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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4
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1answer
104 views

Shapley value: an alternative representation

It is my belief that the more common representation of the Shapley value is given by $$ \phi_i(v)=\sum_{S\subseteq N-i} \frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup\{i\})-v(S)) $$ where $v \in ...
0
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2answers
92 views

Matrix Operation

Let $x$ be a $n \times 1$ vector whose jth element is $x_j$. Show that $A = xx^{T}/x^{T}x$ and $B = I_n - A$ are symmetric idempotent matrices. Note that $x^Tx$ is a scalar (real number)
1
vote
1answer
262 views

SPNE of infinitely repeated game

Let $G$ be a game with finitely many players and $\underline{v}= (\underline{v}_i)$ be the minmax payoff profile. Denote by $G_{\infty}(\delta)$ the infinitely repeated game whose stage game is $G$ ...
2
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2answers
219 views

The 10.4 Problem on Benassy's *Macroeconomic Theory*

This a repost. In my first try at the problem here I was not very clear with my question and I was also troubled by my lack of knowledge of MathJax. And I probably lost the attention of you guys while ...
0
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1answer
214 views

What is the definition of the slope of a linear function in the context of economic graphs?

I only ask this because of the fact that economists tend to plot the dependent variable on the horizontal axis and the independent variable on the vertical, which is opposite to the "normal" way of ...
1
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0answers
101 views

Optimal auction for risk averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...
1
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0answers
106 views

Math model - constrain GDP given different growth rates of industries

ideas needed to model national GDP given different sector growth rates subject to some contraints Given: GDP equations for $n$ industries depend on growth rates and time i.e. $g(r_1,t), g(r_2, t), ...
0
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1answer
237 views

Mathematical Economics - Utility maximization

I am thankful to any hints: What I have: Simple log-utility form: $u = \log c_1 + \beta \log c_2$ Budget constraints: $c_1 + s \leq w$ $c_2 \leq R\; s$ Problem: For utility maximization: $s = ...
0
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1answer
82 views

Portfolio which replicates given payoff

Consider the following payoff function: $$p(S_T) = \begin{cases} 0 & \text{if } S_{T} \leq 70 \\ S_{T}-70 & \text{if } S_{T} \in (70; 100] \\ -S_{T}+120 & \text{if } S_{T} \in (100; 120] ...
0
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1answer
45 views

How to determine the MU in economics?

I currently have a table like so: Hours spent on Activity X | Total Utility 120 220 300 360 396 412 I know that Marginal Utility is calculated use slope formulate (delta Y / ...
0
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3answers
278 views

Arrow impossibility theorem and social choice.

I have read the Arrow impossibility theorem in Foundations of Mathematical Economics(Michael Carter). It is just too difficult to understand. So, does Arrow'theorem mean that there is always a ...
3
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2answers
289 views

Exercise in Mechanism Design

I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution. Exercise. Use the characterization of incentive compatible direct-revelation ...
1
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2answers
1k views

What is the equation representing a constant elasticity of 1?

I'm reading the chapter in my textbook about the price elasticity of demand, and it was pointed out that most demand curves do not represent a constant elasticity of demand - even linear curves like ...
0
votes
1answer
157 views

$\log(0,05)$ is minus, but $\log(0,04999\ldots)$ is plus?

How is this calculated, and why is this? We're calculating fixed-rate mortgage, with following formular: $$ n = 1-\frac{\log(\frac{L\cdot x}{y})}{\log(1+x)} $$ Where: $L$ is the loan size, $x$ is ...
2
votes
1answer
207 views

generalized method of moments and the case when solving linear regression with two error conditions

So, I am slowly getting introduced to generalized method of moments (GMM), but I am getting confused over some issues, and this is one of them: I heard that GMM solves the problem that an estimator ...
0
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0answers
38 views

How to Taylor expand $\ln{1-\exp{-i_t}}$ around i?

my question here is how to Taylor expand around $i$ $\ln{(1-\exp{(-i_t)})}$ to the first order? $i_t$ is a time series variable, $i$ is its steady state. Could anyone show me how to expand it ...
1
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0answers
50 views

How is “index” at an Walrasian equilibrium proved? (in relation to Hopf-Poincare theorem)

So, the index of an (Walrasian/general equilibrium) equilibrium point is determined as the sign of $(-1)^{L-1} \times \det M$ where $M$ is a matrix and $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, ...
4
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3answers
131 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
0
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2answers
48 views

Distance between convex set and non-convex set?

