Impact on interest rates of massive government spending At the last State of the Union address, the President of the United States has reaffirmed his intentions to undertake a massive 1.5 trillion dollar infrastructure spending project over the coming years. If you believe simple macroeconomic models, like the IS-LM model, this kind of government spending should lead to higher real GDP and higher interest rates. I am assuming that, in the real world, this would correspond to a steepening of the yield curve, i.e. medium and long term yields go up. However, looking at yield curve data, I cannot discern any significant uptick in long term yields DoD. Does this mean that either:

the yield curve should not actually respond the way I assumed?
this announcement carried no real information that would have
changed market participants' expectations about the future?
the IS-LM model is too simplistic to even adequately predict the directional
impact of government spending on interest rates?

I realize that I may be abusing the IS-LM model here, as it probably takes a more long term, qualitative view. However, I would think, due to market efficiency, its predictions should be priced into the yield curve immediately.

You could say IS-LM is too simplistic.

The fundamental discrete time consumption-based asset pricing equation says that if $ r_t $ is the risk-free interest rate at time $ t $, we have

$$ \beta (1+r_t) \frac{u'(C_{t+1})}{u'(C_t)} = 1 $$

(I'm dropping the expectation since I will be working with a perfect foresight model later on) where $ u $ is the utility function of any consumer, $ 0 < \beta < 1 $ is a discount factor which measures "impatience" of the same consumer, and $ c_t, c_{t+1} $ is the consumption of the same consumer in time $ t $ and $ t+1 $ respectively. The most elegant form of the relation is with the assumption of power utility $ u(c) = c^{1 - \gamma}/(1 - \gamma) $ with $ \gamma > 0 $, in which case we get (after log-linearization)

$$ r_t = R + \gamma \Delta c_t $$

where $ \Delta c_t $ is log consumption growth and $ R = -\log(\beta) $ is the real interest rate consistent with flat consumption. The intuition is simple: people want to smooth consumption over time. If they expect their consumption in the future to be much higher than their consumption today, they will try to borrow against the better future and push real interest rates up, and vice versa.

Consider a baseline economy with no investment, perfect foresight, and a representative consumer first. (This is unrealistic, but IS-LM is even more unrealistic, so...) In this case, the resource constraint of the economy reads $ Y_t = C_t + G_t $ in every period, where $ G_t $ is government spending. If you assume consumers have utility that is additively separable over time (the asset pricing relation assumes this as well), then the no investment assumption turns their decision problem into a set of one-period decision problems. (This is a technical condition to ensure what I am doing is well-defined.) Define $ D_t $ as the level of output consistent at time $ t $ with $ G_t = 0 $, and let $ m_t $ be the "fiscal multiplier", defined by

$$ m_t = \frac{dY_t}{dG_t} $$

Most models won't give constant $ m $, but if necessary we may linearize the model around somewhere other than $ G_t = 0 $ and make this assumption locally. These assumptions imply $ Y_t = D_t + m_t G_t $ at any time $ t $, so we may express consumption as $ C_t = D_t + (m_t -1) G_t $. Plugging into the relation gives

$$ r_t \approx R + \gamma \Delta d_t + \gamma (m_t -1) \Delta \left( \frac{g_t}{1 - m_t g_t} \right) $$

where $ g_t = G_t/Y_t $ is the ratio of government spending to GDP and $ \Delta d_t = \Delta \log(D_t) $. (Ignore the pole of the denominator - we're working around $ g_t = 0 $.) The fraction $ \frac{g_t}{1 - m_t g_t} $ is increasing in $ g_t $ with $ m \geq 0 $, so we see that the real interest rate at time $ t $ depends on two things: the "natural" growth rate of the economy given by $ \Delta d_t $, and the consumption growth induced by changes in government spending rel to GDP $ g_t $. However, the sign of the effect of an anticipated increase in government spending depends on the value of the fiscal multiplier $ m $. If $ m < 1 $, then government spending crowds out consumption, and rising government spending is associated with low real interest rates, i.e expectations of high spending in the future would push rates down today, and vice versa if $ m > 1 $. Most everyone would agree that the fiscal multiplier in the US economy today is less than $ 1 $, so news of higher spending in the future should be associated with lower real interest rates now, if with anything at all.

It also matters, of course, if government spending affects $ \Delta d_t $ - if we're speaking of an infrastructure investment program, for instance, then we should expect that the change in government spending also leads to a change in the path of $ D_t $. The two effects for such a program act in opposite directions, so the net effect of a government investment program tomorrow on real interest rates today could go either way, even in this simple economy.

I won't work out the model with investment (allowing consumers to invest) here, but the effect of investment is to allow consumers to trade consumption now for consumption in the future in aggregate; so it dampens the impact of a change in government spending on real interest rates (since now consumers have a tool for smoothing consumption in the face of government expenditure shocks).

It's also true that the announcement in the State of the Union address probably carried less information than an announcement in this model, since market participants were already aware of such an infrastructure investment plan for months (maybe a year?), but with a fiscal multiplier less than $ 1 $ it's not easy to say anything about the direction of the real interest rate response.


The 2015 Economics "Nobel": Angus Deaton Angus Deaton is only the sixth economist to not share the "Economics Nobel" (the "Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel") in the last 20 years (i.e. from 1996 onwards).

The Press Release reads:

  Press Release

  13 October 2015

  The Royal Swedish Academy of Sciences has decided to award The
  Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel
  for 2015 to

  Angus Deaton, Princeton University, NJ, USA

  "for his analysis of consumption, poverty, and welfare".

  Consumption, great and small

  To design economic policy that promotes welfare and reduces poverty,
  we must first understand individual consumption choices. More than
  anyone else, Angus Deaton has enhanced this understanding. By linking
  detailed individual choices and aggregate outcomes, his research has
  helped transform the fields of microeconomics, macroeconomics, and
  development economics.

  The work for which Deaton is now being honored revolves around three
  central questions:

  How do consumers distribute their spending among different goods?
  Answering this question is not only necessary for explaining and
  forecasting actual consumption patterns, but also crucial in
  evaluating how policy reforms, like changes in consumption taxes,
  affect the welfare of different groups. In his early work around 1980,
  Deaton developed the Almost Ideal Demand System – a flexible, yet
  simple, way of estimating how the demand for each good depends on the
  prices of all goods and on individual incomes. His approach and its
  later modifications are now standard tools, both in academia and in
  practical policy evaluation.

  How much of society's income is spent and how much is saved? To
  explain capital formation and the magnitudes of business cycles, it is
  necessary to understand the interplay between income and consumption
  over time. In a few papers around 1990, Deaton showed that the
  prevailing consumption theory could not explain the actual
  relationships if the starting point was aggregate income and
  consumption. Instead, one should sum up how individuals adapt their
  own consumption to their individual income, which fluctuates in a very
  different way to aggregate income. This research clearly demonstrated
  why the analysis of individual data is key to untangling the patterns
  we see in aggregate data, an approach that has since become widely
  adopted in modern macroeconomics.

  How do we best measure and analyze welfare and poverty? In his more
  recent research, Deaton highlights how reliable measures of individual
  household consumption levels can be used to discern mechanisms behind
  economic development. His research has uncovered important pitfalls
  when comparing the extent of poverty across time and place. It has
  also exemplified how the clever use of household data may shed light
  on such issues as the relationships between income and calorie intake,
  and the extent of gender discrimination within the family. Deaton's
  focus on household surveys has helped transform development economics
  from a theoretical field based on aggregate data to an empirical field
  based on detailed individual data.

His account in Google scholar provides the following "impact" data

\begin{array}{| r | r |}
  \hline
\hline
  \text {Index name} &  \text{All} & \text{Since 2010} \\
  \hline
\text {Citations}   & 53,873 & 20,851 \\
\text{h-index}   & 92 & 66 \\
\text{i10-index} & 184 & 143\\
  \hline
\end{array}

See for starters the wikipedia article, for the "h-index"

For the "i10-index" it reads:

  The i10-index indicates the number of academic publications an author has written that have at least ten citations from others. It was
  introduced in July 2011 by Google as part of their work on Google
  Scholar.

My question?

Have you used A. Deaton's work in your studies or in your research? How?

Deaton's "Understanding Consumption" (1993) was one of the major sources I used when doing seminar work for my master's. In the end I decided to write my thesis about a different (somewhat related) subject after realising the empirical difficulties with what I planned on doing.

I have since chosen a different research subject, but Understanding Consumption is a book I have spent many hours with. It's a bit dated now, but still a decent read if you want to know about consumption and saving.


What models exists for explaining the price of nonrenewable resources? It seems to me that the seminal model for the price of nonrenewable resources is Hotelling's rule. However, as the Wikipedia article describes, Hotelling's rule has empirically performed poorly. I know there are some variations that seek to improve on the original model, such as by accounting for changes in extraction cost or other factors. However, I am unfamiliar with this literature. Can anybody provide some more detail about variations on Hotelling's rule? Also, what, if any, alternative hypotheses exist to explain the prices of nonrenewables?

Notes:

This questions is related to a previous question about the impact of fracking on oil prices.

Based on my understanding, Hotelling's Rule is a pretty good first step. Like many first steps, it doesn't perform particularly well empirically. I would expect that when technological change is considered, in the very long run, it isn't too bad.

There are a few extensions of this model which are popular.
-Exogenise extraction costs
    The standard idea is that extraction costs increase as the resource moves towards depletion. Something which isn't included in the standard model is that technology is changing, making some previously unprofitable reserves cost effective. At the same time, exploration can uncover more reserves. Whilst the total amount of non-renewable resource doesn't change, the amount available for extraction isn't static.
- Account for substitutes
    The demand for oil for example is constantly growing, but it must remain competitive with substitutes. A rise in prices make substitutes more attractive. This is a phenomenon we have seen with ethanol and into our petrol.

This pdf provides a broad overview of such extensions. It also provides references for further reading. I borrowed heavily from it above. I'll list a few

Chapman, D. (1983). World Oil: Hotelling Depletion or Accelerating Use?Nonrenewable Resources.2(4): 331-339
Extends the Hotelling model to so that the marginal extraction cost is independent of quantity demanded.
Stiglitz, J.E. (1976). Monopoly and the Rate of Extraction of Exhaustible Resources. American Economic Review,66(4): 655-661
Straightforward paper showing that extraction rate under monopoly is lower than under perfect competition.


What does perfectly inelastic demand imply about gains from trade and domestic consumers? Source: p 191, Question 9.7b, 9.7c, Principles of Microeconomics, 7 Ed, 2014, by NG Mankiw

  Consider a country that imports a good.
  True or false. Explain your answer.
  b) “If demand is perfectly inelastic, there are no gains
  from trade.”
  c) “If demand is perfectly inelastic, consumers do not
  benefit from trade.”

