Do policy functions exist for Finite Horizon Dynamic programming problems? I've been looking at the cake eating problem over a finite horizion and have been trying to figure out if we can derive a policy function for such a problem. My work is written below.

sequence form of such a problem is:

$$\max _{c_t,x_{t+1}}U(c_t)=\sum_{i=0}^T\beta^tu(c_t),\ \ \ \ \ \ 1>\beta>0$$
$$s.t. x_{t+1}=x_{t}-c_t$$

supposing that the instantaneous utility function for such preferences are $u(c_t)=\ln(c_t)$ it follows that the function equation for such a problem is:

$$v(x_{t+1})=\max_{x_{t+1}}\{\ln(x_t-x_{t+1})+\beta v(x_{t+1})\}$$

In this case im supposing $T=2$ (we have three periods starting from period zero) and suppose $x_0>0$. Solving the problem from backwards induction we know.
$$v(x_2)=\ln(x_2)$$
it follows from this result that for the previous periods value function we have:
$$v(x_1)=\max_{x_2}\{\ln(x_1-x_2)+\beta v(x_2)\}$$
$$v(x_1)=\max_{x_2}\{\ln(x_1-x_2)+\beta \ln(x_2)\}$$

after taking the first order condition for this result and solving for $x_2$ we get:
$$x^* _2=\frac{\beta x_1}{1+\beta}$$

we know now our value function is properly defined as:

$$v(x_1)=\ln\left(x_1-\frac{\beta x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta} \right)$$

Simplifying:
$$v(x_1)=\ln\left(\frac{ x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta}\right) $$

For our $v(x_0)$ we follow the same procedure before

$$v(x_0)=\max_{x_1}\{\ln(x_0-x_1)+\beta v(x_1)\}$$
$$v(x_0)=\max_{x_1} \left \{\ln(x_0-x_1)+\beta \left[\ln\left(\frac{ x_1}{1+\beta}\right)+\beta \ln\left(\frac{\beta x_1}{1+\beta}\right) \right]\right\}$$

upon solving such an optimization problem we find:

$$x_1^*=\frac{(\beta+\beta^2)x_0}{1+\beta+\beta^2}$$

From this work that I've done above we can see that the policy functions change each period.

Given the solving above do time invariant policy functions exist?

No, they cannot by definition.
To derive a time-invariant policy function, you need to have an infinite horizon problem. This is because the structure of the solution remains the same, no matter when you look at it. Intuitively, this is because any sequence of periods from any time period to infinity looks the same. The technical conditions for this can be found in a number of textbooks (Adda and Cooper, Sargent, as well as Stokey/Lucas).

In your example, the number of periods left determine the solution structure.


What is the advantage and disadvantage to have a high value of USD for USA? What is the advantage and disadvantage to have a high value of USD for USA?

China and Japan always want to keep their currency low as oppose to USA.

When the USD is highly valued, it allows people from the home country, the United States in this case, to purchase goods relatively cheaply from abroad and put pressure on domestic firms to have low prices. From a consumption standpoint, this may be advantageous.

A high value USD may also make goods from the United States more expensive relative to goods from other countries, so the quantity demanded of U.S. goods abroad decreases.

Thus, the high value USD may increase the quantity demanded of foreign goods domestically and reduce the quantity demanded of U.S. produced goods abroad. Together, this can, but not necessarily, have long term consequences on the current account if imports are greater than exports over the long term.

The weak Chinese yuan renminbi allows China to export many goods cheaply and, thus, accumulate a positive current account and become richer relative to the rest of the world. In fact, China's monetary policy has been subject to criticism in the past for actively managing the renminbi's value.


Is GDP the market value of goods consumed or goods produced in a time period? Definitions across various resources state that GDP is the market vaue of all goods 'produced' in a certain time period. But while calculating GDP by expenditure method, consumers' spending is one of the components (GDP= consumer spending + Firm's investments + Government spending + net exports)
Suppose a manufacturer produced goods in Q1 and customers bought those goods in Q2 from an inventory, will Q1 GDP increase or Q2 increase due to this purchase?
Sorry for the rookie question

Goods produced.
In your example, a good is produced in Q1 but sold only in Q2. This adds to Q1 GDP (addition to inventories). For Q2 GDP, there's (i) a subtraction from inventories; and (ii) an addition to consumption—so (i) and (ii) cancel out and there's no net change to Q2 GDP.


Question about Equilibrium Price To find the equilibrium price I understand you set $$ Q^s = Q^d $$
and then solve for p. But as P is not indpendently labeled in $$Q^d$$ I am slightly confused how you would go about writing this equation? Thanks!

So you want to find $\partial p / \partial k$

By implicit differentiation what they mean is let $p=p(k)$ so p is a function of k.

Take your equilibrium condition $Q_s-Q_d=0$ and now differentiate with respect to k.


Differentiating through this integral in banking loans? Working through the math which gives the following profit function for a bank but I can't seem to solve this first order condition. The profit function is defined as follows,

\begin{equation}
\pi_t = p_{k,t-1} R^L_{k,t-1} L_{k,t-1} + (1-p_{k,t-1}) \frac{L_{k,t-1}}{\int_0^1 L_{k,t-1}dk} \tau \theta_{t-1} a_{t-1} - R^D_{t-1} L_{k,t-1} + \mu_t^B \bigg( \int_0^1 \bigg[\bigg(\frac{R_t^L + \eta \frac{1}{\theta_t}}{R^L_{k,t}+ \eta \frac{1}{\theta_{k,t}}} \bigg)^{\epsilon} x_t + \gamma^L s_{k,t-1} \bigg] dj - L_{k,t} \bigg)
\end{equation}

and the FOC they get for $L_{k,t}$,

\begin{equation}
\partial L_{k,t}: p_{k,t} R^L_{k,t} + (1-p_{k,t}) \frac{\tau \theta_t a_t}{\int_0^1 L_{k,t} dk} - R^D_t - \mu_B =0
\end{equation}

I can't seem to make sense of this differentiation of the integral. I have been looking at Leibniz integral rule all day. My understanding was that the following holds when the bounds of the integral are constants,

\begin{equation}
\frac{\partial}{\partial x} \bigg( \int_0^1 f(x) dx \bigg) = \frac{\partial}{\partial x} f(x) dx \Big|_0^1
\end{equation}

Using this rule I can't get what the result showen is. Am I incorrect or is this example incorrect?

Much appreciated.

In generic notation, we examine

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right)$$

Note that we take the derivative with respect to $f$ not with respect to $x$. Applying the standard rules we have

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right) = \frac{\int_0^1 f(x) dx-f\cdot \left[\frac{d}{df}\int_0^1 f(x) dx\right]}{\left(\int_0^1 f(x) dx\right)^2}$$

But

$$ \frac{d}{df}\int_0^1 f(x) dx = 0$$

because this definite integral, having specific values as integration limits, is a number, a constant, not a function. So we end up with

$$\frac {d}{df} \left(\frac {f(x)}{\int_0^1 f(x) dx}\right) = \frac {1}{\int_0^1 f(x) dx}$$

that validates the obtained f.o.c.

Leibniz rule for differentiating under the integral sign does not enter the picture here.


How does investment have a negative impact on the current account (trade balance)? 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”.

I understand how savings, tax revenues and government spending may impact current account, but I don't understand how investment would negatively impact current account, it doesn't seem to be right or even make sense. How does Google investing millions in the local economy contribute to the trade deficit?

Balance of payments is composed of:

the capital account (foreign assets and foreign liabilities)
and current account (includes exports and imports)

One way to define the current account is savings minus domestic investment, $S-I$. When $S<I$, a country's investing has to be financed by capital inflows, or investment from foreign countries. This is a deficit on the current account, making the country a net borrower that must seek matching capital inflows like foreign direct investment, that are a credit on the capital account (capital account surplus).

Whereas, when $S>I$, the excess savings are used to invest in foreign opportunities abroad, creating a current account surplus that makes the country a net lender, giving a capital account deficit.

In relation to the current account, a trade deficit is not identical but is a component of the current account. When there is a current account deficit, a trade deficit happens (imports exceed exports, where imports are not to be confused with investments $I$) because the country does not produce everything it needs and needs to borrow from overseas in order to pay for imports. From Greg Mankiw:

  when we write Y=C+I+G+NX, what is NX? The answer depends on how we define Y. If Y is Gross Domestic Product, then NX is the trade balance. If Y is Gross National Product, then NX is the current account.

Even though a trade deficit sounds bad, standard of living usually increases from a wider variety of imported goods and services, and prices are pushed downwards curbing inflation, but only for so long.


What is the difference between a power law exponent and the Pareto exponent? I use the poweRlaw package in R to fit a power law to my data. I am trying to figure out what is the value of the Pareto exponent.

Assume the probability mass function is defined by:

$$
p(x) = \frac{\alpha-1}{x_{min}} \left(\frac{x}{x_{min}} \right)^{-\alpha}
$$

and the complementary cumulative density function is defined by:

$$
P(x) = \int_x ^\infty p(x’) dx’ = \left(\frac{x}{x_{min}}\right)^{-\alpha + 1}
$$

Is the Pareto exponent $\alpha$ or $- \alpha + 1$ or  $\alpha - 1$?
In most literature, the CCDF is used to describe the income/wealth distribution, and $-(\alpha-1)$ is the slope of the CCDF on a log-log plot, so $\alpha-1$ seems the most intuitive. I'm pretty sure the R library poweRlaw returns $\alpha$ as defined above.
I am using Newman as a reference.

A random variable $X$ has a Pareto distribution with Pareto exponent
  $\theta$ if  $$\text{P}(X>x)=\begin{cases}
 \left(\frac{x}{x_m}\right)^\theta \quad \text{ if } x\geq x_{min}\\  \
 \ 1 \quad \quad \quad  \text{ if } x<x_{min} \end{cases}$$

In this case, the Pareto exponent is $\theta = \alpha - 1$. Remember that $P(X>x)=\int_x^\infty p(x')\mathrm{d}x'$.


Is there a folk theorem for repeated games on networks? Games on networks have been studied extensively, however, I was not able to find a folk theorem for games on networks. Is there one or can it be derived from an already existing folk theorem?

