Questions tagged [leontief]
The leontief tag has no usage guidance.
28
questions
4
votes
0answers
37 views
CES utility with negative values
We know that the CES utility function approximates a Leontief utility function as in the following:
$$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\...
2
votes
1answer
101 views
Cost Minimization of $f(x) = min(x_1,x_2) + x_3$
The following production function is given,
$f(X) = min\{x_1,x_2\} + x_3$
There is a solution here https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-...
1
vote
1answer
88 views
How to find the cost function for perfect complements [closed]
Imagine I got a production function like it :
$$
\min\{x_1, x_2\}
$$
How can I find the cost function?
3
votes
2answers
424 views
What is the returns to scale of the production function q = min {K, L^(1/2)}?
I learned that when there is decreasing returns to scale, the average cost is always increasing.
But the professor told us today that the other way around might not always be true. So if average cost ...
0
votes
0answers
162 views
General Equilibrium: Two Consumers with Perfect Substitutes and Perfect Complements Utility Functions
I am attempting to solve a general equilibrium in pure exchange economy problem where the first consumer has an endowment of $(3, 4)$ and a utility function of $U(x) = 3x + 5y$.
The second consumer ...
1
vote
1answer
121 views
Walrasian demand with a twist of Leontief function
A consumer has the utility function $u(x_1; x_2) = \min(x_1; x_2) + 5 \max(x_1; x_2)$.
Find its Walrasian demand $x^*(p; w)$.
I've tried searching it up when we have two Leontief functions summed ...
3
votes
1answer
342 views
Leontief input output model with column sum greater than 1
In a linear algebra textbook I came across the following question (not included in the answer key):
Consider an open economy with a consumption matrix
\begin{equation}
C =
\begin{...
0
votes
0answers
173 views
how to calculate Leontief demand functions from first order conditions of a CES function when sigma tends to 0?
This question is NOT about how to approximate a CES function to a leontief function.
Knowing that:
$i= good (\begin{array}{*{20}{c}}
{1}&{or}&{2}
\end{array})$
$j= firm (\begin{array}{*{20}{c}}...
1
vote
2answers
317 views
Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$
I know how to solve the two-good case with $u(x) = \min\{x1, x2\}$, but the addition of $x3$ confuses me.
Problem
Derive the demand function $x(p,w)$ from $u(x) = \min\{x1, x2\} + x3$
What I did ...
0
votes
0answers
141 views
How to show that a leontief utility function is homothetic?
usually what i do to show that a utility function is homothetic is by either showing that the function is homogeneous or if the MRS is homogeneous of degree 0.
However the MRS is not defined. So im ...
0
votes
1answer
106 views
Optimal choice for a weird leontief function
Compute the optimal choice for a consumer with the following utility function:
$$u(x_1, x_2) =\max \{\min(2x_1, x_2), \min(x_1, 2x_2)\}$$
I'm familiar with computing optimal choice for perfect ...
0
votes
2answers
703 views
Pareto set with Cobb-Douglas and Leontief preferences
If $U_A(x_A,y_A)=x_Ay_A$ and $U_B(x_B,y_B)=min(x_B,y_B)$ and the total endowments are (8,4), is the Pareto set given by the line joining the kinks of B (black line shown in the diagram)? Shouldn't the ...
2
votes
1answer
234 views
Marshallian demand with Leontif preferences
Consider a utility function on the form $u(q_{1},q_{2},q_{3}) = min\{\alpha ln(q_{1}) + (1 - \alpha) ln(q_{2}), ln(q_{3})\}$
I know that optimal behaviour requires $\alpha ln (q_{1}) + (1 - \alpha) ...
-1
votes
1answer
1k views
Expenditure minimization with Leontief utility
I need to solve the expenditure minimization in a context where $u(x,y) = min\{x,y\}$, i.e. where utility is Leontief.
The minimization problem is
$$\text{min}_{x,y}\,\,p_xx+p_yy \\
\text{subject}\,...
3
votes
3answers
2k views
What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)
I encountered that nobody, even my profs and lecturers so far, has an intuitive way to explain what the Leontief inverse represents. Does somebody here?
As most people here would know, it goes:
$\ ...
6
votes
1answer
94 views
Leontief's Paradox and the amount of capital
Leontief (1953) showed that US exports in 1947 embodied considerably less capital and somewhat more labor than would be required for domestic production of competitive imports. \$2.55 million worth of ...
2
votes
0answers
140 views
Endowment increases but utility reduces
In an Edgeworth economy with two agents and two goods, Player A has endowment $\left(a,\,0\right)$ and utility $u_{A}\left(x_{1},\,x_{2}\right)=\min\left(\gamma x_{1},\,x_{2}\right)$ in which $\gamma&...
1
vote
1answer
676 views
Derivation long run cost function of three inputs with Leontief-like characteristics
Suppose that a firm produces a good using capital, skilled labor, and unskilled labor. Let $K$ denote the amount of capital,$L_1$ unskilled labor, $L_2$ skilled labor. The production function is $f(...
3
votes
1answer
5k views
Leontief function marginal product of labor/capital
Find marginal product of labor of a Leontief production function. For example,
$$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$
MY ATTEMPT
Now as I understand it the marginal product of labor, $MP_L$ ...
1
vote
0answers
22 views
Perfect complement outputs with each output being composed of substitutable inputs
How does one solve the following maximization problem?
$\underset{K_1, K_2, L_1, L_2}{\text{maximize }} min\{K_1 + L_1,K_2 + L_2\}$
subject to $c(K_1 + \mu K_2) + \beta c(L_1 + \mu L_2)$
where $c(...
0
votes
1answer
57 views
why not use $Ap - p$ instead of $p - Ap$?
I am watching this video about the Leontief input-output model: https://youtu.be/hlaBURtSDO8?t=12m26s
One thing I don't understand is: why not use $Ap - p$ instead of $p - Ap$? It seems to make ...
1
vote
4answers
2k views
Weird Leontief production function
I am solving some micro related exercises and I came across this weird Leontief production function: $$Q =\left(\min\{K, L\} \right)^b$$
I am not sure how to solve it. I have to find the inputs' ...
1
vote
0answers
70 views
First Reference Leontief/Perfect Complements
I googled a lot and I'm still to find:
1. In which paper/book/reference do Leontief [production] functions make their first appearance?
Similarly
2. In which paper/book/reference do perfect ...
1
vote
1answer
4k views
maximizing leontief-type utility function
How do I maximize the utility function:
$ U(x,y)= max(ax,ay)+min(x,y) $ , where $ 0<a<1 $ with respect to prices $ p_{x}, p_{y} $ respectively and income $ m $.
I know leontief-type utility ...
7
votes
2answers
1k views
Constant Elasticity of Substitution: Special Cases
Take an $n$-commodity constant elasticity of substitution utility function,
$$U = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho}$$
How do we show the following:
Show that as $\rho \...
3
votes
1answer
466 views
Leontief preferences and 2nd welfare theorem
Does the 2nd welfare theorem hold with Leontief preferences? If not, which of the assumptions does not hold?
8
votes
1answer
6k views
Leontief preferences
I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really ...
26
votes
2answers
34k views
How can I obtain Leontief and Cobb-Douglas production function from CES function?
In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function,
$$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$
(where the ...
