How to estimate a model by pooled OLS when we DO NOT have panel data? The data set is mostly binary with years and location. We have a linear probability model. We DO NOT have panel data. How to estimate this model in R by Pooled OLS? Again, we DO NOT have panel data.

You don't need panel data for pooled  OLS. You can use pooled cross section data which is similar to panel data but it is different data type. In panel data you follow the same units (e.g. individuals, firms, countries, stocks, etc.) consistently over time, whereas in pooled cross-section data you will have 'a time series of cross-sections' where the same units are not necessarily followed.
If you don't have some sort of data collected on multiple units over time you can't run pooled OLS. However, in description you mention the data contain years and locations so this should not be issue as you should have multiple time periods and units and thus at very least the data structure should be pooled cross-section.
In R you can can use the lm function or plm package to run pooled OLS.
PS: You mention that most of the data are binary - if your dependent variable is binary as well, you should probably consider pooled logit/probit model instead of OLS.


Parameter in normal form NE $$\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline \text{Player 1 / 2}& x & y & z \\
\hline X & a, a & a, 0 & a, 0 \\
\hline Y & 0, a & 3,0 & 0,3 \\
\hline Z & 0, a & 0,3 & 3,0 \\
\hline
\end{array}
\end{aligned}$$
Consider that $a \in (0, +\infty)$ I know that if $a=1$ then the NE is $(X,x)$ yet I don't know how to solve for the NE when $a \in (1,3)$ and $a \in(0,1)$ . My attempt was writting the conditions of indifference to find the mixed strategy NE but somehow I still don't arrive to a clear answer. Also I don't now if I should consider all of the possibilities of the support or if there is any shortcut.

Take any $a\in(0,3)$.
Since $a>0$, X is a best response (BR) to x. Since $a<3$, Y is BR against y, and Z is BR against z.
Similarly, x is BR against X, z against Y, and y against Z.
Hence, your only pure strategy NE is (X,x). The latter is also true if $a=3$.
If $a>3$, X and x are strictly dominant strategies such that (X,x) is the unique NE.
Now back to case $a\in(0,3)$.
Suppose 1 randomizes with probabilities $p_X,p_Y,p_Z$, and 2 randomizes with $q_x,q_y,q_z$. If all those probabilities are positive, both must be indifferent over all three options. Let $\pi^i_a$ be the expected payoff of player $i$ playing action $a$.
$\pi^2_x=a, \pi^2_y=3p_Z, \pi^2_z=3p_Y$ and the $\pi^1$ look similar. You need $\pi^2_x=a = \pi^2_y=3p_Z = \pi^2_z=3p_Y$ if 2 shall randomize over all three. Hence, we need $p_Y=p_Z=\frac{a}{3}$, which is a probability as $a<3$. But since $p_X+p_Y+p_Z=1$ we also need $a\leq \frac{3}{2}$. From the same exercise for player 1, we get $q_y=q_z=\frac{a}{3}$ and also condition $a\leq \frac{3}{2}$.
Hence, if $a<\frac{3}{2}$, we have a mixed strategy Nash equilibrium in which both players randomize over all three of their actions with probabilities $q_y=q_z=pY=p_Z=\frac{a}{3}$ and $q_z=p_Z=1-2\frac{a}{3}=\frac{3-2a}{3}$. Such that both of them expect equilibrium payoff $a$. Any unilateral deviation would also lead to payoff $a$.
We also have an equilibrium in which both players only randomize over $Y,Z$ and $y,z$, respectively, all with probability $\frac{1}{2}$. In this case, equilibrium utility of $\frac{3}{2}$ cannot be improved upon by a deviation to $X$ or $x$, respectively, as $a<\frac{3}{2}$, and the other two pure strategies yield the equilibrium utility by construction.
Both these equilibria would not exist if $a>\frac{3}{2}$, as a deviation to  $X$ or $x$, respectively, would be better or the indifference condition could not be fulfilled.
For any $a>0$, there cannot be an NE in which 1 mixes between X and Y and 2 between x and y, because 2 can only be indifferent between x and y if 1 also plays Z, $p_Z=0 \Rightarrow \pi^2_x=a>\pi^2_y=0$.
Similarly, the other combinations (such as X and Z and x and z) would violate the indifference condition.


Can everyone in an economy profit? I earn a salary, but, like many American workers, I don't spend all of it -- I stash away some in the bank. This is a good thing, since I can one day retire (stop producing), and still live comfortably.

Suppose everyone succeeded in this goal. Could such a thing be possible?

Reason for no: everyone can't save money at the same time, because money is only obtained by taking it from someone else. And printing money into the economy inflates it such that it doesn't actually bring inherent value.

I think the best way to answer you is to build a simple model. That's usually what economists do to answer such questions. I hope it will be useful.

Assume there is only one good in the economy, the 'consumption good', that people are all of the same generation or the same age and that one lives only for two periods (youth, and old age).

Then the answer to your question depends on the durability of this consumption good. How long can it remain useful to consumption after being produced ? Or, to think as an economist, is it expensive to keep the good useful ?

If it can for no cost remain useful for more than one period, then it can be rational for all workers to save simultaneously half of the monetary wage they receive as a compensation for their labor in period 1. Then at the end of period 1 firms will sell only half of the output produced by the workers. They will store the rest of it (we have assumed that this storage does not cost much). Then this stock of the consumption good will be available in period 2 and will be sold to the workers, who will buy it with their monetary savings. So the story is exactly the same as it would have been without money, if workers would have stored directly the commodities they produced.

If the consumption good is not durable, or is expensive to store, then this scheme is not possible. In reality, of course, many commodities of first necessity are not durable (goods but also services), and many durable goods are expensive to store. But that's not so much of a problem because we live in a world where generations do overlap, so the young produce for the old. When there is a demographic disequilibrium inside an aging country, then it should be able to trade with a 'younger' country, provided it has accumulated enough 'savings', i.e. valuable durable assets.

Edit. So far I only answered the question : is it possible for every one to live without working at the same time ? But the number of active workers can decrease with time if savings are used to invest in capital goods which make labor more productive.


What are macro-economics simulations horizons and limits possible with exascale computing? What are typical examples in literature of large scale simulations of macro-economies of our time as of 2019 and what will change with the introduction of exascale computing? In terms, are there problems which might become feasible to assess then?

Some example of large macro models would be the the World Banks macro-fiscal model or Smets-Wouters DSGE model.

How will having access to high computing power change the models is very opinion based question I don’t think anyone can predict that but I will indulge you. With increasing computer power it will be definitely possible to make these models more detailed, perhaps even to the point where you don’t just model demand and supply in a given market based on just representative agents but also explicitly modeling interactions of heterogenous individuals and firms and deriving demand and supply from that. Also this is my two cents on the issue but I don’t think that in current macroeconomic modeling lack of computer power would be a main hurdle, I would say lack of available data is far more bigger problem but getting better. In the end no amount of computer power will help if you don’t have enough data to properly parametrize the model.


What is Opportunity Cost? Suppose you have 2 activities, A and B. Doing activity A gives a return of 100 dollars, doing activity B gives a return of -50 dollars. What would be the opportunity cost of choosing activity B? 100 dollars or 150 dollars.

Oxford American Dictionary gives a pretty standard definition of opportunity cost: "the loss of potential gain from other alternatives when one alternative is chosen".

Looking at it one way, the loss of potential gain by picking B is 100 dollars, as that is the gain that would be had from doing A. However, by choosing B, not only are you missing out on 100 dollars, you're also losing 50 dollars by doing B itself. The difference in the gain between the two activities is 100 - - 50 = 150 dollars. This is the foregone value when activity B is chosen, and seems to give the full picture as to what is lost by choosing B. This term "The value foregone.." seems to appear in many definitions of opportunity cost online as well.

Different sites and sources seem to fall on either one side or the other. There are example questions and solutions online which show both methods of approach.

So, in an assignment which requires calculations of opportunity cost, which definition of opportunity cost will get me the marks- the value of the next best alternative, or the difference between the value of the next best alternative and the value of the chosen activity? Or both?

Definition. The opportunity cost (OC) of any alternative is the value you place on the best of the forgone alternatives.

If we adopt the above definition (and I do), the OC of $B$ is the value of $A$, which is simply $\$100$.

With the above definition, the alternative you choose ($B$ in this case) is completely irrelevant when calculating its OC. All that matters is the best of the forgone alternatives.

To illustrate this point, consider this pair of examples:

  Example #1. $A$ and $B$ are the only two alternatives. $A$: "Pay $\$500$ to get back $\$600$ in return." $B$: "Pay $\$500$ to get back $\$450$ in return." The OC of $B$ is the value of $A$, which is $\$100$.

  Example #2. Everything is exactly the same as before, except that now $B$ is changed to "Pay $\$500$ to get back $\$1,000$ in return." The OC of $B$ is again the value of $A$, which is $\$100$.

In the above examples, the value of $B$ changes (from $-\$50$ to $\$500$) but that of $A$ doesn't. And so, the OC of $B$ is $\$100$ in both examples.

An alternative's value is completely irrelevant to its OC. What's relevant is the value of the best alternative forgone.

Notes. "What exactly is OC?"

This seemingly-simple question was discussed in the past (see e.g. Alchian, 1968). Then like many of the most important questions in economics, it was then forgotten/glossed over until Ferraro & Taylor (2005) and the susbequent literature spawned (see in particular the 2016 JEE Symposium).

It all boils down to which definition you use. My view (and I believe also that of Alchian and Buchanan, 1987) is that the above definition is the "most correct" one that "everyone should use". Of course, whoever's marking your homework may disagree.


