Questions tagged [mathematical-economics]

The application of mathematical methods to represent theories and analyze problems in economics.

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38 views

Is there a book or notes that explain all about inflation?

Is there a book or notes or papers (but for easy understanding please) that explain all about inflation? Not at a research level but student level. From what is previously needed to define and ...
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Arrow-Debreu Theorem of Existence: Non satiation

Let $n$ be the number of consumers and $m$ be the number of commodities. The Arrow-Debreu theorem requires closed and convex consumption sets $X_i \subset \mathbb{R}^m$ for all buyers $i \in [n]$. ...
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Calculate Average Annual Real Interest Rate Given Average Annual Inflation

Given that the average annual inflation between the years 1993 and 2012 is 5.6%, what is the average annual real interest rate between 1993 and 2012? I know the formula growth = $(\frac{X_{t+p}}{X_t})^...
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29 views

Calculate average annual inflation

The price index in the year 1993 is 60. The base at 2012 is 100. The annual nominal interest rate between 1993 and 2012 is 66% on average. What is the average annual inflation between 1993 and 2012? I ...
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Economics behind reverse auctions with occasional non-profiting suppliers

I was curious about the theory behind a reverse auction system where some suppliers act in a non-profit manner, namely, that their consideration of extra-auctions benefits of providing their service ...
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1answer
57 views

Proof of Concavity of Cobb-Douglas Function

Given the function $F(\mathbf{x})=x^{a_1}_1x^{a_2}_2 \ldots x^{a_n}_n$ defined on the set $S=\{\mathbf{x}=(x_1, \ldots, x_n) \in \mathbb{R}^n: x_1>0, \ldots ,x_n>0\}$ with $a_1,a_2,\ldots,a_n &...
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253 views

Under what condition is a cost function strictly concave in prices?

Define the unit cost function as $$ c(w) = \min_{z\geq 0} w\cdot z $$ subject to $f(z)\geq 1$. Where $w$ is a vector of input prices, $z$ is the vector of inputs and $f$ is a production function. We ...
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Mathematical question in macroeconomic task

I am currently working on a project where I want to explore how changes in a couple of currencies affects the income for my countrys exported fish. My country (Norway) sells it fish in 4 different ...
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Market clearing price and dual variables in auctions

Suppose the allocation of an auction (or a market) is defined by the solution of a linear program. Then it is known that the associated clearing price is given by the dual variable associated to the ...
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1answer
30 views

Blackwell's Sufficiency Conditions for a Contraction:

I wondered if anyone could throw some intuition on Blackwell's sufficiency conditions: (1) Monotononicty; (2) Discounting; Would be genuinely appreciated to hear both terms expressed in plain English.
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Nested/Recursive Dynastic Utility Functions

I want to find a way of representing a dynastic utility function in which not only the head of the dynasty's utility is dependent on its descendants' utility, but all members of the family tree gain ...
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1answer
45 views

Finding optimum market value of vintage car

I am trying to find first and second order conditions for the following problem: $$P(t) = V(t)e^{-rt}- \int^t_0 me^{-r\tau}d \tau $$ I managed to find the first order condition: $$P'(t) = V'(t)e^{-rt} ...
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1answer
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Is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$?

Very simple question here: say that there are n individuals and each individual $i\leq n$ has a consumption bundle $x_i\in \mathbb{R}^m$ (i.e. there are $m$ types of goods). Suppose that social ...
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Probability transition matrix as a function of the variance in Matlab?

I am working on Probability transition Matrix on Matlab. I Have say 5 points (states) of discretized productivity grid. I would like to have two matrix of probability transition subject to the ...
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Integration by parts with CDF

I am told that the following equality follows from integration by parts: $$\int_{R-k}^{1}(\theta-R)dG(\theta)-G(R-k)k=\int_{R-k}^{1}(1-G(\theta))d\theta-k$$ Where $R>k>0$ and $G$ is the CDF of $\...
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Stationarity, ADF/KPSS, Autocorrelation and Heteroscedastiy

i have a time series which is not stationary due to ADF/KPSS test, but is is in its first difference. So ADF and KPSS tell me it is starionary so it has a constant mean/variance/autocorrelation. But i ...
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Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?