With games on networks I mean games in which the payoff of the stage game only depends on the actions of the direct neighbors in a network. Simple examples for that would be the the majority game (in which the payoff depends on the number of neighbors that play the same action as you are) or a prisoner's dilemma played with each neighbor.

Yes, there are folk theorems for games on networks, depending on information structure and possible communication. Here are some of the most relevant papers:

Ben-Porath, E., & Kahneman, M. (1996). Communication in repeated
games with private monitoring. Journal of Economic Theory, 70(2),
281-297. - public communication, private monitoring
Renault, J., & Tomala, T. (1998). Repeated proximity games.
International Journal of Game Theory, 27(4), 539-559. - complete information, imperfect monitoring
Tomala, T. (2011). Fault reporting in partially known networks and
folk theorems. Operations research, 59(3), 754-763. - partial knowledge of the network and restricted communication
Laclau, M. (2012). A folk theorem for repeated games played on a
network. Games and Economic Behavior, 76(2), 711-737. - private communication

Thanks @Ubiquitous for basically providing the answer.


What is the effect on society of me choosing to make my own shoes, food, or clothing? This question comes from someone who suggested brushing my teeth benefits the economy because it reduces demand for dentists, lowering the price. However, when I think more generally of making my own shoes/clothing my decrease in "clothing-services" leads to an increase in demand for the raw materials, in this case fabric. (I am ignoring the time it takes to learn how to make clothing.) Or suppose everyone simultaneously decided restaurants were unhealthy and started cooking at home; the restaurant industry would collapse, but would agriculture stay the same?

Is the economy the same, worse, or better when a consumer switches from buying a good to buying the raw materials for that good to manufacture it themselves?

The consequences "on society" of you changing any consumption/production choice you make is zero. Assuming that all the markets you interact with (shoes, clothes, etc) are competitive, your contribution to the demand and supply of a good/service is neglectible. Therefore, market equilibrium (price/quantity) will not change.

The story is of course different if everyone, or a "sizable" amount of people does change their decisions. Demand and supply of goods and services will be altered, just as the supply and demand of all the inputs used to produce such goods. Since a lot of goods and services are related in one way or another, either as complements or substitutes, the consequences could be felt accross the whole economy. How far this goes, depend on the number of people changing their actions, and the relation between goods, services and inputs. In short, your actions will have a general equilibrium effect.


Why do government jobs provide better benefits than industry jobs? I've noticed how government jobs in the US are known to have generous pensions, healthcare plans, vacation time, etc. as compared to the corresponding job in the private sector. (I haven't actually done any comparisons specifically, but it's something that many people say, so I am assuming that this belief is true.) Why is this the case?

Someone may answer with, "This is because government workers have unions, which push for better benefits." But why can't the industry employees form unions as well, and thus receive equivalent benefits?

I'll describe in turn five potential explainations:

Risk aversion interacting with the tax code
Is this true when we control for worker attributes?
Lack of incentives for cost control
Solving turnover, training and other HR problems
Political economy: unions and politicians

Risk aversion interacting with the tax code

I've heard it argued (but not empirically substantiated) that because government positions generally offer very low risk of job loss they attract risk adverse people.

All other things equal, relatively risk adverse people like relatively generous insurance and have a greater precautionary savings motive against negative shocks. Because retirement savings, health insurance, and to a lesser extent, life insurance have tax advantages when purchased through the employer, this risk aversion would interact with the tax advantage to induce a high level of retirement savings, health, and life insurance. Disability insurance is not tax advantaged, but employer-run plans do help solve some of the adverse selection problems associated with stand alone disability plans and so they too have an incentive to be employer provided and more generous in government positions for similar reasons.

Is this true when we control for worker attributes?

That all said, I'll echo @rocinante's point that among some sub-populations of workers, it isn't clear that government benefits are particularly extensive. My reading of the Are government employees overpaid? literature is that highly educated or skilled government workers do not typically appear overpaid (in a Mincer regression residual sense) relative to their private sector counterparts, and this includes measures of income and benefits. And elite financial and IT firms are often famous for their lavish benefits of meals, profit sharing, and low cost health care. However, part of the story there is that wages are higher in those fields and benefits are generally increasing in wages (if only because so are the tax advantages). It may be surprising to hear this but the average education and age of a public sector worker is substantially higher than in the private sector, explaining some of the higher wages and higher benefits.

Lack of incentives for cost control

So while it may well be true that, as @kbelder says, public sector employers have less of an incentive to hold down benefits, they have less of an incentive to hold down wages too, at least in conventional wisdom government employee benefits are more different from private sector benefits than the wages are, so that leaves us with a remaining puzzle beyond poor cost containment incentives.

Solving turnover, training and other HR problems

@rocinante's point about turnover is also a good one. We can think of the employee-employer relation as a sort of investment that pays returns while employment persists. Any situation where the initial employer investment (e.g., in training, background checks, customized equipment) is large relative to the immediate payout is one in which the employer is going to want to incentivise the employee to stick around until that investment is repaid. Pensions, which back-load compensation have a powerful incentivizing effect towards retaining staff under these circumstances.

Similarly, when positions require idiosyncratic human capital (extensive expertise and training not useful for other employers), this makes the employee strongly value continuing the employee relationship, and if government employers particularly require this sort of expertise (say maintaining ICBMs) then employment protection benefits may be another efficient way to attract staff without paying higher wages.

Political economy: unions and politicians

Finally, two more reasons from a political economy angle. First, unions may not just get better benefits because they have more power and influence but also because the people who run unions wish greater power and influence. To those ends, higher wages give them less than retirement and benefits programs which require wealth and staffing to be run by the union. Second, benefits, particularly defined benefit pensions, can be opaque in their true costs. Since politicians have to answer in elections to the public, HR strategies that please government employees (like high defined benefit pensions) but appear to be low cost to the public (again like high defined benefit pensions) therefore are relatively attractive to the politicians that administer the government.


list of math intense graduate level microeconomics books? List of math intense graduate level microeconomics books?

Except Reny's book, krep's books, varian's book and mas-collell's book

books from subfields are acceptable, however by math intense i mean most of the books pages contain some level of math

Please categorize the book

Mas-Collel is very popular. It is very math-heavy. Every idea and concept is done pretty rigorously and it pretty well covers everything. The book is a very slow read and is sometimes uneven. Very detailed in some sections and not as good in others. That said, in my experience, it is the graduate microeconomics bible. If you have a question, your answer is somewhere in there.


Financial integration and connection between savings and investment I'm having some difficulty to understand the following sentence:«(...) there's a tight connection between domestic savings and investments at the national level, but very little when we look at the subnational level. Limited spatial correlation is exactly what would be expected if the capital market was doing its job of allocating savings to the most productive investment projects - irrespective of where savings were generated.»

What they're saying is that if there's a high correlation, then there's no financial integration - as at the subnational level?

Any help would be appreciated.

Suppose that you cannot trade capital. I.e., you cannot ship savings from one place to another. In that case, if region A saves 10, region A will invest 10. If region B saves 1, it will invest 1. So, from that example, you can conclude that if markets for capital are closed, the correlation between savings and investment is 1. For completeness, now imagine the free flow of capital situation: region A saves 10, region B saves 1, but it is in region B where all the investment opportunities are, so the regions trade in capital: A invests all its savings in B, and investment in B is then 11, and investment in A is 0. In this second example I and S are therefore negatively correlated ...


Effect of a depreciation on foreign interest rate? According to the interest rate parity condition , if there is a depreciation of the domestic exchange rate, then the domestic interest rate will increase and the foreign interest rate will decrease. Am i Correct?

According to the interest rate parity condition , if there is a depreciation of the domestic exchange rate, then the domestic interest rate will increase and the foreign interest rate will decrease. Am i Correct?

This verbal formulation does not match what interest rate parity states. Note that there are two versions of interest rate parity: covered interest rate parity (an arbitrage condition), and uncovered interest rate parity (an equilibrium condition that appears questionable).

See this question for a description of interest rate parity: link to question I will not repeat the description of interest rate parity, as it is done in that answer.

In both versions of interest rate parity, what matters is expected depreciation, not a historical event of depreciation.

The statement that interest rates adjust in reaction to expected exchange rate movements is also questionable. Short-term rates are largely under control of central banks in floating currencies, and so a more plausible description is that expected currency movements conform to interest rate differentials.


How does the monopoly's deadweight loss affect market surplus and the economic pie? Source: p 312, Principles of Microeconomics (7 Ed, 2014) by N G Mankiw

  Welfare in a monopolized market, like all markets, includes the welfare of both
  consumers and producers. Whenever a consumer pays an extra dollar to a producer
  because of a monopoly price, the consumer is worse off by a dollar and the
  producer is better off by the same amount. $\color{red} {[1.]} $ This transfer from the consumers of the
  good to the owners of the monopoly does not affect the market’s total surplus—
  the sum of consumer and producer surplus. In other words, the monopoly profit
  itself represents not a reduction in the size of the economic pie but merely a bigger
  slice for producers and a smaller slice for consumers. Unless consumers are
  for some reason more deserving than producers—a normative judgment about
  equity that goes beyond the realm of economic efficiency—the monopoly profit is
  not a social problem.

  The problem in a monopolized market arises because the firm produces and
  sells a quantity of output below the level that maximizes total surplus. $\color{green} {[2.]  \text { The
deadweight loss measures how much the economic pie shrinks as a result. }} $ This
  inefficiency
  is connected to the monopoly’s high price: Consumers buy fewer units
  when the firm raises its price above marginal cost. But keep in mind that the profit
  earned on the units that continue to be sold is not the problem. The problem stems
  from the inefficiently low quantity of output. Put differently, if the high monopoly
  price did not discourage some consumers from buying the good, it would raise
  producer surplus by exactly the amount it reduced consumer surplus, leaving
  total
  surplus the same as could be achieved by a benevolent social planner.

On p 311, Mankiw discusses the deadweight loss (hereafter DWL) of a monopoly, which 2 above concerns. So what does 1 mean? It appears to contradict 2. DWL reduces the total market's surplus, and so the size of the economic pie.

