institutional investors for long-dated credit risk free bonds.

These competing demands imply a continued juggling act to balance out issuance at various maturity points.

Finally, governmental debt managers do not want to lock in their portfolio of issued bonds as long-dated instruments, as they can do nothing about them. Unlike mortgages, almost all government bonds cannot be repaid early. (The U.S. issued some 30-year callable bonds in the early 1980s.) If the debt is termed out, and the the government has a fiscal surplus (which happened for a brief period in the late 1990s), the government would end up with no Treasury bills outstanding and a surplus of liquidity. It would have to go into the private markets and lend to private sector entities with that surplus, which the government does not want to do. Hence, they need to have a sizeable amount of short-dated paper that can be run off in such a situation.


Set of consumption over all period is convex in $\mathbb{R}^T_+$? Today in class, the professor said that the set of all consumption $c(S)$ is non-empty, compact and convex subset of $\mathbb{R}^T_+$. i.e. we know $\sum \limits_{t=1} ^T c_t = S$ where $c_t$ is consumption in period t and $S$ is total wealth. The set  $c(S)$ is the set of all T-period consumption plan.
I could understand why $c(S)$ is compact, but I have no clue why it has to be convex and non-empty.
Following the same question, the professor also mentioned that $W(c)$ the summation of all the T-period utility functions denoted by $W(c)=\sum \limits_{t=1} ^T U(c_t)$ is continuous on $c(S)$. I do not understand why $W(c)$ has to be  continuous on $c(S)$
I think this has something to with Berge's Maximum theorem, but I am unable to link it properly. Any help will be very appreciated.

Assuming the following:

$ S > 0 $, $T > 0 $,
$ c(S) = \left\{c\in\Bbb{R}^T_{+} : \sum_{t=1}^T{c_t} = S \right\} $, and
$ U(\cdot) $ continuous.

Let's start with convexity. To demonstrate convexity, we need to show that for any two points $c^1$ and $c^2$ in $c(S)$, any linear combination of the two is also an element of $c(S)$.
Let $\lambda \in [0,1]$, $c^1 \in c(S)$, $c^2 \in c(S)$. Then, we have
$$ \sum_{t=1}^T{\lambda c^1_t + (1-\lambda) c^2_t} $$
$$ = \sum_{t=1}^T{\lambda c^1_t} + \sum_{t=1}^T{(1-\lambda) c^2_t} $$
$$ = \lambda \sum_{t=1}^T{c^1_t} + (1-\lambda) \sum_{t=1}^T{c^2_t} $$
$$ = \lambda S + (1-\lambda)S $$
$$ = S $$
$$ \Rightarrow \lambda c^1 + (1-\lambda) c^2 \in c(S) $$
Since any linear combination of any two points in c(S) is itself in c(S), we have c(S) convex.
For non-emptiness, we just need to show that a single point exists in c(S):
$$ \sum_{t=1}^T{\frac{S}{T}} = S $$
$$ \Rightarrow \left\{ \frac{S}{T} \right\}^T \in c(S) $$
If $U(\cdot)$ is continuous on $c(S)$, then continuity of $W(c)$ follows from the triangle inequality (see e.g., here or here). If we don't know more about $U(\cdot)$, then there's not much we can say about $W(c)$. Berge's Theorem of the Maximum has to do with the continuity of an optimized function, not the sum of functions.


Please help a layman distinguish among sorts of normal goods I'm trying to organize my thinking about differences among types of normal goods, and I'm sure that someone smarter than me has already done the necessary work.
Here's one sort of distinction that I have in mind:
A is willing to spend more money for a better graphics card because it improves his experience watching movies. He is not spending more money to gain any sort of competitive advantage over other users, and his streaming services never change the demands that they place on users' computers.
B is willing to spend more money for a better graphics card because it improves his experience watching movies. He is not spending more money to gain any sort of competitive advantage over other users, but, because other users' purchases affect what sort of demands streaming services place on users' computers, he's in something of an arms race.
C is willing to spend more money for a better graphics card because it confers a competitive advantage playing MMOs. He cares only about the performance of his gaming rig relative to other gaming rigs; he is otherwise indifferent to performance.
Imagine that people like A constitute the entire market for graphics cards. Separately, imagine that people like B constitute the entire market for graphics cards. Imagine that people like C constitute the entire market for graphics cards. Would graphics cards be different sorts of goods in these three scenarios?
A related question: Are there other sorts of goods than these:

club

common-pool

complementary

durable

excludable

free

Giffen

inferior

intangible

luxury

normal

nondurable

nonexcludable

nonrivalrous

public

private

rivalrous

substitute

superior

tangible

Veblen

Would graphics cards be different sorts of goods in these three
scenarios?

No. What we have is different sorts of consumers, with different utility functions. The good is the same, it is the demand function that changes.
In a dynamic setting of course, the preferences of consumers (and which are the predominant ones in terms of demand share) will possibly drive how graphics cards evolve as products.
As for the second question, I am pretty sure that somewhere in the vast and centuries old economics literature someone has defined yet one more subcategory of a good.


Log Utiliy Function Trick I am watching Lecture 3 of Yale's Financial Theory Lecture (by John). At about minute 50 he explains something along this line (with reference to log utility functions).

MUx/Px=MUy/Py

And simplifies down to: 0.75/Px(x)= 0.25/Py(y)

Given that Px(x)+Py(y)=Total Money

He concludes that "The total spent on x relative to 3/4 is equal to the total spent on y relative to 1/4" and therefore a consumer would spend 3/4 of their income on x and 1/4 on y.

How is he drawing that conclusion from the above equations?