Show that $W_t - \int_0^t \xi_s ds$ is forward-measure-Brownian Definitions and stuff:
Consider a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ where

$$T > 0$$

$$\mathbb P = \tilde{\mathbb P}$$

This is risk-neutral measure.

$$\mathscr F_t = \mathscr F_t^{{W}} = \mathscr F_t^{\tilde{W}}$$

where $W = \tilde{W} = \{\tilde{W_t}\}_{t \in [0,T]} = \{{W_t}\}_{t \in [0,T]}$ is standard $\mathbb P=\tilde{\mathbb P}$-Brownian motion.
Consider $M = \{M_t\}_{t \in [0,T]}$ where
$$M_t := \frac{\exp(-\int_0^t r_s ds)}{P(0,t)}$$
Define forward measure $\mathbb Q$:
$$\frac{d \mathbb Q}{d \mathbb P} := M_T = \frac{\exp(-\int_0^T r_s ds)}{P(0,T)}$$
where $\{r_t\}_{t \in [0,T]}$ is short rate process and $\{P(t,T)\}_{t \in [0,T]}$ is bond price at time t.
It can be shown that $\{\exp(-\int_0^t r_s ds)P(t,T)\}_{t \in [0,T]}$ is a $(\mathscr F_t, \mathbb P)-$martingale where bond price dynamics are given as:
$$\frac{dP(t,T)}{P(t,T)} = r_t dt + \xi_t dW_t$$
where

$r_t$ and $\xi_t$ are $\mathscr F_t$-adapted

$\xi_t$ satisfies Novikov's condition (I don't think $\xi_t$ is supposed to represent anything in particular)

Problem:

Define the stochastic process $W^{\mathbb Q} = (W_t^{\mathbb Q})_{t\in[0,T]}$ s.t.
$$W_t^{\mathbb Q} := W_t - \int_0^t \xi_s ds$$
Use Girsanov Theorem to prove:
$$W_t^{\mathbb Q} \ \text{is standard} \ \mathbb Q \ \text{-Brownian motion.}$$

What I tried:
Since $\xi_t$ satisfies Novikov's condition,
$$\int_0^T \xi_t dt < \infty \ \text{a.s.} \ \to \ \int_0^T -\xi_t dt < \infty \ \text{a.s.}$$
$$\to L_t := \exp(-\int_0^t (-\xi_s dW_s) - \frac{1}{2} \int_0^t \xi_s^2 ds)$$
is a $(\mathscr F_t, \mathbb P)-$martingale.
By Girsanov Theorem,
$$W_t^{\mathbb Q} \ \text{is standard} \ \mathbb P^{*} \ \text{-Brownian motion, where}$$
$$\frac{d \mathbb P^{*}}{d \mathbb P} := L_T$$
I guess we have that $W_t^{\mathbb Q}$ is standard $\mathbb Q$-Brownian Motion if we can show that
$$L_T = \frac{d \mathbb Q}{d \mathbb P}$$
I lost my notes, but I think I was able to show using Ito's lemma that

$$dL_t = L_t \xi_t dW_t$$

$$dM_t = M_t \xi_t dW_t$$

From those I infer that
$$d(\ln L_t) = d(\ln M_t)$$
$$\to L_t = M_t$$
$$\to L_T = M_T$$
QED
Is that right?

(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.)

General Fact

Let $W$ be standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$. Consider $(L_t)_{t \in [0,T]}$ defined by
$$
\frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.
$$
In general, $L_t = e^{\int_0^t \psi_s dW_s - \frac12 \int_0^t \psi^2_s ds }$ is a super
martingale. Under some conditions (e.g. Novikov's condition), $L_t$ is a martingale and
one can define a probability measure $\mathbb{Q}$ by
$$
\frac{d \mathbb Q}{d \mathbb P} = L_T.
$$
Under $\mathbb{Q}$, the process
$$
W^{\mathbb{Q}}_t = W_t - \int_0^t \psi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

An informal indication why this is true is as follows. Consider $W^{\lambda}_t = W_t + \int_0^t \lambda_s ds$. By Bayes theorem, $W^{\lambda}$ is a $\mathbb{Q}$-martingale if and only if $L W^{\lambda}$ is a $\mathbb{P}$-martingale. Since

\begin{align*}
d L W^{\lambda} &= L d W^{\lambda} + W^{\lambda} dL + dL dW^{\lambda}\\
                &= L (\psi + \lambda) dt + (\cdots) dW,
\end{align*}
we must have $\lambda = - \psi$, for $W^{\lambda}$ to be a $\mathbb{Q}$-Brownian motion.

Discounted Price as Probability Density

The implicit assumptions are that there is a underlying asset whose price $S_t$ follows
$$
\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t
$$
under the risk neutral measure $\mathbb{P}$. The short rate $(r_t)$ and volatility
$\sigma_t$ processes are adapted with sufficient regularity so that the integrals exist. (For this to be true, the Brownian filtration generated by $(W_t)$ under the risk neutral measure has to be the same as that generated by the physical Brownian motion under the physical measure, so that the Martingale Representation Theorem applies.)

In this Brownian filtration setting, for any time-$T$ claim $X_T$, the risk-neutral dynamics of its price $X_t$ takes the form
$$
\frac{d X_t}{X_t} = r_t dt + \psi_t dW_t.
$$
The process $(\psi_t)$ is the volatility of return of $X_t$, under both the physical and risk-neutral measure.

In other words, the risk-neutral dynamics of the discounted price
$M_t = e^{- \int_0^t r_s ds} X_t$ is given by
$$
\frac{d M_t}{M_t} = \psi_t dW_t, \, M_0 = X_0.
$$
(The discounted price of any $T$-claim must follow a martingale under risk neutral measure, by no arbitrage.)

If Novikov's condition holds, then $L_T = \frac{M_T}{M_0}$ defines a Radon-Nikodym density
$$
\frac{d \mathbb Q}{d \mathbb P} = L_T.
$$
Under $\mathbb{Q}$, the process
$$
W_t - \int_0^t \psi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

In other words, the discounted payoff $ e^{- \int_0^T r_s ds} X_T$ of any $T$-claim $X_T$, normalized by its time-$0$ price $X_0$, can be considered as the Radon-Nikodym density of a measure $\mathbb Q$. Under $\mathbb Q$, the risk-neutral Brownian motion now has drift given by the volatility of return $\frac{d X_t}{X_t}$.

If $(Y_t)$ is the price of a traded asset, then
$ e^{- \int_0^t r_s ds} Y_t$ is a $\mathbb P$-martingale. This implies that $(\frac{Y_t}{X_t})$ is a $\mathbb Q$-martingale.

Forward Measure

The forward measure is a special case of above where $X_t = P(t,T)$ is the time-$t$ price of the zero coupon bond maturing at $T$.
In particular, $X_T = P(T,T) = 1$.
In the expression
$$
\frac{dP(t,T)}{P(t,T)} = r_t dt + \xi_t dW_t,
$$
$\xi_t$ is the volatility of return on zero coupon bond.

(If $(r_t)$ is deterministic, then $\xi = 0$, and the forward measure is the same as the risk neutral measure. The zero-coupon bond is a risky asset only when the short rate is stochastic.)

The corresponding measure $\mathbb Q$ is defined by
$$
\frac{d \mathbb Q}{d \mathbb P} = \frac{e^{- \int_0^T r_s ds} P(T,T)}{P(0,T)} = L_T.
$$
Since
$$
\frac{d L_t}{L_t} = \xi_t dW_t,
$$
it follows from general discussion above that, under $\mathbb{Q}$, the process
$$
W_t - \int_0^t \xi_s ds
$$
is a standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$.

(In the question posted, the martingale $M_t$ should be $\frac{e^{- \int_0^t r_s ds} P(t,T)}{P(0,T)}$. It's the discounted asset prices that are martingales under risk-neutral measure.)

Empirical Comments

The forward measure $\mathbb Q$ has the property that the forward prices form a $\mathbb Q$-martingale.

Suppose $F(t,T)$ is the forward price of the forward contract entered at $t$ with maturity $T$. By no-arbitrage (spot-forward parity, in this case)
$$
F(t,T) P(t,T) = S_t
$$
which, after discounting, is a $\mathbb P$-martingale. So $F(t,T)$ is a $\mathbb Q$-martingale.

Since the forward price
$$
F(t,T) = \frac{S_t}{P(t,T)}
$$
moves inversely relative to the $P(t,T)$.
The forward measure shifts probability mass toward states where
the discounted return of zero coupon bond
$$
\frac{d \left( e^{- \int_0^t r_s ds} P(t,T) \right)}{ e^{- \int_0^t r_s ds} P(t,T)} = \xi_t dW_t,
$$
is high, in such a way that counteracts the movement in $P(t,T)$ and keeps the (conditional) expectation constant.


Why is deflation not considered the opposite of inflation? In this article from 2013 on bitcoin, the author casually states the importance of knowing that "deflation is not the opposite of inflation", without expanding or elaborating. I'm sure that there are differing points of view, but can anyone provide support for this statement and its importance?

To me it seems that deflation is a decline in prices, while inflation is an increase in prices - making them opposites. However, I'm guessing that the author is referring the effects of inflation and deflation not being opposite, or perhaps there are some differences when considering pure currency (the article is about bitcoin) vs. the effect of a currency in the market.

People think that inflation is bad when it gets out of control, say when its more than 3 or 4% a year. The idea is that we all make economic decisions based on prices and its harder to make those decisions when the prices are constantly changing! Moreover, we try to avoid holding on to money and committing to a price for a long time, because we know those become meaningless over time. Also people associate inflation with the ills of fiat money, and other such things.

Deflation would seem to be not a big deal. You hold on to your cash, it buys more things as time passes! But, it turns out to be a big deal. Deflation turns the real interest rate paid by cash into a positive number. And that means that when the fed wants to lower rates a lot to get you to spend and borrow, it can´t lower them below that real rate. So it can't get you to spend and borrow -> economic fragility! Fed is powerless! So people are now scared to death of deflations! Even small ones seem to be hard to turn around. Japan has been flirting with deflation for a long time, unable to escape from its cold grip....

Therefore, to answer the question:
 Inflation and deflation can be defined as opposites: inflation is a positive rate of change of prices sustained over several periods while deflation is a negative rate of change of prices sustained over several periods. However, the effect of inflation and deflation on the economy are not opposites from each other, and policy-maker's concern for one and the other are different, but not the opposite of each other either.


Simultaneity problem: use of Instrumental Variables and test The following model jointly determines monthly child support payments and monthly visitation rights for divorced couples with children:

$$support = d_1 + g_1 visits + d_1 finc + d_1 fremarr + d_1 dist + u_1$$
$$visits = d_2 + g_2 support + d_2 mremarr + d_2 dist + u_2$$

For expository purposes, assume that children live with their mothers, so that fathers pay child support. Thus, the first equation is the father’s ‘‘reaction function’’: it describes the amount of child support paid for any given level of visitation rights and the other exogenous variables finc (father’s income), fremarr (binary indicator if father remarried), and dist (miles currently between the mother and father). Similarly, the second equation is the mother’s reaction function: it describes visitation rights for a given amount of child support; mremarr is a binary indicator for whether the mother is remarried.

a. Discuss identification of each equation.

b. How would you estimate each equation using a single-equation method?

c. How would you test for endogeneity of visits in the father’s reaction function?

I'd like to get some help with the whole question and to have my reasoning on it checked, thank you:

a)identification not possible due to simultaneity (which causes endogeneity)

b)via instrumental variable (possible choice (fremarr and mremarr) or ( dist)

c) my only idea was to estimate visits via OLS then use Hausman test between OLS and IV

Is the approach correct? How can I answer C in an alternative way?

