Questions tagged [leontief]
The leontief tag has no usage guidance.
33
questions
1
vote
1
answer
83
views
how to derive marshallian demand functions from leontief preferences?
For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a ...
- 121
0
votes
1
answer
65
views
Is the leontief utility function homogeneous of degree zero? And if that is true, how can that be prove? [closed]
I have not been able to find a mathematical prove is such statement.
- 1
3
votes
2
answers
260
views
Question for general equilibrium
On the production economy which have 3 good $x$, $y$, $l$ and $2$ consumers(called $1$ and $2$) and two firms(called $X$ and $Y$). Firm $1$ is owned by $1$ and produces only $x$ in function $x=2l$ and ...
- 31
0
votes
2
answers
153
views
Homogeneity of compensated demand for Leontief (perfect complements) function
In my assignment I have a Leontief (perfect complements) function u(x,y)=min(x,2y). Keeping utility fixed, we minimize the expenditure. Since we have a Leontief function, at a fixed level of utility u(...
- 11
0
votes
0
answers
16
views
Calculating Direct and Indirect Resource Use using Input Output Approach
I have been referring to Perman's book titled, "Natural Resource and Environmental Economics" for economy wide environmental modelling. In Chapter 8, Section 8.2 pg.257-259; it talks about ...
0
votes
0
answers
90
views
How to get from a CES production function with inverse elasticity on weights to the special cases Cobb-Douglas & Leontief
I am dealing with a CES production function, and I have attempted some of the "traditional" ways to derive the Cobb-Douglas (logs & l'Hôpital) but I am not sure how to deal with the ...
4
votes
0
answers
67
views
CES utility with negative values
We know that the CES utility function approximates a Leontief utility function as in the following:
$$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\...
- 41
2
votes
1
answer
453
views
Cost Minimization of $f(x) = min(x_1,x_2) + x_3$
The following production function is given,
$f(X) = min\{x_1,x_2\} + x_3$
There is a solution here https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-...
- 51
1
vote
1
answer
518
views
How to find the cost function for perfect complements [closed]
Imagine I got a production function like it :
$$
\min\{x_1, x_2\}
$$
How can I find the cost function?
- 61
5
votes
2
answers
2k
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 ...
- 305
1
vote
1
answer
237
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 ...
- 57
4
votes
2
answers
836
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{...
user24655
0
votes
0
answers
357
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}}...
2
votes
3
answers
412
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\{x_1, x_2\}$, but the addition of $x_3$ confuses me.
Problem
Derive the demand function $x(p,w)$ from $u(x) = \min\{x_1, x_2\} + x_3$.
What I ...
- 21
0
votes
0
answers
272
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 ...
- 47
0
votes
1
answer
139
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)\}$$
I'm familiar with computing optimal choice for perfect ...
- 11
1
vote
2
answers
1k
views
Pareto set with Cobb-Douglas and Leontief preferences
If $U_A(x_A,y_A)=x_Ay_A$ and $U_B(x_B,y_B)=min(x_B,y_B)$ and the total endowments are (8,4), is the Pareto set given by the line joining the kinks of B (black line shown in the diagram)? Shouldn't the ...
- 227
2
votes
1
answer
314
views
Marshallian demand with Leontif preferences
Consider a utility function on the form $u(q_{1},q_{2},q_{3}) = min\{\alpha ln(q_{1}) + (1 - \alpha) ln(q_{2}), ln(q_{3})\}$
I know that optimal behaviour requires $\alpha ln (q_{1}) + (1 - \alpha) ...
- 29
0
votes
1
answer
2k
views
Expenditure minimization with Leontief utility
I need to solve the expenditure minimization in a context where $u(x,y) = min\{x,y\}$, i.e. where utility is Leontief.
The minimization problem is
$$\text{min}_{x,y}\,\,p_xx+p_yy \\
\text{subject}\,...
- 754
3
votes
3
answers
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
1
answer
102
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 ...
- 6,855
2
votes
0
answers
142
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&...
- 31
1
vote
1
answer
793
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(...
- 196
4
votes
1
answer
6k
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, $MP_L$ ...
- 196
1
vote
0
answers
24
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(...
- 11
0
votes
1
answer
59
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 ...
- 113
1
vote
4
answers
3k
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' ...
- 39
1
vote
0
answers
75
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 ...
- 305
1
vote
1
answer
5k
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 ...
- 33
7
votes
2
answers
2k
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 \...
- 6,476
3
votes
1
answer
595
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?
- 31
8
votes
1
answer
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 ...
- 81
31
votes
2
answers
38k
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 ...
- 713