Start here
$$0.75/p_xx= 0.25/p_yy$$

Multiply both sides by $p_xx p_yy$, you get:
$$0.75p_yy=0.25p_xx$$

Add $0.75p_xx$ to both sides, you get:

$$0.75(p_xx+p_yy)=p_xx$$

So:
$$\frac{p_xx}{p_xx+p_yy}=0.75=\frac{3}{4}$$

And you spend the rest of your budget, so $1/4$ of it, on $y$.


Representing these lexicographic preferences by non-continuous utility function? I have the following preferences and I am wondering whether I could represent them by a utility function:

$R_1 \succ R_2$ <-> $R_1\leq Q$ and $R_2 > Q$,

$R_1 \succ R_2$ <-> $R_1 \leq Q$ and $R_2 \leq Q$ and $R_1>R_2$,

$R_1 \sim R_2$ <->  $R_1 > Q$ and $R_2 > Q$

R is positive and real-valued, Q is kind of the "threshold" of R for the first criterion (below Q everything prefered to above Q). These are lexicographic preferences, right?

Can't I represent these preferences by the utility function:

$U(R)=a+bR$ for $R\leq Q$ and positive a,b

$U(R)=0$ otherwise ?

Because now

$U(R_1)>U(R_2)$ <-> $R_1 \succ R_2$

$U(R_1)=U(R_2)$ <-> $R_1 \sim R_2$

So, is it finally possible to represent lexicographic preferences by anon-continuous utility function?

Thanks a lot for help!

Felix

As @HRSE pointed out, your terminology is a bit misleading, and these preferences are not exactly what we refer to when we use the word 'lexicographic'. You might be tempted to consider every $R>0$ as a two-dimensional vector $R=(\mathbb{1}_{R \leq Q},R)$ and to apply the lexicographic order on these vectors, but I would say it is a bit unnatural since the two dimensions of the vector are not independent (you cannot vary the second freely without affecting the first).

That said, the usual observation that lexicographic preferences can be represented by a utility function (but not by a continuous utility function) applies to your case as well. The common feature of both situations is that preferences satisfy standard rationality requirements (weak order) but are discontinuous. You can refer to this question for a formal discussion of these issues:

Lexicographic preference relation cannot be represented by a utility function


Deriving the supply curve In deriving the supply curve how does one create the supply curve like the one we see in textbooks (upward sloping, possible curve), if we are only given a single quantity and price at a moment in time? For example, in the oil industry we are told that the US is supplying x barrels of oil, how can we derive the rest of the curve since we don't know what the price is at x-y barrels...

Cvarza, the supply curve is theoretically derived.

The supply curve of the industry is the horizontal sum of the curves of firms. The supply curve of a firm is the portion of the marginal cost curve with a positive slope. That is, firms act to equal marginal cost to marginal revenue. If the marginal cost is decreasing, that means the firm can increase (marginally) the production (and its profits) till the equilibrium point.

As reference, i recommend the chapter about firms supply in Varian's undergradute book.
As a graduate reference, MasCollel is strongly recomended.


Reactions of the Money Market Here's the image of a money market with two different money supply and demand curves:

Here are some questions regarding money market reactions. I'll always write my proposed reasoning underneath each questions, with a summed up answer at the end of the paragraph, and I need you to check if it's correct.

  1) There's a GDP increase and no monetary policy was implemented. Is a
  movement from point $A$ to $C$ or $D$ to $B$ a logical reaction?

The LM curve formula is $L(Y,i) = Money Demand/Price Level$
If the government has set the money supply to $M_B^S$ and we're in point $A$, then the reaction $A$ to $C$ is correct, because we're shifting the $M_B^D$ curve because of an increase in money demand ($Y$ has a positive influence on the $LM$-equation). Same applies to point $D$, except there we have a higher money supply, so we're moving to point $B$. Answer: Yes.

  2) Are the opportunity costs of money saving higher in point $B$ or $D$?

There's a higher nominal interest rate in point $B$, so saving money in $B$ is more profitable than in $D$, thus the decision not to save money in $B$ is associated with higher opportunity costs. Answer: Point $B$.

  3) Is the money demand in Point $B$ higher than in $C$, because of
  the transaction motive?

Both points lie on the curve $M_A^D$, and any point on that curve has the same amount of money demand, only the nominal interest rate and the money supply are different. Answer: Point $B$ and $C$ have the same money demand. Answer: no.

  4) Can an increase of money supply be interpreted as a movement from
  point $A$ to $C$?

The movement from $A$ to $C$ implies a shift/increase of money demand not money supply. The correct movement is from $A$ to $D$. Answer: No, A to D is correct.

  5) We're in Point $A$ and the households suddenly want to keep less
  money in their savings accounts. Can this be construed as a shift of
  the $M_B^D$ curve to the right?

Consumers now have a higher demand for liquid money, because they don't want to save it but spend it instead. The money demand curve must shift to the right. This leads us from point $A$ to $C$. Answer: Yes, the demand for liquid money is increased.

  6) We're in point $D$ and the nominal interest rate starts to increase.
  What's the money market's reaction?

I'm not sure about this one... the nominal interest rate has a negative effect on the LM curve, thus a higher nominal interest rate will lead to decreased money demand (or a higher price level). But wouldn't that imply a demand curve shift to the left? From the diagram, it's an impossible reaction, as the increase of nominal interest rate is associated with a lower money supply or an increase in demand, no?

All correct up to 6), although you may want to be careful with "$D$ to $A$ is correct" kind of answer, "$D$ to $A$ is consistent" seems a better phrasing - one has no idea how hard it comes out to quantify those shifts, because they impact each other.