To clarify notation, and treating summarily the other variables we have a system

$$y = a_0 +a_1x + \mathbf w_1'\gamma_1 + u_1 \\
x = b_0 + b_1y + \mathbf w_2'\gamma_2 + u_2$$

If $b_1 \neq 0$, then solving for $x$ we obtain

$$ x = b_0 + b_1(a_0 +a_1x + \mathbf w_1'\gamma_1+ u_1) + \mathbf w_2'\gamma_2 + u_2 \\
\implies x = c + \mathbf w_1'\frac {\gamma_1}{1-a_1b_1}+\frac {b_1}{1-a_1b_1}u_1 + \mathbf w_2'\frac {\gamma_2}{1-a_1b_1}+\frac {1}{1-a_1b_1}u_2$$

and  $x$ is correlated with $u_1$ in the first equation, and so it is endogenous (we would obtain an analogous result if we solved for $y$ -it would be endogenous in the second equation). Historically, this is the "original" endogeneity problem, the one realized back in the '20s when looking at market equilibrium, where the assumption of market-clearing created the endogeneity. Namely it is not the usual "omitted variables" endogeneity problem, but the form of endogeneity due to "simultaneity".

So endogeneity of $x$ hinges on $b_1\neq 0$. But running OLS on the second equation and testing whether $b_1=0$ is not valid, because, as said, $y$ is endogenous in the second equation, so the OLS estimation is unreliable, being inconsistent.
This is a bit subtle. We want to test $b_1 =0$, so under the null hypothesis $y$ does not belong in the second equation. But inference will use $y$ itself. This means that if the null is rejected, the rejection is not reliable, because if the null is not true, then $b_1 \neq 0$ and automatically the estimator, t-tests etc become inconsistent.

The question asks to test for endogeneity in the first equation, so the evident thing to do is to search for an instrument, or instruments, for $x$ and then, for example, apply the Hausman test in the first equation.

Assume that we applied the Hausman test in the second equation, as the OP suggested. This would mean searching for instruments for $y$. Here if the null is not rejected, it would mean that $y$ is not correlated with $u_2$. But from this it does not necessarily follow that $b_1\neq 0$, and so it does not guarantee that $x$ is uncorrelated with $u_1$. If the null is rejected, it means that $y$ is correlated with $u_2$. But this could happen for any other reasons, not necessarily due to the simultaneity.
So the endogeneity tests should be conducted on the first equation.

Note that in reality, the Hausman test  in IV estimation has as its null hypothesis that both the OLS and the IVE estimator are consistent. If the null is rejected, it only tells us that at least one of the two is inconsistent. It cannot tell us which one, and it cannot guarantee that not both are inconsistent: The Hausman test is more informative and specific in its conclusions when the null is not rejected, rather than when the null is rejected. (This is a point usually not clearly stressed in the literature, although side remarks and footnotes do exist to that effect).


Question on the Definition of A Numeraire I'm familiar with the idea of a numeraire when prices come in the form $\frac{p_x}{p_y}$, but I'm still trying to figure out what the more general notion of a numeraire is. For instance, if all that I'm given is $p_x,p_y,$ and $m$: what is supposed to be the numeraire there? Is it just 1 unit of currency?

numeraire is by definition some unit of measure by which we measure all other units. In your case, let us say that we would decide to use $x$ as a numeraire. In that case we would divide all prices by $p_x$ so the price for $x$ would become $\frac{p_x}{p_x}=1$ (one $x$ can be worth only itself) and for good $y$ the price would be set as $\frac{p_y}{p_x}$.
For example, let us suppose that 1 apple is worth 2 rocks. In such case if we would decide to make apple numeraire everything would be expressed in apples. Now 1 apple would cost 1 apple and 1 rock would cost 0.5 apples (since 1 apple is worth 2 rocks 1 rock would only cost 0.5). If we would choose to make rocks numeraire then 1 apple would now have price 2 rocks and one rock would have price one rock.
If you want to think of it intuitively, you could consider numeraire to be 'currency' used in your particular problem (in real life money is a numeraire - we value all other goods and services in terms of money, let's say dollars, and one dollar itself is just worth one dollar).


Why does decrease in fixed taxes accompanied by a decrease in Government Spending of the same amount will decrease GDP? I personally don't agree because change in GDP caused by decreased in fixed taxes should be

$\triangle$ fixed taxes $ \times \frac{-b}{1-MPE}$(fix tax multiplier), which turns out to be a positive number while change caused by decrease of Government Spending is

$\triangle$ Government Spending $ \times \frac{1}{1-MPE}$(autonomous multiplier) which is a negative number. Therefore I don't think we can make conclusion. Does anyone know what did I do wrong?

The question basically boils down to whether the inequality below is true or not.
$$\bigg|\Delta G\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\Delta T\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If $\Delta G=\Delta T$, we can divide by either and get
$$\bigg|\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If we multiply each side by $1-MPC$ and we evaluate the absolute value, we get
$$1>MPC$$
This statement is true, so that implies that the first inequality is true. By this inequality, we know that $\Delta GDP_G>\Delta GDP_T$. Since $\Delta GDP_G<0$ and $\Delta GDP_T>0\implies \Delta GDP<0$


Choice between dummy variables and Likert scale in Linear Regression I want to run a linear regression based on the data gathered using a questionnaire. Several of the questions have the following form:
How much do you spend on xyz in a month?
a. Less than \$50
b. \$50 to less than \$100
c. \$100 to less than \$200
d. \$200 or more
This is ordinal data, with multiple categories. I am not sure how to code them.
Should I use code them like a Likert scale (though Likert scale data is interval data instead):
0 for a, 1 for b, 2 for c and 3 for d
Or should I use dummy variables like this:
3 dummy variables for options b, c and d, with the respective dummy variable set equal to 1 if the option is chosen and 0 otherwise. All dummy variables are set to 0 if option a is chosen.
In this case I am concerned that all sense of ordinality is lost and I am treating the options just as nominal data. In this case, I think I would not be able to comment on whether the more you spend on xyz, the more you gain weight for example.
Which one should be used? Or should some completely different coding be used?
I seek to use such variables only as regressors, my regressand is a ratio-scale variable.

In your question you talk about coding so I will address primarily the question of coding.
You should definitely code it using suggested

0 for a, 1 for b, 2 for c and 3 for d

regardless of whether you plan to use it as ordinal variable or categorical variable.
Let me explain, nowadays virtually any language or program will have option to easily turn ordinal, or categorical variables into dummies with virtually zero effort.
In R you can simply use is.factor() function to turn any ordinal variable into separate dummies so you could use:
lm( dep_var ~ ind_var + is.factor(variable), data = XY)

In Stata you can use i.variable to turn any ordinal or categorical variable into dummies, again in Stata you would run something like:
reg depvar independentvar i.variable

Next, coding it as separate dummies is more time consuming (even if just marginally it takes more line of code) than just coding it into one variable.
Consequently, no matter whether in actual model you want to include it as oridinal variable or set of dummies, in most standard languages and programs it will be more efficient (i.e. take less time to code) if you just code it on Likert scale.
As for whether what you should do from scientific standpoint that depends on details of your research. I can imagine you would be being able to defend both including the variable as independent ordinal scale or as a separate dummies. Both of those would work in principle. My sense is that you would likely want to include it as dummies since that will have better interpretation,  but if you have little data and too many regressors then it might be worth while to just have it as ordinal variable.


Is VCG mechanism applicable in reverse auction? If so, how? The VCG mechanism I've learnt is from Roughgarden's Twenty Lectures on Algorithmic Game Theory. Given an auction, we first identify the allocation that maximize the social welfare and then compute the payment for each agent according to the formulas given. In the case of reverse auction, according to this thread, the social welfare in a reverse auction is defined as:
$$\sum_{i=1}^n x_i (v_0 - c_i)$$
where we have one buyer who values the good to be procured at $v_0$ and $n$ sellers who can produce the good at private cost $c_i$, and $x_i$ indicates the outcome(allocation).
I was wondering, in reverse auction, whether or not the VCG mechanism is implemented exactly the same way as in Roughgarden's Twenty Lectures on Algorithmic Game Theory (section 7.2) ?

In general, VCG is also applicable to reverse auction settings. VCG is not even restricted to auction settings and can be used quite generally, see wikipedia for an introduction. If you want a deeper treatment, I recommend Tilman Börger's book (it used to be fully online, maybe there are still copies flying around). In reality, there are often problems associated with VCG, see, e.g., Ausubel and Milgrom writing about "The Lovely but Lonely Vickrey Auction."
The general idea of VCG is that each agent's transfer is their own externality.  In your setting, you cannot simply do that because there is a budget restriction on transfers. Papers with such a knapsack constraint in economics are Ensthaler &Giebe and Jarman &Meisner (JET 2017), who use clock auctions as in Milgrom & Segal (JPE 2020).


Which macroeconomic issues can be considered "curable" and which can't? I like the comparison between economics and medicine. Both study complex phenomena and both are limited in ways to run experiments (though medicine has much more freedom here).

We all know medicine has found a cure for lots of previously dreadful diseases, like plague, flu, TB. Doesn't mean that no one dies from TB now, but it's a problem with medical service in developing world, not with medical science. And there are issues medicine is still struggling with (AIDS, cancer, aging, etc.)

By analogy, can you give examples of macroeconomic issues which are proven to be curable if properly treated and ones not so curable?

Friedman's assessment that "inflation is always and everywhere a monetary phenomenon" seems to have substantial concensus around it empirically: (http://www.nber.org/papers/w1453)

I think this essentially gave us the cure for both hyperinflation (don't monetize government spending or the debt more generally) and deflation (print money or do QE until the price level goes up). That isn't to say we've eradicated these "diseases" because we don't always follow the cure. Or even countries should take the cure. Following the cure can be politically untenable and hyperinflation could well be better than revolution.


Testing for Endogeneity I apologize if this question is very basic. I have the following plain vanilla Instrumental Variable model.

$Y=\alpha+X\beta+\varepsilon$

$X=\delta+Z\gamma+\eta$

$\varepsilon\perp\eta,\quad Z\perp\eta \quad$ true

I am interested in testing $\varepsilon\perp X$, that is, whether X is a valid instrument for the first equation (very informally stated, one might say I want to test whether $X$ is exogenous, or whether I need to instrument $X$ with $Z$)

My idea is: estimate the IV model using 2SLS or GMM, using both $X$ and $Z$ as instruments, and then perform a Sargan/Hansen test. My guess is that the test's power will depend on how strongly $Z$ predictx $X$ (that is, on how relevant of an instrument $Z$ is for $X$). In 2SLS, the first stage will fit perfectly and the test will be basically a test of whether the OLS residual are orthogonal to $Z$. Is this reasoning correct? Is the Sargan/Hansen test a valid test for $
\varepsilon\perp X$?

