Questions tagged [leontief]

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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 ...
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.
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3 votes
2 answers
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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 ...
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(...
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0 votes
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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\...
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-...
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?
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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 ...
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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 ...
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{...
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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 ...
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 ...
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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 ...
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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 ...
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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) ...
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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}\,...
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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 ...
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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&...
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(...
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, $MP_L$ ...
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(...
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 ...
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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' ...
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 ...
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 ...
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 \...
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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?
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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 ...
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 ...
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