Point 6) changes interest rate, which means that money supply, demand and interest rate do not coincide: the equilibrium is lost on the money market. As it is lost with higher interest rate than the equilibrium rate, it means that people want less money than is available. Like every good, this causes money to lose value, which in turns means either inflation on the open market, or a quick reaction via a decrease in supply from the central bank. As the question explicitly ask for the effect on the market, you can suppose the central bank doesn't make a move, and let the market move into inflation. This does happen in cases when the central bank doesn't want to or cannot cut money supply, usually because the central bank is controlled by the government, that needs the money, and led to some disastrous effects in the past. For this reason, usually central banks are relatively independent from governments, and hopefully the situation is addressed by central banks rather than by the market.

Here is an example of a government-dependent central bank having decided not to cut money. Wheelbarrows were commonly used on the way to buy loaves of bread in 1920s Germany...


What is the true source of dynamic(Pareto) inefficiency in OLG models? This question confuses me for some time. My first impression is that Pareto sub-optimality is due to feature that old generations will consume everything without any incentive to trade. But after taking a look at the first example in an modern introduction to economic growth, it's just a disguised version of Hilbert's paradox of grand hotel.

Screenshot in the book:

What is the true source of dynamic(Pareto) inefficiency in OLG models?

Shell 1971 argues (in a ten page paper, so read it!) that the dynamic inefficiency stems from the double infinity of traders and goods, and not the dynamics. This allows us to do the Hilbert hotel switch.

  Therefore, even when all souls are able to transact business in the same
  Walrasian market, the absence of Pareto-optimality persists in the competitive-equilibrium model. This absence of Pareto-optimality is implicit in Gamow's hotel problem. An innkeeper has committed each of the denumerably infinite number of beds on a certain rainy night. A guest asks for a bed when all are occupied, but a bed can be found if the innkeeper requires each guest to move down one bed. In our little chocolate game, the imposed allocation can produce one extra chocolate. In the hotel problem, on the other hand, the innkeeper by imposing an allocation will be able to produce a denumerable infinity of extra beds.

Consider the following static setup.

A countable number of agents $i$, indexed $-1, 0,1, 2, 3, \ldots$.
A countable number of goods $j$, indexed $0, 1, 2, 3, \ldots$.
Agent $-1$ has preferences $u^i(c_0, c_1, \ldots) = c_0$.
Agent $i\ge 0$ has preferences $u^i(c_0, c_1, \ldots) = c_i + c_{i+1}$.
Agent $i\ge 0$ is endowed with one unit of good $i$. Agent $-1$ has nothing.

Then autarky (everyone consumes what they were endowed with) is a Walrasian equilibrium (even though everyone can meet at the same time). This Walrasian equilibrium is not Pareto efficient (for the same reason as always). The fact that people can meet and trade does not alleviate the inefficiency.

Note that in the static setup, introducing money cannot help (because who would be so dumb as to hold money after the static trade?).

  The fragility of intergenerational social contracts shows up in our example. If the $k$th generation repudiates the money of its elders, it can avoid passing chocolate backward. Furthermore, the $k$th generation can gain seigniorage by printing new money. Under this monetary reform the reforming generation may have chocolate passed to it, but itself passes none backward. If all souls had met in an atemporal Walrasian market, the monetary reform would have been revealed to be a nonequilibrium solution. In the actual dynamic world, "monetary reform" merely leads to frustration of the $(k - 1)$th generation's expectations. The most notable instance of such frustration is when the peasants' sons leave home for the city and fail to provide adequately for their parents' old-age.


Aggregate efficient production and division of labour How to resolve the following seeming contradiction? Mas-Colell, Whinston, Green p. 148 Prop. 5.E.1 says that the efficient aggregate (coordinated) production is just the sum of individually optimal productions. But Adam Smith argued that division of labour allows more efficient production than the sum of individual productions.

Of course Adam Smith's argument is informal, so it is not clear what assumptions differ from Mas-Colell. It could be that Smith envisioned production units with market power, externalities or some other inefficiency and the inefficiency somehow disappears with aggregation.

I would say, it is more Proposition $5.\text{F}.1$ on page 150, in combination with $5.\text{E}.1$ that leads to, as the authors write discussing the two,

  "...if a collection of firms each independently maximizes profits with respect to the same fixed price vector $p >>0$, then the aggregate
  production is socially efficient. That is, there is no other production plan for the economy as a whole that could produce more output using no additional inputs."

Note that no assumption of identical suppliers is made here, or any other restrictive assumption about technology or the production set. The crucial assumption is the "price-taking" behavior assumed for all firms. Or, "no-market-power". Or, "all firms are equal": doesn't Perfect Competition sound a lot like the market version of Communism (not Socialism)?

P.S.: Externalities could also create issues here, of course.


Uses of convex analysis in Economics I'm taking kind of a crash-course in convex analysis to complement my mathematical skills and was wondering if anyone knew about nice ways in which this kind of tools were used in Economics.
To be more precise, some of the things I've seen so far are not strictly in the area of convex analysis but are very related, like dual spaces, weak topology, subdiferentials and Hahn-Banach theorem.

The only example I know of is the duality between the UMP and EMP in consumer theory (and of course the firm maximization and cost minimization problems). I also think that Hahn-Banach is used in the proof of the first welfare theorem.

Has anyone here used this kind of mathematical concepts in their work or has seen any interesting recent usage of them?

A partial answer: convex analysis is extensively used in axiomatic decision theory, at least in its recent developments. Most of these papers focus on individual behavior. You can have a look for instance at the following papers on ambiguity-averse preferences:

"Maxmin Expected Utility with Non-Unique Prior" (Gilboa & Schmeidler)
"Ambiguity Aversion, Robustness, and the Variational Representation of Preferences" (Maccheroni, Marinacci & Rustichini)
"A Smooth Model of Decision-Making under Ambiguity" (Klibanoff, Marinacci & Mukerji)
"Ambiguity in the Small and in the Large" (Ghirardato and Siniscalchi)

Here is a paper that applies convex analysis to a model of trade under ambiguity aversion: "Subjective Beliefs and Ex-Ante Trade" (Rigotti, Shannon & Strzalecki).