Please explain if this Best Answer is wrong? It claims that b) and c) are true. Yet my own work below shows a surplus of $+B$ after imports $\implies$ Gains from importing $\implies$ B and C are false.
Please feel free to correct my graph, but please also explain the intuition.
I wish to develop intuition, rather than be mechanically engrossed in graphs.

@VicAche this is incorrect. While the consumer may be willing to pay a large amount, consumer surplus is defined as the willingness to pay minus the price she pays. So if the willingness to pay is infinite as you correctly asserted, then the consumer surplus is also infinite. See my answer to this question.


Why may printing trillions of dollars not lead to inflation? By definition, inflation should be affected by the increase of the money supply. During the pandemic, there have been numerous huge monetary policies executed, e.g., quantitative easing (QE), injection liquidity into the market by increasing the monetary base.
A lot of people say that this will lead to inflation and you have to save your wealth into rather deflationary assets, like gold or bitcoin. However, a lot of economist argue that the picture is way more complex than this. For instance, a supply shock should lead to inflation, but high unemployment should keep it down.
Even if inflation would not be significant in the short term, or significantly higher than the target inflation, why wouldn't there be a huge inflation in the short to mid term?
I know once the economy is healthier, central banks can raise interest rates, and that would slow down the economy and would cause deflation. But, still, this kind of huge money printing could only be cured by very radical interest rate increases, which may not happen at all.

An important subtlety
In the literature, the words "inflation/deflation" are often used to describe "any increase/decrease in the price of a good for any reason". If some kind of crisis causes the value of stocks to drop by 10% and a monetary scheme like QE is implemented to bump the stocks back up by 10%, then some would say that the inflation is zero.
However, it is clear that the stock prices are now 10% higher than they would have been without the injection of money via QE. We could say that the price is inflated relative to a case without such intervention. From this viewpoint, any expansion of the monetary base must be inflationary, since it will always raise the prices of some assets/goods/services relative to the price without the intervention.
One could argue that the 10% drop reflects a real change in the state of the economy (people are more interested in having the funds to see themselves through an immediate crisis than to invest long term), and that injecting more money into the system to prop up the value of stocks that nobody would want to buy in the absence of QE is unnecessary.
To answer your question in short, if one uses the broad definition of inflation, it is indeed possible to do the exact right amount of QE to make sure the price in dollars and cents of assets do not change as a result of the crisis in the short/medium/long term. If one uses the more narrow definition, QE and similar interventions are always inflationary.


Show that strict preference ordering is not reflexive How do i show that the strict preference (>) ordering is not reflexive. I've tried to prove this by asymmetry but I am not sure this is the way to do it. Any help? Thank You.

Just think about the definition of $\succ$ (in terms of $\succeq$) and the result follows immediately.

Proof:
Suppose $x\succ x$. Then, by definition, $x\succeq x$ and $x\nsucceq x$, which is impossible. Hence, $x\nsucc x$ which means that $\succ$ is not reflexive.


Is the price-quantity diagram fundamentally wrong? I have wondered about the following diagram:

which shows a relationship between Price (Preis) and quantity (Menge). The supply (Angebot) curve is rising with the amount; the demand (Nachfrage) is decreasing.
As a guy with some mathematical intuition, I am wondering about many things here:

My teacher told me I could only read from price to quantity. Since we are speaking of bijectives functions; this is wrong.
If someone is willing to buy 1 billion "Goods", and my production cost is lower than what he offers - I will produce 1 billion "Goods" for them (assuming that I can) after I tried to get a better price - since I will make a profit after all. Therefore this diagram makes no sense; the amount the producer offers is only determined by the production cost, the production ability and the demand of the goods. The producer will always try to satisfy all demands as long as they make a profit. Naturally, they will try to bargain about the price and sell to the highest bidder in the first place.

So how do you explain this?

The graph you've depicted is just very generic supply and demand intersection, under the generic/introductory assumption that an equilibrium price exists.

Even at this level of abstraction, the demand curve is perfectly flat for a price taker, so it's not always the case that those two functions (supply or demand) are always a bijection between price and quantity. That should address some of your confusion/question (1).

Furthermore, some introductory econ texts are bad at explaining that even a this level of abstraction, the market and an individual firm may have completely different price-quantity graphs; the graph you've shown is valid for a competitive market as a whole, but not for individual producers in it. Here's standard textbook slide of the better variety, which does make this distinction.

  Demand curve faced by an individual firm
  is a horizontal line

  Each firm is so small that its sales have no
  effect on market price. As a result, each
  regards market price as given.

  Demand curve faced by whole market is
  downward sloping

  Shows amount of goods all consumers will
  purchase at different prices

Your main question (2) involves a production cost, which does not appear depicted anywhere on that graph. The production cost isn't necessarily the same thing as the producers' price-output curve.

Even in the case where the market consists of only one supplier (a monopoly), so when the (producer) firm and industry graphs coincide, one needs to plot additional functions for the marginal return and marginal cost to start talking about profits. Even in this case, the discussion isn't trivial, e.g. it depends whether the monopoly can engage in price discrimination (sell to different consumers at different prices).

As for your specific scenario in (2):

  If someone is willing to buy 1 billion "Goods", and my production cost is lower than what he offers - I will produce 1 billion "Goods" for them (assuming that I can) after I tried to get a better price - since I will make a profit after all.

This is closer to a how a price-taking producer (aka perfect-competition producer) optimizes output. Remember from the first diagram, that the demand from its perspective is flat, so that entails a linear total revenue (or, equivalently, a constant marginal revenue):

Note that the demand isn't even included in this graph except implicitly as the slope of the total revenue. However, since it follows that in order to maximize profits MC = MR = P (again under the perfect competition assumption), then the marginal cost is equal to the externally set price (for a price-taking producer), so the intersection of the price (demand) and MC gives the quantity:


Question about Locally non-satiated preferences If a consumer has locally non-satiated preferences, which of these 2 bundles is preferred and why?

Bundle A: (1,3)

Bundle B: (4,2)

This is what I've reasoned from my very limited understanding of LNS preferences.

The distance between the two bundles has to be less than the radius of a ball around one of the bundles for the other bundle to be preferred. A ball around bundle A can't have a radius larger than 1, thus bundle B isn't preferred. However, a ball around bundle B can have a radius large enough so that Bundle A is within the ball. So bundle A is preferred.

I'm very unsure of this reasoning and I'd appreciate it if anyone could correct me where I'm wrong or help improve my answer.

Consider a locally non-satiated preference represented by the utility function
\begin{equation}
u(x,y)=-(x-4)^2(y-2)^2
\end{equation}
It is easy to verify that $u(4,2)=0>-9=u(1,3)$. Therefore $B$ is strictly preferred to $A$.
Likewise, you can construct a locally non-satiated preference, one represented by $$u(x,y)=-(x-1)^2(y-3)^2,$$ under which $A$ is strictly preferred to $B$.

What local non-satiation does is that it rules out indifference "zones", so that we're left with indifference curves. Hence, with local non-satiation alone, it's impossible to tell whether one alternative is preferred to another.
The following figures from Jehle and Reny (2011) illustrate preferences with local non-satiation (Fig 1.3) and without it (Fig 1.2).

Axioms 1 through 4' are completeness, transitivity, continuity, and local non-satiation, respectively.


How are assets distributed by types of organisations? How wealth, assets or capital (measured in terms of revenues, assets, equity, etc.) is distributed among the various types of organizations (e.g. State, public companies, private conpanies, individuals) existing in an economy?

The Flow of Funds (Financial Accounts of the United States) should have what you need.

For example, table  B.1 Net National Wealth Table Description Table (Billions of dollars; amounts outstanding end of period, not seasonally adjusted) might be very close to what you want.


Trigonometric Cost Function I've been reading on producer theory and came up with a ridiculous question. Has anyone tried to model costs with a trigonometric function? would it work with the assumptions we need? Thanks!

There is a long tradition of using flexible functional forms for cost, utility or production functions. This can be done using a (low) order series expansion. Usually, these are Taylor expansions. For example the translog is a second order polynomial in log prices.
Of course, one could also use a Fourier expansion, which would give a function in terms of cosine and sine functions.
See for example the appendix of Griffin, Montgomery, Rister,(1987) "Selecting Functional Form In Production Function Analysis," Western Journal of Agricultural Economics


How to get this Production Function in Growth Rates I'm struggling to understand how Khan & Reinhart (1990) go from the next production function.
$$y=A f(K,L,Z)$$
Where $y$ is the production of the economy, $A$ is a variable which contains the technological change, $K$ is the capital stock of the economy, $L$ is the labor (work) of the economy, $Z$ is a vector of variables containing usual covariates.
Into the following result as an expression of growth-rates.
$$\frac{dy}{y} = [A \frac{\partial y}{\partial K}]\frac{dK}{y} + [A \frac{\partial y}{\partial L}  \frac{L}{y}]\frac{dL}{L} + [A \frac{\partial y}{\partial Z}\frac{Z}{y}]\frac{dZ}{Z} + \frac{dA}{A}$$
The information is that $y,K,L,Z$ are variables (redundant I know but I want to make it clear), however, in the article, they mention $A$ as assumed to grow an exogenous rate (I assume that $A$ is also taken as a variable in the first equation). They don't make clear anymore transformations, simply "growth terms".
The last expression is mentioned by the authors to be "growth terms" of $y=Af(K,L,Z)$. but I lost it that way. Usually for expressing growth rates I simply use logarithms and differentiate over time. but this approach is not clear to me.
My intuition is that $y$ was calculated with the total differential of the form of
$$dy=\frac{\partial y}{\partial K}dK + \frac{\partial y}{\partial L}dL + \frac{\partial y}{\partial A}dA$$
and then it was divided by $y$. However, I never saw the total differentials while I was studying, so I don't know if $A$ is multiplying $f$ and we may need to proceed with the product rule of the derivatives? or what is actually the process to get the result of $dy/y$ as they get.
Reference:
Khan, M. S. & Reinhart, C. M. (1990) Private investment and economic growth in developing countries World Development, Volume 18, Issue 1, January 1990, Pages 19-27, DOI: https://doi.org/10.1016/0305-750X(90)90100-C

Given
$$Y = Af(K,L,Z)$$
it follows that
$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$
where dotted expression are time derivatives and dividing with $Y$ it follows
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{1}{Y}\right] \dot L + \left[A \frac{\partial f}{\partial Z} \frac{1}{Y}\right]\dot Z ,$$
and dividing and multiplying with $L$ and $Z$ in relevant summands it follows that
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{L}{Y}\right] \frac{\dot L}{L} + \left[A \frac{\partial f}{\partial Z} \frac{Z}{Y}\right]\frac{\dot Z}{Z} ,$$
and then terms in square brackets are defined as $\alpha_1,\alpha_2$ and $\alpha_3$ being respectively

The marginal productivity of capital $\alpha_1 = A \frac{\partial f}{\partial K}$
The elasticity of output with respect to labor $\alpha_2 = A \frac{\partial f}{\partial L} \frac{L}{Y}$ and
The elasticity of output with respect to other factors  $\alpha_3 = A \frac{\partial f}{\partial Z} \frac{Z}{Y}$

In the text the authors write $\partial Y/\partial X$ with $X\in \{K,L,Z\}$ instead of
$\partial f/\partial X$ which must be a typo. For example it is hard to convince yourself that with $Y = A f(K,L,Z)$ the marginal productivity of capital is $A \frac{\partial Y}{\partial K}$ and not instead $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.
Obviously the typo is of no consequence since the terms are parameterized for the purpose of estimation.


