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Questions tagged [production-function]

A function whose value is the produced quantity associated with a given vector of factor inputs. The production function represents the technology available to the firm.

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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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How to derive firm's cost function from production function?

I recently learned how to solve the following type of problem using the method of Lagrangian multipliers: Given a consumer with utility function $u(x,y)$, wealth $w$, prices $p =(p_x,p_y)$, budget ...
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13 votes
2 answers
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Solow Model: Steady State v Balanced Growth Path

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model: $$ Y = K^\beta (AL)^{1-\beta} $$ I have been asked to derive the steady ...
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12 votes
5 answers
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Why is the Cobb-Douglas production function so popular?

As relatively novice quantitative analyst/ Cost analyst, Ive been asked to estimate the level of a given organizations productivity more than once, and then forecast for the next couple of periods. ...
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How do we estimate production functions?

In a standard economics education we learn about production functions, indicating an output as a function of a given input of capital and labour. An average model looks like this: (1) $F(L,K)=L^{...
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7 votes
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Deriving the translog production function

Ive been having difficulty deriving the translog production function defined as: $$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$ I know ...
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1 vote
1 answer
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Returns to Scale Microeconomics

Are there any production function $f(x_1,\ldots,x_n)$ that is having decreasing returns to scale, given that the marginal product in every input $i$ in the function $f$ is constant?
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CES Production Function with $\rho>1$

In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the ...
Sher Afghan's user avatar
7 votes
2 answers
2k views

CES production function estimation

Introduction There are different ways of estimating the parameters of a production function. For example, single-equation and system equation techniques are both possible. Another difference among ...
Graeme Walsh's user avatar
5 votes
2 answers
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Transformation Function

In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
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Why labour, capital, and output levels cannot be pinned down in perfect competition?

Consider a firm producing with the following technology: \begin{equation} Y = AL^{\alpha}K^{\beta} \end{equation} Assuming that factors are paid their marginal contribution to output, it can be ...
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Lessons From Successfully small island economies

What economic and development lessons/strategies can developing Caribbean countries learn from successfully small island nations like Singapore?
user2960's user avatar
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Combining functions that satisfy the Inada conditions

Suppose $f: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R}^n \to \mathbb{R}$ are functions that satisfy the Inada conditions, and also $$ \forall i: \lim_{x_i \to \infty} h(\mathbf{x}) = \infty. $$ ...
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Why does marginal cost (derivative of total cost) differ from variable cost at each level?

Why does the marginal cost equation (as the derivative of total cost equation) make predictions of variable costs that are very different from costs calculated using the Total Cost equation? Marginal ...
Bryan Gentry's user avatar
4 votes
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How to econometrically identify perfect complements in production?

The production $$f(x_i,...,x_n)=\min\{x_i,...,x_n\}$$ is pretty straight forward and usually with smaller size data sets and can usually be picked up on rather quickly in an intuitive sense. ...
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4 votes
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Deriving long-run cost function

I'm a bit unsure about how to derive a long-run cost function. Suppose my production function was $X(L, K)=L^a K^b$, where $a+b>1$. I'm thinking about doing the following, but I'm not sure it's ...
pril's user avatar
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3 answers
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How was the Cobb Douglas function derived?

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the ...
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3 votes
1 answer
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Homothetic production function and Profit Function

I know that homothetic production function implies that cost function is multiplicatively separable in input prices and output, and it can be written as C(w,y)=h(y)C(w,1). Can some one help me derive ...
Sher Afghan's user avatar
3 votes
1 answer
367 views

Kimball (1995) Specification of Final Good Production

Kimball (1995) defines production of the final good ($Y$) with intermediate goods $y_l$ in his equation (1) as $$ 1 = \int_0^1 G\left(\frac{y_l}{Y}\right) dl $$ with $G(1) = 1$, $G'(x) > 0$ and $...
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How to show the production function is concave in K and L but not strictly so?

Suppose we have a production function with constant returns to scale. Let us denote it by $F(A,K,L)$ where $A$ is the technology, $K$ the capital and $L$ Labor. Further assume the first partial ...
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3 answers
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Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?

There is a market economy with technology given by: $$Y = K^\alpha L^{1-\alpha} \tag{1}$$ Firms behave competitively and input prices are: $$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})...
Tecon's user avatar
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2 votes
2 answers
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In a box diagram, why does efficiency locus lie on one side of the diagonal, if both sectors haves constant returns to scale function?

The following is what I understand, so far. If we measure labour in the $x$-axis and capital in the $y$-axis, the slope of diagonal of the box is the capital-labour ratio $K/L$ in the economy. Let $A$ ...
Shaikh Ammar's user avatar
2 votes
1 answer
227 views

Arguments for Concavity or Quasi-concavity

I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions. For example the production ...
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2 votes
1 answer
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Finding the conditional input demand function

Find the conditional input demand function and cost function for the given production function $$f(a,b,c,d)=\min\{ a,2b\} + \max\{3c,4d\} $$ In The solution, The production function is defined as $f(x,...
studentp's user avatar
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2 votes
1 answer
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Confusing on the CRS Property of CES Function

Say a CES function is that $$Y = A\left[\alpha K^{\rho}+ \beta L^{\rho}\right]^{\frac{1}{\rho}}$$. Clearly this function is constant return to scale whatever the values of $\alpha$ and $\beta$ take. ...
Alalalalaki's user avatar
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1 vote
1 answer
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Mixed Partial Derivatives in Profit Function

$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$ Question 1: Using the first order conditions, we get: $x = \frac{pa}{w_x}$ $z = \frac{pb}{w_z}$ What do we call these Input demand functions as a ...
CormJack's user avatar
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1 vote
1 answer
162 views

Homotheic Function Definitions

There are a number of different definitions of Homothetic functions i have come across. I have used each of them to prove that a function $f(x, y) = x^a y^b$ with $a+b > 0$ is homothetic. But i ...
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China's Prosperity

This is my first time doing this but with respects to economic prosperity. What are some of the tools/strategies that Chinese leader such as Deng Xiaoping would have used to stimulate and enhance ...
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1 vote
1 answer
980 views

Nested CES Production Function

If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a,...
user10158324's user avatar
0 votes
1 answer
800 views

How to prove that a concave production imply that the input requirement sets are convex?

According to page 7 of this slide, "A convex production set Y implies that the associated input requirement set V(y) is convex". How can one go about proving it?
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1 answer
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Homothetic Functions and Monotonic Transformations

Using the following definition of a homotheic function (taken from my Mathematical Economics course pack). A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is homothetic if it has the form: $f(x,y) = q(...
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0 answers
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how is output elasticity different from marginal product of a factor input?

marginal product has been defined as the addition to total product given the employment of one more unit of a factor input. output elasticity has been defined as the percentage change in output given ...
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0 votes
1 answer
64 views

Production function involving profit maximisation

​Hi, I don't get how the answer of d is deduced in this question because I don't think I made any mistakes in my calculation and have also used all the information given. After knowing L is 800, I ...
onetwothree's user avatar
0 votes
1 answer
229 views

Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations

There is one agent with utility function given by: \begin{equation} U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1} \end{equation} With budget constraint: \begin{...
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