Consider the following two examples:

There is a consumer who is willing to pay 50 dollars for a good. (Reservation price is 50.) There is a seller who is willing to sell for 30 dollars. (Marginal cost is constant 30.) If they make a deal at any price $p$ the total surplus they enjoy will be $(50 - p) + (p - 30) = 20.$ The individual surpluses need only be non-negative because otherwise one party would probably not consent to the deal. The price that is finally reached depends on their bargaining power. With a monopoly all the bargaining power is on the seller's side, so the price would be $p = 50$. Total surplus is still 20. So there's no loss.

However if there are two consumers, one with a reservation price of 50 and the other with 31 then at price 50 only one of them (the former) will buy the good. At price 30, both of them would buy the good and total surplus would be $(50 - 30) + (31 - 30) + (30 - 30) = 21$. But the monopoly's profit is 20 dollars if it sets $p = 50$, and 0 dollars if it sets $p = 30$. So it will choose the former. Because of uniform pricing, a good that would increase the total surplus is unsold, creating deadweight loss.


How can I model my problems with math? I want to ask a question about mathematical economics. When I read an article about economics, I see lots of mathematical equations. I can solve them without any help. But I can't create my own mathematical equations. For example, this guy created his own equation. How can I learn this skill? I know how to solve some advanced equations like differential equations. But I want to create my own problems' equations. Please give me some advice.

The best way how to learn how to create new models is to see how other people create new models and learn from that. If you are already good at math then you could learn this by observing how models are created from some graduate textbook focused on theory such as MWG Microeconomic Theory.
Seeing how models are derived will help you in two ways:

It will help you to see what is the thinking process of an author that motivates model development.

Most new models are created just by tweaking old ones. In order to do that you already need to have an excellent knowledge of the existing models.

Next, the point of models is to create a simplification of a reality which will allow you to think about problem analytically. So when you try to create new model you first need to find some new problem, that has not been modeled yet, and then try to capture it in mathematical form.
For example, let us take equation from that linked Math.SE question as an example and let us think about how it was derived. First we can observe that people in cities seem to be more productive, they usually earn more money even when controlled for difference in price levels, also commercial and industrial activity seems to be located at or around various cities. Cities are by definition places where many people live (each country might have different definition of what city is but usually it will be defined in terms of demography and mainly population), so how could we model such situation?
Well a natural and logical way is to assume that peoples productivity $P$ depends on both some innate productivity $P_0$ and then has also some component which depends to proximity of other people so we can add term that tells us how productivity varies with proximity to other people $\int_APf(r)dA$ which will then gives us the expression derived by user Daniel:
$$P=P_0+\int_APf(r)dA $$
Moreover, this can also serve to illustrate my previous point. Now that we learned about this model we can  also try to tweak it and in process come up with a new model. For example, what else does productivity depends on? I would guess that pollution will also be one thing that affects productivity. And in turn what does level of pollution depend on? Well historically again proximity to other people, cities are more polluted than villages. So based on this observation we can come up with 'new' equation:
$$P=P_0+\int_A Pf(r) - Pg(r)dA $$
where $-Pg(\cdot)$ will be term that captures how productivity is negatively affected by pollution which will again depend on proximity to other people. So we created new model from the old one (although I am not sure if this model already does exist or not in the literature, perhaps we just rediscovered some already existing model but for didactic purposes this serves as an example).
So in conclusion in order to learn how to make new models you should observe how other people create new models. Then you should find problem you are interested in and try to translate the model into math like in example above. Also this can usually be done just by tweaking the old models, most theoretical research does not start from scratch but builds on some preexisting work.


How can I prove $U(x) = [_1_1^+_2_2^]^{(1/)}$ is equal to Cobb-douglas Utility function when $\rightarrow0$ This is the question, I have problem with part b, I don't know what function should I use to reach the result
thanks in forward

Hint: find $\ln(U(x))$ and then use L'Hopital's rule to find the limit as $\rho\rightarrow0$.


Demand Curve Equation Explanation I've been reading (trying to brush up on my economics knowledge) and came across the equation: $q_D = a + b(P)$.  The variable $a$ is explained as "factors other than price affecting demand."

I read up on non-price determinants of demand (e.g. income changes, preference changes, etc); but, what I've not been able to satisfactorily understand is how $a$ is calculated from those determinants.

So in the equation: $q_D = 6 - 0.5(P)$, what does $6$ represent and how was it determined?

In this demand curve, $a=6$ means that the autonomous demand for this good is of 6 units. That is, if the market price of that good was $0$, then people would demand 6 units.

It is reasonable to think that a positive income shock, or a shift in preferences making this good more desirable would make $a$ higher. I don't think there's any deeper meaning here, just that a downward sloping demand curve requires a positive intercept or else quantity demanded would be negative at any price $P$.


How to define best value? I'm trying to define which hotel offers the best value.

Let's say we have two hotels - A and B.

For A, you pay $10 a night and the rating for the hotel is 9.8.

For B, you pay $8 a night and the rating for the hotel is 9.6.

So for A, you pay $2 more and get one with a higher rating of 0.2. But is it the better value?

How would you solve this?

Thanks!

This depends on the person you are asking.

Suppose an individual's utility function only depends on hotel rating scores and money (which is used to purchase other things). Then it is completely consistent for two individuals to prefer A=(9.8,m-10) to B=(9.6,m-8) (and vice versa). Thus, there is no value metric for all individuals. However, if you focus on one individual, you could ask how much cheaper/costlier you would need to make each hotel such that the individual would be exactly indifferent to A. This idea is captured by the concepts of equivalent and compensating variation. However, for this you would need to know a lot about each consumer.

A simpler measure of value (but less meaningful to each individual) would be to measure the proportion $p(A \succsim B)$ of individuals which would prefer A over B. To each individual, this number would of course be meaningless. Notice also that the numbers obtained need not be consistent across hotels. Indeed, $p(A\succsim B)> .5$ and $p(B \succsim C)>.5$ does not imply $p(A \succsim C)>.5$. This follows from the cyclicality of pairwise majority voting. Arrow's impossibility theorem tells us that under some reasonable assumptions, there exists no aggregation procedure of ordinal preferences. Thus, unless you are willing to make stronger assumptions on preferences (than most economists are willing to make), it will be impossible to measure the value of hotels.


How can difference equations with an infinite summation be represented in matrix form? I have derived the microeconomic foundations of a dsge model and I've obtained the IS and NKPC. I would like to represent them in matrix form to study the system. However the problem is that both equation have in infinite summation. How can i proceed? Do I need to approximate, if possible, the infinite sum to a finite term?
\begin{equation}
 \tilde \pi_t = E_t \sum^{\infty}_{T=t} (A)^{T-t} \left[ B  x_{T+1} + C \tilde \pi_{T+1} \right].
 \end{equation}
\begin{equation}
{x}_{t}=\tilde{E}_{t} \sum_{T=t}^{\infty} D^{T-t}
 \left[F{x}_{T+1}-\sigma\left(\tilde{\imath}_{T}-\tilde{\pi}_{T+1}\right) \right]
\end{equation}

Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (If you're not familiar with the lag operator I suggest you to read the first section of the page linked above.):
We have the optimal price level equation (equivalent to your NKPC but less developed):
$$\tilde{P_{t}}=\theta\tilde{P_{t-1}}+(1-\theta)(1-\beta\theta)E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$$
Now, we can see that the difference between one period of time might be represented by:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k(\tilde{MC}_{t+k}-\beta\theta\tilde{MC}_{t+k+1})$$
In terms of lag operator we could factor $\tilde{MC_t}$:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k\tilde{MC_t}(1-\beta\theta L^{-1})$$
What this tells us is that we can use the same series lagged by $-1$ (same as forward by 1) to eliminate future periods from the original one. Let $S_t=E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$, then (omitting expectations operator):
$$S_t-S_{t+1}=
$$
\begin{bmatrix}
\tilde{MC}_{t}+(\beta\theta)\tilde{MC}_{t+1}+(\beta\theta)^2\tilde{MC}_{t+2}+(\beta\theta)^3\tilde{MC}_{t+3}+...\\
-\beta\theta\tilde{MC}_{t+1}-(\beta\theta)^2\tilde{MC}_{t+2}-(\beta\theta)^3\tilde{MC}_{t+3}-(\beta\theta)^4\tilde{MC}_{t+4}-...
\end{bmatrix}
$$=\tilde{MC}_t$$
Look that we managed to reduce the infinite sum into a single expression, now to replicate this in our prices equation, we need to get something similar to $S_t-S_{t+1}$ from $S_t$ only, taking advantage of lags because if we just sum $-S_{t+1}$ in both sides of the equation we would've done nothing. Then you can check that:
$$S_t-S_{t+1}=(1-\beta\theta L^{-1})S_t=\tilde{MC}_t$$
With that in mind, we just need to multiply both sides of our prices equation by $(1-\beta\theta L^{-1})$, and develop the corresponding algebra for arriving to:
$$\tilde{P_t}-\beta\theta E_t\tilde{P_{t+1}}=(1-\theta)(1-\beta\theta)\tilde{MC}_t-\theta^2\beta\tilde{P}_t+\theta\tilde{P}_{t-1}$$
Which allows us to make better analysis by getting rid of the series, and therefore being able to write in terms of matrices and vectors that are more "readable". I'm pretty sure you can do this same process with your equation that is in term of inflation (instead of prices). Hope this works.


Why did the inverse correlation of stocks and commodities stopped working in 2008? Usually, the commodities asset class is considered being inversely correlated with other asset classes, due to its nature of an input for manufacturing (if the prices of commodities go down, the manufacturers can produce for less, and have a larger profit, and vice versa).

Therefore, why during the stock market crash of 2008, the commodities also went down, by far? What happened macro-economically to stop this inverse correlation from working?

The problem in your question is the "and vice versa".

To give a very simplistic view, the problem in the 2008 crisis was not on production side - companies had enough to produce, but no-one to sell to. Given this, excess commodities found no buyers, even as the price lowers. The inverse correlation of stocks and commodities supposes full capital use, which is not the case during the 2008 crisis.

Globally, the correlation between commodities and other stocks is around 0, though during full capital use periods it does tend to be negative, there is usually a positive correlation during crisis.