Why do we need at least as many instrumental variables as endogenous regressors to identify parameters in 2SLS? As the title says. Why do we need at least as many instrumental variables as endogenous regressors to identify parameters?

In standard linear regression model
$$y = x^\top \beta + \epsilon$$
with exogeneity $\mathbb E[x\epsilon] = \mathbf 0$ you have $K$ parameters because $\beta$ is $K \times 1$ and you have $K$ equations
$$\mathbb E[x\epsilon] = \mathbb E[x(y - x^\top \beta)] = \mathbf 0 \Leftrightarrow \mathbb E[xy] = \mathbb E[xx^\top]\beta,$$
where $\mathbb E[xy] = \mathbb E[xx^\top]\beta$ is a system of linear equations. Only if the matrix $\mathbb E[xx^\top]$ is invertible it follows that $\beta := \mathbb E[xx^\top]^{-1}\mathbb E[xy]$.
For each endogenous variable in $x_k$ you lose one identifying equation because for the endogenous variable $x_k$ it does not hold that $\mathbb E[x_k\epsilon] = 0$. So for example with 2 endogenous variables you would be trying to identify $K$ parameters using only $K-2$ equations which amounts to trying to solve a system of $K-2$ linear equations for $K$ unknowns. So assuming that you know how systems of linear eqautions work it should come as no surprise that you need to find at least two instruments. This would allow using the true equation $\mathbb E[z_k (y - x^\top \beta)] =0$ instead of the untrue $\mathbb E[x_k (y - x^\top \beta)] =0$ for each $k$ (which amounts to assuming $z_k$ is exogenous while $x_k$ is not).


How can I calculate with the average of the consumption in an RBC framework? In the following open-economy RBC model consumption level ($C_t$) is not the most important for the representative households, but the difference of the consumption level and the earlier level of the average consumption ($\overline{C_{t-1}}$):
$$H_t=C_t-b \overline{C_{t-1}}$$
$$0<b<1$$

The consumer's utility function:
$$E_0 \sum\limits_{t=1}^{\infty}\beta^{t-1}\left( \frac{H_t^{1-\sigma}}{1-\sigma}+\frac{L_t^{1+\eta}}{1+\eta}\right)$$

After solving the Utility Maximization Problem, deriving the First Order Conditions, here's the Euler-equation:
$$\beta E_t \left(\frac{C_{t+1}-b \overline{C_{t}}}{C_t-b \overline{C_{t-1}}}\right)^{-\sigma}(1+r_t)=1$$

Question: I've got no idea what I should do with the average of the consumption ($\overline{C_{t}}$) term. How can I calculate with this? I need to find the steady state, and do the log-linearization, but I don't understand the substance of this variable.

Consider the risk free steady state. In that situation the Euler equation becomes:

$$ \left(\frac{C_{t+1}-b \overline{C_{t}}}{C_t-b \overline{C_{t-1}}}\right)^{-\sigma}=\frac{1}{\beta \cdot (1+r)}$$

Which implies

$$ \Rightarrow  \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] = \frac{\ln[\beta] + \ln[1+r]}{\sigma}$$

Assume that in the non-stochastic setting that growth is constant of the form:
$$ C_{t+1} = \gamma \cdot C_{t}$$

Note that by definition
$$ \overline{C_{t}} \equiv \frac{1}{t}\sum^t_{\tau=1} C_{\tau}$$
Under a constant growth rate
$$ \Rightarrow \overline{C_{t}}=\frac{1}{t}\sum^t_{\tau=1} \gamma^{\tau-t}C_{t} = \frac{C_t}{t} (1+\frac{1}{\gamma-1}) = \frac{C_t}{t} (\frac{\gamma}{\gamma-1})$$
$$ \Rightarrow \overline{C_{t+1}}= \frac{C_{t+1}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \frac{C_{t}}{t} (\frac{\gamma}{\gamma-1}) = \gamma \overline{C_{t}}$$

$$ \Rightarrow  \frac{\ln[\beta] + \ln[1+r]}{\sigma} = \ln[C_{t+1}-b \overline{C_{t}}] -\ln[C_t-b \overline{C_{t-1}}] $$
$$ = \ln[\gamma C_{t}-b \gamma \overline{C_{t-1}}] -\ln[C_t-b \overline{C_{t-1}}] = \ln[\gamma]$$

Therefore the one period growth rate in consumption ($\gamma$) in the non-stochastic model is $$ \gamma = e^{\frac{\ln[\beta] + \ln[1+r]}{\sigma}}  = [\beta(1+r)]^\sigma$$


What exactly is purchasing power parity (PPP)? I have some questions regarding what exactly PPP is. In my understanding, PPP is essentially the value that a currency can purchase within an area. For example, although the USA might have more money in the international area, the total amount of monetary value can purchase more goods in China due to the low price of living there.

The problem I have with this is that it seems that value is subjective. For example, it might be cheaper to purchase labor in China than in the US however it is much more costly to purchase an iPhone there. How you decide which is more valuable?

If what I am saying makes no sense at all, please explain what PPP is.

What you are describing is basically purchasing power. Purchasing Power Parity (PPP) claims that your purchasing power is the same across countries.

PPP states that, if say a pound of bread costs 4 US$, you can take that amount, exchange it to Chinese yuan, go to China and buy exactly the same amount of bread. (i.e. you have the same purchasing power in both countries)

3 factors play a role: the price (of bread) in the US, the price in China, and the Exchange rate. There is no subjective factor here.

Even though I don't fully understand what you mean by "purchasing labor", make sure that you compare goods, or baskets of goods, only (PPP does not say anything about wages or production costs).

And be sure to remember that PPP does not necessarily have to hold in reality.

Hope this helps!


What does it mean by saying someone is "effectively risk averse/loving"? Recently I am reading a paper by Ortner & Chassang (2018) on corruption control. It is a nice paper to read, and the idea is kinda cool.

The game is as follows. There are 3 players, a principle, a monitor, and an agent. The agent can choose to engage crime, and the principle offers a wage structure to the monitor in order to deter crime (for details, please refer to the paper if you are interested). One of the results says that, when the monitor's private cost of misreporting is strictly concave (convex) over the support, the agent is effectively risk-loving (risk-averse).

However, I do not quite understand the concept of being "effectively risk averse/loving" that they mentioned. I never saw this before. What does it mean? Does it simply mean that it is beneficial for the agent to be risk-loving?

Thanks for your clarification in advance. Please kindly provide some reference if there is any.

"Effectively" has two definitions:

1: in such a manner as to achieve a desired result.

2: actually but not officially or explicitly.

O+C are using the second definition here. This is because the risk averse/loving behaviour that is being exhibited does not come from the concavity of the agent's utility function, but instead indirectly, from the concavity of the c.d.f of the random cost $\eta$ that the monitor will have to pay if they misreport information. So the agents are not "officially" risk averse/loving but as a result of the concavity/convexity of the monitor's cost cdf they will behave as a risk averse/loving agent would when the cdf is linear.

Hope this helps!


Why doesn't the trade balance equation take into account the value of products sold by the U.S.? $$NX ≡ (S-I) + (T-G)$$
  Read this as “the trade balance (net exports) is identical to the sum of net savings (savings minus investment) and the government’s budget surplus or deficit (taxes minus spending).” This is not a theory of what causes what; it’s an identity, different ways of expressing the same “what”.

Let's say the U.S. had a $NX$ or trade deficit of $500$ billion dollars and overnight the U.S. forced countries around the world to buy $1$ trillion dollars of U.S. military equipment giving the U.S. a trade surplus of $500$ billion. Does this mean that the equation: $NX ≡ (S-I) + (T-G)$ doesn't take into account non-macroeconomics factors and therefore is wrong?

If foreign countries suddenly bought $\$1$ trillion worth of stuff, presumably the US would supply this out of existing inventories (i.e. stocks of goods).

Before anything else happens, this represents a reduction of investment in the US and your equation would continue to balance since the increase in $NX$ would be offset by a reduction in $I$ (which has a negative sign in your national accounts identity).

What would happen next is probably that the manufacturers would make more stuff to bring inventories back towards previous levels, increasing investment. They and their workers may pay taxes on their income.  They may import some more stuff.  What is left may add to savings.  But throughout this process the national accounts identity will remain true.


Can demand side restrictions explain the productivity paradox (Solow paradox)? Productivity paradox https://en.wikipedia.org/wiki/Productivity_paradox is the name for the slow productivity growth (i.e. slow GDP growth) of the developed economies despite the robotics, AI, automation and digitization.
Productivity paradox presents uneasy interpretation of the effects of 4th Industrial revolution to the techoptimists.
Productivity paradox can be interpreted as the supply side constraints/bounds and those are very bad per itself, yet - it is in stark contrast with low-inflation (even negative inflation) regime of the current economy, with the possibility of governments to finance stimulus and Covid-19 idleness without raising inflation. Such low inflation regime generally proves that there are no attained supply side constraints, that there is idle capacity.
Such interpretation - about idle capacity - proves that demand side bounds determine the low inflation, low growth and hence - low productivity? So - can we really say that technology has created ample supply side capacities (and hence - potential productivity growth, I mean - total productivity of technology and humans, human productivity growth can be exhausted already and that is why Artificial Intelligence and robotics almost fully determine the future productivity growth) and only demand side restrictions puts bounds on growth and hence - the productivity doesn't show up in macroeconomic figures?
Of course - I should explain the term "demand side restrictions". Some may argue that there are none: if there is capacity for growth, then investors/workforce will use it, they will earn money and they will fully spend it - either for consumer goods or for capital goods (bank deposits is example how money is spent for capital goods) and hence - the growth should happen. And if growth doesn't happen then that is entirely due to the capacity restrictions, e.g., due to the low productivity. But I don't know - whether one can consider the consumer and capital goods as being the same. Maybe inequality can create "demand side restrictions": if money concentrates in the hands of the wealthy, then they are investing them in capital goods and hence in the supply side capacity building. If the money goes to workers and to all other people as the social assistance of Universal Basic Income, then it is being spent and hence the potential productivity is realized and becomes the actual productivity growth that appears in macroeconomic figures?
So - are there any demand side restrictions (e.g. to wealth concentration) in the period "2010-2020+" in the advanced economies and can such demand side restrictions determine that the potential productivity (e.g. due to the technological achievements) is not realized and doesn't show up in the macroeconomic figures as the actual strong productivity growth. Can this be the explanation of the productivity paradox? If now - what is wrong with this reasoning?

