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5 votes

Question for general equilibrium

To find efficient allocations in this economy, we can first determine the production possibility frontier (PPF) which is given by the line segment $\dfrac{x}{2}+y=100$ where $x\in[0,200]$. Pareto ...
Amit's user avatar
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5 votes

Weird Leontief production function

This is not a weird case, but a Leontief production function which is not homogeneous of degree one, but homogeneous of degree $b$. You can see this if you use the connection between a C.E.S. ...
Alecos Papadopoulos's user avatar
4 votes
Accepted

Cost Minimization of $f(x) = min(x_1,x_2) + x_3$

While not said explicitly in the question I am guessing from the Langrangian function you set up that the problem you intend to solve is $$\min_{x_1,x_2,x_3} p_1x_1 + p_2x_2 + p_3x_3 \\[8pt] h(x_1,x_2,...
Jesper Hybel's user avatar
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4 votes

What is the returns to scale of the production function q = min {K, L^(1/2)}?

To understand what is the issue here, try dutifully to examine all possible sub-cases in the production function. The production function is $$Q_0 = \min\{K_0, L_0^{1/2}\}$$. Consider cases A. $K_0 &...
Alecos Papadopoulos's user avatar
4 votes
Accepted

What is the returns to scale of the production function q = min {K, L^(1/2)}?

You have a Leontief production function and in optimum you will always have $K=\sqrt{L}=q_1$. Now increase both inputs by factor $k>1$ and you arrive at $k*K > \sqrt{kL}=q_2$ where the first ...
Bayesian's user avatar
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4 votes
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Walrasian demand with a twist of Leontief function

Why don't you just plug in some values for $x_1$ and $x_2$? Start with something like $x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$. Then increase $x_1$ or $x_2$ and let $x_1=2, x_2=1$ and find ...
Bayesian's user avatar
  • 5,290
4 votes
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Leontief function marginal product of labor/capital

Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at $\bar{K}$. Then, the optimal choice of capital and labour is given by: $$\frac{L^*}{a}=\frac{\bar{...
luchonacho's user avatar
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4 votes
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What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)

Something you didn't mention is that $x = Ax + y$ means that to produce one unit of $x$, you use $A$ unit of $x$. E.g. you need electricity to produce electricity. Under the condition that $1>|A|...
keepAlive's user avatar
  • 1,425
4 votes

maximizing leontief-type utility function

You can re write $u(x, y)$ as \begin{eqnarray*} u(x, y) = \begin{cases} ax + y & \text{ if } x > y\\ ay + x & \text{ if } x \leq y\end{cases} \end{eqnarray*} When one plot the ...
Amit's user avatar
  • 8,476
4 votes

Weird Leontief production function

We are given the production function $Q = \left(\min(K, L)\right)^\beta$. Cost minimization problem of the producer is defined as finding the labor-capital combination that minimizes the cost of ...
Amit's user avatar
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3 votes
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Pareto set with Cobb-Douglas and Leontief preferences

I believe you are correct. The points for which Ya = 0 and \begin{equation} 0\le Xa \le 4 \end{equation} will be all Pareto efficient points. Proof: Consider an allocation like (2,0). The ...
Rayhan Ali's user avatar
3 votes

Pareto set with Cobb-Douglas and Leontief preferences

The bottom right origin is actually not in the Pareto set. At that point, $(x_A,y_A)=(8,0)$, so $U_A(x_A,y_A)=0$. Similarly, $(x_B,y_B)=(0,4)$, so $U_B(x_B,y_B)=0$. As an example, $B$ could give ...
ExpelledFromParadise's user avatar
3 votes

Is the leontief utility function homogeneous of degree zero? And if that is true, how can that be prove?

Let $x$ and $y$ be the quantities of the two goods and $t>0$ and $U$ a standard Leontief utility function. $U(tx,ty) = min\{tx,ty\}$ Since $t>0$, it can be factored out of the min $U(tx,ty) = t \...
Nicolas Torres's user avatar
3 votes

Weird Leontief production function

No problem. $$Q =\left(\min\{K, L\} \right)^b$$ It just means that first you compare $K$ and $L$ and your quantity $Q$ will be equal the lower one to the power $b$. Example: $b=2$, $K=3$ and $L=7 \...
snoram's user avatar
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3 votes
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How to find the contract curve when an agent has Leontief utility?

In my opinion, your solution is correct. The contract curve must be such that $x_{1B} = x_{2B}$, otherwise it could be possible to increase the utility of one consumer without decreasing the utility ...
BakerStreet's user avatar
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3 votes
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How to find the contract curve for a funky utility involving the min operator?

