Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
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questions
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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 ...
4
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2answers
218 views
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 $i \in [n]$. ...
0
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0answers
16 views
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})^...
0
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1answer
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 ...
2
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0answers
20 views
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 ...
2
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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 &...
3
votes
2answers
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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35 views
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 ...
2
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0answers
16 views
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 ...
1
vote
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.
1
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2answers
53 views
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 ...
2
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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:
$$P'(t) = V'(t)e^{-rt} ...
4
votes
1answer
55 views
Is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$?
Very simple question here: say that there are n individuals and each individual $i\leq n$ has a consumption bundle $x_i\in \mathbb{R}^m$ (i.e. there are $m$ types of goods). Suppose that social ...
0
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0answers
15 views
Probability transition matrix as a function of the variance in Matlab?
I am working on Probability transition Matrix on Matlab. I Have say 5 points (states) of discretized productivity grid. I would like to have two matrix of probability transition subject to the ...
0
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1answer
47 views
Integration by parts with CDF
I am told that the following equality follows from integration by parts:
$$\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 $\...
0
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1answer
23 views
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 ...
3
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0answers
104 views
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 ...
1
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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 ...
0
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1answer
31 views
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}}...
2
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1answer
49 views
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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1answer
31 views
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 +\...
1
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1answer
50 views
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:
$$\mu_t = (1+\theta_1 + \theta_1^2+...
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0answers
27 views
Interpretation of market condition given by relation between elasticities
The total market demand is given by the sum of the two destinations ($e$ and $f$) given to the product.
Given these conditions:
Demand: $d(p) = e(p) + f(p)$ where:
$d(p) > 0$, $d'(p) \leq 0$;
$e(...
1
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0answers
15 views
Measure of urban-ness or city density
I have a dependent variable I want to examine in relation to a measure of city density. This means city population as well as geographical spread. Ideally, it would also include a measure of ...
0
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1answer
75 views
Uses of simple interest
Recently I started to read again Brealey & Myers's "Principles of Corporate Finance" (an older edition I hold it since I attended to the university course on Economics and Corporate ...
1
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0answers
30 views
graph of dependent income
I would need help with the following problem about consumer theory. Let us say that $X$ is the amount of days at the sea and $Y$ is the amount of days on the cottage. We have some utility function $u(...
5
votes
1answer
157 views
Derivation of Surplus in Paul Romer's paper on “mathiness”
In this paper by P. Romer https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.p20151066
I'm wondering the Surplus $S$ was derived.
By using the given condition I found that $$q_0=m^{-\tfrac{1}{a+b}}N^{-\...
3
votes
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 ...
3
votes
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?
2
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1answer
49 views
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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0answers
37 views
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 ...
2
votes
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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0answers
35 views
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 ...
1
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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=\...
1
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0answers
27 views
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 ...
0
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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,...
3
votes
1answer
79 views
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)$...
1
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1answer
32 views
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 ...
0
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2answers
55 views
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); ...
1
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1answer
34 views
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-\...
2
votes
1answer
27 views
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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0answers
25 views
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-\...
0
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2answers
78 views
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 ...
5
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0answers
94 views
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,...
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), ...
0
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0answers
28 views
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 ...
0
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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?
0
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1answer
27 views
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 ...
0
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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 ...
2
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0answers
22 views
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, $x$ ...
