17 votes

Prove the sample variance is an unbiased estimator

For a shorter proof, here are a few things we need to know before we start: $X_1, X_2 , ..., X_n$ are independent observations from a population with mean $\mu$ and variance $\sigma^{2}$ $\mathbb E(...
  • 561
10 votes

How to show that a homothetic utility function has demand functions which are linear in income

I think what you need is that if $U(x,y)$ is homothetic then $$ \forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt \frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial ...
  • 27.3k
9 votes
Accepted

Prove the sample variance is an unbiased estimator

I know that during my university time I had similar problems to find a complete proof, which shows exactly step by step why the estimator of the sample variance is unbiased. The proof I used can be ...
8 votes

Books friendly to self-studying Industrial Organization

While taking Industrial Organization I remember working with: Strategies and games: theory and practice by Dutta Introduction to industrial organization by Cabral Industrial organization: theory and ...
7 votes

Self-teaching 1st year PhD Microeconomic Theory

My prof is always telling us, if we want to pursue PhD level Econ in the future, we should master the full content of the following book: Microeconomic Theory. Andreu Mas—Colell Michael D. Whinston ...
7 votes
Accepted

Books for microeconomics and macroeconomics for selected topics

From Varian (7th edition): Consumer behavior: at least chapters 1–6; preferably also chapters 7, 8, and 12. Perfect competition and Theory of production: chapters 15, 16, and 18-23. Monopoly price ...
  • 16.6k
7 votes

Book recommendations: Introduction to economics

It's great that you're developing an interest in economics. I would suggest Mankiw's Principles of Economics to start with. I believe it meets both your requirements and covers the two major areas of ...
7 votes
Accepted

Properties on conditional demand correspondence from the textbook of Mas-Colell et al

Let $z_1$ and $z_2$ be $\geq 0$ and solution to $$\min_z \{w^\top z\lvert f(z)\geq q\}$$ then clearly $f(z_1)\geq q$ and $f(z_2)\geq q$ and since $\{z\geq 0\lvert f(z)\geq z \}$ is convex it then ...
  • 313
7 votes
Accepted

MWG 8.B.7 - Any strictly dominant strategy must be a pure strategy

Fix any $\sigma_{-i}$. Assume $\sigma_i$ is strictly dominant but not a pure strategy. Let $X$ be the support of $\sigma_i$. Since $\sigma_i$ strictly dominates all pure strategies $s_i\in X$, we have ...
  • 4,909
6 votes

Interest rate rule in monetary DSGE model

I've just solved this problem. First of all, your solution does not make too much sense, as in a simple interest rate rule it must hold that the sum of all coefficients must be greater than one. In ...
  • 148
6 votes

a risk lover agent preferences and the preference of risk natural agent may be the same

Don't commit the cardinal mistake of equating preferences with choices. In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that $...
6 votes
Accepted

Moral hazard vs hidden information.

Hidden information concerns characteristics that are unobservable by one side of the market. For example, a consumer's willingness to pay, a worker's productivity, the quality of a used car all fall ...
  • 14.7k
6 votes
Accepted

Is it right to derive social marginal benefit by adding individual prices instead of quantities?

Vertical summation of the individual marginal benefit curves is the correct way to find social marginal benefit if the camera system, so far as the two stores are concerned, is a public good. ...
  • 7,444
6 votes
Accepted

Finitely repeated Prisoner’s Dilemma with switching cost

A couple hints. Regarding the lower bound on $\epsilon$: What happens if deviation occurs at stage $T$? In other words, there is no opportunity for your so-called "punishment stages". ...
  • 14.7k
5 votes
Accepted

Understanding subscripts in first order conditions of dynamic optimization problems

In an intertemporal maximization problem, we seek to find the optimal sequence of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical"...
5 votes
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What are the prerequisites to study Mathematical Economics?