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can ...
2
votes
1answer
121 views

function with a continuum of inputs (economic application)

In economics, we often use real-valued functions of the following type: $$U (x_1, x_2)$$ $x_1$ and $x_2$ are the quantities of two goods (real numbers). It is straightforward to work with this kind ...
0
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1answer
228 views

Factoring a constant into a graph's edge weights for triangular arbitrage

I wrote a program which finds negative weight cycles in a graph to find triangular arbitrage opportunities, using the bellman ford algorithm. The basic principle is this, given three currencies ...
1
vote
1answer
93 views

Economic Elasticity: where elasticity-equation come from?

I know the equation for economic elasticity is: $$\varepsilon = \frac{\%\,\Delta Y}{\%\,\Delta X}\frac{X}{Y} = \frac{\partial Y(X)}{\partial X}\frac{X}{Y} = \frac{\partial \log(Y)}{\partial ...
0
votes
1answer
22 views

Change in the price of an item based on a group's need for the item.

Okay so this is likely to be confusing to read, please bear with me. The base price of an item is 1440. There are only 6 of these items and there is a population of 20. 10 of the population want or ...
5
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1answer
250 views

When do $\epsilon$-Nash equilibrium strategies converge to Nash equilibrium strategies?

Suppose I have a game on $n$ players and a sequence of strategy profiles $(s_1^{(1)},\dots,s_n^{(1)}), (s_1^{(2)},\dots,s_n^{(2)}), (s_1^{(3)},\dots,s_n^{(3)}), \dots$. Each ...
3
votes
1answer
201 views

Existence of asymmetric equilibria in the dollar auction game

Consider a game in which an auctioneer sells one dollar to the highest bidder. The high bidder wins the dollar, but every bidder pays their bid. Concretely, assume that there are two bidders ...
3
votes
1answer
123 views

Wealth indicator function for bidder agent logic

I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others). Given: Total number of competitors $n$ Amount of all ...
0
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1answer
3k views

L'hopital's rule in deriving Cobb-Douglas function from CES production function

$$ \ln(Y) = \ln(A) + \frac{\ln[\alpha K^\gamma + (1-\alpha) L^\gamma]}{\gamma}$$ can be taken to the limit by applying l'Hôpital's rule: $$\lim_{\gamma\rightarrow 0} \ln(Y) = \ln(A) + ...
1
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0answers
226 views

Econometrics Simultaneous equation Indirect Least Squares and Two Stage Least Squares

I still can't figure out this problem. PLEASE HELP! (1) $F_t = a_1 + a_2.C_t + a_3.P_t + e_t$ (2) $P_t = b_1 + b_2.F_t + b_3.S_t + b_4.I_t + u_t$
5
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0answers
272 views

Open Problem in Fixed Point Theory [Prize]

This open problem appeared on the bulletins of Evans Hall at Berkeley this week. I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post ...
1
vote
1answer
319 views

Random walk serial correlation

Given a model $$Y_t =b_0 + b_1 \cdot X_t + b_2 \cdot Z_t + e_t,$$ where the error term $e_t$ follows a random walk form of serial correlation $e_t = e_{t-1} + u_t$. Further assume $u_t$ has zero mean ...
3
votes
1answer
81 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
0
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2answers
664 views

Endowments & Utility Function to get Demand Function

We have two people, $A$ and $B$, A has $200$ units each of both good $X$ and $Y$ and $B$ has $100$ units each of both good $X$ and $Y$. $A$ has tastes providing a utility function such that $u(X,Y) = ...
0
votes
1answer
53 views