(i) I think your idea makes sense. Under the null, $[X,Z]$ is orthogonal to $\varepsilon$. Under the alternative, $X$ is correlated with $\varepsilon$. (ii) Your statement that it's "basically a test of whether the OLS residual are orthogonal to $Z$" is exactly what I think. (iii) Your thought about the power depending on the relevance of $Z$ also makes sense. If $Z$ is irrelevant, then the moment restrictions $E[Z_i' (y_i - \alpha - X_i \beta)] = 0$ hold for any $\beta$ (due to the uncorrelatedness of $Z_i$ and $X_i$) so the OLS residuals should appear to be uncorrelated with the instruments although the OLS estimator is inconsistent. I think this test is closely related with standard methods explained below, although I have not done any derivation in this regard.

(iv) As you would be aware, there are textbook methods to test endogeneity. Stata implements some. See help ivregress postestimation. You will see estat endogenous there. There, for the 2SLS estimation, OLS and 2SLS can be compared; or $y$ is regressed on $[X,\hat{\eta}]$, after which the statistical significance of $\hat\eta$ is tested.


Current Account vs Net Exports In Mian, Sufi and Verner (2017), the authors look at the impact of an increase in household debt on both net exports and the current account, and find a significant negative coefficient in the regression specification employed (see Table V).

Why would you look at both the current account and net exports? Wouldn't net exports be the only item of interest in the current account? Why would one care about the impact on transfers, for instance?
I wonder about this further, because they don't include the current account in their working paper version here.

Indeed, their theory relates primarily household debt to net exports. This is the main focus of their working paper. However, as a referee, I would try to find out more and ask whether the change in net exports comes from imports or exports, or whether household indebtedness could also affect international transfers. A household may use debt to invest abroad and get dividends and interests in return. Are there no other margins that could be affected? This type of request is quite common from referees and may explain the differences between the working paper and the final version.


Why is national debt bad if the central bank can keep buying bonds? When I read about national debt, it's usually about how debt is bad because with too much debt, debtors would demand higher interest rates. Eventually the government can't pay the interest, defaults, and the economy collapses.

However not discussed is why the central bank can't just step in and keep lending money to the government. It's not like the central back doesn't already do that - according to this article, 27.6% of the US national debt is currently to another arm of the government. But if the central bank can just keep buying government bonds, the government can also force the interest rate on its debt as low as it desires (just get the central bank to buy more bonds), not to mention deficit spend to its heart's content. And if the government is able to do that, then national debt isn't a problem at all!

It seems to me that there's only a problem if 1) the country doesn't have a central bank or 2) you're using another country's currency (e.g. for members of the Eurozone). Am I missing something?

Yes.

The central bank (say the Fed in U.S.) can purchase bonds, but notice:

The Fed is not going purchase t-bonds and t-bills directly from the Treasury. It will only purchase or sell bonds in the market. ("Open market purchase" or "open market sell.")
Open market purchase is not only the buying of bonds, but also the creation of money. (You sell bonds to the Fed, and the Fed will pay you.) Increase in money supply means inflation. Obviously, the Fed can't buy too much bonds, or the U.S. will have to deal with disastrous hyperinflation.
Most central banks have some degree of independence in determining what to do. The Fed won't purchase more than they want, and the government is not able to force them.


Subgame perfect equilibrium and expected profit I am so confused because I cannot set up the monopolist's profit maximization problem.What I did is the following one:

Any help will be appreciated. Thank you.

After obtaining the loan, entrepreneur has two possible strategies to choose from :

Strategy $a$ : Expected payoff from strategy $a$ is $\frac{3}{4} (12000-R)$
Strategy $b$ : Expected payoff from strategy $b$ is $\frac{1}{4} (16000-R)$

He'll choose strategy $a$ if $\frac{3}{4} (12000-R) \geq \frac{1}{4} (16000-R)$, and strategy $b$ if $\frac{3}{4} (12000-R) < \frac{1}{4} (16000-R)$. Or simply, he'll choose strategy $a$ if $R \leq 10000$, and strategy $b$ if $10000 < R \leq 16000$.

Bank will maximize it's expected profits by choosing $R$ taking as given the entrepreneur's strategy. Therefore, Bank's expected profits equal $\frac{3}{4}R - 1000$ if it chooses $R \leq 10000$, and $\frac{1}{4}R - 1000$ if it chooses $10000 < R \leq 16000$. Clearly, expected profits of the bank are maximum at $R^*=10000$. Observe that entrepreneur is willing to pay more for the loan than $R^*=10000$ since his expected payoff is $\frac{3}{4}(12000 -10000) = 1500 > 0$, but the bank  will not charge more than 10000 because that would change the entrepreneur's choice of strategy to $b$ which is a more risky choice and yields lower expected returns to the bank.


Converse in supermodularity and single-crossing Suppose I denote $$f(x,t)=g(x,t)+h(x)$$ where $X\in \mathbb{R}$, $t\in [0,1]$ and $h(x)$ is any function.

I want to prove that if the single-crossing property holds in $f(x,t)$, then $g(x,t)$ must be supermodular over $X\times T$.

I know in general supermodular means single-crossing but the converse is not true, but my guess adding a random function $h(x)$ should guarantee single-crossing property will show supermodularity.

I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\cdot,\cdot)$ will have increasing differences, which implies the single-crossing property for $g$ (this can be quickly seen by letting $h$ be identically zero). But I don't see how this is enough, since single-crossing doesn't imply supermodularity. Maybe I'm missing something, though.


Whats the name of the famous hostile takeover where the winner payed less than the loser In my studies I learned about a bid war ending in a hostile takeover where the winner ended up paying less than the loser offered. It is a very nice demonstration of elements of game theory so I would like to reread that again and to recommend it to a friend.
Unfortunately, I cannot find it in the internet regardless of what search terms I try. I will state the details I remember. I would be very glad if the answer could tell which companies were included so I can research the interesting details.
Here we go:

Two companies, A and B, competed about to acquire a third one.
The bid war happened at least 10 years ago, maybe more.
The bid war ended in a hostile takeover.
Company A offered a fixed amount. I am not sure, but that offer might have been valid only if enough shareholders accepted it.
Company B offered a higher amount regardless of enough shareholders accepting it. However, if a certain number of shares were reached, the offered price was reduced linearly.
B's offer was dominating A's offer. B won the hostile takeover.
In the end, shareholders received less for their shares than what A offered initially. (This is the crazy thing about this bid war).

I was unable to translate that list in sufficient search terms. Does it ring a bell for someone?

With the help of a former fellow student I was able to find out some details.
The anecdote was about the following three companies:

Macy's, Inc.
Robert Campeau
Federated (covered in the article about Macy's)

Some news coverage exists:

https://mindyourdecisions.com/blog/2011/09/20/when-game-theory-backfires-a-case-study-of-robert-campeaus-takeover-bid/
https://archive.fortune.com/magazines/fortune/fortune_archive/1990/06/18/73686/index.htm

In short, Macy's offered \$67 per share contingent on getting a majority of shares. Campeau offered \$74 per share if less than half was tendered, otherwise $\frac {0.5}{x} * \\\$74 + \frac {x - 0.5}{x} * \\\$60$ with $x$ the fraction of shares offered to Campeau. It seems as if the bidding war continued a bit after this scheme. The prices given in the blog above differ, I am sticking to these from the lecture.
I am very grateful for having had the opportunity to attend Prof. Kirste's lectures at the Universität Rostock. They were full of impressive details like this gem here.


Why does a currency devaluation make a country more attractive for foreign direct investment? The author of this article says "[the devaluation of the yuan] may also have impact on [foreign direct investment (FDI)] if China becomes a more attractive destination vis-a-vis India". Why is this?

I understand that the following sentence then says "investors would go there where with the exchange rate he will get more kick for his dollar", but what makes China a more attractive destination for FDI than India following the devaluation of the yuan?

A depreciation in the Yuan will result in an increase of Chinese exports as where countries compete on price for goods like steel and tires the demand is very elastic. A small change in price will lead to a large shift in the quantity demanded. India knows this and after a succession of poor last few months exports will stand to lose out even more now that the Yuan has devalued.

Also now that China has changed its exchange rate, China could be leaning to a more flexible exchange rate (which the article suggests) opening up for an increase in capital mobility which investors look favourably upon. The devaluation of the yuan has also decreased the price of Chinese assets thus making it cheaper to invest.

It is likely that the further reduction of the price of Chinese exports will cause the currency to appreciate once again as demand for Chinese goods increases. Some investors will take on Chinese assets and currency, predicting the currency will appreciate in the future, resulting in a nice profit.

Finally as India's export reliant sectors struggle to compete with supplying major industries around the world,  the drop in market share will be swallowed up by China resulting in more attractive Chinese businesses (which may be open to FDI). India is also struggling to supply its domestic industries as it cannot compete on prices. This uncertainty around India and its lacklustre performance will likely see FDI being pulled out and reinvested into more lucrative opportunities.


What would be the "text-book" result of governement taking on massive debt immediately before a significant recession Supposing, (entirely hypothetically, of course ;) ), that the UK government had taken on massive future debts (by spontaneously giving 2-3% of GDP to its citizens, in GBP, say.) immediately before a significant recession.

What would be the simplistic prediction of the impact on:

Interest Rates
Inflation

over the following, say, 20 years?

Obviously such situations are incredibly complex and there are numerous ways that the outcomes could be changed. But I want to get a feel for what the "base line" or "default" expectation might be.

As asked, this question is too vague and open-ended. But the following statements seem safe.

The effect of a fiscal stimulus is to raise growth and expected growth. All else equal, the central bank would be expected to raise interest rates, and inflation would be slightly higher, but still should remain near target (or else the central bank is going to have to explain why it is failing in its inflation-targeting).
A recession is a reduction in growth. Inflation normally falls, and the central bank cuts rates.

The effects are in different directions, hence the net of the scenario is ambiguous.


Monetary Base vs. Stock price link: https://research.stlouisfed.org/fred2/graph/?g=3big
I have watched this video where he explain this chart.
Basically he shows high correlation between Monetary Base and the Stock Market. But I cannot understand why he multiply the Wilshire for 47 and the "Federal Funds Rate" for 250 ?
UPDATE 2021:That correlation was true only for that time.

The multiplication factors are there so that they can be charted and be seen on the same axis as the St. Louis Adjusted Monetary Base.

The factor of 47 means that the closeness of the match between that and the Wilshire 5000 in 2000-2015 is easiest to see, by overlaying the curves in that period.


