Questions tagged [economics]

For questions regarding the mathematical analysis of economic models and problems. This includes questions about the formulation or solution of models from microeconomics or macroeconomics.

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33 views

Financial math savings problem

At the end of every month I deposit a 1000 dollars into my bank account for a period of 15 years with a nominal interest rate of 14% per annum with half-yearly compounding that falls under a 15% tax. ...
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37 views

Stuck on a derivation and proof for economics.

I am trying to solve this question but it seems I keep getting stuck. If: $r^n_t - r^{n*}_t = E_t(\Delta y^n_{t+1} - \Delta y^{n*}_{t+1})$ Prove that: $E_t \sum^\infty _{i=0} \left(r^n_{t+i}-r^{n*}_{t+...
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2answers
63 views

What is it to solve an equation forward?

I'm reading a book in Monetary Economics and I don't understand a step. I have this expression: $$ \dfrac{\lambda_{t}}{P_{t}} = \beta \left( \dfrac{\lambda_{t+1} + \mu_{t+1}}{P_{t+1}} \right) $$ And ...
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1answer
22 views

Complete Market / Replicating a portfolio

Attempt So Far: 1) First Part: I have shown that the market is arbitrage-free since the only possible portfolio for which $V_1^h\geq0 \ $ given that $V_0^h=0 \ $ is $h=(0,0,0)$ and this clearly ...
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Express function $Q_t=(\delta L_{t}^{-\rho} + (1-\delta) K_{t}^{-\rho})^{-1/\rho}$ in growth rates

Does anyone have a clue, how to express this equation (CES production function) step-by-step in growth rates: CES in levels form: $Q_t=(\delta L_{t}^{-\rho} + (1-\delta) K_{t}^{-\rho})^{-1/\rho}$ ......
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4 views

Purpose of Expectation in the Denominator of Gittins Index

I am self-teaching "Multi-armed Bandits and the Gittins Index Theorem" (notes here). On slide 27, the Gittins index is defined as $$ G_i(x_i) = \sup_{\tau\geq 1} \frac{\mathbb{E}\left[\sum_{...
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2answers
37 views

If $Q*P$ is constant, why is $\frac{\Delta Q/Q}{\Delta P / P}$ constant?

The title says it all; I don't quite get why $Q*P$ being constant implies that $\frac{\Delta Q/Q}{\Delta P / P}$ is constant. Or does it? This is about iso-elastic demand curves - why is it that $Q*P=...
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1answer
18 views

Finding Marshallian Demand without Lagrange?

I need to find Marshallian demand for goods x and y (in terms of $P_x, P_y,$ and $I$) with the following utility function: $$U(x,y) = x + 10y - y^2$$ and general budget constraint $$I = P_xx + P_yy$$ ...
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3answers
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Why is $\partial f(x)/\partial x_i$ homogeneous of degree $k-1$ if $f(x)$ is homogeneous of degree $k \geq 1$?

It's as (simple) as the title says, but I can't figure out why that is. How can I show (I think I need small, tiny steps) that $$\frac{\partial f(tx)}{\partial (tx_i)}t=t^k\frac{\partial f(x)}{\...
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24 views

Environmental economics problem: how to think to find law of motion and objective function?

I'm reading an interesting book of mathematical economics, written by a german professor, it contains many theory chapters and exercises, but not all of them have the solution at the end of the book. ...
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21 views

How to model the impact of fungicide resistance on yield

I'm not sure if I am in the right area or not but I am really struggling to figure out how to fungicide resistance into a cost model. The variables I have currently are yield potential disease ...
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1answer
116 views

Replication invariance of weighted sums

Consider the following function: $\phi(\langle x_1,x_2,...,x_n \rangle) = \alpha_1 x_1 + \alpha_2 x_2 + ... + \alpha_n x_n$, where $x_1, x_2, ...$ are all real numbers such that $x_1≤x_2≤...$ and ...
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1answer
20 views

Proof Doubt - Scale Elasticity Formula (Microeconomics)

My professor presented two concepts of elasticity in my last microeconomics class: Product Elasticity: $\mu_i=\frac{\partial f(x)}{\partial x_i}\frac{x_i}{f(x)}$ Scale Elasticity: $\mu(x)=\frac{d\ln(f(...
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29 views

Lagrangian optimisation problem

If labour $L > 0$ and capital $K > 0$ and they are used to produce 2 goods (1 and 2) so we have $$ L = l_1 + l_2 , $$ $$ K = k_1 + k_2 , $$ and all $l_i$ and $k_i$ are non-negative. Production ...
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How do I find what cost level(Y) is correct for my target marginal cost(Y) per result(X), without using a table?