Capital accumulation would still appear as investment GDP. Its possible theres other applications for this spending that would result in higher gdp. For example, GDP spiked in ww2 because military spending happens to be very efficient at inflating GDP.
So it is intuitively plausible that potential output is hidden by rising investment- the closest equivalent in economics is baumol cost disease, where the low productivity sectors gain more wages as growth progresses. However, the example you are asking about, in technology, is not that related to macroeconomic investment. The overall infrastructure level in energy production and distribution drives the output of everything- so there isnt much tradeoff there, consumption is made possible by shared infrastructure used for both consumption and investment.
Technology likewise is shared infrastructure- you could use a semiconductor plant for either consumer or business goods- so theres only a limited extent to which it would matter in practice. But there could indeed be Baumol like factors which cause resources to be allocated in a way that results in lower output. Baumol phenomena are almost the opposite of what you are describing but they are the closest thing in economics to the idea that growth may be "hidden" through factor reallocation.


Where does capital come from? Sorry if this is a birds and the bees type question.  Where does capital come from?  And I am not talking human capital (which to my mind is skills basically).  I suppose I'm taking about financial capital.

So, I think one can create personal capital by choosing not to consume, i.e. save.  Also I think retained profits are another (and I was wondering perhaps the ultimate) source of capital.

If we look at the capital structure of a company then bank loans, corporate bonds and equity capital are all invested, retained profits are internally generated capital.  External corporate capital, to my mind, comes from household savings.

I know there is the external sector but can we simplify the problem by excluding this?  Let us consider the closed economy only.

I suppose I'm asking for a theory of capital.

I know Marxists have a view and they are welcome to give an updated view (please no re-hash of nineteenth century economic theory, bring it up to date please).

I also think that the Austrian school is critical of Keynesians because the former says the latter have no theory of capital.

Need illuminating in the capitalism facts of life.   Thanks.

I wish there was a tag 'meta-economics' or 'philosophy of economics' to describe this question.

Material capital is any durable good that is used as a factor of production and, by virtue of being durable, it is gradually consumed in production over a maximum possible duration of a length that is determined by (i) how much a unit of capital is used and (ii) the depreciation rate of a unit of capital. Capital forms by labor and savings (which is in accordance with the labor theory of value) and/or capital forms by chance (which is in accordance with the marginal theory of value).

Imagine, for example, a solitary person that is stranded on a tropical island. To stay alive, that person must eat and, on the particular island, imagine that virtually every bit of edible biomass is a coconut. In order to be able to consume any coconuts, coconuts must be produced. Without any tools, this person must climb a palm tree to obtain a coconut and, once back on the ground, then must find a way to crack it open. If the person has obtained enough coconuts to last long enough for the person to take a day off from climbing trees then the person, by virtue of their coconut savings, can afford to leisure. Yet, perhaps the person decides not to leisure and, rather, decides to produce tools (i.e. to labor) to make coconut production easier. Perhaps the person fashions a pole out of bamboo to knock coconuts from the tree so that the person doesn't have to keep climbing trees, which reduces the amount of work required to produce coconuts and, therein, makes coconut production that much more productive by virtue of the bamboo tool, which is a form of material capital (as it can be used repeatedly to produce coconuts, which cannot be used repeatedly). As Abraham Lincoln (1861) once said, "Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed." Yet, capital also forms by chance. To break open the coconuts, the person might use a rock and, perhaps, if there were no rocks on the island that could be used to crack open coconuts then, thanks to misfortune, the person might starve.

Capital cannot be discussed without also discussing tools and technology; they are inseparable. A tool is when capital and a technology coalesce. For example, a makeshift wooden spear is a hunting tool. It is a coalescence of wooden material (capital) and a hunting technology. The wooden object of which the spear is made is material capital, which depreciates at a rate at which the wooden material’s maximum strength decreases. Knowledge about how a wooden object can be used as a spear is a hunting technology. Such technical knowhow is cognitively embodied (in some form or another) in the living body of the tool user, and it is expressed in the body’s behavior, which is behavior that determines how the material object is used (and effectively renders it a tool). This theory has support in the life sciences. (Alfred Marshall [1922] once wrote, “Economics [is] biology broadly interpreted.”)

The phenotype of an organism extends beyond the individual organism, as expressions of an individuated organism’s genes are not contained within that individual organism. Gene expression can be manifest in the behavior of its kith and even in the behavior of another species entirely. “Genes in orchids express themselves phenotypically [as] changes in bee behavior, which result in [a] transference of pollen grains containing those same genes” (Dawkins 1984). The extended phenotype of any individuated organism is the gene expression that extends beyond the embodied physiology of that individual, and it is “less purely biochemical than . . . the conventional local phenotype” (Dawkins 1984).  Among the many kinds of phenomena that can be considered to be of an extended phenotype are artifacts like fabricated shelters and tools. “Caddis larvae live in houses which they themselves build out of stones [or] other material. The form of the house is determined by the behavior of the builder, which is presumably influenced by the builder’s genes” (Dawkins 1984). The genome of a builder is the energetic scaffolding of the behavior of the builder (Hoffmeyer 2014), and such behavior, which is gene expression, determines the form of what the builder builds.

The coalescence of capital and a technology as a tool is a process of semiosis.  A tool is a sign of capital signifying a technology, which is a notion that comes to be more clear in reference to Ferdinand de Saussure’s semiological conception of the (dyadic) sign. The makeshift wooden spear is a sign of capital signifying a technology. The wooden spear is a hunting tool, and a tool is a sign. The wood of which the spear is made is material capital, and capital is a signifier. The knowledge about how the wooden object can be used as a spear is a hunting technology, and a technology is a signified. The wooden spear, which is a hunting tool, is a sign of wooden material capital signifying a hunting technology, and the arbitrariness of the signifier speaks to the old adage that anything can be used as a weapon. Tools are to technology are to capital as signs are to signifieds are to signifiers.

Returning to the topic of capital, material capital extends from human capital (e.g. knowledge, skills, & innovation) and financial capital extends from material capital. Financial capital is part of financial infrastructure, which is a tool by which other tools (and material capital commodities) can be obtained by supplying one's saved income (or positive profit in production) rather than one's labor. Human capital is phenotypic, which means there is a genetic component. As Ronald A. Fisher once wrote, "In organisms of all kinds the young are launched upon their careers endowed with a certain amount of biological capital derived from their parents." (Fisher 1930). Fisher considered the inheriting of household wealth by offspring from their parents to be an extension of a more generalized inheritance process stemming from the inheriting of genes by the offspring from the parents, which is a notion that foreshadowed the Dawkinsian theory of the extended phenotype. Fisher was a protégé of the political economist Leonard Darwin, a son of Charles Robert Darwin. Charles Robert Darwin notably employed the Malthusian population principle from classical political economics (see Malthus 1798) as the architecture for designing his theory of natural selection (see C. Darwin 1860) and, as Karl Marx (1867) once remarked, "Darwin has interested us in the history of nature's technology, i.e., in the formation of the organs [that] serve as instruments of production for sustaining life." Indeed, Charles Robert Darwin's grandfather, Erasmus Darwin, had written of an "Economy of Vegetation" (1791) and, prior to that, Carolus Linnaeus described all life as an "Economy of Nature" (1749). Indeed, "The ultimate subject matter of biology and economics is
one, viz., the life process" (Daly 1968).

NOTE: In answering this question, I've borrowed heavily from two draft works of mine that are on my Academia.edu page (i.e. I have not plagiarized whatsoever): https://independent.academia.edu/AaronHRubin
The following two links are to the two works of mine from which I'd borrowed:

https://www.academia.edu/28416634/A_Brief_Historical_Survey_of_Economic_Thought_and_the_Life_Sciences

https://www.academia.edu/29823064/LifeEconomyEnergyandSemiosis_-_working_paper.pdf


Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good? Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good?

So for example, let us say that the government covers part of your medical costs, and in response, private clinics raise their prices because they estimate that they can charge what they were before and then-some, proportional to the added subsidy provided by the government.

That's a standard case of a Consumer Price subsidy. Its effect will be that of market expansion: higher quantity at higher equilibrium price, irrespective of whether it will be lump-sum (a fixed monetary amount) or a fixed proportion of the price.

A) LUMP SUM SUBSIDY
Consider a monopolist. The monopoly faces a downward-sloping demand curve, say

$$q_d = a -b(p-S) = (a+bS) - bp$$

where $S$ is the lump-sum subsidy. Here the demand schedule shifts upwards by $bS$ maintaining the same slope. Profits are (assume linear costs for simplicity)

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [(a+bS) - bp]$$

and the first order condition  for a maximum gives

$$(a+bS) - bp -b(p-c) = 0 \implies p^* = \frac {a+ bS +bc}{2b} = p^*|_{S=0} + \frac {1}{2}S$$

Profit-maximizing price is increasing in the lump-sum subsidy. Now substitute this into the demand equation

$$q_d^* = (a+bS) - b\frac {a+ bS +bc}{2b} = \frac {a-bc}{2} + \frac{b}{2}S = q_d^*|_{s=0} + \frac {b}{2}S$$

Quantity demanded also increased. So the lump-sum subsidy leads to market expansion.

B) PROPORTIONAL SUBSIDY
Here we have
$$q_d = a -b(1-s)p,\;\;\; 0\leq s<1$$
The demand schedule changes slope (becomes more "flat"), maintaining the same intercept.