In the preface, Takayama writes that the book was written with the intention to keep the prerequisites to a minimum: elementary calculus and matrix algebra. Perhaps he was exaggerating a little, but ...
5 votes
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Homogenous of degree one in utility function.

The way you show that $v(p,m)$ is homogeneous of degree one in $m$ is correct, but the reason why this implies that, $e(p,u)$ is homogeneous of degree one in $u$, is not very precise in your argument. ...
5 votes
Accepted

Mixed Nash equilibrium

(i) $A$ & $B$ If player 1 play $A$ with probability $p$ and $B$ with probability $(1-p)$, where $0<p<1$, then player 2's expected payoff from playing $D$ is $4p+4(1-p) = 4$ $E$ is $6p + 2(...
  • 5,525
5 votes
Accepted

separable utility function and cross price effect

The following argument assumes that we are dealing only with interior solutions. Let $(x_1,x_2,x_3)$ be an optimal demand bundle at prices $(p_1,p_2,p_3)$ and income $m$ and assume that $(x_1,x_2,x_3)$...
5 votes
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An overview of 4 books for an undergrad course in Mathematical Economics

All four of those books are widely used at the undergraduate level. They are written with the intention to be text-books of a class, so they should be good for self-study. I recommend you to read 1 ...
  • 4,168
5 votes
Accepted

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

The assumption $dm =0$ says that we examine the behavior of the consumer under a fixed nominal income, and this is something interesting to study, because it aligns to a large degree with the observed ...
5 votes
Accepted

Why there is no supply on monopoly markets?

The argument is that the monopolist's decision is based on the demand curve (in effect matching marginal total revenue to marginal cost) so is not independent of the demand curve, and in that sense ...
  • 4,694
5 votes
Accepted

Stochastic AK model derivation

I think you may substitute it directly as mentioned? $$v(A,k) = \max log(c) + \beta E[v(A',k')|A]$$ Applying the value function expression (note period change): $v(A',k') = \frac{log(k')}{1-\beta} + v(...
  • 515
4 votes

Prove the sample variance is an unbiased estimator

Let's improve the "answers per question" metric of the site, by providing a variant of @FiveSigma 's answer that uses visibly the i.i.d. assumption (showing also its necessity). We want to prove the ...
4 votes

Books for microeconomics and macroeconomics for selected topics

As far as I remember, Varian's book is aimed at second year undergraduates who are normally studying a micro 2 module (or something similar). As a result, he does tend to assume some prior knowledge ...
  • 171
4 votes
Accepted

Question on real exchange rate

Note that I am not 100% sure. But in my understanding, we have Year 1 Price for a product in the US : $p_{US}=v$ \$ Exchange rate: $x$ pesos for $1$\$ Price of the product in the Philipines: $p_{...
4 votes

How can I draw an edgeworth box with 3 different goods?

I assume what you're asking based on your comments is: "How can I visualize indifference curves for 3 goods?" I can think of three options: 1) Use a tool like Matlab, or its open-source equivalent, ...
  • 394
4 votes

Resources to improve my economic intuition

While textbooks are the best way to learn the material (MC=MR etc), here are some suggestions for improving your intuitive understanding of economics. Books The Undercover Economist by Tim Harford ...
4 votes

Deriving long-run cost function

You're right. Divide Eq (1) by Eq (2): $$ \frac{a L^{a-1}K^b}{bL^aK^{b-1}} = \frac{aK}{bL} = \frac{w}{r} ~~~\Rightarrow~~~ L = \frac{ar}{bw}K \tag{4} $$ Now use this in Eq. (3) $$ C = wL + rK = \...
  • 1,206
4 votes
Accepted

Optimization problem with Kuhn-Tucker conditions

Player $1$'s utility maximization problem is the following : $$\max_{0 \leq t_{12} \leq T} \ \ \left(T-t_{12} + t_{12}t_{21}\right)^a \left(w(T-t_{12})\right)^{1-a}$$ An equivalent way to solve the ...
  • 5,525

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