Beyond models of ambiguity aversion, virtually all recent work in axiomatic decision theory makes use of convex analysis, and applies its tools to study various phenomena: regret aversion (Sarver, Ergin), cost of thinking (Ortoleva), random choice (Gul, Pesendorfer)... Please tell me if you want more precise suggestions.

For the mathematical part, a very good reference is Convex Analysis by Rockafellar (1970). It is cited by most of the papers above ;-).


The phase diagram for the simple RBC model I'm reading these notes in the RBC model. In page 14, the author, tries to explain the direction of the arrows, in the space $(\tilde C_t,\tilde K_t)$ (tilde refers to proportional deviations from steady state) , to explain the dynamics.

If I look at equations (47) and (49), the ones that are used to derive the isoclines (equations (50) and (51)), I do not understand how to see the directions  Up/Down, Left/Right...

Any help would be appreciated.

There are two tricks to safely construct a phase diagram, as regards the dynamics off it.

First, solve for the "isoclines" as weak inequalities rather than as equalities. This method makes also clear the dynamics off the isocline.

Consider a standard capital accumulation equation

$$K_{t+1} = (1-\delta)K_t + F(K_t) - C_t \tag{1}$$

We want to obtain the isocline, but also the dynamics off it. Re-write as

$$K_{t+1} - K_t  = -\delta K_t + F(K_t) - C_t$$

Now require that

$$K_{t+1} - K_t \geq 0 \implies -\delta K_t + F(K_t) - C_t \geq 0$$

$$\implies C_t \leq F(K_t) - \delta K_t \tag{2}$$

With equality, $(2)$ is the expression for the isocline. The inequality tells you that for capital to tend to increase ($K_{t+1} - K_t > 0$), consumption must be lower than the level indicated by the isocline. Etc

The second trick is already used in $(2)$: although this is the capital difference equation, I wrote it as though it is an expression determining consumption. In this way, one can look at the phase diagram and for both difference equations, treat the variable in the vertical axis as the "dependent" variable of the two isocline functions. This is the most natural to the eye, and helps avoid the up/down, right/left confusion that can result if one tries to rotate in its mind the phase diagram, for one of the two equations. It also makes easier to determine the slope/shape of the two isoclines.


Unconstrained Optimization:Why is there no "profit style" function in consumer theory? When being exposed to your equations for profit maximization you have an equation of:

$$\pi=pf(x)-c(x)$$

with this you can solve for the optimal input(s) $x$ from the first order conditions and some basic algebra.

I was wondering why in consumer theory we don't solve for the bundle that maximizes consumer surplus using a similar type of formula.

The motive behind this question is that if such a formula exists it would simplify the calculations for utility maximization in consumer theory. The profit function is convenient because it incorporates the constraint, however we can't seem to do this in consumer theory.

Why does such a function not exist?

Edit: I'm interested in understanding why we don't have a function which circumvents the use of the Lagrangian in consumer theory. I.e. Why don't we solve to maximize consumer surplus

The "profit" function for consumer choice does exist; it is the Lagrangean:
\begin{equation}
\mathcal L(x;\lambda)=u(x)+\lambda[c-g(x)]\,,
\end{equation}
where utility $u(x)$ is to be maximized subject to budget constraint $c\ge g(x)$.

Note that $\max_x \mathcal L(x;\lambda)$ is an unconstrained optimization problem just like $\max_x\pi(x)$.


How do I predict Macroeconomic indicators?Or are there any free resources where I can get the predicted values? I am building a time series forecasting model in which I am considering the macroeconomic indicators as predictors.I wanted to ask 2 things

How do I get the future values?I have seen trading economics and some other sites but they are paid. Also if I want predict indicators on my own how should I go about that?

All the data for indicators are given in two forms on govt websites - seasonally adjusted and seasonally not adjusted. I am using seasonally adjusted data for my model building.Is that correct?

3.I am building a product demand forecasting model. I have chosen 20-30 macroeconomic indicators and other exogenous variables. I wanted to understand how do I check with explanatory variables will help me in forecasting the demand. How do I compare two time series .Can I build a separate regression models for each indicator with dependent variable as demand and then select variables which are significant into my final forecasting model. Or can I use Cross Correlation or Granger's causality approach to determine significant predictors?
Any help would be appreciated

I will give a partial answer.

I assume you mean “forecast values.” Forecasts generally come from private firms that do the forecasts, and are typically expensive. There are some surveys that are in the public domain (Philadelphia Fed Survey of Professional Forecasters). You can generate your own using a model, but quality depends uponyour model.
It is normally easier to work with seasonally adjusted data. However, the data are passed through a filter, and distortions can occur. (For example, the pandemic of 2020 has caused spikes in data, and seasonal adjustment can make those spikes worse.)
The last question is open-ended. People spend their entire careers in finance and at central banks building models.


How product leaks before announcements affect a company economically? It happens specially for well known companies (e.g. Apple, Garmin, etc.) that some features of their products leaks into the media before the announcement. I was wondering what are the economical effects of these leaks on the company and how strong are they?

Opinion

If the leak is close to a release date, I would say most of these are made on purpose, to feed expectations, leading to greater attention and publicity (and sales). I don't see how this can be negative for the company. In technological firms, most of the features are patent-protected so they cannot be copied anyway. However, if the leak were to happen long before a release, this could indicate rivals potential direction of own developments (either copying them or moving away/further than the competitors).

Research

Maybe the question is related to the literature on knowledge leakage, which relates to R&D, knowledge management, knowledge sharing, etc.