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [a -b(1-s)p]$$

and the first order condition  for a maximum gives

$$a -b(1-s)p -b(1-s)(p-c) = 0 \implies p^* = \frac {a+ b(1-s)c}{2b(1-s)} = \frac {a}{2b(1-s)} + (c/2)$$

Equilibrium price is (non-linearly) increasing in the proportional subsidy. Substitute this into the demand schedule to get

$$q_d^* = a -b(1-s)\frac {a+ b(1-s)c}{2b(1-s)} = a -\frac {a+ b(1-s)c}{2} = \frac {a-bc}{2} + \frac {bc}{2}s$$

Here too equilibrium quantity is increasing in the subsidy.

For a sector like Health services, where increased market access is of concern (in some countries) due to societal values (and/or cold calculations as regards the material benefits to society in having more healthy members), and where "quantity demanded" is not based so much on desires but rather on not-negotiable needs (illnesses, accidents, etc), the rationale for such subsidies is that the quantity increased will mainly come from new "consumers", previously excluded from the market due to the pre-subsidy equilibrium price in combination with their income level.


Can an irrational function be a utility function? Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.
I know rationality over $\preccurlyeq$ does not imply a utility function. But rationality and continuity over $\preccurlyeq$ implies a utility function. But what about the reverse direction?
For example, $u(x) = sin(x) + 1$, is not rational, but is continuous, is it a utility function?
In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?
My Answer A utility function is the representation of a preference relation $\preccurlyeq$. All preference relations are by assumption (or definition), rational. Given a function, if there does not exist any rational preference relation, then it must not be a utilty function.

I don't particularly understand the question.
Start with any function $f:X \rightarrow \mathbb{R}$.
Define $x \succeq y$ if $f(x) \geq f(y)$.
We get a rational preference over $X$.
By the way $\sin(x) + 1$ is a perfectly valid utility representation.


What does an individual's demand function describe? Now, this seems a super easy and straightforward question, right? From wikipedia:

  The demand curve is the graph depicting the relationship between the price of a certain commodity and the amount of it that consumers are willing and able to purchase at that given price.

To me, this reads as follows: for an individual, the demand function answers, for all values of $p$, the question: "if the price is $p$ per unit, how many units $q(p)$ would the individual buy?"

As an example, let's consider my personal demand curve $q = 10-p$ for a certain good. I would interpret that, also with the above definition, as follows: if the price is 10/unit or more, I purchase nothing. If it is 7/unit, I'll purchase 3 units.

The big point here is that this seems to imply, that I'd pay 7/unit for each of the 3 units. Is that correct? Things would be different if I were faced with a situation in which I had to pay more for the first 2 units. In that case, I might decide to buy less than 3 units. And I wouldn't be contradicting my demand function. At least, not, if it were as I interpreted the definition.

Alternative: it seems less close to the text, yet much more convenient to the mathematics, to interpret the demand function in the following way. The demand function answers, for all values of $q$, the question: "if an individual already has $q-1$ units, how much would he be willing to pay for unit $q(p)$?" (or, as an integral, "if an individual already has an amount $q$, how much would $p(q)$ be, with $p(q)dq$ the amount he's willing to pay for an additional amount $dq$?").

Thanks!

The demand function does not say how much he's willing to pay for an additional amount dq. The demand function says: given the price of good is $p$, you will purchase $Q(p)$ of them. I think your confusion lies in the fact that it is possible to charge different prices for each unit of good. Notice that our theory is assuming one price, not multiple.

Moreover, the notion that consumer surplus measure the difference between what you paid and what you are willing to pay is completely false. Suppose the demand function is $Q(p)=-5p+100$. Suppose further that the price is $5$, so you would be purchasing at a quantity $75$. The consumer surplus is: $$CS=\int_0^5Q(p)dp=\int_0^5 -5p+100dp=562.5.$$ If interpreting it as the difference between what you paid and what you are willing to pay, we know that you are willing to pay $562.5+5\times75=937.5$ for $75$ units of them. In other words, you are willing to accept a price $937.5/75=12.5$. However, plugging this number into our demand function, we find that $Q(12.5)=37.5$, i.e. you are only willing to buy 37.5 units of the good, which clearly contradicts to our interpretation of consumer surplus.

For a better treatment of consumer surplus, I would recommend you to consult advanced texts on microeconomics theory such as Mas-Colell's or Varian's more advanced book.


Is there a proof for composite commodity theorem? I have been reading Economics and Consumer Behavior by Angus Deaton and John Muellbauer, specifically reading up on Composite Commodity Theorem, which states:

  if prices move in parallel to each other, then the corresponding group of commodities can be treated as a single good. (page 120 in the book)

This is a pretty mind blowing result (well at least for me), however im yet to find an actual proof for the theorem in the literature.

is there a proof for this result?

There is this nice expository note:
Carter M., 1995, "An expository note on the composite commodity theorem," Economic Theory, 5, 175-179
and an interesting generalization by
Lewbel, A., 1996, "Aggregation without Separability: A Generalized Composite Commodity Theorem," American Economic Review, 86, 524-543.

EDIT:
One reason why it is difficult to find a proof of the theorem, is that the so called "composite commodity theorem" does not really have the mathematical status of a theorem.
It is more a principle directly following from the reparameterization below.
Let $x\in \mathbb{R}^J_+$ denotes the disaggregate demand system, which depends on the price vector $p \in \mathbb{R}^J_+$, and the budget $b>0$. The relative prices are denoted by $\alpha = p/\textbf{p}$ where the aggregate price index is $\textbf{p} \in \mathbb{R_+}$. If $\alpha $ is a vector of constants, we can simplify the demand system such that it depends only upon the aggregate price $\bf{p}$:
$$ x(p,b) = x(\alpha \mathbf{p},b) \equiv \mathbf{x}(\mathbf{p},b).  $$
Note that now $\mathbf{x}:\mathbb{R^2_+} \rightarrow \mathbb{R}^J_+$.
If one is also interested in aggregating the elementary quantities, it is (for instance) possible to define the aggregate composite quantity as:
$$ \mathbf{X}(\mathbf{p},b) = \alpha^T\mathbf{x}(\mathbf{p},b),  $$ where $ \mathbf{X}:\mathbb{R^2_+} \rightarrow \mathbb{R}_+$. In this case, the elementary and aggregate expenditures are the same, in the sense that: $$\mathbf{p}\mathbf{X}(\mathbf{p},b)=p^Tx(p,b).$$


Solow model and economic interpretation of $\dot{K}=sY(t)$ I am reading Robert Solow's 1956 paper, entitled 'A Contribution to the Theory of Economic Growth'. I am trying to understand the economic interpretation of the differential equation $$\dot{K}=sY(t),$$ which is located on page 67 (Solow denotes this equation by $(3)$). I have understood each term as: $s$ is the proportion of saved output, $Y(t)$ is the production function at time $t$, and $\dot{K}$ is the rate of increase of capital stock with respect to time $t$. Although, I am having trouble understanding what this equation actually means (I am a novice at economics).

One interpretation:
Think about the equation above in GDP accounting terms. We know that in an economy, Total Savings ($S(t)$) = Total Investment ($I(t)$) (standard GDP accounting exercise).
How is this relevant here?
The right hand side is the Total savings in the economy: $\underbrace{s}_{\text{savings rate}}\times\underbrace{Y(t)}_{\text{total output}}$.

The left hand side is related to the total investment in the economy: The increase in capital stock ($\dot{K}$) in the economy is the difference between investment ($I$) and depreciation ($\delta K$). That is, $\dot{K} = I(t) - \delta K(t)$.

Replacing the identity $S=I$ then yields:
\begin{align}
\dot{K} &= S(t) - \delta K\\
&=sY(t) - \delta K
\end{align}


Log-linearization of Euler equation with an expectation term There are a few online resources available to help with log-linearization (e.g., here  or here). However, log-linearization where an expectation is involved
is a little tricky because the log can't simply "pass through" the expectation operator.
Could somebody help with the algebra in this example?

I have the Euler equation (equation 1)
$$
1 = E_t \left [ \left \{ \delta \left (\frac{C_{t+1}}{C_t} \right )^{-1/\psi}
\right \}^\theta
\left \{ \frac{1}{1 + R_{m,t+1}} \right \}^{1 - \theta} 1 + R_{i, t+1}
\right ]
$$
where $\theta = ( 1 -\gamma)/(1 - 1/\psi)$. I'm trying to derive an expression for
the risk-free rate and an expression for the equity premium. How should I go about doing this?

It seems from the second link above that I should start by replacing the
variables of interest like so $C_t = c e^{\tilde C_t}$. Then following the steps given,
it seems like I should arrive at (equation 2)

\begin{align}
1 = E_t \left [
\left \{ \delta \left (\frac{\tilde C_{t+1} + 1}{\tilde C_t + 1} \right )^{-1/\psi}
\right \}^\theta
{}
\left \{ \frac{1}{(1 + R_m)[\widetilde{(1 + R_{m,t+1})} + 1]} \right \}^{1 - \theta} \cdot \\
\cdot
[(1 + R_i)[\widetilde{(1 + R_{i, t+1})} + 1]]
\right ].
\end{align}

But where do I go from here?

EDIT:

I have copied equation 1 directly from the notes that I have. It is probably the case that the term on the right, $1 + R_{i,t+1}$, should be in parentheses, $(1 + R_{i,t+1})$. In my initial attempt at log-linearization I have treated it this way.
In equation 2, I have followed the steps in the instruction that can be found in the second link at the beginning. So, $R_i$ and $R_m$ without time subscripts are these values in the steady state.
$R_m$ is the return on the market portfolio and $R_i$ is the return on asset $i$.

EDIT 2:

Thanks for the useful comments.
So, from what I have gathered so far, I should get something like this:

\begin{align}
1 &= E_t
\left [
  \delta^\theta (1 - \frac \theta \psi (\tilde C_{t+1} - \tilde C_t )
  (1 + R_m)^{\theta - 1} (\theta - 1)
  \left (
    1 + \tilde R_{m,t}  \frac{R_m}{1 + R_m}
  \right ) \right .\\
  & \left . \, \cdot
  (1 + R_i)
  \left (
    (1 + \tilde R_{i,t} \frac{  R_i}{1 + R_i}
  \right )
\right ]
\end{align}

Then this would imply that the risk-free rate is found as follows:

\begin{align}
1 &= E_t
\left [
  \delta^\theta (1 - \frac \theta \psi (\tilde C_{t+1} - \tilde C_t )
  (1 + R_m)^{\theta - 1} (\theta - 1)
    \left (
    1 + \tilde R_{m,t}  \frac{R_m}{1 + R_m}
    \right )
  (1 + R_f)
\right ] \\
1 &= E_t[ m_{t+1} (1 + R_f)] \\
\frac{1}{E_t[m_{t+1}]} &= 1 + R_f.
\end{align}

Is this correct? And now, to finish the question, how would I find the
equity premium?