How will the economic effects of Brexit impact a middle-class American family? On the news it was stated that the UK leaving the EU has already caused trillions of dollars of loss to the world economy. While the newscaster made it sound very serious, I don't have any international finances at all, and my employer only operates in North America. It seems to me that the ripples will be so small by the time they reach me that I wouldn't even notice, but I don't know much about the global economy.

Is the buying power of the USD likely to fall? Will certain categories of goods become more expensive than others? Will employment rates change significantly? Interest rates on savings and/or debt?

Will my pocketbook really notice over here in the USA?

There will be many effects, but most of them are too small or too difficult to measure for you to notice anything.

Some examples of the bigger effects:

If you have a pension or own any stocks (hopefully you do), you will have been negatively affected - even though you don't have any 'international' finances. Since the world economy is highly integrated, losses in the UK are correlated with losses in the U.S., Japan, and other world markets. For example, after the news of Brexit broke, the S&P (one of the US stock market indices) fell roughly 100 points. The markets tend to move together (the reason behind this is another question in of itself), so any changes in Britain will affect Americans as well. We're still not sure about the long-term economic effects of this, since we don't know exactly how/when/if the U.K. will exit, but the short-term uncertainty is causing investors to be nervous.
British things will be cheaper! The pound fell, meaning that it's cheaper for Americans to buy British things, and more expensive for the British to buy American things. If you wanted to visit the U.K., it's going to be relatively cheap. The simplest effect is that the fall of the pound will positively affect U.S. consumers of British goods, and negatively affect U.S. exporters who export to the U.K..
There may be some effect on your work, but it's impossible for you to know (which is part of the problem. Although your employer may appear to be completely local, it's important to remember that since we're so globalized, there probably will be some effect at your company. For example, one of your
company's clients may purchase services from a consultant in the
U.K, and this might affect your client, in turn affecting you. It's
this kind of domino effect and the uncertainty that Brexit has
caused that has so many politicians and economists worried about
what might happen.


Finding Bertrand equilibrium with no demand function explicitly given Two cell phone companies A and V are located at the extremes of a line of
length one and transportation cost t = 4. Consider an initial situation where both firms offer a generic phone that consumers value the same (except obviously
for the transportation cost.) Marginal cost is zero.

*N = 800 denote the total population,
show that profits for each firm will be 1600

How do I interpret the demand function for this example?

This is not a Bertrand equilibrium. Altough both firms set prices and they do so simultaneously, the products are not perfect substitutes because the travel distance differs for most consumers.

What this seems to be is the second stage of a Hotelling model. The usual assumptions are that a continuum of buyers are spread out uniformly along the line and that every buyer's reservation price is very high compared to the transportation cost, so in equilibrium all buyers will be served. The total quantity of buyers may be scaled to one for simplicity.

Given these assumptions you can derive the demand function each firm faces.


What does a central bank guarantee against its currency? During the Bretton Woods system, currencies were issued against gold. In contemporary times, what is the real thing against which currency is issued? What does the fiat currency actually guarantee?

Most central banks do not provide guarantees on their currency to the public.  In fact it can be argued that historically they were never fully guaranteed as most central banks issued more receipts for gold (or other assets) than they had, so the receipts were never fully redeemable in the aggregate.

The American Fed was stressed during the great depression and had to go off the gold standard for the public in 1933.  The international gold standard was retained and dollars were redeemable for gold for a while.  This too was "overbooked" and when the rightly suspicious French started to mass redeem dollars for gold, the Americans had to get off the international gold standard as well in 1971.

Today the dollar is not redeemable for anything.  You can exchange paper dollars for coins or electronic dollars, but that is just changing the form of the same thing (the monetary base).

Because you can go to jail if you don't pay your taxes in dollars, it can be said Americans are forced to purchase dollars and thus provide extrinsic value to the dollars though public tender laws (granted still different from intrinsic value but still important).

Pegged currencies are a different matter and are the closest modern equivalent to perhaps what you are asking.  With a peg, a country must acquire a certain amount of an external currency to maintain a peg.  It could be thought of indirectly that the country maintaining the peg is guaranteeing a redemption of their local currency to an external one.  The idea is to somewhat stabilize the local banking systems which need a lot of artificial stability to survive.  Granted this doesn't work well as pegs often do run out of "reserves" and suffer "runs" just like a gold standard.


Multicollinearity in Transformed model: GLS Hi I am reading Hayashi's econometrics and have got stuck in the review question 1.5.1. The question asks us to prove that the multicollinearity assumption of the CLRM is satisfied by the transformed data matrix CX where C is (n by n) invertible matrix and X is an (n by k) matrix with rank = k. We have to show that the rank of CX = k. I know this is a theorem in matrix algebra and we can prove that rank CX = Rank X = k. But Hayashi suggests in the hint to the question another way to prove it and says that we must show that for all non-zero vectors c it must be that Xc is not equal to zero. Now, I could easily show this from the the fact that X has a full rank. But I get stuck in showing the final result that rank (CX) = rank (X) = k. Any suggestions on how to go about this, will be very helpful. It might something very straightforward that I am missing. Thanks a lot in advance. P.S. In case you are interested in looking at the question it is on page 59 of chapter 1.http://press.princeton.edu/chapters/s6946.pdf

For every $k \times 1$ vector $\mathbf c\neq \mathbf 0$, we have that $\mathbf X \mathbf c \neq \mathbf 0$.
Choose $\mathbf e_i$ to be a standard basis $k \times 1$ vector having $1$ in the $i$-th position, and $0$ everywhere else. $\mathbf X\mathbf e_i \neq \mathbf 0$ is a $n \times 1$ vector, and equal to the $i$-th column of $\mathbf X$.
So in block matrix notation we have that

$$\mathbf X = \left[ \begin{matrix} \mathbf X\mathbf e_1\;\; ...\;\;\mathbf X\mathbf e_k\end{matrix}\right] $$

Note that the set of vectors $\mathbf X\mathbf e_i$ also form a basis.
Now,

$$\mathbf C \mathbf X =\left[ \begin{matrix} \mathbf C\mathbf X\mathbf e_1\;\; ...\;\;\mathbf C\mathbf X\mathbf e_k\end{matrix}\right]$$

Since $\mathbf C$ is non-singular (and so of full column rank), and $n \times n$,it follows that the $n \times 1$ vectors $\mathbf C(\mathbf X \mathbf e_i) \neq \mathbf 0$. Moreover they are linearly independent by virtue of the full set of basis vectors used. So $\mathbf CX$ is of full column rank.


Why does $\varepsilon_{x,p_x}^H =-s_y \sigma $? Suppose I have two goods $x$ and $y$ and their associated prices $p_x$ and $p_y$. Income $m$. $x^H$ is Hicksian demand and $x^M$ is Marshallian demand.

Slutsky Equation:

$$\frac{\partial x^M}{\partial p_x} = \frac{\partial x^H}{\partial p_x}  -\frac{\partial x^M}{\partial m} x^M $$

Elasticity Version:

$$\varepsilon_{x,p_x}^M = \varepsilon_{x,p_x}^H - \eta_x s_x$$

$$s_x = \frac{p_x x}{m}$$

My professor claims $$\varepsilon_{x,p_x}^H =-s_y \sigma $$, but didn't give a proof even when asked. Nor could my TA provide one.

$\sigma$ is the elasticity of substitution

My Question:

Can someone show why $\varepsilon_{x,p_x}^H =-s_y \sigma $?

First we need to define the elasticity of substitution $\sigma$. This can be a difficult and confusing concept. (If you want to have your mind blown, look at Table 2 in this survey by Stern, which classifies no fewer than 10 notions of the elasticity of substitution!).

That said, the formula that your professor mentions is true only when there are two goods, so I'm going to assume we're restricting ourselves to the two-good case - and there, defining the elasticity of substitution is (for the most part) unambiguous. I'm first going to define it as (minus) the inverse elasticity of the ratio of marginal utilities of $x$ and $y$ to the ratio of their quantities, holding the overall level of utility constant at some $u$:
$$\sigma\equiv - \left.\left(\frac{\partial \log(U_x/U_y)}{\log(x/y)}\right)^{-1}\,\right|_{U=u}$$
This measures the curvature of the indifference curve between $x$ and $y$: the higher $\sigma$ (i.e. the more substitutable $x$ and $y$ are), the closer the indifference curve is to a straight line.

Now, Hicksian demand minimizes expenditure subject to being on an indifference curve. This involves setting the ratio of marginal utilities equal to the ratio of prices: $$U_x(x^H,y^H)/U_y(x^H,y^H)=p_x/p_y$$
Hence we can reinterpret $\sigma$ as (minus) the elasticity of relative Hicksian demands with respect to relative prices:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x/p_y)}\tag{1}$$
Since the denominator has only the relative price $p_x/p_y$, it doesn't depend on how we change this relative price (say, by raising $p_x$ versus lowering $p_y$). Let's assuming that we're just raising $p_x$, and then use your $\varepsilon$ notation for elasticities for simplicity:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x)}=\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x}\tag{2}$$
At this point, we bring in a simple identity for Hicksian demand, namely that the sum of elasticities of goods with respect to some price, weighted by their shares, is zero (proof included below):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
Since $s_x=1-s_y$, we can rewrite (3) as
$$(1-s_y)\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=s_y\cdot(\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x})+\varepsilon^H_{x,p_x}=0\tag{4}$$
But the term in parentheses in (4) is just the elasticity of substitution $\sigma$, as we showed in (2). Rearranging, we've proven your professor's statement
$$\varepsilon^H_{x,p_x}=-s_y\sigma$$
as desired.

Proof of (3). We know that holding income constant, Marshallian expenditure must remain constant after any change in prices. The elasticity of spending on $x$ with respect to $p_x$ is $1+\varepsilon^M_{x,p_x}$ (combining price and quantity changes), and the elasticity of spending on $y$ with respect to $p_x$ is $\varepsilon^M_{y,p_x}$. Weighted by initial shares, these must sum to 0:
$$s_x(1+\varepsilon^M_{x,p_x})+s_y\varepsilon^M_{y,p_x}=0$$
Substituting the elasticity Slutsky equation, we get
$$s_x + s_x(\varepsilon^H_{x,p_x}-\eta_xs_x)+s_y(\varepsilon^H_{x,p_x}-\eta_ys_x)=0$$
which, once we use the fact that $\eta_xs_x+\eta_ys_y=1$ (as income expands, total spending expands proportionately), has a $-s_x$ canceling out the $s_x$ at the front, and reduces to just (3):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
as desired. This is a very important identity governing Hicksian demand: it says that as we change prices, to first order the cost of Hicksian demand at the old prices stays constant (this is related to it being "compensated" demand: locally, it would cost the same at the old prices).