I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving in Matlab and Python and could not find any) so that I cannot get an explicit expression ...
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1answer
48 views

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 ...
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Labor demand and supply curves in terms of elasticities

I would like to ask for your help to derive some expressions. Let the labor demand curve be described by: $N^D = N^D(W/P, \bar{K}), \quad N^D_{W/P}= \frac{1}{F_{NN}}< 0, \quad N^D_K = -\frac{F_{NK}}...
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1answer
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Economic interpretation: IS curve contra GDP in equilibrium

I have in a problem shown that for the IS curve (Y) contra the GDP in equilibrium ($Y^*$) it applies that: $\frac{\partial Y}{\partial G}>\frac{\partial Y^*}{\partial G}$. Where G is public ...
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Variance in the Context of a AR (1) Model

I wondered if someone could help me in terms of the required algebraic steps from expressions (3) - (4), for the the moving average representation of the AR (1) below? Would be appreciated. $$y_t=a +\...
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1answer
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Recursive Substitution in Time Series

Would appreciate some guidance on a matter of recursive substitution, where we have the AR model: $$y_t = \alpha +\theta_1y_{t-1}+ u_t$$ And $$E(y_t)= \mu_t$$ Where: $$\mu_t = (1+\theta_1 + \theta_1^2+...
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Interpretation of market condition given by relation between elasticities

The total market demand is given by the sum of the two destinations ($e$ and $f$) given to the product. Given these conditions: Demand: $d(p) = e(p) + f(p)$ where: $d(p) > 0$, $d'(p) \leq 0$; $e(...
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Measure of urban-ness or city density

I have a dependent variable I want to examine in relation to a measure of city density. This means city population as well as geographical spread. Ideally, it would also include a measure of ...
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Uses of simple interest

Recently I started to read again Brealey & Myers's "Principles of Corporate Finance" (an older edition I hold it since I attended to the university course on Economics and Corporate ...
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graph of dependent income

I would need help with the following problem about consumer theory. Let us say that $X$ is the amount of days at the sea and $Y$ is the amount of days on the cottage. We have some utility function $u(...
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1answer
157 views

Derivation of Surplus in Paul Romer's paper on “mathiness”

In this paper by P. Romer https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.p20151066 I'm wondering the Surplus $S$ was derived. By using the given condition I found that $$q_0=m^{-\tfrac{1}{a+b}}N^{-\...
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4answers
90 views

Why should $dp_2=dm =0$ in this problem?

I am studying Essential Mathematics for Economic Analysis and in chapter 12 problem 10 there is a problem that I can solve but I do not understand why the solution works. The problem asks us to ...
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1answer
57 views

Interpolation of Gini Index?

Is there a conventional curve used to interpolate the Lorenz curves that model the distribution for calculating the Gini coefficient? Is it polynomial? But why? Which degree?
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1answer
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Set of consumption over all period is convex in $\mathbb{R}^T_+$?

Today in class, the professor said that the set of all consumption $c(S)$ is non-empty, compact and convex subset of $\mathbb{R}^T_+$. i.e. we know $\sum \limits_{t=1} ^T c_t = S$ where $c_t$ is ...
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Deconvolution in economics

Here it says In mathematics, deconvolution is an algorithm-based process used to enhance signals from recorded data. Where the recorded data can be modeled as a pure signal that is distorted by a ...
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1answer
96 views

If production function is concave, then demonstrate that profit function will also be concave

Show that concavity of firm's production function implies concavity of its profit function. (Hint: For a concave function, first order conditions gives the vector that maximizes the function) ...
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The Envelope Theorem, Some Workings/Substitution

I have an application of the envelope theorem below, and would love some feedback on the steps taken to get from (2) - (3) and (3) - (4) and (4) - (5) The overall question is: 'derive the marginal ...
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1answer
53 views

How does explanatory variables in variance affect unconditional variance of GARCH(1,1)

I have a question about the unconditional variance of a GARCH process, where exogenous explanatory variables are included in the variance. The usual GARCH models the variance using: $$\sigma^2_t=\...
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Can anyone help me derive saving from the OLG model? (Just check to see if my start is okay please)