Let's ignore for the moment the existence of the expected value. If this was a deterministic set-up, linearization through taking logs would be straightforward, and without the tricks of the links the OP provided. Taking natural logs on both sides of the first equation we obtain:

$$0 = \theta \ln \delta-\frac {\theta}{\psi}\ln \left (\frac{C_{t+1}}{C_t} \right )-(1-\theta) \ln(1 + R_{m,t+1}) + \ln (1 + R_{i, t+1}) \tag{1}$$

Set

$$\hat c_{t+1} = \frac{C_{t+1}-C_t}{C_t} \Rightarrow \frac{C_{t+1}}{C_t} = 1+\hat c_{t+1} \tag{2}$$

Also, note that it is standard approximation to write $\ln (1+a) \approx a$ at least for $|a|<0.1$. Usually this is the case with growth rates and financial rates so we obtain

$$0 = \theta \ln \delta-\frac {\theta}{\psi}\hat c_{t+1} -(1-\theta) R_{m,t+1} + R_{i, t+1} \tag{3}$$

which is a clear dynamic relation that links the three variables present.
If in the model, the steady-state is characterized by constant consumption and constant returns, then at it we will have $\hat c_{t+1} =0$ and so the steady-state relation will be

$$R_{i} = - \theta \ln \delta + (1-\theta) R_{m} \tag{4}$$

But we did all these ignoring the expected value. Our expression is $E_t\left[f\left(C_t, C_{t+1},R_{m,t+1},R_{i,t+1} \right)\right]$, not just  $f\left(C_t, C_{t+1},R_{m,t+1},R_{i,t+1} \right)$. Enter first-order Taylor expansion of $f()$. We need a center of expansion. Represent the four variables simply by $\mathbf z_{t+1}$ (it doesn't hurt that a variable with $t$-index is present in $\mathbf z_{t+1}$). We choose to expand the function around $E_t(\mathbf z_{t+1})$. So

$$f\left(\mathbf z_{t+1}\right) \approx  f\left(E_t[\mathbf z_{t+1}]\right) + \nabla f\left(E_t[\mathbf z_{t+1}]\right)\cdot \big(\mathbf z_{t+1}-E_t[\mathbf z_{t+1}]\big) \tag{5}$$

Then

$$E_t\left[f\left(\mathbf z_{t+1}\right)\right] \approx f\left(E_t[\mathbf z_{t+1}]\right) \tag{6}$$

Obviously this is an approximation, i.e. it has error, even if only because of Jensen's inequality. But it is standard practice. Then we see that all the previous work we did on the deterministic version, can be applied in the stochastic version inserting conditional expected values in place of the variables. So eq. $(3)$ is written

$$0 = \theta \ln \delta-\frac {\theta}{\psi}E_t[\hat c_{t+1}] -(1-\theta) E_t[R_{m,t+1}] + E_t[R_{i, t+1}] \tag{7}$$

But where are the steady-state values? Well, steady state values in a stochastic context are a bit tricky -are we arguing that our variables (which are now treated as random variables) become constants? Or is there another way to define a steady-state in a stochastic context?

There are more than one ways. One of them, is the "perfect foresight steady state", where we forecast perfectly a not-necessarily constant value (this is the concept of "equilibrium as fulfilled expectations"). This is for example used in Jordi Gali's book mentioned in a comment. "Perfect-foresight steady state" is defined by
$$E_t(x_{t+1}) = x_{t+1} \tag{8}$$

Under this concept, eq. $(7)$ becomes eq. $(3)$ which is now the "perfect-foresight stochastic steady state" equation of the economy.

If we want a stronger condition, saying that variables become constant in the steady-state, then it is also reasonable to argue that, again, their forecast will eventually be perfect. In that case, the steady-state of the stochastic economy is the same as that of the deterministic economy, i.e. eq. $(4)$.


Why are ecology and economics seemingly disjunct topics? The words ecology and economy are very similar.  Ecology derives from the words οἶκος + λογία, i.e. the study of the house/household.  Economy derives from the words οἶκος + νόμος, i.e. the laws of the house/household.  In modern usage, ecology studies, among other things, movement of materials and energy through living communities (e.g. Wikipedia).  Economy, in modern usage, is defined by Wikipedia as An economy or economic system consists of the production, distribution or trade, and consumption of limited goods and services by different agents in a given geographical location.
An interdisciplinary field is called ecological economics, which is the treatment of the economy as a subsystem of the ecosystem (again according to Wikipedia).

I've often wondered by ecologists and economists seem, often quite literally and physically, separated fields.  Ecological economics considers the economy as a subsystem of the ecosystem.  Why is this not standard?  How can the economy ever not be a subsystem of the ecology, and why is not all economics "ecological economics"?  Or is it?

I think you are right to say that an economy is always a subsystem of a larger ecosystem, but it needs not always be modeled as such.

It may be interesting to have a precise and elaborate model of the solar system taken in isolation, independently of its interactions with the rest of the galaxy, other galaxies, etc. In very much the same way, there are many good reasons to build a model of an economic system without accounting for how it fits in a more general ecosystem.

The same kind of question applies to many other topics. One could for instance ask "why partial equilibrium and not always general equilibrium?", "why small open economies instead of replicating the actual size structure of the global economy?", etc. I guess the only answer is that the smaller, less general models are considered useful in one way or another.

This is not to say that the opposite endeavours (the ecological economics attempts to model economic systems as part of an ecosystem) are worthless, quite the contrary. But there is only so much you can do and there is always a trade-off (in terms of estimability, tractability, additional assumptions, you name it ) to increasing the generality of a model.


Regular annual pension I am supposed to solve the problem:

A 24-year-old man decides to invest 200,000 euros at a 7% annual interest rate to bring him a regular annual pension from 31 to 50 years inclusive. What will be the pension?

What I did was that I used the formula for the future formula of pension:

$$FV=A\frac{\left ( 1+i \right )^{n}-1}{i}\left ( 1+i \right )=200,000\frac{\left ( 1+0,07 \right )^{7}-1}{0,07}\left ( 1+0,07 \right )=1851960.514$$ and then I divided it for twenty years which is 92598.02

But that is incorrect. The correct solution is 28331.

Can someone tell me, where I made a mistake?

Firstly, the formula you have used for FV is equivalent to that shown here. What it calculates, with your numbers, is the value after 7 years of investing 200,000 annually at a compound rate of 7% (as a rough confirmation, 1,851,960 is somewhat more than 7 x 200,000).  Instead, you need the value after 7 years (or 6 years if the man is already at the end of his 24th year) of a one-off investment of 200,000 (call that V).

Secondly, the simple division by 20 takes no account of the interest earned during years 31-50. To allow for this, you need the present value of an annuity formula (source):

$$P\Bigg[\frac{1-(1+i)^{-n}}{i}\Bigg]$$

Set this formula equal to V and treat the annual (periodic) payment P as an unknown to be found.


How does hedging using futures work? I can't understand how it makes sense. From what I've read, corn producers (for example) sell their corn at current price but deliver the corn later, in order to protect themselves from decline in corn prices.
On the other hand, those who buy corn agree to current price in order to protect themselves from potential increase of corn prices.
But in the end, one of them is going for profit from this (corn producer if prices fall, buyer if prices raise).
Let's say price of corn rose. Essentially, the producer of corn didn't protect himself - he just lost money on that trade.

The only way I see this work for the producer is if there is >50% chance that price of corn will decline, but if that's the case then buyers of corn wouldn't agree to that trade.

So in the end, doesn't it all even out? Sometimes the producer will make money sometimes he will 'lose'.

Producers of corn may not be able to afford a decrease in prices.

Remember that farmers are (usually) not speculating and not “investing”. They are producing a product that they need to offload in order to make money. They have zero interest in uncertainty (again, assuming that most are not speculating). The future price they can receive today may be better than the spot price in 6 months, or it may be worse - you really have no idea, but at least you can lock it in, so that you know what to plan for.

Remember that farmers have two uncertainties about their product when laying the first seed: 1) quantity and 2) price. By hedging the price on some quantify you can reduce your risk significantly (and hence also your upside) which may be far more appealing then the risk of losing everything.


How to make sense of so many conflicting theories and ideas in the field with an open mind? Is every school of economic thought as valid as the rest? 'm a junior economics major and lately I've been in a sort of crisis because I feel like I'm not sure what economic ideas I believe in anymore. Don't get me wrong, I love reading anything that has to do with the subject. Thing is, I just feel like, for somebody as skeptical as me, it's tough to remain convinced of anything I learn. I used to blindly agree to mainstream econ when I first started studying this, but then I was introduced to different schools like Marxism, the Austrian School, New Keynesianism etc. and it made me realize things are not as simple as I supposed they were. Now, however, I feel like I don't have the tools to discern what is good and bad economics. Sure, I know marxist theory or austrian theory are not exempt of criticisms, but neither is orthodox economics. And what's worse is that I feel I'm not capable of criticising them on my own: I always depend on somebody else's analysis of the state of modern economics, his/her interpretation of Marx and Von Mises, etc etc.; so I don't know how much to trust their opinions either.

And I look around me and it seems as if everybody I study with just CHOOSES what to believe in (Austrian/Keynesian/New Classical/etc) instead of deeply analyzing all theories and accepting what seems to be the most rational school of thought for them.

So in summary, what is there to do for me? It's like there's a counterpoint for every argument and I'm not capable of sorting out on my own which is more reasonable. Is there anything I could read to find out what different methodologies each school of thought uses, where exactly do they differ and how scientifically valid are their theories?

I'm not sure if I'm asking for too much, but I'd honestly appreciate any kind of response.

This is how I view it. At the very core of economics, is each and every human on this earth making decisions he or her thinks are in his or hers best interest. In real life, we do not calculate utility before making a decision or compute any expected values; we do what feels right at the moment.

Taking the cross-section of human decisions, economists try to make sense of it all, and ultimately deduce some theory that resembles a law of physics to shed light some basic insight into the human condition.

This is a mighty hard task; near impossible. Its amazing to me we've made any progress at all, yet here we are.

As a field of knowledge, economics will never explain the world perfectly in whole, because the nature of what we study is far from perfect and certainly stochastic. There will never be a unified theory of economics, but that does not mean the small pieces of insight each economics paper tries to uncover is trash. It is still valuable to use what the data can tell us, to the best of our ability, to guide our future decisions, and analyze the decisions we've made in the past.

If anything, its interesting as hell.


Non monotonic supply curve Supply curves are usually non decreasing. For example in a electricity market, providing more capacity means turning on new production units with higher costs, see e.g. the following representation:

My question is, in what context would it make sense to have a non monotonic supply curve?