One recent paper  mentions some of the costs:

  Knowledge leakage has various negative effects on
  firms,
  including loss of revenue, damaged reputation, loss of productivity
  and costs arising from breached confidentiality agreements
  (Ahmad et al., 2014). It also decreases bargaining power and even
  creates new competitors for the original knowledge owner
  (Baughn et al., 1997). Therefore, when an employee leaks knowl-
  edge that is rare, inimitable and non-substitutable, recovery from
  such leakage can be very challenging (Ahmad et al., 2014; Barney,
  1991, 1996).

The same paper states that :

  Firms traditionally attempt
  to protect their innovation-based competitive advantages with
  intellectual property rights (IPRs), trademarks, copyrights or other
  knowledge-protecting  mechanism

I suggest you look into those references if you want to get further insights.


Simple hawk-dove type super-game We have the following super-game:
\begin{array}{cc}
  & a_1 & a_2  \\
a_1 & 0,0 & -1,1 \\
a_2 & 1,-1 & -2,-2
\end{array}
I want to show that both the trigger (or grim) strategy and the mutual punishment, give the cooperative solution $(a_1,a_1)=(0,0)$ for the discount factor $\delta>\dfrac{1}{2}$. In the mutual punishment case, if players have observed $(a_1,a_1)$ or $(a_2,a_2)$ in the preivous round, they choose $a_1$. In case they have observed $(a_1,a_2)$ or $(a_2,a_1)$, they choose $a_2$ for one round and then we asumme that they cooperate again.

My thoughts for the second is (player 1 (p1) is the line player and player 2 the column player (p2)):
Strategy: For as long as we cooperate we both gain a zero payoff, so the value function is:
$$V_1^{C}= 0 + 0 \cdot \delta + 0 \cdot \delta^{2} + ... = 0$$
Suppose that p2, deviates unilaterally from the cooperation strategy in round 1 by playing $a_2$, so both players in the second round play the mutual punishment strategy, that is, $-2$ looses for everyone. From then onward, we assume that they both play according to the cooperative strategy, namely, the value function is:
$$V_2^{NC}=1 - 2 \cdot \delta + 0 \cdot \delta + 0 \cdot \delta^{2} + ... = 1   - 2 \cdot \delta$$
They will cooperate if:
$$V_1^{C}>V_2^{NC}\Rightarrow \delta>\dfrac{1}{2}$$
In the case of grim/trigger strategy, the players stop to cooperate in perpetuity, how does the solution differs?

Note: The game is the hawk-dove type super-game.

Finally, I believe it is easy to answer it. Well, in case where some player will not cooperate and the triger strategy is enabled, we have the following soloutions:
Soppuse that p2 does not cooperate in the first round so, he will gain a payoof of 1, but in the second round p1 plays $a_2$ and he won't cooperate with p2 in perpetual. Thus, we have that p2 will play $a_1$, because by playing $a_2$ he increases his looses and the value function is:
$$V_2^{NC}=1-1\cdot\delta-1\cdot\delta^2-1\cdot\delta^3+...=1-\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{\delta}{1-\delta}$$
Hence, they will cooperate if:
$$0>1-\dfrac{\delta}{1-\delta}\Rightarrow \delta>\dfrac{1}{2}$$
Differently, if both players choose trigger strategies, so p2 also chooses $a_2$, then both will make looses of -2 and the value function is:
$$V_2^{NC}=1-2\cdot\delta-2\cdot\delta^2-2\cdot\delta^3+...=1-2\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{2\delta}{1-\delta}$$
and hence they will cooperate if:
$$0>1-\dfrac{2\delta}{1-\delta}\Rightarrow \delta>\dfrac{1}{3}$$


In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium? Where I can find a simple proof for this fact?
For example, a trivial bimatrix game with alternating move has the following payoff matrix:
\begin{array}{|c|c|c|}
\hline
& 1 & 2 \\ \hline
U & (0,0)  & (0,0)\\ \hline
 L&   (0,0)& (0,0)\\ \hline
\end{array}
Then all of the pure and mixed strategies are trivially the equilibrium strategies.
I guess that, if the game structural is so complicated that it becomes impossible for the players to solve the game, then this complete information game becomes effectively like an incomplete information game. But I am not sure how to rigorously describe this.

This statement is wrong. Consider Alternating Matching Pennies with imperfect information (the follower doesn't observe the leader's move). The strategic form of this game is just the classical (simultaneous-move) Matching Pennies Game and the unique NE has both players mixing.


Finding Marshallian demands for Leontief production function with different powers Attempting to solve a practice problem, not sure if I'm heading in the right direction since my solution seems pretty messy. Given the following utility function,

$u(x,y)=min\{x^{1/2},2y\}$, find the Marshallian demands.

My answer:

Since Leontief is perfect complements, must be the case that $x^{1/2}=2y$, substituting this into a budget constraint yields the following:

$p_x \times x + p_y \times y = w$, where w is total income. Taking $x^{1/2}=2y$ and squaring this yields $x=4y^2$. Subbing this into constraint would give:

$p_x \times 4y^2 + p_y \times y = w$, at this point I applied the quadratic formula and got a demand function for y as follows,

$$y = \frac{-p_y \pm \sqrt{p_y^2 + 16 p_x w}}{8p_x}$$

This seems messy to me, I figure I can rule out the minus side of the quadratic seems that would imply y is negative. Even if some confirmation that this is the right approach would be appreciated.

Thanks!

As Amit said, your answer is correct. As Toby said, you can rule out negative output.

There is no much insight to add as an answer really. However, I was also fairly puzzled that such a simple problem gives a complex answer. As it turns out, you get a linear demand (on income) only if the exponents on both components is the same.