What is Return to Consumption Asset In a paper I was reading, a budget constraint for the Euler equation for the consumption asset was described as:-

$$
W_{t+1} = (W_t - C_t) R_{c, t+1}
$$

Here $ R_{c, t+1}$ is the 'return to the consumption asset'. What does this mean?

Paper: Volatility, the Macroeconomics, and Asset Prices

On the previous page (introducing equation 2.2) the authors state that "consumption asset" means "wealth portfolio", so $R_{c,t+1}$ is the return to the wealth portfolio, which makes sense from the equation you quote.

  the log return to the consumption asset (wealth portfolio), $r_{c,t+1}$.


Counter example for strong and weak Pareto optimality We know that strong Pareto efficient is equivalent to weak Pareto efficient if we have continuous and strongly monotone preferences.
Please give me an example which we don’t have continuous and monotone preferences, so this statement doesn’t hold.
I think that the statement holds for quasi linear utility function.
But I cannot find the counter example.

There are two goods and two consumers, no production. The aggregate endowment is strictly positive. Consumer 1 cares only about the amount of good 1 they consume, with more being better. Similarly for consumer 2 and good 2. Preferences here are continuous and monotone but not strictly monotone. The allocation in which consumer 1 consumes the entire aggregate endowment is weakly Pareto optimal but not strongly so.


How new money flow work in country? I want to find out how money flow works in the country.  Imagine I leave in country X. Country has 100$ in general, 50 dollars have a government, 50 dollars have peoples. My part is 10 dollars, this means that other peoples have 40 dollars,  I am an entrepreneur,I've made goods on 10 dollars, peoples, buy it, they have spent , 20 dollars for my goods, they have less money on 20 , Then I pay tax from this amount , for example 5 dollars, let's do the math

General - 100$
government - 50$
me - 10$
peoples - 40$

I've earned 20$ , peoples spend 20 dollars, I've paid 5 dollars tax

general - 100$
government = 55$
me - 25 dollars
peoples = 20 dollars

country pay salary to peoples to return this 20$ to peoples back

general  = 100 dollars
government = 35 dollars
me = 25 dollars
people = 40$

This mean that governent (budget) , have less money, and even If I will pay in future some tax from next iteration , it will not cover 20$ dollars which they paid to people, how then government compensate it ? Or I've misunderstood the process ?

Your example is faulty

You have 10 and spent it on Inputs (Let's assume just wages)
Now you have 0 and people have 50 (40 + 10 wages you paid them)
People buy your goods for 20. You have 20 and people have 30
You pay a tax of 5. You have 15, people have 30, govt has 55
Govt pays salaries (Only to its employees). You have 15, people have 50, Govt has 35

In this case, the govt ended up with less money than it started with. This is called a "Fiscal Deficit" when the govt spends more than it earns. A deficit can be funded in many ways
 - loan from international institutions
 - Monetary financing (Printing Money)

But a deficit is a choice the govt makes. It is by no means inevitable if the govt has budgetary discipline. It just needs to keep its expenses controlled w.r.t its income


Derivation of demand for intermediate goods in DSGE model I'm self-studying Herbst-Schorfheide book on DSGE estimation and having troubles replicating their steps in the derivation of demand for final goods firms (Ungated model is here, page 3: http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=1361&context=soe_research). Their final goods producers maximize profit function:

$\Pi_t = P_t (\int_0^1 Y_t(j)^{1-\nu} dj)^\frac{1}{1-\nu} - \int_0^1 P_t (j)  Y_t(j) dj$

and get the demand:

$Y_t(j) = (\frac{P_t(j)}{P_t})^{-\frac{1}{\nu}} Y_t$

My trouble is that I cannot understand derivation steps: how FOC looks like and why? Help is much appreciated!

So I've seen this sort of thing where you basically differentiate with respect to each Y_j so the integral drops out (seen this in moral hazard chapter from
MWG and Romer's endogenous growth model).

The foc becomes:

$P_t(Y_t(j)^{(1-v)})^{v/(1-v)}Y^{-v}-P_t(j)=0$

This gets the result.


Utility Possibility Frontier with two consumers and 3 commodities I am trying to solve the following problem:

Consider a pure exchange economy with three commodities and two households with individual endowments:

$$e_1=(1,2,3), e_2=(3,2,1),$$

respectively, and utility functions

$$u_1(x_{11},x_{12},x_{13})=x_{11}+2x_{12}+3x_{13}$$ and $$u_2(x_{21},x_{22},x_{23})=3x_{21}+2x_{22}+x_{23}$$
respectively.

Which of the following is the Utility Possibility Frontier?
Options:

A. $\displaystyle \max\left\{ u_1+\frac{u_2}{2}, u_1 + u_2, \frac{u_1}{2} +u_2 \right\}  = 32$

B. $ \displaystyle \max\left\{ u_1+\frac{u_2}{3}, \frac{3}{4} u_1 + \frac{3}{4} u_2, \frac{u_1}{3} +u_2 \right\}  = 24$

C. $\displaystyle \max\left\{ u_1+\frac{u_2}{3}, u_1 + u_2, \frac{u_1}{3} +u_2 \right\}  = 24$

D. None of the Above

My attempt:

I tried calculating the Pareto optimal allocations by taking two commodities at a time. I found out that between good $1$ and good $2$, the Pareto efficient allocations are where:
$$x_{21}= 4 \quad \text{or} \quad x_{12} = 4$$

Similarly, between good $2$ and good $3$, I find that:

$$x_{22}= 4 \quad \text{or} \quad x_{13} = 4$$
are  Pareto optimal.

Similarly, between good $1$ and good $3$, I find that:

$$x_{21}= 4 \quad \text{or} \quad x_{13} = 4$$
are  Pareto optimal.

Also I observe that the preferences among good $1$, good $2$ and good $3$ of individual 1 are:

$$ \text{good } 3 > \text{good } 2 > \text{good } 1 $$

And those of individual $2$ are:

$$ \text{good } 1 > \text{good } 2 > \text{good } 3 $$

Now I consider the following allocations:

$$((x_{11},x_{12},4), (x_{21},x_{22},0))$$

And the fact that
$$x_{11} + x_{21} = 4$$
$$x_{12} + x_{22} = 4$$

I find that:

$$u_1 = x_{11}+2x_{12}+3x_{13} = x_{11}+2x_{12} + 12$$

$$u_2 = 3x_{21}+2x_{22}+x_{23} = 3x_{21}+2x_{22}$$

$$\implies u_1 + u_2 /3 = 16 + 2x_{12} + 2/3 x_{22}$$
$$\implies u_1 + u_2 /3 = 16 + 2/3(3x_{12} + x_{22})$$
$$\implies u_1 + u_2 /3 = 16 + 2/3(2x_{12} + 4)$$

$$\implies u_1 + u_2 /3 \le 24 $$
because $x_{12} \le 4$.

Similarly I consider the allocation:

$$((0,x_{12},x_{13}), (4,x_{22}, x_{23}))$$

And find that:

$$ u_1 /3 + u_2 \le 24 $$

I am stuck here. I am not able to see how I can take it from to the options given. Please drop hints as to how I can.

Thank you for reading this.

The utility possibility frontier (UPF) plots the maximum total combination of utilities that can be achieved, given the preferences and total resources. To fix ideas, let's suppose we are plotting $u_1$ in the $y$ coordinate and $u_2$ in the $x$ coordinate. The easiest way to find the UPF is to start with a single agent, say agent 1 and give all the resources to him, so allocation $(4,4,4)$ his utility will be 24, so that point, $(0,24)$, in the plane is definitely part of the UPF (by the way this calculation already precludes option $a$). Now, think that you are going to decrease agent $1$'s utility in order to increase agent $2$'s utility. Since we are characterizing the frontier, you want to think what is the best way to move some resources from player $1$ to player $2$. After some thought, it is obvious that the best is to take away some of good $1$ from player $1$ since he only loses $1$ util while the other player gains $3$ utils. Therefore, the point $(3,23)$ is also part of the $UPF$ note that the line connecting this two points is $u_1+\frac{u_2}{3}=24$, (unfortunately, this realization does not narrow the possible answers, since $b$ and $c$ share this equation as its first argument).

Now, this point was found by moving of the good 1 from agent 1 to agent 2, however, there are only 4 such units, so after agent 2 has all units of good 1, in order to keep increasing her utility, the most efficient transfer is to take away from agent 1 some of the units of good 2 he enjoys. Here the loses from agent 1 equal the gains from agent 2, so we should have this defined by a line with equal coefficients: $\alpha u_1+\alpha u_2=C$ both options have this format, so we only need to make sure which one is correct given that the constant is 24.

Note that the point where $x_1=(0,4,4)$ and $x_2=(4,0,0)$ is part of this line and if we compute $u_1(0,4,4)+u_2(4,0,0)=32\neq 24$, but $\frac34 u_1(0,4,4)+\frac 34u_2(4,0,0)=24$, as desired. You can conclude that the answer is $b$.

The $\max$ operator ensures that the graph has the correct "kink" points when the type of good you are transferring from one agent to the other changes.


Which of the following explains that the price increase of oil cause the price of natural gas to increase? A) Oil is inferior good while natural gas is normal good.

B) Both of them are normal good.

C)They are substitute goods and the higher price for oil increased the demand for natural gas.

D)They are substitute goods and the higher price for oil decreased the supply for natural gas.

In my personal opinion, the answer should be C. We can eliminate A and B first as there's no indication of income change. For D think since substitute good refers to the customer side so I think we should also remove D. Is there a better explanation ?

I think your answer is OK.

A doesn't connect prices in any way. In a two-goods economy and given that wealth doesn't change... you may give it some more thought but you need to add extra assumptions.
B tells us nothing on why such change would take effect. It may just as well be that oil is more valued.
C is the right answer
You need more assumptions for D, e.g. why the oil price increased. At least in the really short term, though, supply isn't a factor.