I am not an expert in micro theory. This answer is based on my limited knowledge.
Usually you don't. If the supply curve is drawn by stacking all the units of supply according to their marginal cost, as what happened in your figure, then the supply curve is monotonic by definition.
However, while the same goes to demand curve, I have actually seen non-monotonic demand curve in Becker 1991. In that paper, Becker shows that you can have non-monotonic demand curve if there is social influence so that a consumer's demand depends on demands by other consumers.
In analogy, I conjecture that if there is some externality of production so that a firm's marginal cost depends on other counterparts to some degree, there might be non-monotonic supply curve. But I don't know such a paper. I expect some other experts could give a real example from the literature.


Does elasticity make sense for different levels of prices? If not, how can we fix that? Let's say we know that our customers can't make sense of a raise in 5% of the price. It's too little for them to notice. However, an increase above 10% is easily noticed. If this is true, different level of prices impacts the customers in different ways and the formula we learned at college is not enough for real-world problems. Additionally, 10% of increase in something that costs 10 dollars is different than 10% of increase in something that costs 100 dollars, right?
So, how can we deal with this type of problem? How can we deal with price in real-world? How can we get elasticity in this situation?

The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as $EL = \frac{df(x)}{dx}\frac{x}{f(x)}$ for some function $f(x)$ (see Essential Mathematics for Economic Analysis by Sydsæter, Hammond & StrØm - which is an undergraduate textbook used at many econ departments).
For example, for demand function given by: $$Q= a - bp$$ the elasticity will be given by:
$$\epsilon = \frac{dQ}{dp}\frac{p}{Q}= -b \frac{p}{a-bp}$$
so for $a=100$ and $b=10$ it will be:
$$\epsilon = - \frac{p}{100 - 10p}$$
So for example at price 5  dollars the elasticity (in its absolute value) will be 0.1 at price 9 dollars elasticity will be 0.9 and so on. Hence, elasticity can change when prices change and you don't need any new concepts. There is a special type of demand functions where elasticity will always be constant but then there are infinite number of possible demand functions and in most of them it will change with price.


Why do economists disagree so much? I'm coming at this from a scientific point of view, having had no formal training in economics - most of what I know about it was from self-studying an undergraduate economics textbook.

My question is, why do economists disagree so much? This doesn't happen in the sciences. For example, for the question "will a driver in a car with these safeguards survive a collision at 100km/h", most scientists will agree on an answer. It might be conditional on various things (e.g. age of the driver), but most scientists will come to the same conclusion.

However in economics, it's different. Faced with a question such as "should we provide more fiscal stimulus for the economy", it's possible that one big group of economists says "yes", and then another big group says "no". To some extent this might be because of personal interpretation of what is desirable. However I would expect that economists should still be able to say something like "if you provide fiscal stimulus, this will happen, and it's up to you to judge whether the consequences are desirable", and yet it seems there's no consensus on what will actually happen. It doesn't help that when I see economics debated in the media, both sides advance what looks like reasonable arguments.

Does economics have predictive power? If so, why can't economists simply tell policymakers what to do? If not, what's the point of economics (it might as well be astrology)?

Some areas of economics have more consensus and predictive power than others. Most economists would agree on the effects of trade barriers, could fairly accurately forecast the effects of a price change given a good demand estimate, would come to the same conclusion about the effects of allowing a merger between two large competing firms, know how asset prices will react to a change in interest rates, etc.

The effect of a fiscal stimulus is one of the most complicated questions in economics because you are asking about the effect of stimulating a system with millions of moving parts (people and firms) and many dimensions (consumption/saving, employment/work, trade, investment, innovation, ...). This is a long way from the simple, closed systems for which natural science is able to give sharp predictions. In fact, when you look at parts of the natural sciences that deal with similar levels of systemic complexity, the overall (lack of) predictive power looks similar to that in economics:

although principles of ecology are well understood, it is almost impossible to accurately predict what the final effect of, say, introducing a new species into an ecosystem will be
the theory of evolution is probably one of the greatest scientific theories of all time, but can only be used to make the vaguest predictions about the future
medics cannot accurately predict the onset of a wide range of diseases—only diagnose them ex post
models of future global temperature increases have wide error bars
weather forecasts more than one or two days into the future have wide error bars

The other thing to note is that fiscal stimulus is a highly politicised topic and economists often do a (frustratingly) poor job of keeping politics out of the debate. If you read the economic literature, you will find a story similar to that for your car crash: any outcome is possible but some look more likely, depending on the exact circumstances. Given this ambiguity, politicians of different persuasions have no difficulty finding an economist willing to ignore this nuance and take a politically expedient position (just as conservative politicians can always find scientists willing to play down anthropogenic global warming). But that is less a failing with economics than with economists.


Should selling price depend on product quality or on work to produce the product if both not in positive correlation? I am a Natural Science student and read a little about economics. I am not sure this question is related to economics or not, decided to give it a try.
Recently, I am looking for printing service. From more than one service providers, I got the details that, printing a page is more expensive than photocopying a page. I asked them why is that so, and they said because the quality from printing is better than the quality of photocopying (photocopying inevitably causes a little dislodgement, cannot be 100% same as the original). They confirmed that the inks and papers used for both are the same. This is what I replied: ''Raw materials required for both are equal, printing needs way less work than photocopying each page one by one, so I actually expect the price for printing to be at worst the same as photocopying if printing is not cheaper than photocopying.'' And I didn't hear back from them.
Who is right here?
If the shoppers are right and printing should cost more than photocopying, do you really want to penalise your customer for reducing you work?
If I am right and printing should cost less than photocopying, do the shoppers really want to give higher-quality product to customers at lower price?
Is there any economics theory for this and other similar everyday occurrence?

No one is "right" here, there is rarely a "should" in economics, it is not about morals.
When making a purchase, you are usually willing to pay an amount $x$ larger than the price $p$ which is larger than the cost $c$ it took to make the product/provide the service. You are arguing that price should only depend on cost. The store is arguing that if you are willing to pay more, they can charge more. Unless they have competition who undercuts them, they are right.
In the theoretical model of perfect competition, where there is a single type of product and firms face very strong competition the price would equal the average cost in the long run. I would argue the model does not apply here, competition is not perfect (if I am wrong, go to the competition). In case of imperfect competition it is normal to charge people whose demand is less elastic w.r.t. price more.

What you can do: pay them or find a better offer. If you are convinced that you are right, found a rival printing firm, undercut them and put them out of business.


Structure of Non-profits and Moral Hazard When one looks at a Non-Profit Organization there are three main things people notice.

1.They are an Income tax exempt organization

2. They are primarily financed by donations
3.They are heavily reliant on volunteer labour

what systems are in place to see that surplus income is not redistributed among stakeholders or among board members?

Although they are non-profit organizations, they still submit legal documents such as statement of cash flow, statement of financial position, etc.

These contain detailed information of the spending of such organizations.


Per capita income growth rate is difference between income growth rate and population growth rate? We know: $PCI = \frac{Y(Income)}{P(population)}$ and that
$PCI$ growth rate is $\frac{\Delta PCI}{PCI}$.

How is this equal to the difference between income growth rate and population growth rate?

In other words how is $\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) = (∆Y/Y) - (∆P/P)$ ?

Knowing that:
$$\frac{\Delta PCI}{PCI}=
\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) $$

If we rearrange the terms on the right hand side:

$$\frac{\Delta PCI}{PCI}=\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

In order to get your last equation we would have to log the variables, for logged growth rates. This makes things easier because we can think of the variables in terms of percentages.

Logging the variables we get
$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right)$$

in more formal notation used in macroeconomics:

$$\hat{PCI}=\hat{y}-\hat{p}$$

hope this helps.


Why use the income approach to approximate GDP if it relies on the value produced by the expenditures approach? To preface, I'm taking a macroeconomics class and we're learning about measuring GDP. The textbook says that there are two main ways to measure GDP: the expenditures method and the income method.

The expenditures method can be represented like so:

$GDP = C + I_g + G + X_n$

where $C$ is personal consumption expenditures, $I_g$ is gross private domestic investment, $G$ is government purchases, and $X_n$ is net exports.

The income method is more complicated, and looks like this:

$GDP = I_n - I_f - C_c + s$

Where $I_n$ is the national income, $I_f$ is the net foreign income, $C_c$ is the consumption of fixed capital, and $s$ is the statistical discrepancy.

$s$ is calculated by subtracting the expenditures method from the income method, in order to make them the same.

What is the point of having the income method if it relies on the expenditures method? I get that it would be useful for statistics, but why use both?

The definition of the expenditure measure of GDP is:
$$GDP_{EXP} \equiv C + I_g + G + X_n$$

The definition of the expenditure income measure of GDP is:
$$GDP_{income} \equiv I_n - I_f - C_c$$

As I understand it, $s$ is not part of it. $s$ is defined as the difference between the two measures:
$$s \equiv GDP_{EXP} - GDP_{income}$$

Apparently it is also possible to define it with the RIPSAW approach:
$$GDP_{income} \equiv R + I + P + S + A + W$$
R = rents, I = interests, P = profits, SA = statistical adjustments (corporate income taxes, dividends, undistributed corporate profits, and W = wages.

And yes, it is useful to calculate GDP with multiple methods. Among other things, it reduces noise in GDP estimates:

  Two often-divergent U.S. GDP estimates are available, a widely-used
  expenditure-side version $GDP_E$ , and a much less widely-used
  income-side version $GDP_I$ . We propose and explore a “forecast
  combination” approach to combining them. We then put the theory to
  work, producing a superior combined estimate of GDP growth for the
  U.S., $GDP_C$ . We compare $GDP_C$ to $GDP_E$ and $GDP_I$ , with particular
  attention to behavior over the business cycle. We discuss several
  variations and extensions

Improving U.S. GDP Measurement: A Forecast Combination Perspective Aruoba, Diebold, Nalewaik, Schorfheide, and Song (2012, ungated here)


Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$? Marginal costs MC is defined as $MC=\frac{dC}{dq}$. Taking into account that $C=wL+rK$,

$$MC=\frac{dC}{dq}=w\frac{dL}{dq}+r\frac{dK}{dq}$$

Recall that marginal product of labor $MP_{L}=\frac{\partial q}{\partial L}$ and marginal product of capital $MP_{K}=\frac{\partial q}{\partial K}$.

Question: is the following correct

$$\frac{dL}{dq}=1/\frac{\partial q}{\partial L},\;\frac{dK}{dq}=1/\frac{\partial q}{\partial K}$$

which implies

$$MC=w\frac{1}{MP_{L}}+r\frac{1}{MP_{K}}$$

If no, then no need to read further.