How to build (and maximize) this equation

I'm trying to solve an economics problem but I cannot figure out how to build the equation system, or how to find the maximum in a piecewise function. A simplified version of the function would be ...
0
votes
1answer
75 views

Find the transaction cost-adjusted expected return of the stock

Let $W^b_i$ denote the weight of stock i in the existing portfolio and $W^a_i$ denote the weight of stock i in the new portfolio to be created. Let $c_i$ denote the transaction cost of stock $i$. If ...
1
vote
2answers
335 views

Geometric series to calculate price

I decided to add my extension to this question as a new question here. I am trying to represent the following as a geometric series equation: ...
0
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1answer
107 views

Using limit argument with non-continuous social-choice functions

This question is related to another question of mine Invariance of strategy-proof social choice function when peaks are made close from solution, and it revolves around the use of limit arguments with ...
6
votes
1answer
146 views

Invariance of strategy-proof social choice function when peaks are made close from solution

A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ...
1
vote
2answers
283 views

Solving for steady state in macro model, probably just simple calc problem…

I am building a macroeconomic model and I am having trouble calculating the steady state. GDP in the model is determined by Y(L,B,K) = x*L+y*B+z*g*K where (x,y,z) are known constants, L is the ...
0
votes
1answer
24 views

Finding the most profitable option

Right now, I have 1€. And I know exactly how to invest that euro to make profit, but I have 3 options: Invest the euro, and win exactly 1€ (I'd have 2€ then) Invest the euro, and earn some money ...
0
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1answer
75 views

Compound discount

I am trying to construct an equation for calculating a price, that takes into account compound interest of 10%. 1 item = $10 2 items = $19 3 items = $27.1 so ...
1
vote
1answer
59 views

Concept of efficiency in auctions

I have some confusions about the concept of "efficiency" in auction theory. One interpretation is that an auction is efficient if it maximizes the social-welfare. But social-welfare is not well ...
2
votes
1answer
304 views

What kinds of sets are reasonable to place on the continuum?

Warning: I don't know anything about set theory so I wouldn't really know how to spot an existing answer if it were around. Suppose I want to model some economic good or product. I would like to ...
4
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2answers
117 views

Does Arrow's Theorem apply when choosing a single best candidate?

According to Wiki, Arrow's Impossibility Theorem proves that we cannot create a social welfare function that obeys unanimity, non-dictatorship, and IIA. However, in real elections, we want to choose ...
0
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1answer
173 views

Minimizing total cost function

In today's test (question c) I had to minimize equation $(3)$ and solve for N*. I did it through deriving, setting to $0$ and solve for N (no doubts about that). My question is, in this image it ...
0
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1answer
102 views

Calculating the bank multiplier.

The question is as follows: In a simple close economy, banks are required to maintain a liquidity ratio of 8%. An additional £15 billion of currency is deposited in the banking system. Calculate the ...
1
vote
1answer
250 views

Relation between quasilinear utility and quasilinear function

When we say Quasilinear utility, it is known that function is linear in numeraire. It is expected to be linear in one argument and hence it can be called *quasi*linear. Can any one tell me if there is ...
0
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1answer
166 views

Quasilinear utility functions

Is the utility function below quasilinear? $U(X,Y)=XY+10Y$ I know that an equation of the form $U(X,Y)=f(X)+Y$ is quasilinear but I'm not sure about functions of the form $U(X,Y)=f(X,Y)+Y$.
1
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1answer
22k views

Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$. Derive the ...
0
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1answer
349 views

Deriving a demand curve intermediate microeconomics

The exercise says as follows: Zac consumes only pizza and Chianti in fixed proportions. 2slices of pizza per 1 glass of chianti. income is $100. Derive demand functions for pizza and Chianti. How do ...
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1answer
203 views

Preference Relation and Utility Function - Problem with inductive proof

I have a problem with an inductive proof of the following result. Theorem: If $X$ is a finite set, a binary relation $\succ$ is a preference relation iff there exist a function $u:X\rightarrow R$ ...