I have a data set of Cost(Y) and Results(X). Using this data I've calculated the marginal cost per results for the different cost levels. Marginal cost per results is calculated as: (this cost - ...
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1answer
47 views

TIME INTEGRAL PROBLEM: Taking an integral of a growth-rates-form equation with respect to time

I am trying to solve the following problem as described here: https://www.adb.org/publications/total-factor-productivity-testing-growth-models (pages 4-5) http://digamo.free.fr/macombie98.pdf (pages ...
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1answer
40 views

Wages Regressed on Education and Experience: Estimate Effect of Changing Parameter?

I'm confused about the last question. I will quickly go through the beginning parts, so you can skip to question 3 at the very bottom as it may not be necessary. Estimate linear regression for log ...
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28 views

How to find elasticity

I have come by this problem in microeconomics. The question is "is the good q1 an inferior-, normal- or luxury good?" How do I go about this problem. I don't understand how to actually solve ...
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2answers
75 views

A fair way to choose multiple winners without giving anyone a prize that they do not want

I'm trying to organize a raffle that has two prizes — prize X and prize Y: the $N_X$ X-fans want prize X and don't want prize Y, the $N_Y$ Y-fans want prize Y and don't want prize X, the other $N_{...
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Derivation of exponential function (minor help)

This is basic maths but I'm just not seeing the connection here. How is 2-3 equation derived? $$e^{-\gamma(c_{t-1}+K_{t-1}+\nu_{t})}=E_te^{-\gamma(c_{t-1}+K_{t-1}+\nu_{t}+K_t+\nu_{t+1})}$$ $$e^{\gamma ...
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Calculating value of the own-price elasticity of demand

Diana has a linear demand function for good q, whose price is 𝑝: q = 20 −2𝑝 What is the value of the own-price elasticity of demand at 𝑝=5?
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24 views

Sequential Voting Game

Consider the following model of sequential vote buying. A political leader P needs to convince 2 out of a set of 3 committee members or legislators (called L1,L2,L3) to support her policy project. ...
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33 views

Risk aversion case, how can I solve this?

There are two consumers, 1 and 2. Both are expected utility maximizers and 1’s Bernoulli utility function is denoted by u while 2’s is denoted by v. The utility function v is such that v'>0 and v&...
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2answers
54 views

Is $f$ always positive if $f'>0$, $f''<0$?

Is functions $f$ always positive if $f'>0$ and $f''<0$? I intuitively can think that this is true but I want more details about the reason why this satisfies or not. Thank you.
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50 views

Handling Reserve in Vickrey Auction

I'm trying to learn more about auction theory and after studying Vickrey and first price auctions I wanted to try including a reserve price (in a Vickrey auction), but something seems to be going ...
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1answer
16 views

Interpret input quantity matrix multiplied by the transpose of a price matrix

I have two $77 \times 8$ matrices where rows represent firm ids: input quantity matrix, $X$, where columns represent quantity types (e.g., grain, seed, and chemical). price matrix, $W$, where ...
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57 views

Derivative of an integral function - chain rule

I have this problem in one of my econ classes, and google is not helping much in finding satisfying answers. So, lets say I have to write the first derivative of $f(x) = \Big[\int_0^1 x^{\frac{n-1}{n}}...
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32 views

Equilibrium for Implicit Best Response Functions

I am dealing with a problem in economics, specifically game theory, where $n$ agents have a best response $x_i$ given by an implicit function, as described below. We have $ i \in \{ 1, 2, ..., n \}$, $...
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1answer
29 views

Solve Normal form Nash eq. (pure and mixed), especially the probabilities

can someone please help me to solve this question: Consider the normal form game: i) For Player 1 the strategy B is strictly dominated by C. ii) For Player 2 the strategy E is strictly dominated by F....
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43 views

Basic reality check on a possible error on a paper

In this paper I may choose between a job that offers $\$12,000$ per year, and another that offers $\$2,000$ per month. Let’s say the jobs are identical in all other respects: I would then choose the ...
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30 views

Hamilton–Jacobi–Bellman equation under exponential discounting

If we want to choose $\mathbf{u}(t)$ in order to minimize $$ J(\mathbf{x}(t),t) = \int_{t}^{T} g(\mathbf{x}(s),\mathbf{u}(s),s) \, \textrm{d}s + h(\mathbf{x}(T),T), $$ subject to $ \dot{\mathbf{x}}(t) ...
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0answers
40 views

Proof that $E(\hat{\beta^{'}}\hat{\beta})=\beta^{'}\beta+\sigma^2\sum_{i=1}^k\frac{1}{\lambda_i}$

I am having some difficulty proving the result above. I am considering $\hat{\beta}$ as the ols estimator of $\beta$ $(Y=X\beta +\epsilon)$ with $Var(\epsilon \mid X)=\sigma^2I$ and $\lambda_{i}$ a ...
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2answers
115 views

Solution of a functional equation $f(\alpha x+\beta)=\left( f(\alpha+\beta)-f(\beta) \right) f(x) + f(\beta)$

I am looking for an accessible proof of the following claim (which comes up in social choice theory). The proof is supposed to follow from a theorem on page 150 of Aczel's 1966 Lectures on functional ...
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24 views