Basic Solow Growth Model: Stability Proof I am reading through McCandless "The ABCs of RBCs" this summer to get a preview of what I need to know for the coming Fall semester. It did not take long to find a statement that I can easily accept but cannot prove. On Page 9 after deriving the steady state condition of $(\delta + n)\bar{k} = \sigma A_0 f(\bar{k})$ in a zero-technological growth regime (where $\delta$ is depreciation and $n$ is growth rate of the labor force), the author says that "the stability of the positive stationary state can be seen from the equation $$ k_{t+1} = g(k_t) = \frac{(1-\delta)k_t + \sigma A_0 f(k_t)}{1+n}$$ Notice that between 0 and the positive $\bar{k}$, the function $g(k_t)$ is above the 45 degree line, so that $k_{t+1}$ is greater than k_t." He provides a standard-looking Solow model state diagram where I can verify this graphically, but not analytically.

I'm trying to prove every statement in the book in order to better familiarize myself with the details of macroeconomic theory before diving deeper, but I am absolutely stumped as to how I would prove that $k_{t+1} > k_t$ when $0 < k_t < \bar{k}$ and vice versa. I first tried manipulating the capital motion equation to prove it directly, and I could not reach a proof. The next strategy I have been trying to employ is to differentiate the capital motion equation and prove that its derivative is less than 1 at the steady state point, but it fails:

$$
\frac{\partial k_{t+1}}{\partial k_t} = \frac{(1-\delta) + \sigma A_0 f'(k_t)}{1+n} > \frac{(1-\delta) + \sigma A_0 f'(\bar{k})}{1+n} = \frac{(1-\delta)+(\delta+n)}{1+n} = 1
$$

Does anyone have an alternative strategy I can follow? It's been driving me nuts for two days.

For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is

$\dot{k} = s f(k) - \delta k = \phi(k)$

Then we have

$\frac{\partial \phi}{\partial k} = s f'(k) - \delta = \frac{sf'(k)k - \delta k }{k}$.

In steady state (i.e., $\dot{k} = \phi(k^{\ast}) = 0$), we have $\delta k = s f(k)$, hence

$\left. \frac{\partial \phi}{\partial k} \right|_{k=k^{\ast}} = \frac{s f' k - s f(k)}{k} = \frac{s}{k} f(k) \left[ \frac{k \cdot f'}{f(k)} - 1 \right]$

Since both $\frac{s}{k}$ and $f(k)$ are always positive, the sign of that expression hinges on the term in brackets. But remember that $\frac{k \cdot f'}{f(k)}$ is the output elasticity of capital, which is smaller than 1 if $f$ is concave in $k$, which means $\frac{\partial \phi}{\partial k}$ at $k^{\ast}$ is negative. Therefore by theorem 3.2.1 in Zhang (2005), the steady state in the Solow model is stable.


Shapley Value - Any Real World Applications? The Shapley Value has been around for seven decades now. It is intuitive, tractable, and has many desirable properties. To my surprise, however, its actual applications in the real world are not readily available (as defined by a simple Google search). Most of the "applications" of the Shapley Value I could find are theoretical/hypothetical in nature. For example, the Investopedia entry lists the pricing of airport runway usage and marketing analytics as two applications without reference to actual real world cases.
My question is: Are there any real world applications of the Shapley Value in pricing transactions and/or allocating profits/costs among cooperating parties?
More specifically, I'm looking for answers in any of the following forms:

"As documented in [news/professional/academic] Publication X, Company A used the Shapley Value (or its variant) to price its transaction with Company B and Company C."
"As documented in [news/professional/academic] Publication X, Company A used the Shapley Value (or its variant) to allocate its [R&D/marketing/overhead] costs among its subsidiaries B, C and D."
"The empirical study by Author et al. (20xx) in Publication X found that
the profit/cost sharing practice within a [firm/industry/multinational enterprise] is consistent with the prediction of some model based on the Shapley Value."
"The empirical study by Author et al. (20xx) in Publication X uses [a structural estimation/an empirical strategy] that is based on the Shapley Value."

Please do provide links/references to the sources. Thank you.

Note 1. While I'm primarily interested in the application of the Shapley Value in a business context, pointers to its use in other domains such as politics would also be appreciated.
Note 2. I found a paper by Littlechild and Thompson (1977) where the authors applied the Shapley Value in calculating the prices of airport runway usage based on actual data from the Birmingham Airport. It is unclear, however, whether those prices were subsequently adopted/implemented by the airport authorities.

Allegedly, Barclays uses (or once used) Shapley allocation. Quoting Mauro Cesa:

The basic objective of every bank is to find an optimal business strategy that maximises return on capital (ROC). To this end, banks will allocate more capital to desks that generate the highest ROC, while those with lower ROC receive a smaller share of available capital. This is an intuitive and seemingly sensible solution.  But the authors of this month’s paper, Reduced-form capital optimisation, argue that ranking business units by ROC might not result in an optimal allocation of capital. The approach ignores the correlation between businesses and will deliver the optimal allocation only if the correlation is zero – an assumption for which there is little evidence in the real world.  The problem is further complicated by the Basel III capital rules, which are primarily based on two ratios: risk-weighted assets (RWAs) and leverage balance sheet (LBS). The minimum capital requirement for each legal entity under the same parent bank is the greater of the two.  To optimise its ROC, a legal entity must hold RWA and LBS capital in equal measure. The selection of RWA or LBS, as directed by the greater-of-the-two rule, introduces non-linearities that are difficult to deal with in an optimisation context.  “This is a huge, unsolved problem for banks,” says Yadong Li, managing director of quantitative analytics at Barclays, and one of the authors of the paper. “The bank as a whole ultimately needs to deploy the resources to different business units. This is a key part of senior management’s job.”  Dimitri Offengenden, a Tel Aviv-based managing director of quantitative strategy at Barclays, joined the bank in 2017 specifically to work on the capital allocation problem. He consulted with Li, who has worked on related issues in the past.  “I suggested we look at Shapley allocation, which is a natural way to solve the greater-of-two problem” says Li.  Shapley allocation is well known in game theory and has been widely applied in economics and business decision making. It attributes a value to the contribution of each agent in a system where agents co-operate and share the costs and gains of their activity. In essence, it measures the marginal contribution of each agent by observing the difference the agent’s presence or absence makes to the activity. Based on that, a portion of the shareable pie is allocated to each agent. In the case of a bank, the agents are individual business units and the pie is the overall capital stock.  Offengenden, Li and Jan Burgy, a quant strategist at Barclays, realised that regressing the overall allocation of capital on the selection of RWAs and LBS obtained from the application of the Shapley method resulted in an almost perfect approximation.

Mauro Cesa, Game theory plays well for capital management, Risk.net, September 30, 2019.

Yadong Li, Dimitri Offengenden, Jan Burgy, Reduced form capital optimization, May 15, 2019.


What's the significance of the triangular region for calculating the deadweight loss? I am familiar with the concept of deadweight loss but I can't deduce the significance of the triangular region we consider deadweight loss as.

I mean what's the issue with taking the deadweight loss as the area under ABCD instead of CDE or for that matter CDF (F being the mid point of DE) or does that particular area represents some quantity from economic perspective that I might be missing out on?

I suspect your chart could look more like this:

Initially with the price and quantity at E, the combined surplus is the triangle to the left (between the orange line and the lower green line to the left of E, i.e. the pink and grey areas).

But with for example a tax, the new equilibrium has consumers paying the price at C, so the new combined surplus is still between the orange line and the lower green line, but only to the left of C (i.e. the pink area).

So the deadweight loss is the triangle CEF (i.e. the grey area) which is the loss in surplus associated with the reduced quantity.  If all three lines are straight, and the green lines are parallel then this is the same area as the triangle CDE, but not necessarily otherwise.


Pareto Improvement with Monopolies This may be one of the more 'elementary' questions on this site.. But I really can't wrap my head around it and a search on the web hasn't yielded much.
Given that Pareto efficiency is defined as when the allocation cannot make any party better off without making anyone worse off, it seems that something like breaking up monopolies is in fact not a Pareto improvement, as it's making the consumers better off by making the monopolist worse off?
I'm inclined to say that antitrust measures are (theoretically) a Pareto improvement, because that's moving closer to the allocative efficient level in a perfectly competitive market. However I don't know how to incorporate the definition of Pareto improvement into this.
Would be extremely grateful for any help!!
Many many thanks,
Shine

First of all good question. I tried myself on that one, but if any other member of this wonderful site has additional input please also answer :)
In a monopol we know there exists a consumer who would be willing to pay a price for an additional unit of the good that is higher than the additional cost to produce that unit. Possibility of Pareto improvement: monopolist produces one additional unit and receives marginal cost from consumers. Giving away one unit for marginal costs does not make the monopolist worse off in the first case. But if the monopolist wants to sell an additional unit, he must lower the price not only for the last unit, but also for all remaining units. This is the result of one critical assumption: no perfect price discrimination. But from the perspective of a social planner one could produce an additional unit and find a trade where $MC \leq  P$ by allowing the monopolist only to change price for its last produced unit. In absence of such a planner the monopolist falls back to the old logic.
This leads us to the definition of market failure: individual rational behaviour leads to collective irrational outcomes.
Pareto optimality is also a very narrow definition, take for example the Kaldor Hicks criteria which states: An economic policy measure is welfare increasing, if in the society as a whole, increases in benefits outweighs the losses of benefits. This simply means winners could compensate losers, or in other words the sum of gains needs to be greater than the sum of losses (independent from individual changes). This holds true for the destruction of the monpoly because sum of consumer and producer surplus will increase. The pareto criterium is often criticised to favourite the status quo, while Kaldor Hicks creates more possibilities for restructuring in exchange for violating the methodological individualism.


When do workers versus capital owners share of income increase? The share of total income obtained by workers rather than capital owners is for obvious reasons of interest. Assuming that the economy can be desribed by an aggregate production function
$$Y = F(K,L)$$
where $K$ is capital input and $L$ is labor input the total expenditure on labor and capital is given as
$$pY = rK + wL,$$
where $p$ is output price, $r$ is user cost of capital and $w$ is the wage (perfect competition is assumed on all markets). Labors share of total revenue or expenditure is therefore given as
$$s_L := \frac{wL}{pY} = \frac{wL}{rK + wL},$$
depending only on $w,r,L,K$.
The elasticity of substitution is defined as
$$\sigma := - \frac{d\log(K/L)}{d\log(r/w)},$$
describing how the capital to labor ratio changes given a change in the ratio of user cost of capital to wage.
Since both labor share of expenditure and elasticity of substitution depend on the variables $w,r,L,K$ this suggest the question:
How does changes in labors share of income $s_L$ as a result of changes in relative price $(r/w)$ relate to the elasticity of substitution $\sigma$?