Value flow in economy - redefinition credit and money? There are voices that there are already too much debt in the world (government, personal, government debt) and that can be the cause of the crisis. From the other hand the the only solution offered by the governments and the central banks to the possible (forthcoming or not) crisis is to create more debt. So - if consumers take loans and spend them, then manufacturers can produce and the GDP grows. But if consumers are prudent and save money, then manufacturers can not sell produce and can not continue operations, GDP falls and recession emerges.

If this vignette is true, then how sustainable growth can happen? Or is this vignette the example that there always should be boom/bust cycles? Or does my vignette contain some paradox, some oversimplification, some mistake? My question is - is my vignette correct or does it contain mistakes and what are those mistakes?

One can try to solve the paradox in vignette by saying that consumers create value as well (labor) and that they can spend the money earned with their labor without taking loans and so the sustainable GDP can exist. If that is so, then why governments or other entities are trying to boost credit instead of boosting labor? Some can say that "boosting" labor can be quite cruel - e.g. the governments can put into force laws that pressure citizens to accept any work however bad paid it is (i.e. if there are no betters jobs available in the economy). But from the other side one can argue that government can increase the skill level, the creativity level, the entrepreneurial spirit of the citizens and so they can create new businesses, new ideas, be worthy employees for businesses that pay high wages. So - such boosting of labor can be beneficial for the all (citizens, businesses, budgets; such boosting does not require increase of the debt level). Why such beneficial boosting is not happening? Why central banks does not suggest such boosting and why everything is talking about increase quantitative easing and debts?

Your question is very broad so it is hard to answer but I will try to do it

Can there be sustainable growth without debt? Yes! Most models of economic growth don't even have debt in them. Most economists believe that growth comes form increase in factors of production like capital accumulation and from increases in avaiable technology. Debt can also play role in promoting growth by extending credit to entrepreneurs who are poor, but in most macro models it does not play role in long term growth.
The vignette that you described holds only in the short run. In the short run it is possible to boost economic activity by 'supporting' loans by lowering interest rate, and in short run savings can exaggerate recession, but this does not hold in the long run (at least not in standard textbook macro models). Regardless, you should not confuse the short run dynamics with long run economic growth. Unless there is some hysteresis effect, which is still controversial, business cycle does not necessary affect the economic growth it just happens around it. Even in presence of hysteresis economic growth would still occur but business cycle could made it permanently lower than it could have been.
Value does not come from labor - that is an archaic classical economics theory that was since disproved countless times and does not square well with empirical observations. Value is subjective. This being said boosting labor supply definitely increases GDP because going back to the point 1. it increases the avaiable stock of factors of production, although unless you can do it consistently it will lead to level increase in GDP not really to sustainable growth.
Every single advanced country I can think of has some programs that are trying to improve labor force participation. For example, many countries have earned tax credit, unemployment offices that help people find job, vocational training programs, lifelong learning programs etc. So the "boosting" of labor supply is definitely happening in most countries - in fact the only exception I can think about are some defunct states like Somalia which are in state of anarchy. Of course some states do this better than others, but that's completely different topic.
It is not a job of central bank to conduct or even suggest labor market policies. Central banks job is to conduct monetary policy and bank/financial supervision in accordance with its mandate. This mandate differs across the world but most central banks have mandate that is similar either to the Fed's or ECB's mandate. The Fed has dual mandate of price stability and full employment, but full employment here means of resources in the economy not just labor and its done through the monetary policy. ECB has only single mandate of price stability so it is not its job to even care about labor supply, except to the extend it can have some influence on price stability. In short central banks talk about quantitative easing and debts because that is their job. It is simply not in their competence to conduct some policies on labor market even if they would want to - that is job of labor department or ministry of finance/economics depending on what country we are talking about.


Do countries have to run on deficits? What is their possible economic future? Do countries have to run on deficits? Is their future to behave and act like big companies that produce goods and services?

One of the main reasons for the general tendency to run deficits is political economy. Governments are elected for a limited period and try to stay in power by benefiting their electorate. An alternative explanation is that high debts reduce the scope for the subsequent government. So there are good reasons to impose some restrictions to the deficit into the constitution.

I find the comparison to big companies misplaced. Governments fulfil different purposes than companies. Many of their 'products' are public goods (public infrastructure, education, defense, social security) which cannot be provided by private actors due to external effects. A strictly profit-oriented view of public services therefore falls short of what governments are supposed to do (which does not say there is probably lots scope for efficency-enhancing in most administrations). As a sidenote, running huge deficits is not uncommon to big companies, especially in digital services. As long as creditors are convinced these debts will be paid, companies can be run on losses for quite a while.


The underlying force for creating specialized areas in the cities of developing economies Big cities in developing countries often have streets or quarters that specialize in selling a particular type of good like: books, musical instruments; or giving a specialized service like: haircuts, selling coffee. A prominent example would be the capitals like Hanoi or Bangkok.

This almost never happens in cities in developed countries. Barcelona, Spain for example has its services, shops and even supermarkets distributed uniformly along its blocks.

How to explain it in simple terms?
Is there any economic model that could effectively describe and explain the phenomenon?

This is an interesting observation that has produced a lot of theoretical and empirical research in industrial organization. In fact this phenomenon also occurs in developed economies, for example with gas stations. It is also not limited to location, the broader questions being why competitors in many markets tend to produce very similar products when they could also choose to differentiate themselves more strongly. A special case of the question is also often applied to politics, where the question is why different parties tend to be very similar to each other (most of the time).

Most simple explanations are based on an old and simple game theoretic model by Harold Hotelling called the Hotelling model or linear city model. Applied to politics, this model is behind the predictions of the median voter theorem.

You can find a very good explanation of the reasoning behind the model in the links below:

http://mindyourdecisions.com/blog/2008/03/25/game-theory-tuesdays-hotelling%E2%80%99s-game-or-why-gas-stations-have-competitors-nearby/
https://en.wikipedia.org/wiki/Location_model#Hotelling.27s_Location_Model
https://en.wikipedia.org/wiki/Hotelling's_law


Why are the concepts of short-run and long-run equilibrium exclusive to aggregate supply and demand and macroeconomics? Shouldn’t there also be a concept of short vs. long-run equilibrium in microeconomics? After all, short-run equilibrium describes the state of the economy when wages and prices are still adjusting and when worker misperceptions are affecting quantity-supplied of labor; and long-run equilibrium is the state of the economy when worker misperceptions have been corrected, and wages and prices have adjusted to their final levels.

Doesn’t this happen in microeconomic situations? If not, why not?

The concepts of short-run and long-run equilibria certainly feature in microeconomics.  Indeed, they have done so for over a century: see Marshall's Principles of Economics (1890).


What happened to the Zimbabwe cash dollar after the introduction of Dollarization? Let's say, before dollarization, I owned 1 million Zimbabwean dollars in cash and the next day dollarization went into effect.
So in dollarization, the current Zimbabwean dollars will be invalid and only USD will work in the economy. So what happened to my 1 million Zimbabwean dollars that I had in cash.
Did the banks exchange it for USD or did something else happened?

Normally in order to full dollarization to occur there will be some conversion rate at which debt denominated in old currency is converted into new one. For example, according to the World Bank definition of full dollarization:

All government and private debt under full dollarization is denominated in dollars, and both public and private accounts must be converted to dollars. To make the conversion, countries must set the rate at which old debts, contracts, and financial assets will be converted to dollars.

However, I suppose technically the rate 1 Zimbabwean dollar =  0 U.S. dollar would be possible. This seems to be the case according to Mpofu (2015) [emphasis is mine]:

Zimbabwe is viewed as a dollarized
economy given the dominance of the currency among
the other currencies used in the country and the fact
that the Zimbabwean Government conducts all its
business using the US dollar. In the same month that
the multi-currency regime was introduced in 2009,
the Zimbabwean government suspended the use of
the Zimbabwe dollar as legal tender and decreed that
all wages, prices of all goods and services and
transactions be done in U.S. dollars. They further
decreed that all Zimbabwe dollars be removed from
circulation with no compensation provided to
currency holders, a decision that was viewed as
indicating that the currency had no store of value.

This suggest that in the case of Zimbabwe simply the value of Zimbabwean dollar vis-a-vis US dollar was set to zero. However, I briefly looked over several papers discussing this episode (Noko (2011)]3,  Kairiza (2009), Kramarenko, Engstrom, Verdier,Fernandez, Oppers,  Hughes, ... & Coats,(2010), because this question caught my interest, but none of them seems to explicitly state what happened to private debt. This issue aside the sources mention that US dollars were already widely circulating in the Zimbabwean economy even without government sanction, so my guess is that most likely already most debts were denominated in US dollars already.
PS: An interesting tidbit is that actually Zimbabwe did not just dollarize they adopted multicurrency regime where besides dollar, the rand, euro, pound, metical, and kwacha also became official tenders but dollar was without doubt the most prominent one.


CES preferences intuition I asked this question on math.stackexchange but deleted it from there and brought it here.

I had a question about Constant Elasticity of substitution type preferences of the form: $$U=\int_{0}^{1}(c(\omega)^{\rho}d\omega)^{\frac{1}{\rho}}$$
 where the paramter $\rho$
  governs the degree of substitutability between goods. Here, $c(\omega)$
  represents consumption of good $\omega$
  which exists on the unit interval. As such, this type of preference specification aggregates over consumption of different goods.

My question is as follows. I have always thought of integrals of the form: $$I=\int f(x)dx$$
  as approximating sums of areas of infinitesimally small rectangles (in terms of their base) and heights being determined by $f(x)$
 . In the case of the example above, what really is $c(\omega)$
 ? Is it a function?

It is my impression that the correct expression for "Dixit-Stiglitz" preferences is

$$U=\left(\int_{0}^{1}c(\omega)^{\rho}d\omega\right)^{\frac{1}{\rho}}$$

which then can be seen as a continuous incarnation (in [0,1]) of, say

$$\left (\sum_{i=1}^na_i\omega_i^{\rho}\right)^{\frac{1}{\rho}}$$

with $c(\omega_i) = a_i^{1/\rho}\omega_i$.

In other words, a definite (Riemann) integral is indeed conceived as a sum of infinitesimally small rectangles, but it can also be seen as the continuous incarnation of a sum.

A formal link between an integral and a sum is provided by the Euler-MacLaurin formula.