Consider the more general case:

$$ U(x,y) = \text{min}\{ax^\alpha,by^\beta\} $$

Optimality requires (* notation for optimal values omitted)

$$ ax^\alpha = by^\beta\ $$

Replacing one of these in the budget constraint leads to:

$$ p_x x + p_y \left(\frac{a}{b}\right)^{\frac{1}{\beta}} x^{\frac{\alpha}{\beta}} = w $$

Here we see that only when $\alpha=\beta$, demand will be linear (on income). In this case:

$$ x = \frac{w}{p_x + p_y\left(\frac{a}{b}\right)^{\frac{1}{\alpha}}} $$

It seems an algebraic solution for the general (unrestricted) case cannot be derived, as the polynomial you get might have irrational powers.


Calculating elasticity for $y^2e^{x+\frac{1}{y}}=3$ I want to calculate elasticity of y with respect to x  for $$y^2e^{x+\frac{1}{y}}=3$$
My attempt:
I calculate $y'$ using:
$$y'= -\frac{f_1'(x,y)}{f_2'(x,y)} = - \frac{y}{2-y}$$
I calculated this using implicit differentiation that can be used where $f(x,y)=c$, in my case $f(x,y)= y^2e^{x+\frac{1}{y}}$ and $c=3$.
Then elasticity should be just
$$y'\frac{x}{y} = -\frac{x}{2-y}$$
But I was told by professor that is a wrong answer  and that correct answer is: $\frac{xy}{1-2y}$.
I don't understand where I have made a mistake and how come its  $\frac{xy}{1-2y}$. Please can someone help me to understand it?


CES function estimation For a paper I was using the micEconCES package to estimate the CES production function for a country at the aggregate. For a two-input function with capital and labour I used for the variables the Perpetual Inventory Method to construct aggregate capital, labour hours and GDP for output. I used methods provided in the package to do the estimation. But now I am kind of unsure, what I have to check the data for f.e. autocorrelation, multicollinearity etc. or do I even have to check for that?

You should always validate statistically your model, running various model diagnostics. It appears you work with time series (single country?).

Here, certainly autocorrelation may be an issue. After all, your series for capital is by construction an autoregressive equation with a forcing factor (investment).

Software packages usually warn for signs of multicollinearity.

Also, a serious issue could be that of endogeneity of regressors.


Won't countries be in trouble if all sovereign debtors demand payment? I'm trying to understand this r/explainlikeimfive analogy. I rectified typos, and simplified some  names and numbers.

Won't Cal demand  his principal back at some time in the future?

Similarly, won't countries' sovereign debts demand their principal?

For example, China holds much US treasuries and US sovereign debt. If China thinks COVID-2019 will cripple the US and wants to park its money else where, won't China demand all of its principal from the US so that it can buy Australia or New Zealand sovereign debt instead?

'Country A owes \$10  to country B, and has agreed to pay 2% interest annually. Call this Debt X.
Country B owes \$15  to country A, and has agreed to pay 1% interest annually. This is Debt Y.
Investor A is willing to purchase Debt X for \$12, because he values the steady rate of return, and believes country A is very likely to pay its debts. He isn't as sure about Debt Y, but he is willing to pay \$16 for it. Investor B thinks that investor A is going to change his mind about Debt Y and value it more highly, so he is willing to pay \$17 for it today.
Debt is traded constantly and it's valued differently. Just because 2 countries owe each other similar amounts of money, doesn't mean the value cancels out in any meaningful way.
Think of debt this way. You and your three brothers all own a house together. Each year you take up a collection of \$1000 a head for \$~4k annual upkeep on the house. One year, a tree falls on the roof, and the house needs another \$4000 bucks. Everyone could just chip in 1k each, but one brother is having a tough year, so instead, you loan the house \$4k, with the expectation that it will pay you 1% interest a year.
Your brothers agree, and over the years, this comes up a few more times. You even use debt to finance installing a new hottub. You now own \$20k in debt from the house, 1 brother owns \$5k, and for some weird reason, your neighbor Cal wants to purchase the debt from you. (He really thinks your household is trustworthy, and wants somewhere to park \$20k for 1% interest).
Now C owns \$20k of debt from your house, so C owns 80% of all your house's debt. Would you say C owns your house now? Well of course not, your house is likely worth many times more what C has claim to. If someone in your house is super worried about the situation, they could take up a collection in the house to buy the debt back from C, but it's not a popular idea, because most people can think of better things to do with their money than pay off C so they don't have to make the small yearly interest payments.

The principal has to be repaid eventually, but with majority of formal debt instruments such as government bonds the creditor can’t just demand repayment whenever he  wants. Most debt instruments specify the date, and if not date, then conditions when and how the principal is being paid back.

For example, a standard 10 year bond will pay its principal in 10 years not a day earlier, unless you convince the debtor to voluntarily agree to pay it sooner).

The above works for sovereign countries too. A 10 year government bond will have on it printed that principal is paid back exactly after 10 years.

Now this is outside economics, but technically sovereign nations are in state of anarchy. So I can imagine one country could just decide to say to another — pay now or we will invade you/nuke you etc! But aside some military intervention, you can't legally force the country to pay.

In fact any government can, in principle, default on its debts anytime it wants. The only constraint on sovereign default is that if you constantly default on your debt, nobody will want to lend you money in the future, or if only at punitively high interest. Hence on sovereign level, governments as debtors have much better bargaining power than just individuals.

Government can be forced to pay its debt only through military action, or in some cases by diplomatic pressure (for example EU could threaten Greece by revoking its euro-zone or EU membership, US could threaten countries by economic sanctions etc.). Aside from that, there is nothing other country can do to get its money back if debtor nation says no.

Based on 1, and 2 you probably already know the answer. No that’s not possible. China could patiently wait for bonds to expire and then not buy new US bonds, but instead buy Australian or New Zealand bonds. Of course, since US government issues a lot of debt, bonds expire probably every month and are replaced by new ones, so China could start to refuse to buy new US bonds sooner than later.

