7
One of the most important consideration for using CARA utility is tractability. CARA utility and Gaussian errors yield a certainty equivalent completely described by a simple function of the mean and variance of the Gaussian distribution..
It also turns out that maximizing expected utility in this setup is equivalent to maximizing the certainty equivalent -...
6
In my honest opinion the best book in this regard is fundamental methods of mathematical economics by Chiang and Wainwright.
It starts at the very basics and then goes all the way up to dynamic optimisation, also covering linear algebra, matrices, advanced calculus topics such as implicit functions and equation solving at all levels.
Combined with a fine ...
5
I don't believe it is lower semicontinous.
Let $w = (0,\dots,0)$, $p \in \mathbb{R}^n_+$ be any vector such that $p_1 = 0$ (the first coordinate being 0).
The allocation $x=(1,0,\dots,0) \in B(p,w)$.
Define the sequence $p_n = p + (\frac{1}{n},0,\dots,0)$ and $w_n = (\frac{1}{n},0,\dots,0)$. $w_n \rightarrow w$ and $p_n \rightarrow p$.
For any $x^n \in B(...
5
If you have time and patience, "Foundations of Mathematical Economics" by Michael Carter is great. The book consists mostly of exercises that let you, broken down in manageable steps, prove many landmark results in mathematical economics on your own. Mathematics is not a spectator sport and this book will give you a great workout. If you do the ...
4
The set need not be bounded. To see this, just take $U$ to be constant. Then the set will either be empty or equal to $\mathbb{R}_+^n$.
It is also possible that no minimum exists. Let $n=2$, $p=(0,1)$, $v=1$, and $U$ be given by $U(x)=U(x_1,x_2)=x_1\cdot x_2$. Clearly, $px=0$ is only possible if $x_1=0$, which would lead to a utility of $0$. But for every $\...
3
One which I have found to be a great level for someone who is slightly shaky in fundamentals is Carl P. Simon and Lawrence Blume's Mathematics For Economists. It is slightly pricey but has stood up as a main pillar IMO for mathematical concepts in economics.
The one for which I used in my undergrad and then used again in my preparation for grad school was ...
3
One approach could be the following. For a $(p_n,w_n)$ in the sequence and $x \in B(p,w)$ define:
$$ \alpha_n = 1 \text{ if } p_n x \le w_n$$
and
$$ \alpha_n = \frac{w_n}{p_n x} \text{ if } p_n x > w_n$$
Then define:
$$ x_n = \alpha_n x$$
Here $x_n$ equals $x$ if $x$ is in the budget $B(p_n,w_n)$. If not, then $x_n$ is the radial projection of $x$ onto ...
3
Some good textbooks that focus on mathematics for economics are:
Essential Mathematics for Economic Analysis by Knut Sydsaeter, Peter J. Hammond, Andres Carvajal and Arne Strom - not technically graduate level but it has some topics that go beyond 'simple calculus'
Further Mathematics for Economic Analysis by Knut Sydsaeter - this is text that is fully on ...
2
If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,
$$ x c'(x) = k c(x).$$
That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is ...
1
What the authors seem to be doing in the footnote is imposing a condition stating that if intergenerational transfers tend to infinity (say for the case of children of billionaires) there will be no growth in wealth from human capital investment.
This condition is here to just communicate that the parent views allocation between human capital of the child ...
1
I assume $x_i$ represents the quantity and belongs to $\mathbb R_{+}$. You can form the constraints as follows:
$$
x_i \geq0 \quad\forall i \in [3] \\
\sum_{i=1}^3p_ix_i \leq I
$$
You can simplify the objective by noting that for the utility to be the maximum, $x_2 =x_3$. Try to reason why this is true. Hence, the final problem becomes,
$$\
\max_{x_1, x_2, ...
1
One of the simplest specification I can think of (and for which the first order condition can be solved analytically in $L$) is:
$$
y=\left\{
\begin{array}{ccc}
L^{\alpha} & & L\leq L_{e} \\
L_{e}^{\alpha}+g\left( L-L_{e}\right) & & L>L_{e}%
\end{array}%
\right.
$$
with $g\left( L-L_{e}\right) =(L-L_{e})^\beta$ and $\alpha\geq1$ and ...
1
I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose.
You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.
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