From the shares equation, we obtain:
$$
1 + \frac{rK}{wL} = \frac{1}{s_L}
\to tk = \frac{1 - s_L}{s_L}
$$
where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$.
then taking logs gives:
$$
\ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L)
$$
Then take the derivative of this expression with respect to $\ln(t)$:
$$
\begin{align*}
-\sigma &= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\
&=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right)
\end{align*}
$$
Where we defined $\gamma$ to be the elasticity of the labour revenue share with respect to the $r/w$ ratio.
Solving for $\gamma$ gives:
$$
\gamma = \left(\sigma-1\right)(1 - s_L)
$$
which means that $\gamma > 0$ if $\sigma < 1$. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when $r/w$ increases.


Microeconomic rationale for UIP (uncovered interest rate parity) The key idea behind UIP is that as for all common financial instruments, the "law of no free lunch" should also hold for currencies. However it differs from traditional replication-based no-arbitrage conditions like CIP in that there is no obvious argument for why the exploitation of opportunities when it does not hold true necessarily depletes all such opportunities, thereby making "free lunches" rare in a world of rational investors.
Concretely, the currency of a country with a relatively higher interest rate needs to gradually depreciate to offset the positive earnings potential. What mechanism drives this gradual depreciation?
I've heard that there are models on this issue but have not been able to locate the literature.
Thanks for all suggestions!

There is an arbitrage argument for UIP because under UIP forward rate must be equal to expected exchange rate so the same arbitrage arguments from CIP can be extended to UIP (under its strict assumptions). You will find it in most international macroeconomics or finance textbooks. For example, according to Nelson Mark's International Macroeconomics and Finance:

If foreign exchange participants are risk neutral, they
care only about the mean value of asset returns and do not care at all
about the variance of returns. Risk-neutral individuals are also willing to take unboundedly large positions on bets that have a positive
expected value. Since $F_{t} − S_{t+1}$ is the profit from taking a position in
forward foreign exchange, under risk-neutrality expected forward speculation profits are driven to zero and the forward exchange rate must,
in equilibrium, be market participant's expected future spot exchange
rate
$$F_{t} = E_{t}(S_{t+1}). (1.6)$$
Substituting (1.6) into (1.2) [result given from covered interest parity where instead of expected exchange rate you have forward rate] gives the uncovered interest parity condition
$$ 1 + i_t = (1 + i_t^* )\frac{E_t[S_{t+1}]}{S_t}. (1.7)$$
If (1.7) is violated, a zero-net investment strategy of borrowing in one
currency and simultaneously lending uncovered in the other currency has a positive payoff in expectation. We use the uncovered interest
parity condition as a first-approximation to characterize international
asset market equilibrium, especially in conjunction with the monetary
model (chapters 3, 10, and 11). However, as you will see in chapter 6,
violations of uncovered interest parity are common and they present an
important empirical puzzle for international economists.

The in text square brackets contain my clarifications. So as the above passages show there is an arbitrage argument in UIP. You can find further sources and more detailed explanations in the textbook itself.


Why are New Zealand houses so badly insulated? New Zealand is relatively new country.

It essentially was founded in 1840 in a contract between the native Maori people and the British Government.

For some reason a large chunk of the housing stock in New Zealand is badly insulated and drafty and cold, compared to houses in Europe for example.*

I don't understand how a country that was essentially built in the modern era, has built such bad houses.

What economic factors contribute to this?

*Here's a good paper about the subject. PDF.

Insulation is more of a luxury than a necessity in the oceanic climate. If you don't have insulation in a continental climate, you'll die. In NZ, it will be uncomfortable, but you'll be ok.

Dunedin may provide arelevant counter argument. It gets below freezing often there. I don't know about insulation rates.

My second answer is: We're just tough mate. Harden up.


Why are real interest rates so non-volatile compared to investment? Intuitively I would guess that real interest rates are highly related to investment.

But there is a very low interest rate volatility, compared to investment:

log investment is about 20 times more volatile than real interest rates. The standard deviation of interest rates is about 0.15 of that of log gdp, while investment is about 3 times as much.

What explains this massive difference?

In most countries, interest rates are actively managed by the government or central bank in pursuit of various policy objectives (some closely linked to investment, some, like stable foreign exchange markets, not).  Investment in turn is determined by a wide range of factors other than interest rates.  Basically I don't think that your assumption of a close link is valid.


Jordi Gali Euler Equation Beta Derivation euler equation in gali book, I don't understand

" β " transformation.

Jordi Gali book, page 42

There is no explanation gali book

the notes which are prepared by Drago Bergholt (Page 6)

explain FOC for "Ct"  (2.13)
and
(2.18) explain Euler equation

Writer uses FOC for "Ct" and FOC for "Ct+1" to form euler.

and I expect to different " β " for "Ct"  and for "Ct+1" in (2.18)
But there is only one " β " in (2.18)

My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and
for this (0.99) need to βt+1 is less than βt. Is it possible ?

Sincerely

There is no trick in that Euler equation. In the New Keynesian model, the Euler equation for consumption is derived from the first order condition for $B_t$, the bond holding. You have to pay attention to the fact that bonds appear in the budget constraint at two moments in time $t$ and $t+1$. Thus you have to compute the foc as
\begin{gather}
B_t:\beta^t\lambda_tQ_t-\beta^{t+1}\lambda_{t+1}=0
\end{gather}
Then substituting $\lambda$ using the foc for $C_t$ and simplifying the betas, you obtain
\begin{gather}
Q_t=\beta E_t\{\frac{U_{c_{t+1}}}{U_{c_t}}\frac{P_t}{P_{t+1}}\}=0
\end{gather}

EDIT: In the NK model, household maximizes its utility function choosing $C_t$, $N_t$ and $B_t$. Thus, you have to compute the FOC for these three variables, i.e.
\begin{align}
B_t:&\beta^t\lambda_tQ_t-\beta^{t+1}E_t\{\lambda_{t+1}\}=0 \\
C_t:&\beta^tU_{c,t}-\lambda_tP_t=0 \\
N_t:&\beta^tU_{n,t}-\lambda_tW_t=0
\end{align}

In order to compute the consumption Euler equation, you need to substitute $\lambda_t$ from the $C_t$ foc into the $B_t$ foc. Then rearranging and simplifying $\beta$ as $\beta^{t+1}/\beta^t=\beta$ you get the equation you wrote above.

You get only one $\beta$ because it is constant over time, it is not index by $t$ as the other variables, it is a constant. Thus whatever time is $C$, you will have the same $\beta$. I think it is well explained in Gali too.


Is Russia in a deep economic crisis? Premise: I see that the site is for professional and academic economicists, and I'm neither. However,  since I've been participating to other SE sites for some time now, I consider SE a trusted source, and reading here I think my question is on-topic, thus I'll take my chance and ask.

I was talking about Russia at work, and a coworker started saying that Russia is, and has been for a lot of time, in a deep economic crisis, nearly on the verge of bankrupcy. This guy is used to pretending he knows all about everything, thus I don't trust him. But, come to think of it, I noticed that I know very little about Russian economy: I don't know if it has a large national debt, who's the owner of that debt (Russians or other states?), whether it has a positive or negative trade export, etc.

On one hand, I guess Russian economy must have taken a blow from the failing oil prices, because lower prices => less profits for exporters like Russia (right?) and also because cheaper oil => less market for gas, which Russia also exports. However, shouldn't this be compensated by higher oil exports because of the lower price? I'm getting a bit confused here. Also, sanctions against Russia must take a toll on its economy.

On the other hand, Russia is considered a big international power. It has a lot of natural resources, and one of the strongest armies in the world. It seems weird to me that a nation "on the verge of bankrupcy" could mantain such a military power. Is my coworker really mistaking Russia for old Soviet Union? Or is he right, and Russia is actually down to its feet in a dramatic crisis?

The Russian economy has been in an recession since 2014, when oil prices plumited. The Russian debt to GDP ratio is still very low though:  below 20%, compared to over a 100 for the United States. This is because Russia had a huge stash of cash before the recession, which has shrunk from over 91 billion in 2014 to 32 billion. If this trend continues at some point they will have to start borrowing money to pay for expenses. But I don't see them going bankrupt anytime soon, because their debt to GDP ratio is so low so investors will lend them money against low interest rates.


Expectation augmented Phillips Curve Currently studying Intro to Macroeconomics, and faced Expectations augmented PC. In my textbook it is not clear how expected inflation affects unemployment. My view so far, is that expected inflation move the Phillips Curve upwards. For the same rate of Unemployment corresponds greater inflation in the short-run. However, unexpected iflation reduces unemployment in the short-run until labour market and companies absorb the unexpected increase of inflation.

In a New Keynesian framework, expectation-augmented Philips curve is the following:

$$ \pi_t = \beta \;\mathbb{E}\pi_{t+1} + \alpha x_t$$
where $\beta$ and $\alpha$ are constant, $\pi_t$ stands for the inflation rate (actually long-linear approximation of it but doesn't matter intuitively) at time $t$, $\mathbb{E}\pi_{t+1}$ is inflation rate at time $t+1$ anticipated at time $t$,  and $x_t$ is the output gap -gap between natural level of output at time $t$ and the actual output at time $t$-.

As the equation above suggests, provided a constant $\pi_t$,  an increase in the exptected inflation at time $t+1$ leads to a decrease in the output gap at time t, in other words, total production at time goes down -so unemployment increases-. In this model to obtain a relation between expectations and the current economic decisions, we make some assumptions on technology and preferences. In particular, we assume that firms set nominal prices based on the expectations of future marginal costs, thus output gap today depends on marginal cost of production at the current period and at the following periods.


Why does an increase in money in circulation increases price levels? I'm a physics student who likes economics very much. I'm trying to get a simple explanation for inflation. I hope that there is one that doesn't involve people willingly rising prices. Thank you very much!

new money is created; it belongs to somebody.
that somebody can either save/invest the money, or spend it on consumption.
provided at least some of the money is spent on consumption, the demand for consumer goods increases.
in a competitive market, this increase in demand results in an increase in the price level. (Note that the sellers are not "willingly rising prices" because, in this simple model of competition, sellers take the price a given natural consequence of the interaction of supply and demand).

You might also find it interesting to read about the quantity theory of money.