How can you find all parameter estimates are statistically significant but without performing any sophisticated calculations? I have a question that contains a list of parameter estimates and their related standard errors. It then goes on to ask that I explain how I know all parameter estimates are statistically significant at a 5% significance level, but without performing any 'sophisticated calculations'. Does this mean I have to compare the standard errors to the estimators or do I find their t-values?

Normally an exam question like that can be solved by simply calculating the $t$-statistics which can be simply done as:
$$t_{\hat{\beta}} = \frac{\hat{\beta}-\beta_0}{se(\hat{\beta})}$$
where $\hat{\beta}$ is the estimate, $\beta_0$ is the assumed value of beta under null (usually 0) and $se(\hat{\beta})$ standard error of the coefficient and then compare it to $5\%$ critical value (for large $n$ and two sided hypothesis approximately $1.96$).
However, you should ask your teacher/supervisor to clarify what is considered 'sophisticated calculation' - some might consider even subtraction and division sophisticated. An approximate rule of thumb that requires even less calculation is to see if the coefficient $\hat{\beta}$ (and assuming null hypothesis assumes $\beta_0=0$) is twice as large as the standard error.


Are goods that are wasted counted into the GDP of a nation? If x apples are produced in an economy, and only x - y are consumed (y apples get rotten at different stores in the country), is the value of these wasted apples considered as a part of GDP?

It may depend on when they are wasted:

yes if a consumer purchases them, as it does not matter whether they are eaten or thrown away rotten from the consumer's cupboard (except on the consumer's waistline and perhaps therefore on other purchases);
no if a producer fails to sell them and writes them off as they rot in the producer's stores, as they have a value of zero
probably not if a retailer buys them from the producer but writes them off as they rot in the shop, as they count towards GDP from the producer's accounts but negatively towards from the retailer's accounts (either as negative profit in the GDP income calculation, or as consumption of intermediate goods in the GDP production calculation, or simply not counted in the GDP expenditure calculation)


Is Ireland setting itself up for a debt crisis? I'm just curious about some things I read in a banking report. It says more than 20% total bank loans in Ireland are Non-performing loans which are likely to default.

  "With the NPL ratio for households at 26% for non-financial
   corporations - 44% for SMEs - and 44% for commercial real estate."

Then it goes on to say that Credit to non-financial corporations in 2015, was 267% of GDP. The current value of the bad loans is around $50 billion.

Is this a bad situation, or do the numbers make it look worse than it actually is?

Your figures seem to come from a European Central Bank analysis published in September 2016 for the position at the end of 2015

Different people get different numbers, but the order of magnitude is probably credible

But this sort of number has been coming down compared with previous years.  For example IMF publishes the following graph


Equality of ordinal vs. cardinal utility under transformations - who made the distinction first? In the context of preferences among items or choices under certainty, we can consider ordinal value function (viz., utility functions) to be equivalent under monotonously increasing transformations $u(x)\equiv F(u(x))$ for strictly increasing function $F$, but cardinal value functions only under linear transformations, i.e., $u(x)\equiv a u(x)+b$ for positive $a$ and constant $b$.

For adding a reference to an article in the philosophy of economics, I'd like to know who showed this the first time and where? Von Neumann and Morgenstern (1947)? Debreu (1960)? Or is this just "folklore"?

I'm aware of the history of utility in economics, the cardinal approaches of 19th century and the switch to ordinal representations by Pareto, Samuelson, etc. But I can't find a good "first" reference for the above distinction.

It appears to be the Theory of Games and Economic Behavior (1944) by John von Neumann & Oskar Morgenstern. I have the 1953 edition which is counted as "3d", but by reading the included introductions to the 2nd and to the 3d editions, it appears that nothing of substance has been changed in chapter 3, where I locate the issue of equivalence up to linear transformations.

In sub-chapter 3.4 the authors discuss in general the issue of equivalence under transformations for any system of quantities, mainly from physics. They end this sub-chapter by writing (bold my emphasis)

  "One may take the attitude that the only "natural" datum in this domain (utility) is the relation "greater," i.e. the concept of
  preference. In this case utilities are numerical up to a monotone
  transformation. This is, indeed, the generally accepted standpoint in
  economic literature, best expressed in the technique of indifference
  curves. To narrow the system of transformations it would be
  necessary to discover further "natural" operations or relations in the
  domain of utility. Thus it was pointed out by Pareto (in V. Pareto, Manuel d'Economie Politique, Paris, 1907, p. 264) that an equality
  relation for utility differences would suffice; in our terminology it
  would reduce the transformation system to the linear
  transformations. However, since it does not seem that this relation is
  really a "natural" one—i.e. one which can be interpreted by
  reproducible observations—the suggestion does not achieve the
  purpose."

In sub-chapter 3.5 they start by writing

  "The failure of one particular device need not exclude the possibility  of achieving the same end by another device. Our
  contention is that the  domain of utility contains a "natural"
  operation which narrows the system  of transformations to precisely
  the same extent as the other device would  have done(*). This is the
  combination of two utilities with two given  alternative probabilities
  $a, 1 — a, (0 < a < 1)$ as described in 3.3.2. The  process is so
  similar to the formation of centers of gravity mentioned in
  3.4.3. that it may be advantageous to use the same terminology."

(*) Note: The "failure of one particular device" refers to the suggestion of Pareto mentioned in the previous passage. So what they are saying here, is that, apart from ordinal preference, there is another "natural operation" related to utility (but not the one suggested by Pareto), that will nevertheless achieve the same purpose as Pareto's suggestion, i.e. to narrow the admissible transformations to linear ones only.

And which is this "natural operation"?

  "...The combination of two utilities with two given  alternative probabilities $a, 1 — a, (0 < a < 1)$"

which ushers us into the world of expected utility.

The details come next, together with Appendix "The Axiomatic Treatment of Utility" at the end of the book.

PS: By rejecting Pareto's suggestion, the authors make clear that expected utility is not cardinal utility, since what they reject is the existence of "an equality relation between utility differences". This rejection preserves fundamentally the ordinal nature of expected utility.


Why are oil prices increasing in India despite countervailing trends in the international market? Crude oil price is decreasing in the international market. But in India, the price of petrol and diesel is gradually increasing. Why?

To repeat EnergyNumbers's comment as an answer to this question, the following chart from the St Louis Federal Reserve shows the increase in the crude oil price in US Dollars in the first half of 2016.

Other factors also affect local diesel and gasoline prices and timings, but this increase will have contributed to the local change.


Connection between low demand for US Treasuries and expectation of inflation The expectations for inflation and fear of accelerated rate hikes on the part of the Fed seem to be driving the stock market correction in the US this February, 2018, and as a result of the jobs report on Friday, February 2.

As a measure of inflation, the 10 year Treasury notes yields are inching higher, close now to 2.9 percent.

  I want to ask what is the connection between inflation expectations and the "tepid demand" for Treasuries in the most recent auctions (10 year notes).

Presumably yields at the auctions are perfectly fine-tuned to meet supply and demand. Therefore, if the issue was simply an expectation of higher yields in the near future, this would have been factored in at the time of the auctions.

At this point the fiscal hole of $1 trillion created by the recently enacted tax cuts is already known. Presumably, economists still see US Treasuries as risk free. And there is the "flight-to-safety" effect, which would anticipate investors fleeing the stock market to seek refuge in US Treasuries, and is seemingly absent here...

  So what is behind the "tepid demand"? Is it in some way connected to the main issue of inflation concerns at a technical level, or is it a parallel topic (e.g. institutional trust)?

I am not asking for an opinion-based answer, but a beginner's account of the technical connections between these concepts to the extent that this is possible.

There’s a few points to cover in this question.

The description of market events in newspapers is not necessarily reliable. The job of a newspaper editor is to sell newspapers. Readers of the business press like to hear that government borrowing raises interest rates, and so the editors give them that story - a lot. Financial market analysts are paid to give good forecasts, and their explanations are often quite different. (That assertion is based on my experience of writing and reading interest rate commentary.) In this case, the newspaper account may be an accurate reading, but you must keep in mind the prior knowledge that they blame practically every tick up in yields on fiscal policy.
Treasury bond auctions are announced in advance, with the amounts fixed. (They move around money market auctions to deal with fluctuations in cash.) the sizes are based on a lot of consultations with the Primary Dealers and investors, and internal analysis. However, since they fixed the quantity, the price (yield) has to swing. If investors think the yield is too low, it will affect the bids at auction. Going into the auction, there is “when issued” trading, which is based on where dealers think the auction will clear. If they are wrong about demand, actual yields will not end up where the “when issued” pricing was, and the whole curve moves. (Dealers will end up receiving more bonds than they expected, and so will have to trade out of their larger-than-desired duration position.) This dynamic is behind the legitimate stories about tepid auction demand.
The best approximation of inflation expectations is to look at the spread between conventional bonds and TIPS (inflation-linked Treasury securities). The spread roughly equals the future rate of inflation that gives the two securities equal returns. E.g., if the 10-year nominal yield is 4%, and the 10-year TIPS yield is 2%, an inflation rate of 2% results in the two securities having the same return. (One can argue that there are risk premia that drives this breakeven inflation rate away from the true expectation.) There is no guarantee that inflation expectations (by this measure) and nominal interest rates move in the same direction, so one should not use the concepts interchangeably.
The usual explanation for the linkage between higher future deficits and higher nominal interest rates is that “all else equal”, a relaxation of fiscal policy would result in higher nominal GDP (with higher inflation). The presumption is that the Federal Reserve would raise interest rates to keep inflation near its implicit target (offsetting fiscal policy). The expected change in the policy rate affects nominal bond pricing. The exact mix of what happens depends on the specifics of the change to fiscal policy, and your beliefs how fiscal policy works. In any event, all the bond market participants care about is what the Fed does. Otherwise, stories that increasing fiscal deficits raise interest rates run into the counter-example of modern Japan. The debate about that effect is highly political. See also this question: link to question on interest rates and deficits


Log of dummies variables I am having a bunch of dummies variables and some of them when trying to get the scatter plots they are collapsed on the horizontal line. It is hard to see any relation. I know from theory they should show some correlation with the key variables. Winsorizing did not help much. Is it correct to get the log of the dummies only for those specific dummies? What do you people think on this?
I attach an image from stata

A dummy variable is a binary variable, that does need to be logged. You can use them as they are. Also, if the variable is set to 1 for a given result and zero otherwise, by taking logs you will turn your variable to 0 and a missing value because the log of 0 does not exist.

update: Instead of a plot, why not simply calculate pairwise correlation coefficients. Here is an example using Stata

. sysuse auto, clear
(1978 Automobile Data)

. * Create a dummy variable
. gen byte big= weight>3020

. ta big

        big |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |         35       47.30       47.30
          1 |         39       52.70      100.00
------------+-----------------------------------
      Total |         74      100.00

. * Correlate big with price
. pwcorr big price

             |      big    price
-------------+------------------
         big |   1.0000
       price |   0.3469   1.0000

Both are positively correlated.