If yes, then, consider profit maximization of a firm.

$$\max_{L,K}pq\left(L,K\right)-wL-rK$$

FOC:

$$\begin{cases}pMP_{L}=w\\pMP_{K}=r\end{cases}\Rightarrow\begin{cases}MP_{L}=\frac{w}{p}\\MP_{K}=\frac{r}{p}\end{cases}$$

Therefore,

$$MC=w\frac{1}{MP_{L}}+r\frac{1}{MP_{K}}=w\frac{1}{w/p}+r\frac{1}{r/p}=p+p=2p$$

The result is wrong for sure. I wonder, at what step of derivation I made a mistake?

Question: is the following correct ?

  $$\frac{dL}{dq}=1/\frac{\partial q}{\partial L},\;\frac{dK}{dq}=1/\frac{\partial q}{\partial K}$$

In general, no. Since $q= f(L,K)$ is a multivariable, single-valued function, then by the implicit function theorem applied on the implied equation $H = f(L,K)-q=0$, what we can say is only that

$$\frac {\partial L}{\partial q} = -\frac {\partial H/\partial q}{\partial H /\partial L} = -\frac {-1}{MP_L} = \frac {1}{\partial q /\partial L}$$

Essentially, here we treat $q$ as a univariate function (by taking partial derivatives, we keep all other variables fixed).

But total derivatives are another matter.
For $q = f(L,K)$, we have the implicit relation $L = L(q,K)$ and so by taking the total differential and divide throughout by $dq$ we get

$$\frac {d L(q,K)}{d q} = \frac {\partial L(q,K)}{\partial q} + \frac {\partial L(q,K)}{\partial K}\frac {dK}{dq}$$

But $\frac {\partial L}{\partial K} \neq 0$, because we implicitly keep $q$ constant in order to calculate this partial derivative, and so by changing $K$ we have also to change $L$. The other possibility would be that $dK/dq$ is zero. If we do not have such a production function (or we are not in a range where such a thing holds), we obtain

$$\frac {dL}{dq} \neq  \frac {\partial L}{\partial q} =  \frac {1}{\partial q/\partial L}$$


Showing marginal product of capital is independent of the scale of production The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.

I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.

It's either I'm not partial differentiating correctly or the whole theory is wrong.

I don't see anything wrong with what I've done but why are they not the same?

There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).

To compute the MPK, we must differentiate the production function with respect to the current level of capital: $\partial F/\partial K$.

But in your final line, you are not differentiating with respect to the current level of capital (which is $\tilde{K}=\lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/\lambda$ of the current amount of capital.

If we compute the derivative with respect to $\tilde{K}\equiv \lambda K$ instead of just $K$ then everything works as it should:

$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$

This does not depend on $\lambda$ so MPK is indeed independent of the scale of the economy. QED


Demand Function And Inverse Demand Function Representation in Books Demand Function tells us how demand for a good varies as price varies. Inverse Demand Function is inverse of the demand function. Consider Q=210-3P , This is clearly demand function . So on X axis we should put price and on Y axis we should put Demand i.e quantity. But in all the books in demand function Price is Y axis. Shouldn't that be inverse demand function?(P=70-Q/3)

Any time demand function is specified as function of price $Q(p)$ and we plot it using convention that price $p$ belongs on y-axis and quantity $Q$ on x-axis the plot shows the inverse demand function. However, ‘inverse demand function’ is quite a mouthful and hence it will often be just abbreviated as demand.
Furthermore, there would be nothing necessarily wrong with plotting demand function with quantity being put on y-axis and price being put on x-axis. The graph would be equally valid but it would go against established convention.
Generally in science the convention is to put dependent variable on y-axis and independent variable on x-axis. In the case of demand the price and quantity are actually jointly determined and hence both variables could be considered dependent. However, this convention got established by Alfred Marshall in his influential Principles of Economics which was extremely popular economic textbook, and if you actually go over the book you will discover that Marshall often thought of price being determined by quantity  and consequently in his own mind at least he was just using the standard scientific convention of putting dependent variable on y-axis and independent on x-axis. Marshall book was so influential that the way how we plot supply-demand in present day textbooks is practically the same way.
There is actually already an excellent post discussing this notation on the history of science and mathematics stack exchange by user EconCow (the post also gives greater detail of the Marshall’s own position and further resources). As the user shows Marshall was not even the first one to plot price on y-axis and quantity on x-axis, and there also were in past economists that did exactly the opposite plotting. To be more specific the user provides examples for the following economists and ways how they arranged their graphs:

Augustin Cournot (1838). Price on horizontal axis.
Karl Rau (1841). Price on vertical axis.
Jules Dupuit (1844). Price on horizontal axis.
Hans von Mangoldt (1863). Price on vertical axis.
Fleeming Jenkin (1870). Price on horizontal axis.
...
William Stanley Jevons (1871, figure). Price on vertical axis.

The post I linked even shows the pictures of these graphs so I encourage you to have a look at it.
As you can see from the above at the begging people plotted demand in different ways. However, after Marshall's principles of economics profession decided to favor his convention and we continue using it to this day.


Global Macro Investing textbooks and reading list What is a good reading list for Global Macro investing? What does Bridgewater, Bervan Howard, Soros, AQR etc. use to teach their staff about macroeconomic investing?

Let us assume a top-class education at the undergraduate level in economics, maths, and economic history so as to avoid the common recommendations of not very useful introductory texts.

Note: Global Macro Investing involves expressing views on commodities, rates, FX, Real Estate, Equities and other asset classes using ownership, futures, or options to find relative value and generate alpha on a global basis.

For example: Short USDTRY on asset outflows from turkey due to geo-political risk and US tightening cycle compared to TRY printing/easing.

There are quite a few books in the market on global macro strategies, the following could be useful:

Bernad Baumohl (2012) The Secrets of Economic Indicators: Hidden
Clues to Future Economic Trends and Investment Opportunities
Greg Gliner (2014) Global Macro Trading: Profiting in a New World Economy

Also try reading Burstein's book


Price Level and Value of money When the price level increases in the economy,

we generally know that the value of money decreases.

However, what is the definition of money here?

Is it just currency? Or does it include assets, such as stocks and bonds?

Because if the price rises, due to the interest rate effect people sell their bonds and stocks. So the creditors charge a higher interest rate. Then the means that the value of assets becomes higher right?

Could someone explain this logic clearly? Thank you!

We need to keep two things seperate: the unit of account, and instruments.

The unit of account is a currency unit, like the U.S. dollar. When the price level (e.g., CPI) rises, the cost of goods measured in that unit of account rises.

We then have monetary instruments, like $20 bills, or bank accounts. Their value in the unit of account are unchanged. Therefore, they drop in value versus goods and services (as expected).

The prices of stocks and bonds may go up or down, for any number of reasons. Like other things, their value is measured in the unit of account. Therefore, it is unclear what relationship there will be between the price level change and financial asset prices. For example, CPI inflation has been close to a 2% annual rate since the Financial Crisis, and stock indices have generally risen much faster than that.

As for interest rates, the relationship is: bond price up, yield down (and vice versa).


How to find mixed optimal strategies in this zero-sum game? I'm trying to solve this problem from last year final exam in game theory:

Consider the zero-sum game $G=(X, Y, g)$ where $X=Y=[0,1]$, and $$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(1-2 x)\}$$
Find a mixed optimal strategy for each player. (Hint: one can consider mixed
strategies of player $1$ which plays $x=0$ with some probability and $x=1$ with the remaining probability).

My attempt:
Let $\sigma$ be a mixed optimal strategy in which player $1$ play $x=0$ with probability $p$ and $x=1$ with probability $1-p$. A mixed optimal strategy $\tau$ of player $2$ is a probability measure on $[0,1]$.

Then I'm stuck to proceed. Could you please help me finish this exercise? Thank you so much!

First, a caveat: I'm on the job market this year in the midst of the couple weeks when calls are rolling in. Hence, this seemed like a nice way to kill some time (semi-)productively. This is also a disclaimer in case I've made an error :)

Now, let's look at the one you're suggested to try for, where player $1$ chooses $0$ with $p$ and $1$ with $1-p$. Again let player $2$ choose $G$ with full support.

If player $1$ chooses $0$ and player $2$ chooses $y$, Pl $1$ gets $$\int_{0}^{1}yg(y)dy = \mu$$
and if he chooses $1$, he gets $$\int_{0}^{1}(1-2y)g(y)dy = 1 - 2\mu$$
where $\mu := \mathbb{E}_{G}[y]$
Hence, for indifference we need $\mu = 1/3$. Ah but we also need this to be robust to any other deviation. Namely, we need.
$$\frac{1}{3} \geq x\int_{0}^{x}(1-2y)g(y) + (1-2x)\int_{x}^{1}yg(y)$$
for all $x \in [0,1]$.

Knowing that $\mathbb{E}_{G}(y) = 1/3$ for $G(y) = \sqrt{y}$, let's guess that that is the solution. The right-hand side reduces to
$$\dfrac{2x^\frac{3}{2}-2x+1}{3}$$ which is maximized at $x = 0$ and $x = 1$ and equals $1/3$ there, as required.

Finally, for player $2$, she is indifferent over any $y$ if $$p(-y) - (1-p)(1-2y)$$ does not depend on $y$. This holds if $p = 2/3$.

Thus the equilibrium is given as follows: player $1$ chooses $0$ with probability $2/3$ and $1$ with probability $1/3$; and player $2$ chooses cdf $G(y) = \sqrt{y}$ on $[0,1]$. The value of the game is $1/3$ for player $1$ and $-1/3$ for player $2$.


Logically inconsistent optimality condition I'm reading a book on monetary policy, and it defines the representative household's consumption as $$C_t = \int_0^1 C_t(i) di$$

so, basically a sum over the consumption of good $i$.

Then later on, it says that the consumption of each good is identical $$C_t(i) = C_t(j) = C_t$$

This equation makes no logical sense. How can $C_t(i) = C_t$ for all $i$? $C_t$ is DEFINED as an integral sum over all those $C_t(i)$. They cannot be identical.

That's like saying 4 = 2 + 2, and then say 4 = 2.

There is $C_t$ the number and $C_t()$ the function. This is already so in your first equation. Your second equation says that the function is constant over $[0,1]$. In which case it is true that $C_t$ the number equals $C_t()$ the function's value times 1.

The same thing with a different notation:
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di.$$
If there exists a constant $c_t$, where for all $i$
$$f_{C,t}(i) = c_t.$$
then
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di = \int_0^1 c_t di = c_t. $$