Question assumptions: Consider the effect of a capital tax on the OLG model. The government imposes a capital tax rate at the rate $\tau\in[0,1)$ and pays all the tax revenues back to the old in a ...
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1answer
42 views

Competitive equilibrium with production

Consider an economy with four goods, two individuals and two firms. Firm 1 produces good $x$, firm 2 produces good $y$. Consumers' utilities are $u_1(x,y,z,w)=\min\{x,2y\}$ and $u_2(x,y,z,w)=\min\{2x,...
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1answer
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homothetic functions in economics

Are there any results in economics that require function to be homothetic? The textbook I am using (Essential Mathematics for Economic Analysis) says that function is homothetic when " $f(x)=f(y)$...
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1answer
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Would some help me fill in missing steps from a textbook exercise left for the reader

Question: Suppose $C_t=(1-s)Y_t-\lambda G_t$ where $s>\sigma$ as in the basic Solow model. Out of the government expenditure , proportion $\phi$ is invested in public capital formation. Hence we ...
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Can we model risk with only probability?

Sorry for the confusion! I am adding an example to see if it helps: For example, consider a gamble A, with payoffs {a,b,c,d}, whose probability of each payoff being realized is equal (so 25% each); ...
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1answer
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Would someone be able to help me solve capital per capita in the steady state (check my work)

Question: Suppose $C_t=(1-s)Y_t$ where $s>\sigma$ as in the basic Solow model. Solve for capital per capita in the steady state. $Y_t=K^{\alpha}_tL^{1-\alpha}_t$ $Y_t=C_t+I_t+G_t$ $K_{t+1}=I_t+(1-\...
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1answer
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Are there any analysis of what is the net benefit of NASA program to US economy?

I am looking for some study that tries to calculate whether the NASA program on a net basis benefits the US economy. The program is very expensive, but it generates lots of patents, research, jobs and ...
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Can someone help me prove that the CES function is also a Cobb-Douglass function [duplicate]

I would like some assistance with a problem that I have showing a CES function is also a Cobb-Douglass utility function. Question: we have a CES function: $Y=A[\alpha K^{((1-\sigma)/\sigma))}+(1-\...
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How to find $\phi$, that denotes the correlation of signals among informed traders?

Since I do not have an answer on Quantitative Finance in my question I cross-post here the problem to tag some other categories The following assumptions are part of the paper of Back, Chao and ...
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Certainty equivalence when the utility is semi-continuous instead of continuous

Let $U:\mathbb R^2\to\mathbb R$ be a utility function. If $U$ is strictly increasing and continuous, then it is well known that for any $(x_1,x_2)$ there exists a certainty $(c,c)$ such that $$U(x_1,...
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1answer
38 views

Find equilibrium price using excess demand function

Consider an economy with two agents. There are two goods, x and y. Agents' preferences are Leontif ones as follows: $u_1(x,y)=\min(x,4y)$ and $u_2(x,y)=\min(x,y)$. Initial endowment for 1 is (2,4), ...
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What are the values of the Laffer curve for Argentina?

According to this article, (spanish only) Argentine companies pay 106% of taxes for their revenues Supposedly the Laffer curve is different from country to country. Which are the tax rate values at ...
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1answer
34 views

What are the advantages of using demand systems estimation? How is this different from individual demand estimation for product?

I am unable to understand how is demand systems estimation is different from simple demand equation estimation. What's the best resource for getting a history line of demand systems?
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1answer
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Slope vs elasticity of demand function - Is it the same thing?

Do the slope of a linear demand function and the elastisicy of demand coincide when we use specific preferences for pricing. As a paradigm, if we consider the case of CARA normal preferences, by ...
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1answer
23 views

Economic and Statistical Significance Of Coefficient

I want to know If I am comprehending these terms correctly Summarizing the difference between economic and statistical significance of coefficients (Describing the terms, process of assessing each ...
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Why does the following approximation hold?

In Essential Mathematics for Economic Analysis, on page 426 there is a model on gains from search where profit function is given as: $$\pi (t) = [p^0 -p(t)]x^0 -wt$$ $t$ is time, $p$ price, $x$ ...

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