. * Add a other dummy and compute new correlations

    pwcorr price foreign big

             |    price  foreign      big
-------------+---------------------------
       price |   1.0000
     foreign |   0.0487   1.0000
         big |   0.3469  -0.5682   1.0000


Why do we need a restriction of a game to prove the given statement? Consider a game $G$. We have to prove that is $s$ is a Nash Equilibrium of $G$, then it is also a Nash Equilibrium of the game formed by removing strictly dominated strategies of $G$.

I looked at the proof for this in https://homepages.cwi.nl/~apt/stra/ch3.pdf (the proof is described in first two pages of it)
They use a restriction $R$ ($R_{i}=$ (possibly empty) set of strategies such that $R_{i} \subseteq S_{i}$)  of a game at every point of the proof.

My question is what good does it do to take the $R$ of the game instead of directly taking the set of all possible strategies for all the players $S$?

It is because the set of strictly dominated strategies changes as we eliminate strategies.

The point of the proof is to show the result for ANY $R$ such that $R \rightarrow_{S}R'$.

Look at Example 7 in the next page of the paper.

In $S$, there is only one strictly dominated strategy, and we remove it and get $R$ (hence $S \rightarrow_S R$).
Now look at $R$. There is a strictly dominated strategy that was not one before (in $S$), removing it get $R'$ (hence $R \rightarrow_S R')$.


Creation and distribution of money from scratch in a country (USA as example) Suppose in an alternate history, 21st century USA operated on the barter system... no system of money is used. It  has all the technology of the 21st century (computers etc). Just no money. The knowledge of "what money is" and economics is there, but for whatever reason they've chosen to stick with barter.

Now the government decides to introduce money. What would be the steps to create money, get it into circulation and then get people to use money instead of barter? ie: how do we get the current american financial system from scratch.

I ask this question because I've tried to understand money creation, but I'm always a bit puzzled because it starts with a money system already in place... I want to understand how it would be created from scratch.

The way in which colonial governments forced indigenous tribes to use money was to impose a tax that had to be paid in money. For a textual reference, this is described in “Understanding Modern Money: A Key to Full Employment and Price Stability” by L. Randall Wray. (This concept is discussed quite often in Modern Monetary Theory, and articles are probably easily found using a web search.)

The way the locals could get money is via providing labour at a fixed wage (or goods at a fixed price). They could trade amongst themselves to allow those with a surplus of money to distribute it to those who did not get money this way. This has the effect of forcing monetary transactions into the economy.

As was typical for colonial powers, policing was brutal to ensure that the taxes would be obeyed.


Does rioting impair the economy? In this article, the author looks at some of the indirect effects caused by the rioting in Ferguson, Missouri after the jury's decision not to indict the police officer.  He explains that the rioting has hurt the economy (both short-term and long-term) in Ferguson and other communities where rioting is taking place.

Are there actual numbers that support this conclusion?  Also, can this conclusion be expanded to say that rioting impairs the economy?

There are no numbers to support the problem specific to Ferguson because it happened recently, and not enough time has passed to assess any effect.

The inflammatory rhetoric of the site you linked to probably stems from this paper http://www.nber.org/papers/w10243 that tries to quantify the effects of the race riots in the 1960s. The authors themselves concede the speciousness of their conclusions by acknowledging that the data they used are not detailed enough to identify the precise mechanisms at work.

A more thorough study that examined the riots in India between the Hindus and Muslims concluded that the motivation for the riots was primarily economically based. Where there was economic growth, riots decreased. Moreover, rioting in one region had no effect on the rioting of neighboring regions. https://openknowledge.worldbank.com/handle/10986/5432


Why don't we draw National Income Curve(45 deg. line) simultaneously with graph for Saving Function? When we talk about Consumption Function , we draw the 45 degree line of Income or Aggregate supply curve simultaneously.

But while graphing saving function , we don't show Income curve (45 deg. line) on same graph .

Why is it so ?

The role of 45 degree line while showing a consumption function is that the 45 degree line translates the values on x-axis to equal values on y-axis. The consumption function is drawn on an income and aggregate expenditure plane. Hence the 45 degree line shows consumption + savings which is aggregate output or aggregate supply or income.

The other way to look at it is when you add up consumption function and saving function on the income- expenditure plane you will get a 45 degree line.

You can show the 45 degree line along with the saving function and the difference between 45 degree line and the saving function will reflect the consumption expenditure.

Another intuition that you will get is that the saving function will never intersect the 45 degree line which is representative of the fact that there is some consumption at the 0 level of income and rate of increase in savings is less than the rate of increase in income which is very obvious.


The effect of inflation differentials on the relative prices of goods across countries I am struggling to understand the answer to this question (the answer was given as C):

  If inflation in South Africa is lower than that in the United States (US), US exports are becoming relatively _____ in South African markets. This shifts the demand curve for US dollars to the _______ and the supply curve of US dollars to the ________.

  A. cheaper, left, right

  B. more expensive, right, left

  C. cheaper, right, left

  D. more expensive, left, right

  E. cheaper, left, left

I selected B. Firstly, I think  that if you have to pay more for US exports because of inflation, they are more expensive compared to before. Secondly, using the equation E = RER x P*/P (where E is the exchange rate, RER stands for 'real exchange rate', P* is the foreign level of prices, and P is the domestic level of prices) I think that the rand depreciates and the dollar appreciates, which implies that the demand for dollars increased (presumably because more dollars are needed to pay the higher price), and the supply of dollars decreased (presumably because traders see that they should hold on to dollars until a later date, given the appreciation caused by increased demand).

During an inflationary period the value of the US Dollar has not increased, but decreased. From within the US, goods are becoming more expensive because each US Dollar is worth less.

Inflation is believed to be caused by an oversupply of money; if there is an oversupply of US Dollars, it will lead to a decrease in its value. If inflation is increasing at a higher rate in the US than in South Africa, the Dollar will be relatively cheaper, making it more desirable. Thus, an increase in the demand for the US Dollar, and a decrease in its supply.


BLP using micro data I am currious to know if there has been any work of estimating the BLP model with micro data. Could you give me a reference list.

Yes, the same authors Berry, Levinsohn, Pakes have written a second paper that uses both macro and micro data to estimate demand for automobiles as a function of the characteristics of the car.

"Differentiated Products Demand Systems from a Combination of Micro and Macro Data: The New Car Market"

http://dash.harvard.edu/bitstream/handle/1/3436404/Pakes_DifferentiatedProducts.pdf


Is this non-separable utility? On this presentation, the last slide is titled "Non Separable Utility", and the preferences given are

$$ \frac{\left(c^\gamma (1-n)^{1-\gamma}\right)^{1-\sigma}}{1-\sigma}$$

However, I can log-transform them as

$$ \log\left[ c^{\gamma(1-\sigma)} (1-n)^{(1-\gamma)(1-\sigma)} (1-\sigma)^{-1}\right]$$

which are clearly separable. Did I miss something?

You are right, these preferences are separable. A way to see it is to notice that
\begin{equation*}
U(c,n) \geq U(c',n) \Rightarrow U(c,n') \geq U(c',n')
\end{equation*}
for any $n,n'$, and
\begin{equation*}
U(c,n) \geq U(c,n') \Rightarrow U(c',n) \geq U(c',n')
\end{equation*}
for any $c,c'$.
They are even additively separable, as your log-transformation shows.


How to work out average income based on square meter of property rented Is it possible work out the average income of a household if one knows the rental cost per square meter of property?

Example: Edinburgh, Scotland = £2,125.00 per square meter.

I hope I have posted this in the right SO platform.

Thanks

It is only possible if you also know the average fraction of income that is spent on housing in that area... and even then only if you assume either that rental costs per square foot are similar to ownership costs per square foot, or that the rent/buy decision is uncorrelated with income.

So: not really. Rental costs are correlated with income, but there's a lot of error.


Lee and Saez (2012): Pareto-Improvement? I'm interested in the following quote that came up in this earlier answer.

  Second, when labor supply responses are along the extensive margin
  only, which is the empirically relevant case, the co-existence of a
  minimum wage with a positive tax rate on low-skilled work is always
  (second-best) Pareto inefficient.A Pareto improving policy consists of
  reducing the pre-tax minimum wage while keeping constant the post-tax
  minimum wage by increasing transfers to low-skilled workers, and
  financing this reform by increasing taxes on higher paid workers.
  Importantly, this result is true whether or not rationing induced by
  the minimum wage is efficient or not.

How are higher paid workers not worse off, given the increased taxes they have to pay?

(Note that this answer implicitly makes reference to the specific model in Lee and Saez.)

Short answer: the increased taxes on high-skilled workers exactly offset the higher real wages they obtain from a decline in the minimum wage for low-skilled workers.

Longer answer: Suppose that I'm the government, and I decide to lower the minimum wage $\bar{w}$. The direct effect will make low-skilled workers currently earning the minimum wage worse off, while making high-skilled workers better off. (The fall in low-skilled wages means an increase in high-skilled wages.)

Furthermore, a lower minimum wage means that fewer low-skilled workers will be rationed out of the labor market. These workers will be better off.