Are Parameter Estimates Unbiased If the Dependent Variable is a Per Capita Rate Based on Approximate Population Data? Suppose we want to estimate a model:

$Y_i  =  \beta_0 + \beta_1X_i + \epsilon_i\tag{1}$

where the $Y_i$ are averages or rates per capita for geographical regions or zones, so that if $Z_i$ is the regional aggregate variable of interest and $P_i$ is population:

$Y_i  =  Z_i / P_i\tag{2}$

If all the variables are accurately measured, and if there exist $β_0^*$ and $β_1^*$ such that for all i:

$E[Y_i - \beta_0^* - \beta_1^*X_i]  =  E[\epsilon_i]  =  0\tag{3}$

then OLS estimation of model (1) should yield an unbiased estimate of $\beta_1$.

Suppose however that estimates of $P_i$ are only approximate.  This is in practice likely since the values of $P_i$ will almost certainly be obtained or derived from census data, leading to several possible sources of error:

error in the original census data;
extrapolation from census data using assumptions on population
trends where data are required in respect of a later date;
the regions used in the model may not correspond to census areas, eg
concentric rings around a central point.  An example is a
travel cost valuation study reported in Herath (1999) (1) where
populations in the range 5,000 to 35,000 for concentric ring zones
are all stated in exact multiples of 5,000, suggesting that the
figures are only very approximate.

The errors in $P_i$ will obviously ‘infect’ the values of $Y_i$, but not in a straightforward way, since the absolute effect on $Y_i$ of a given error in $P_i$ will depend on the size of $Z_i$ (and if $Z_i = 0$ there will be no error in $Y_i$).

Question: Given errors in $P_i$, will OLS estimation of model (1) yield an unbiased estimate of $\beta_1$, and if not, under what additional conditions would the estimate be unbiased?

Reference

Herath, G (1999) Estimation of Community Values of Lakes: A Study of Lake Mokoan in Victoria Australia  Economic Analysis & Policy  29(1)  Table 1 p 37

What about assuming a multiplicative error process and then using logs?

Say the data generating process were a little different:
$Y_i  =  \beta_0 \cdot  X_i^{\beta_1}\cdot E_i$

If $Y_i  =  Z_i / P_i$ in truth but all we could really observe was:

$\hat{P}_i = P_i \cdot \Gamma_i$, where $P_i$ was the true value and $\Gamma_i$ was a strictly positive measurement error.

We' be estimating the following equation in practice:

$\hat{Y}_i  = Z_i / \hat{P}_i =  \beta_0 \cdot  X_i^{\beta_1}\cdot E_i$

take the logs of both sides (lower case letters are logs of individual variables):

$y_i  = \beta_0 + \beta_1 \cdot x_i + \epsilon_i - \gamma_i$

If we define a variable $\xi_i = \epsilon_i - \gamma_i$ then we can have an equation $y_i  = \beta_0 + \beta_1 \cdot x_i + \xi_i$ that looks very much like the one you wrote above.

If $\xi_i$ satisfies the same relationships with the log variables that $\epsilon_i$ does with the level variables it seems like everything should still work.


Why does falling global bond yields signal coming deflation? This article suggests that, "Bond markets are signalling something very nasty coming down the road at us – an all encompassing, worldwide deflation. "

I cannot understand why bond yields falling signals deflation. Is it because falling bond yields indicates a weakening global economy? Or is the transmission mechanism more direct?

Bond yields falling from their current near-zero position will place them in negative yield territory. Negative bond yields are deflationary by definition.
Paragraph 3, sentence 5 of the article says:

With Bank Rate already close to the floor, and some UK bond yields now in negative territory... [emphasis added]

To understand why negative bond yields are deflation markers, consider the example of negative Asian and European central bank rates described in the same article:

This is because a negative rate is effectively just a tax on the banks, forcing them to pay for the privilege of holding reserves with the central bank.
Someone has to shoulder the burden of this tax; either borrowers pay more, or depositors get less, or profits go down, and if it is the latter, then it damages the banking sector’s ability to rebuild capital, crimping credit availability accordingly. Whichever it is, the end result is a monetary tightening, rather than the loosening intended.

So you can think of the effect of negative central bank rates (and, similarly, negative bond yields) as having the inverse impact as quantitative easing which creates an inflationary effect.
This page explains the relationship between prices and money supply as follows:
$$P=\frac{MV}{Y}$$
where $P$ represents prices, $M$ is the money supply, $V$ is the velocity of money and $Y$ is the real GDP.
Therefore,
$$\Delta P = \Delta M \cdot \frac{V}{Y}$$
and for constant $V$ and $Y$,
$$\frac{\partial P}{\partial M} = \frac{V}{Y} > 0$$
So an effective restricting of the money supply caused by negative rates/yields via the mechanism above described will tend to put downward pressure on prices (a/k/a deflation).
Edit:
In the comments to this answer, OP asked about the relationship between negative interest rates and negative bond yields.
Whereas central banks set rates by policy, long term bond yields are determined by the market (i.e., supply-demand equilibrium). An inverted yield curve signals recession and implies rates are set too high relative to market expectations. If central banks set rates in positive territory when long term bond yields are negative, they will de facto force the yield curve to invert. Although this sometimes happens, it is usually counterproductive to the policy goals of the central banks (in most environments) and, therefore, typically avoided.


Why is income tax so high? (UK) Granted, I expect this question could be easily answered, but I cannot find a definitive reason as to why income tax is pinned at such a high rate in the United Kingdom.

I'm not much of an economist, but I do understand the function of national insurance, and I do understand the function of a tax to fund the various essential organs of the state (Monarchy, Government, Parliament, Intelligence Services,and Defense and Protection forces). However, I don't get why, if I earn around 40k (Sterling), a large portion, around 15% (which is around 5-6k), is taken for income tax.

In the UK, government spending is 38.5 per cent of GDP.  To oversimplify, that means that for every £1 spent on private goods, 63p needs to be raised in taxes.  At present, the UK raises a little over a quarter of that from income tax.  One of the reasons income tax is a significant component is that it is progressive, unlike other major sources such as VAT and excise duties.

It is also worth mentioning that the categories of government expenditure you mention are relatively small.  Half of all government spending goes on pensions, healthcare and education.  The categories you mention are marginal by comparison.


If Eurodollar never leaves U.S., then what do we mean by bring eurodollar on shore We already know that Eurodollar never leaves U.S. But then what do the media mean when they say bringing U.S. dollar offshore (eurodollar) onshore? Isn't it always onshore? (Let us omit the real physical cash)

Also, the Fed website says that both Eurodollar and Fed funds liability do not have reserve requirement. My question is how can banks in the U.S. hold eurodollar liability? I mean, eurodollar is defined to be U.S. dollar deposited off-shore or IBF. So doesn't that mean U.S. based banks cannot have Eurodollars by definitions? And since eurodollar does not leave the U.S., its borrowing in Eurodollar market in its off-shore branch is actually simply a transfer of Fed reserve from the lender's bank's U.S. bank to the borrower who is a U.S. based bank. Then the liability of the U.S. based borrower shows up as time deposit due to its off-shore branch. So I think this is how banks borrow from Eurodollars market. But the above question remain.

Any comment is appreciated.

http://libertystreeteconomics.newyorkfed.org/2015/05/the-eurodollar-market-in-the-united-states.html#.VjvTKIQt82Z

Eurodollars can and do leave the US. They can be anywhere in the world: any bank outside the reach of the US Federal Reserve (henceforth "the Fed"). That's what makes them Eurodollars: they're US dollars in deposit accounts beyond the reach of the Fed.

That money is said to be "brought onshore" when it is transferred from a deposit account outside the reach of the Fed, into an account that is within reach of the Fed and its regulation / protection. This changes it from being a eurodollar to a dollar.

A US-based bank can have offshore banking subsidiaries and activities, and that's how it can hold eurodollars.

The whole point of a eurodollar is that it takes it outside the reach of the Fed.


How to derive the measurement equation for the state-space representation of a DSGE model? DSGE models, after log-linearisation, have a state-space representation.
In this representation, in most papers, the measurement/observation equation is simply stated.

I'm wondering how one deduces that equation, or if it's something ad-hoc, how does one proceed, or what guidlines should we use?

Any help would be appreciated.

Hi: I can't speak for DSGE models specifically but, in more standard "rational expectations" econometrics, the measurement equation usually comes from some assumed linear relation between the dependent variable and the expectation of some other variable. For example, one might have

$y_t = \beta x^{*}_{t} + \epsilon_{t}$

where  $x^{*}_{t}$ is the expectation of $x$ at time t so $E(x_t|t-1)$. Note that $x_t$  is often not observable so some formulation for how the expectation is generated must be assumed.  So, for example, if one assumes the adaptive expectations hypothesis for the expectation, then this implies the following relation for $x^{*}_t$:

$x^{*}_t = x^{*}_{t-1} + \gamma(x_{t-1} - x^{*}_{t-1})$

Putting the two relations together, one ends up with an exponential smoothing relation for $y_t$ which would then be the measurement equation:

$y_t = \beta \times \gamma \sum_{j=0}^{\infty} (1-\gamma)^{j} x_{t-j-1} + \epsilon_t$

Almost all of above is taken from Harvey's "Econometric Analysis of Time Series" which I highly recommend not for DSGE models but for RE type econometrics. I assume that something similar is done in DSGE models but I'm not absolutely sure because I have no experience with them. Still, this may help some.


Can quantative easing be summarized as "The state prints cash to pay its debt to private sector"? I find that most sources describe quantitative easing in an overly complicated way that obscures what it basically means.

As far as I understand a central bank is a public entity  owned by a state, so I don't see much point in distinguishing a state entity from a central bank entity.

Therefore, I tend to find that the traditional descriptions like  "the central banks buys government bonds from private banks in order to inject money in the economy" are ridiculously  convoluted and funny.

As far as I understand, if I have a government bond, it is a promise that the state will pay me back some money at some point. So when I'm hearing "the central bank buys government bonds", all I'm hearing is actually "the government is printing cash from nothing to pay me right now, and destroy its debt".

Is my interpretation correct? Why isn't my interpretation the standard official intuitive explanation of quantitative easing? Is it just for newspapers and bankers to sound "serious"?

Altough you are correct saying that the state "own" central banks, they should be independent from their government. This gives freedom to monetary policymakers from direct political or governmental influence in the conduct of policy.

Given that, I would say that your interpretation is a bit exaggerated and maybe biased somehow.

Quantitative easing (QE) is a monetary policy in which a central bank purchases government securities or other securities from the market in order to increase the money supply. This is made with newly created bank reserves, but doesn't mean that new currency (bills and coins) has been printed. In theory, with more liquidity, the domestic demand would increase and eventually promote economic growth. Then, the excess of liquidity is absorbed by the growth and inflation would remain low.

The problem is if the economic growth don't happen, there will be inflation.

Using a more informal language, it's correct to say that the government is creating money in the present, hoping that this fresh money will increase economic growth in the future.