Guarantee an interior solution for profit maximisation

A function $c: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$ is is said to be a cost function if The value of function $c$ at $y = \textbf{0}$ is $0$: $c(\textbf{0}) = 0$ $c$ is continuous on the ...
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1answer
20 views

Microeconomics Doubt about Convexity and Satiation [$u(x_1,x_2)=-x_1^2-2x_2^2+2x_1x_2-10x_1+40x_2$]

I am having some difficulty with this question: $$\text{Prove that } u(x_1,x_2)=-x_1^2-2x_2^2+2x_1x_2-10x_1+40x_2 \text{represents a strictly convex preference and has a global satiation point}$$ I ...
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2answers
73 views

Find interest rate in a leasing problem

A firm has entered into a 3-year lease agreement for an audio broadcasting system valued at $\$18'000$. The contract calls for 8 deferred payments each four months period of $\$2'803.33$ each and a ...
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1answer
41 views

profit / standard derivation calculation

$$\begin{array}{c|c|c|} & \text{Stock A} & \text{Stock B} \\ \hline \text{ Profit} & 14 \% & 21\% \\ \hline \text{standard deviation} & 15\% & 26\% \\ \hline \end{array}$$ ...
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1answer
45 views

Using L'hopital rule [closed]

Bare with me, but could someone help me with L'Hopital's rule? $$u(c,l)=\frac{[c\cdot \textrm{exp}(v(l))]^{1-\sigma}}{1-\sigma}$$ How can we show that it's $\lim\limits_{\sigma\rightarrow 1}=\text{log ...
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0answers
75 views

Seemingly simple convex analysis problem

A function $v: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$ is is said to be a valuation function if The value of function $v$ at $x = \textbf{0}$ is $0$: $v(\textbf{0}) = 0$ $v$ is continuous on the ...
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1answer
33 views

Help with density functions in economics - easy double integration

I've already posted in the econ.stack, since the model I’m having trouble with is an econ one (link). However, I feel the nature of my doubt is more about math than the actual econ model behind it, so ...
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1answer
28 views

Why is convexity necessary in monotonic binary relations?

First, let's start with the definition of a monotone binary relation: Definition: Let $C \subset \mathbb{R}^n$ be nonempty and convex. A binary relation $\succeq$ on $C^m$ is monotonic if for any $(...
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1answer
46 views

Transformation of a normal distribution

I got these 2 models for hourly wage for 2 periods: The hourly wage for period 1 is normally distributed with mean $µ$ and variance $σ^2$ so $Y_1 \sim N(\mu,\sigma^2)$. And the hourly wage for period ...
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24 views

Total Differential Exercise: Production Function in Growth Rates.

I'm struggling to understand how Khan & Reinhart (1990) go from the next production function. $$y=A f(K,L,Z)$$ Where $y$ is the production of the economy, $A$ is a variable which contains the ...
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1answer
28 views

Why does $ x\mathcal{R}y \iff |x| >|y| \text{ or } (|x|=|y| \text{ and } x>y) $ not allow a utility function representation?

Suppose we have the preference relation $\mathcal{R}$ defined on the interval $[-1,1]$ by $$ x\mathcal{R}y \iff |x| >|y| \text{ or } (|x|=|y| \text{ and } x>y) $$ How do you show that this ...
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1answer
59 views

General solution of $a^x = b^x\cdot(1-c) - c$

I am doing some simple models to estimate the cost of having unrealized taxed investments, but I keep running into equations that have the form of: $$a^x = b^x\cdot(1-c) - c$$ Or even, $$a^x = b^x\...
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1answer
11 views

Determine the linear production function given relationship between labor and capital productivity

Determine the linear production function in the form $Q=aK+bL$ if it takes $3.5$ labor ($L$) hours to produce what a machine ($K$) can do in one hour? I was given the answer is: $Q=3.5 \times K+L$ But ...
4
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1answer
87 views

How to check the consistency of OLS estimator in macroeconomic models

Problem: We have a model $$C_t = a + b Y_t + e_t$$ and $$ Y_t = C_t + I_t$$ It's known that $Cov(I, e)$ is zero. A student estimates the following model: $$C_t = a + b Y_t + e_t$$ Are the estimators $\...
2
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1answer
35 views

Differential Equations - How to solve a logistic growth model with the added mean

For the general logistic growth model that can be applied to biology and economics $\frac{dP}{dt} = gP(1-\frac{P}{K})$ I know to separate the variables and divide by $P(1-\frac{P}{K})$ resulting in $...
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19 views

Pricing of forwards contracts

Of the courses I am taking in college this semester, two are Financial Mathematics and Derivatives. In each course, we learn different formulas to calculate the forward price of a forward contract. ...
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1answer
21 views

What is the difference in the equations listed here

This is a question in economics where we create a model for GDP per capita. GDP per capita can be caculated in such formula where you take GDP/population. This formula (GDP/population) can be broken ...

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