As the government, I want to turn this into a Pareto improvement - which it currently isn't, because workers who are already earning the minimum wage are hurt by the decrease. I try the simplest possible offset: I adjust taxes so that everyone's after-tax wage rate is exactly the same as before. The combination of this tax rate change and the minimum wage decrease means that everyone's welfare is unchanged, except for some low-skilled workers who were previously unemployed and now can work. These workers are better off - hence overall we have a Pareto improvement. Great!

There's only one catch: I didn't verify that this policy was feasible for the government. Maybe the proposed change in tax rates would violate the government's budget constraint.

This is where some slightly more involved logic comes in, and it's useful to think about the infinitesimal case for simplicity. Before the policy change, output is produced by a mix of low- and high-skilled workers, who I'll call the "old" workers. The minimum wage increase leads to additional low-skilled workers entering the mix; I'll call them the "new" workers. These new workers produce additional output and earn income. At the margin, though, (pretax) wage equals marginal product - so when an infinitesimal number of new workers is added, they increase output by exactly as much as they draw away in earnings, and the total earnings of the old workers are unchanged. There's a redistribution in earnings among the old workers, toward the high-skilled and away from the low-skilled, but this is just a zero-sum redistribution, and the government can use taxes to reverse a zero-sum redistribution while maintaining a balanced budget - so this part is fine.

All we need to worry about now is the budgetary impact of the new low-skilled workers. But this is simple: we're initially taxing low-skilled work at some rate $\tau_1$, and the net budgetary effect of more low-skilled entry will be positive (even after the infinitesimal decrease in $\tau_1$ that's part of the reform) as long as $\tau_1>0$. New workers are good for the budget as long as their tax rate is positive.

So that wraps it up: we see that there is a potential Pareto improvement involving a decrease in the minimum wage and an offsetting change in tax rates, and this policy change is feasible as long as the initial tax rate on minimum wage work is positive: $\tau_1>0$. Lee and Saez emphasize this point (in conjunction with their other results) to demonstrate that in their model, the minimum wage can only be rationalized as a policy that's complementary to a labor subsidy $\tau_1\leq 0$. They really like this purported complementarity, and it's one of the most popular current arguments in favor of minimum wages.

(I happen to think that the way they prove this complementarity, which relies on the assume-a-can-opener assumption of efficient labor market rationing, is incredibly silly. But that's a different part of the paper, not directly related to your question here.)


Overview: Crises and their causes There have been many business cycles in the US over the past 50 years (although less so since the great moderation). Many of us youngsters are too young to remember many of these.

Is there some summary article / plot containing these, including their causes? For example:

1990: Gulf war, drop in oil supply
2000: crash of internet related stocks
2008: crash of mortgages/housing market in the US, amplified through banking sector

Manias, Panics, and Crashes

by Charles Kindleberger

(Full book pdf)

An easier reading compared to Reinhart and Rogoff. It informally explains how crises occur and presents some famous examples.


What is the long term outcome of inflation? I was just reading how \$1 from sometime in the mid-1800s is worth a little over $30 today.

This got me wondering, where does this go in the long term? In the year 2400, will a loaf of bread be $200? Will an average house be tens of millions? What about the year 2700?

My understanding is that inflation is a universal economic phenomenon. Has a currency ever lasted long enough to know where it leads in the very long term? If not, are there theories predicting what might happen?

Inflation is not a fixed number, so there is no way to predict the value that a loaf of bread or an average house will be in 2400 or 2700. Inflation is the result of a complex macroeconomic process rooted in the behaviour of individuals, and people's expectation of future inflation. The central bank is a key role player, so the integrity of that institution will determine how stable inflation is.

When there is high inflation, in the long run governments often just divide the numerical value of the currency by, for example, a million, in order to simplify the currency for the population. That doesn't change how much real wealth people hold, since it is done across the board. There is a common understanding that, for example, 1 million Zimbabwen Dollars yesterday is the same thing as 1 Zimbabwen Dollar today, so everyone catches on as to what happened.


Why is instantaneous utility of current period discounted? Consider a two period model of consumption.

I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the next period.

Could someone give me an intuition behind this result?

Below, the derivation is presented.

$$ \max_{c_{t},c_{t+1}}u_{t}\left(c_{t},c_{t+1}\right)$$

such that

$$c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}$$

FOC: (MU denotes marginal (aka instantaneous) utility)

$$\begin{cases}
MU_{t}=\lambda\\
MU_{t+1}=\lambda\frac{1}{1+r_{t+1}}\\
c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}
\end{cases}\Rightarrow\frac{1}{1+r_{t+1}}MU_{t}=MU_{t+1}$$

What you see as
$$
\frac{1}{1+r_{t+1}}MU_t = MU_{t+1}
$$
can also be seen as
$$
\frac{1}{1+r_{t+1}}MU_t = \frac{1}{1}MU_{t+1}
$$
where $1+r_{t+1}$ and $1$ are the respective prices of current and future money measured in future money. So basically what is happening is that you calculate how much utility an additional unit of future money would buy you in the current and in the future period. In optimum these quantities are equal. You can also rewrite the equation into the form of the familiar MRS condition:
$$
\frac{MU_t}{MU_{t+1} } = \frac{1}{\frac{1}{1+r_{t+1}}}.
$$


Does the Prospect Theory value curve change according to reference wealth size? After reading "Thinking Fast and Slow" and half a day of internet search on this topic I resort now to asking this question here.

Prospect Theory seems to only attribute value to changes in wealth. The reference point in wealth is only used to calculate the change in wealth. The change is then valued according to a value function where losses weigh heavier than gains and both have diminishing marginal utility (see figure below). It seems to me however that the absolute value of the reference point matters, but is ignored.

The diminishing marginal utility of wealth from expected utility theory is still a valid concept, as is the utility of change of wealth from prospect theory. In my view they should be combined. Losses weigh heavier than gains and the size of your reference wealth influences how you weigh the loss/gain.

In other words, this is a typical value function in Prospect Theory:

I think it needs a third axis for the reference wealth value. Higher reference wealth values lead to flatter curves while lower reference wealth values lead to taller curves. A rich person has a different value function than a poor(er) person. They value the loss (or gain) of 500 euro differently.

To illustrate this think of the following "Thinking Fast and Slow"-style problems:

Anna had 1500 euro yesterday. Today she wakes up with 1000 euro.
Ben had 20000 euro yesterday. Today he wakes up with 19500 euro.

It seems to me Anna and Ben would not attribute the same (negative) value to their loss. Prospect Theory seems to claim they would. They incur the same loss (-500 euro) with the same (weighted) probability and value it using the same value function. I haven't found an explanation of Prospect Theory where absolute wealth influences the value function.

In Thinking Fast and Slow, Kahneman does allure to an influence of your reference wealth (page 284):

  All bets are off, of course, if the possible loss is potentially ruinous, or if your lifestyle is threatened.

But this seems a sidenote covering an exception rather than a intrinsic part of the theory.

Does prospect theory ignore the influence of reference wealth size on the value curve? And if it does, why is that valid to do?

You are certainly right that Kahneman and Tversky neglect the importance of levels in their model (even though they do acknowledge the importance of levels informally, as in the quote you reference). However, this situation has now been rectified by Kosegi and Rabin, who include both 'levels' utility (i.e.'consumption utility') and 'gain/loss' utility (they also generalise Prospect Theory to handle many goods, and endogenise the reference point).

As I understand it, the point of the Anna/Ben story is that marginal utility is a decreasing function of wealth. I would think that this is adequately captured by Kosegi and Rabin, assuming that we specify that levels utility is a strictly concave function of consumption, as is standard.

Of course, a different route would be to allow the curvature of the Prospect Theory value function to depend on wealth (as you propose). However, I'm not sure what this would buy us above and beyond the Kosegi and Rabin approach. Moreover, the theory would still be missing the rather obvious fact that people do care about absolute levels of wealth, at least to some extent.


What's the relationship between Output elasticity and Returns to scale? https://en.wikipedia.org/wiki/Output_elasticity

"If the production function contains only one input, then the output elasticity is also an indicator of the degree of returns to scale. If the coefficient of output elasticity is greater than 1, then production is experiencing increasing returns to scale. If the coefficient is less than 1, then production is experiencing decreasing returns to scale. If the coefficient is 1, then production is experiencing constant returns to scale. Note that returns to scale may change as the level of production changes."

So, the above is clear for 1 input. But what if the production function has two ore more inputs? Capital, labor, materials, etc? Can anyone show the proof?

Define the production function $Q=f(x_1,..,x_k)$, where $x_i$ denotes the ith input.
Next recall that the total differential of output can be written as
 $\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\Delta x_i$
 Now we are interested in returns to scale, which is the change in output due to the multiplication of every input with a constant. We can write: $x_{1i}=ax_i$, for every x where a is a constant and the subscript 1 means the new x due to the multiplication. It follows that:
 $\Delta x =x_{1i}-x_i =ax_i-x_i=(a-1)x_i$
 Substitute this last result into the total differential. We obtain:
   $\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ (a-1)x_i$
  which can be rewritten as
$\Delta Q = (a-1)\sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\ x_i $
 Now recall that the output elasticity with respect to an input i can be written as
$\epsilon\ = \frac{\partial Q}{\partial x_i}\ \frac{x_i}{\ Q}$
rewrite this to the expression
 $\epsilon\ Q = \frac{\partial Q}{\partial x_i}\ x_i$

Substitute this result into the last version of the  total differential which becomes:
$\Delta Q = (a-1)\sum_{i=1}^{\ k}\ \epsilon\ Q = (a-1)Q\sum_{i=1}^{\ k}\ \epsilon$
Rewriting this we obtain the multiplicator of the output associated with multiplication of every input with $a$:
 $\frac{Q_1}{Q} = (a-1) \sum_{i=1}^{\ k} \epsilon +1$
Where $Q_1$ is output ex post.

From this last expression it can be immediatly seen that constant returns to scale ($ax \implies aQ$) follow from the sum of elasticities being equal to 1, increasing returns to scale from it being greater than 1 and decreasing when it's smaller than 1.

The graph is plotted for a=2. In red all points which correspond to decreasing returns to scale, in green for increasing returns to scale. Further, it can be seen that
 $\sum_{i=1}^{\ k} \epsilon = 1 \implies \frac{Q_1}{Q} = a$