Given a pure-exchange economy: $u_1(x_1,y_1) = \min(3x_1+y_1,x_1+3y_1)$, $u_2(x_2,y_2)= x_2^{0.5}y_2^{0.5}$ Total Endowments of X and Y are $\omega^X > 0$, and $\omega^Y > 0$, respectively. ...
Amit's user avatar
  • 8,476
2 votes
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Leontief's Paradox and the amount of capital

Leontief did not use total capital stock for his calculation of capital/labor requirements in the export/import sector. He measured 'requirements per million dollars of exports and imports ...
hrrrrrr5602's user avatar
2 votes

Derivation long run cost function of three inputs with Leontief-like characteristics

Your reasoning that $L_1=L_2^{1/3}$ is valid for any $K$. Indeed if this equality does not hold you can lower the cost by reducing the excess input of either skilled or unskilled labor. Thus you can ...
Oliv's user avatar
  • 3,242
2 votes

why not use $Ap - p$ instead of $p - Ap$?

$p$ represents total production. $Ap$ represents the intermediate goods and services used in production, i.e. intermediate consumption. So $p-Ap=(I-A)p$ represents net production, i.e. output minus ...
Henry's user avatar
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2 votes
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how to derive marshallian demand functions from leontief preferences?

The utility function looks like this: $(big)^2 - (small)^2$ Since $small$ is something that takes away utility, you want $small = 0$. Otherwise, you’re spending some money in getting unhappier, ...
Nicolas Torres's user avatar
2 votes

Optimal choice for a weird leontief function

One needs to go case-by-case and arrive at a utility function with branches. To get you started, if $x_1 < x_2/2 \implies \min(2x_1,x_2) = 2x_1$, but then also $\min(x_1,2x_2) = x_1$. Therefore in ...
Alecos Papadopoulos's user avatar
2 votes

What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)

The series <I + A + A^2 + A^3 + ... + A^N> converges on the Leontief inverse (I-A)^(-1) as N approaches infinity. In this format, I can be thought of as initial demand for a given product, A ...
Tom Howells's user avatar
2 votes

What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)

What intuition can the mathematical concept of an inverse (function, matrix) have? In a single-input monotonic production function $F(x) = q \implies x= F^{-1}(q)$ the operator $F^{-1}$ is the ...
Alecos Papadopoulos's user avatar
2 votes
Accepted

How to solve a Leontief Production function?

Short Run Cost Minimisation problem: \begin{eqnarray*} \min_{l\geq 0} & \ wl +rk \\ \text{s.t. } & y\leq \min(\sqrt{l},\sqrt{k}) \end{eqnarray*} where $y\geq 0$, $k\geq 0$, $w>0$, $r>0$ ...
Amit's user avatar
  • 8,476
1 vote

Homogeneity of compensated demand for Leontief (perfect complements) function

Given a utility function $u:\mathbb{R}^L_+\rightarrow\mathbb{R}$, price vector $p\in \mathbb{R}^L_{++}$ and target level of utility $\mu\in\mathbb{R}$, expenditure minimisation problem is defined as ...
Amit's user avatar
  • 8,476
1 vote

Homogeneity of compensated demand for Leontief (perfect complements) function

Since $x({\mathbf p},u)=u$ and $y({\mathbf p},u)=u/2$ do not depend on prices ${\mathbf p}$, homogeneity of degree zero in prices is trivially satisfied for this special case: $x(t{\mathbf p},u)=x({\...
VARulle's user avatar
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1 vote

Marshallian demand with Leontif preferences

So, I have not really worked out the maths behind solving this. I prefer to take a shortcut and just compute it numerically. Here is what I have. These are the contour surfaces and the budget ...
erik's user avatar
  • 721
1 vote
Accepted

Expenditure minimization with Leontief utility

The Lagrangian method wouldn't be of any use, because Leontief function is not differentiable at the point of optimality/kink. However, you can consider the following approach. We know that for $U(x,...
superhulk's user avatar
  • 525
1 vote

Leontief input output model with column sum greater than 1

You are right when saying that mathematically in the cited theorem the condition of column sums being less than 1 is not an "if and only if" condition and thus exceptional circumstances are ...
Atilio Morillo's user avatar
1 vote
Accepted

Leontief input output model with column sum greater than 1

In terms of if and only if statements according to Peterson & Olinick (1982); A substochastic matrix A is productive if and only if $I-A$ is nonsingular. In substochastic matrix the sum of ...
1muflon1's user avatar
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