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For examples of real-valued function defined on subsets of $\mathbb{R}^2$, consider: Cobb Douglas: $u:\mathbb{R}^2_{++}\rightarrow\mathbb{R}$, where $\mathbb{R}^2_{++}=\{(x,y)\in\mathbb{R}^2|x>0, y>0 …

$k(t)=\dfrac{K(t)}{L(t)}$ Taking log both sides, we get $\ln k(t)=\ln K(t)- \ln L(t)$ Differentiation with respect to $t$, we get $\dfrac{1}{k}\dfrac{dk}{dt}=\dfrac{1}{K}\dfrac{dK}{dt}-\dfrac{1}{L}\df …

In order to determine the labor supply function, we need to solve the following optimization problem: \begin{eqnarray*} \max_{l, L, x} & \ x^{0.5}+l^{0.5} \\ \text{s.t. } & px \leq w(T-l) + m \\ \tex …

Proposition 1. For $a>0,b>0,c>0,d>0$, if $\dfrac{a}{b}=\dfrac{c}{d}$ then $\dfrac{a+c}{b+d}=\dfrac{a}{b}$. Proof is immediate. Proposition 2. For $a_i>0,b_i>0$, where $i\in\{1,2,\ldots,n\}$, if $\dfra …

Utility function $u(x_1, x_2) = x_1^2x_2$. Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$. Here are the demand functions for $x_1$ and $x_2$: $$x_1(p_1, p_2, m) = …

WLOG let $u(0)=0$ and $u(100)=60$. Therefore, $u(40) = \dfrac{60}{2} = 30$. By concavity of $u$, $\dfrac{u(180)-u(100)}{80} \leq \dfrac{u(100)-u(40)}{60}=\dfrac{1}{2}$. This gives an upper-bound on $u …

I think there is some mistake in your calculations for Hicksian Demand. Here is what I have got for the two-commodity case: Given that $u:\mathbb{R}^2_+\rightarrow\mathbb{R}$ defined as $u(x_1,x_2)=-( …

We solve the utility maximization problem of the individual whose utility function is $u(c, l) = c - \frac{l^{1+\gamma}}{1+\gamma}$ to get the supply function. The problem can be written as: \begin{eq …

Let $P = \alpha A + (1-\alpha) B$ where $A$ and $B$ are returns (random) from the two assets, and $P$ is their portfolio. Variance of portfolio $P$ can therefore be written as \begin{eqnarray*} \sig …

If $S_Y$ is the amount saved by $Y$ in the first period, and $T_X$ is the amount transferred by $X$ to $Y$ in the second period, then $X$'s payoff as function of his choice $T_X$, taking as given $Y$' …

I think you are trying to find a feasible allocation that maximises the sum of the utilities of the two individuals. So we can write the objective function as: \begin{eqnarray*} \max_{x,y} & \ xy^5 + …

To show that $\min(f(x,y), g(x,y))$, consider the upper level set $$P^a = \left\{(x,y) \in \mathbb{R}^2 | \min(f(x,y), g(x,y)) \geq a\right\}$$ We'll show that this is a convex set for every $a\in\mat …

Consider the maximisation problem : $$\max_x f(x) \text{ s.t. } g(x) \leq c$$ Note that If $f$ is quasi-concave and $g$ is quasi-convex, then the set of solutions to the above problem is either an em …

Check that the following utility function also represents the same preference as the one in the question: $u(x_1,x_2)=(x_1+1)(2x_2+1)$ Solving $\max_{(x_1,x_2)\in\mathbb{R}^2_+} (x_1+1)(2x_2+1)$ subje …

As you can see in this post, that there are also "corner" solutions to this problem under some conditions. These are solutions where $x_1=0$ or $x_2=0$. For this reason, you may use Kuhn-Tucker (KT) c …