Also, usually when people want to get rid of a debt, they won’t wait for maturity and for repayment of the principal. They will take the bond to bond market and sell it at market price to get money now. That creates a trouble for country because now it has to set lower price or higher interest on its newly issued bonds due to increased supply, but whether this becomes a problem depends on plethora of factors unrelated to the original debtor and creditor countries, such as how big is the demand for the debtors bonds etc.


If porn is free on the Internet, then how do porn stars actually make money? On the Skeptics Stack Exchange, porn is estimated to generate 57 billion dollars a year in America.  However, I've never once paid for porn, and I don't know of any other friends, classmates, students who pay for porn either.

What I am missing?  Is there porn that's targeted at wealthier, paying customers that I am not aware of?

Seems like porn is in great demand, but there's also a great supply of Internet porn, too, and thus anyone can consume porn for free these days...

Using Google is also free, yet the firm is quite profitable, so a possible solution could be ad revenue.

Another solution could be that your sampling is off. E.g. most urban Democrats do not know many rural Republicans and vice-versa. It is similarly possible that people who do pay for porn do not move in your social circle.

There is also an incentive issue to your sampling. This is a subject most people probably do not wish to discuss. As such you can reasonably doubt if you are getting honest answers. If one were to answer your question in the affirmitive, he would incur not only the stigma of watching porn, but also the stigma of having paid for it, which as you say is kind of strange given its free availability.


Kuhn Tucker optimization problem Assume that the player I select $x_i \ge 0$ at constant cost level $c>0$

The payoff function for player i is

$$v(x_i,t)-cx_i$$

where $t$ is technology parameter.

The function v(.)is twice continuously differentiable increasing and strictly concave in $x_i$.

$v(0,t)=0$

$\partial v(0,t)/\partial _i>c$

$\partial v(x_i,t)/\partial _i<c$

I want to maximize this payoff with respect to $x_i$. But the question strictly emphasize to use Kuhn Tucker method and to state and discuss the slackness conditions.

And I need to find the solution of this problem, say $x^*$

My solution:

$$ L= v(x_i,t)-cx_i +\mu [x_i-0]$$

First order condition

$$(\partial v(x_i,t)/\partial _i)-c+\mu=0$$

Kuhn Tucker condition

$$\mu [x_i-0]=0$$ for $\mu \ge 0$

Case 1: $\mu \ge 0$

Then, $x_i=0$

Case 2: $\mu = 0$

Then,  $x_i=0$
$$(\partial v(x_i,t)/\partial _i)-c=0$$

However, the question gives that $$(\partial v(x_i,t)/\partial _i)-c<0$$

This is a contradiction I think. Therefore, I don’t think my solution is true. And I cannot write the $L$ function with two constraints but I know that Kuhn Tucker method requires at least two constraints.

Please share your ideas with me. Thanks a lot.

Here is the Lagrangian function for the stated optimization problem:
$$\mathcal{L}(x_i, t) = v(x_i, t) - cx_i + \mu x_i$$

Necessary conditions for optimality:

$$\frac{\partial\mathcal{L}}{\partial x_i} =\frac{\partial v}{\partial x_i} - c + \mu = 0 $$
and
$$x_i\geq 0, \ \mu \geq 0, \ \mu x_i = 0$$

To solve it, consider the following cases :

$x_i > 0$

$x_i > 0 \rightarrow \mu = 0 \rightarrow \dfrac{\partial v}{\partial x_i} - c = 0$

If there exists $x_i^* > 0$ such that $\dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=x_i^*} - c = 0$, then $x_i = x_i^*$ solves the problem.
$x_i = 0$

$x_i = 0 \rightarrow \mu = c - \dfrac{\partial v}{\partial x_i}$

If $c - \dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=0} \geq 0$, then $x_i = 0$ solves the problem.


OLG - Government Budget Constraint Consider a two period OLG model where each young agent recieves an endowment of $w$ units of the single commodity good in the 1st period of his/her life, and nothing in the second period.

Each period the government consumes the fraction $\bar{g}$ of the total endowment. $\bar{g}$ is i.i.d with mean $\bar{g}$. The government's expenditures are financed  by lump-sum taxes, money and one-period discount bonds. $M_{t}$ and $B_{t}$ are the supplies of money and nominal bonds at the end of period $t$. The government's budget constraint is then:

$\frac{B_{t-1}}{p_{t}} + \bar{g}w = \tau_{y}(t)+\tau_{o}(t) + \frac{M_{t}-M_{t-1}}{p_{t}} + \frac{B_{t}}{(1+i_{t})p_{t}}$

$\tau_{i}(t)$ is the lump-sum tax where $i$ is either the old or young generation.

My only problem with this budget constraint is the $\frac{B_{t-1}}{p_{t}}$ component. I understand that it is the current real obligation but I don't understand why there is no interest component. I am self-studying OLG models so my reasoning might be flawed but here is how I see this constraint:

The RHS is the government's revenue in period $t$, which consists of tax revenue from the young and the old, seigniorage and new bond issuance. The LHS is the expenditure of the government in period $t$. It can spend the income on consumption ($\bar{g}w$), to pay off the existing debt ( $\frac{B_{t-1}}{p_{t}}$) but what about the interest on the existing debt?

I am used to seeing government constraints in the form of:

$g_{t} + r_{t-1}b_{t-1} = t_{t}+(b_{t}-b_{t-1}) + h_{t} - \frac{h_{t-1}}{1 + \pi _{t}}$

This is almost identical to the author's budget constraint only that it includes an interest component. What am I missing here?

For reference, the budget constraint was obtained from Aiyagari and Gertler (1985)

The interest component is already included. It was just their way of writing out the budget constraint that confused me. If anyone is interested:

$\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}} = \frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}} - \frac{B_{t}}{p_{t}} + \frac{B_{t-1}}{p_{t}}$. This is simply the interest on existing debt and the face value of existing debt (on the expenditure side). And real bond issuance on the income side. Simple rearranging and you have it.


How does the self-corecting mechanism put the economy back to long run equilibrium following a negative shock on the stick-wage SRAS? According to some lecture notes, apparently it is possible for the economy to return to long run equilibrium if via the self-correcting mechanism if there is a temporary shock to the stick wage (horizontal) SRAS. I believe the situation can be graphed as below (SRAS shifting to SRAS1).

The problem is I don't see an explanation as to how it will return to long run equilibrium ($Y_f$), other than to let labour markets adjust itself over long periods of time??

The SRAS wage price being bumped up temporarily means that employers are forced to pay more for labour. From the graph, you can see aggregate demand is much lower at the higher wage price (SRAS1) however the stickiness of wages & employment means that employers have been forced to pay more in wages but cannot immediately adjust the 'supply' by firing employees. Likewise workers will not likely be willing to accept pay cuts, so in the short run the market will be pushed off equilibrium. However, given a little more time, the firms will find ways to let go of employees, bringing the market towards point Y1. This is 'automatic' in the sense that workers are expecting wages of SRAS1 and firms only want to pay for Y1 amount of labour. Once the employees have been laid off to reach Y1, there will be unemployed workers who are willing to work for less (since otherwise they get nothing and we know the position of the LRAS of labour implies a larger available work force than point Y1) and their negociation with firms will push the market back to equilibrium at the lower wage price.

To give a real life example of this, (I know it's micro and this is a macro effect but it helps to build an intuitive sense of how this works) I know of a factory that had a strong union that pushed continuosly for wage increases despite the factory being the least efficient and cost-effective in the firm. So when they threatened to strike, the firm gave in to their wage demands however the firm also invested to build capacity in the other more efficient factories. Doing so took about 2 years.

Finally, once the firm was sure it could handle losing the output from the 'bad' factory, it gave an ultimatum, the workers would have to be more efficient at a lower wage or it would shut the factory down. The workers refused, and the factory closed. Over the next year, the workers realized that nobody was hiring for the wages they wanted and after a little over 12 months most of the workers were hired back at the lower wages the firm wanted in the beginning and the factory re-started. From start to finish it took 3 years for the wage to 'equilibrate' at this firm.


Understanding Envelope Theorem Can someone please verify whether or not my knowledge is accurate.

From what I understand, the intent of the envelope theorem is to make a shortcut from indirect utility to the expenditure function. That being said, what you can do once you have obtained your indirect utility, is derive it in terms of I (income) and this will give you the expenditure function (hicksian demand). Is this correct or am I totally off?

The Envelope theorem is a general mathematics result says that you can differentiate a value function with respect to a variable without implicitly differentiating the maximum argument variable.

Example: $V(p_x,p_y,w)=U(x(p_x,p_y,w), y(p_x,p_y,w))+\lambda(p_x,p_y,w)[w-p_x-p_y]$

If I want to see how utility at its optimal level changes when wealth changes, I can simply differentiate this equation with respect to wealth explicitly:

$V_3=\lambda$ - so the marginal utility of wealth is simply the Lagrange multiplier.

As you can see I did not differentiate anything else, even though wealth does appear in U. This is simply because if you did do that, those terms would drop out =0 as a result of the FOC

The relationship between indirect utility and expenditure is from the 'duality' between the two. The reason why $V(p,e(p,u))=w$ stems from the fact that the optimal arguments for both will be equivalent when the value of utility achieved in the utility max problem is the same utility level that we set as $U=\bar u$ in the expenditure minimization problem.

If you do the FOC for both utility max and expenditure min you get the tangency conditions:

$p_x/p_y=MRS$ in both cases. So if the target level $\bar u$ is what we achieve in the utility max problem, then the marshallian and hicksian demands will be the same. Then the value function for the expenditure min is:

$e(p_x,p_y,\bar u)=p_xh_x+p_yh_y=p_xx+p_yy=w$ and since $\bar u=V$

$e(p_x,p_y,V)=w$

Similarly,
$V(p_x,p_y,w)=U(x,y)=U(h_x,h_y)=\bar u$ and since $w=e$

$V(p_x,p_y,e)=u$

You can find a lot more by searching for 'utility duality'.


How net capital outflow = net exports? How net capital outflow is equal to net exports suppose I export a car to US and in return i got few thousand dollars then instead of acquiring any asset abroad i decided to bring money back to my home country by exchanging dollar with local currency. Can any one clear my confusion?

Suppose you are Japanese and you sold your Japanese-made car for say $\$15,000$

You now own $\$15,000$ of a foreign financial asset you did not have before, whether a debt from a US company or individual or US cash or a bank deposit. This is what you are describing as a capital outflow from Japan, though I would call it a financial outflow

You then sell this asset for say $\yen 1,650,000$ which you can use in Japan

You may have acquired these yen from some other Japanese person or company.  They now hold the foreign financial asset, so this does not have a further effect of the international financial position
You may have acquired these yen from some US person or company, or from an entity from a third country.  They now hold fewer financial Japanese assets, so your subsequent negative financial outflow (wiping out your original positive outflow) is balanced by the negative financial inflow (from a Japanese perspective) of the US or other foreign entity reducing its Japanese assets. This negative financial inflow is equivalent to a positive financial outflow


Is it possible to have a preference relation that is complete but not transitive? I've been doing my own reading on non-rational preference relations.
I'm currently under the impression that transitivity follows as a direct result of completeness of preferences. However my (much more advanced colleagues) have told me this is not necessarily the case.
I'm still to understand how this can be. What would such preferences look like?

I'm quite surprised nobody has picked the obvious one:

I prefer rock over scissors.
I prefer scissors over paper.
I prefer paper over rock.

Complete, definitely not transitive.