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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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304 views

Is Artificial Intelligence a completely new (and underestimated) production factor?

Accenture research defines Artificial Intelligence as completely new production factor along labour and capital. Is such notion acceptable by economists? As far as I understand then every asset of AI ...
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Fisher Separation with negative interest rate

In class we discussed Fisher Separation which states that the investment decision is independent of the financing decision. The optimality conditions are that MRS = MRT = (1+i) (i = interest rate). ...
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How do you determine if the production function has decreasing returns to scale?

How do you determine this for the production function $f(k,l) = k^{1.4}l^{0.5}$ ? So far, I have found the marginal product of both labour and capital however, the marginal product of labour is ...
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the difference between cobb-Douglass and leontief production technology [closed]

whatis the differences between cobb-Douglass and leontief production technology
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Does the marginal product need to be diminishing to have a diminishing average product?

We have the rule 'If the marginal is less than the average, then the average declines'. So if our production function for an input x and output y is concave, does that mean that MP must be ...
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Some doubts about netput vectors

I have started studying producer theory on my own and there are some confusions. We know that a production plan is $y=(y_{1},y_{2},y_{3}....y_{L})$ where $ y_{i} $ is an output if its greater than $0$...
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Is it possible to derive the marginal product of an input using a transformation function?

I'm using a transformation function $F(\cdot)$ to describe a production set $y = (x, z, L, K)$, where $x$ and $z$ are private goods denoted by positive numbers, $L$ is labour input, $K$ is capital ...
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conditional factor demand functions for capital and labor [closed]

I don't how to put maths in here so using an image:
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304 views

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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281 views

Absolute Value of Total Factor Productivity in an Aggregate Cobb-Douglas Production Function

A standard formulation of the Cobb-Douglas production function (e.g. here) is: $$Y=AK^{\alpha}L^{\beta}$$ Have there been estimates, at aggregate level for any large economy, of the absolute value of ...
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Constant Returns in a Production Function $\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$ ($R$ = Resource)

In his 1977 article (from which has developed a considerable literature on the Hartwick Rule for maintaining long-term constant consumption given depletion of non-renewable natural resources), ...
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Returns to scale - Constant Function

Suppose we have a production function $f(z)=2$. I am asked to determine whether the function exhibits increasing, decreasing, constant or no returns to scale. For $t>0$, $f(tz)=2$. I'm not sure ...
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How to find a firm's cost function based on its production function [closed]

The question A firm’s production function is given by $$q=F(L,K)=L^{1\over{4}}K^{1\over{4}}$$ find the firm's cost function $C(w,r,q)$, What I know so far I'm aware that the Technical rate of ...
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The factor elasticities from a Cobb-Douglas function in Romer's macroeconomy book

Good night, I'm reading the Romer's macroeconomy book in the page 42, "A complication" section title. The begin of third paragraph say: This is not a general property of production functions, ...
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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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Slope of a production function

Let $F(K,L)$ be a production function with variables $K$ for capital and $L$ for labor. The slope of the $F(\overline K,L)$ ($K$ taken constant) is defined as the marginal product of labor ($MPL$) ...
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Derive the Cost Function TC(Q)

Suppose $F(K,L)= 50L^{\frac{1}{2}}K^{\frac{1}{2}}$, the wage is $w = 5$ (euros) and rent is $r = 20$ (euros). What is the cost of producing $1000$ units? Derive the cost function $TC(Q)$. I know ...
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Capital - augmenting and energy-augmenting technical progress

I've read a number of papers on factor - augmenting technical progress and directed technical change. Authors do not note about the implications of factor - augmentation in sufficient details. Assume ...
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Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates?

Suppose you have a neoclassical production function with N-inputs $F(x_t^1,...,x_t^N)$ All input factors grow in continuous time with constant, but not identical growth rates $g^j$. Assume $g^1 \leq ...
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Estimating elasticity of substitution in nested CES functions

I have aggregate data on $L_t, K_t$ and $X_t$, and want to estimate elasticity of substitution parameters, $\gamma$ and $\sigma$ for these factors. Assuming the production function takes the following ...
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1answer
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Contract curve for firms with linear utility functions

I am attempting to find a contract curve for a production economy with two linear utility functions. Normally, I would find the point where the Marginal Rate of Technical Substitution were equal for ...
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CES: Production function: Elasticity of substitution $\sigma = 1/(1 + \rho)$

I have to prove that $\sigma = 1/(1 + \rho)$ for the CES production function: \begin{align} q = (l^\rho + k^\rho)^\frac{1}{\rho} \end{align} I found out that I need to solve the following equation: \...
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Robinson Crusoe Production Economy [closed]

Robinson Crusoe’s preferences over coconut consumption, C, and leisure, R, are represented by the utility function U(C, R) = CR. There are 48 hours available for Robinson to allocate between labor and ...
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Examples of technical progress

I'm wondering if anyone can give me some intuition about technical progress. I can buy the idea that factors improve in quality over time. So 5 workers and/or 5 machines in 1990 can produce more than ...
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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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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 ...
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Is a production function bilinear?

I believe the following is the multiplicative property of bilinearity: $$ Y=F(K,AL) $$ $$ c_1 F(K,AL) = F(c_1 K, AL) $$ $$ c_2 F(K,AL) = F(K, c_2 AL) $$ But when we have multiplied through the ...
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Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8

Letting $f_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x_i$ is a scalar, commodity $i$ in the bundle) show that, $\sigma_{ij}(\mathbf{x})\...
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Marginal Product of Capital in the Solow Model

In the classic form of the Solow Model: $$ Y=K^\alpha (AL)^{1-\alpha } $$ Describe circumstances in which the marginal product of capital could rise over time, at least for a temporary period. I've ...
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Solow Model, Growth rate of K/L and Y/L in steady state

I have been given the following setup: $$ Y=K^\theta (AL)^{1-\theta }$$ Where Y = Output, K = Capital, L = Labour and A = Productivity. $$ \frac{\dot{L}}{L} = n $$ $$ \frac{\dot{A}}{A} = g $$ The ...
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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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Inverting price functions and their conditions/intervals

If I have demand functions For $P<15$: $$ Q(P) = 700-40P $$ For $P>15$: $$ Q(P) = 400-20P $$ If I invert them, I get the price functions For $Q<100$: $$ P(Q)=20 - (1/20)Q $$ For $Q>...
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How does a positive capital shock cause an increase in land price?

I am trying to determine the exact theoretical mechanism for a positive capital shock to create an increase in land price. Assuming output is a function of land ($L$), labour ($N$) and capital ($K$) ...
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1answer
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Aggregating CRS Production Functions

If thera are two firms and both of them have constant returns to scale production function. Will the aggregate/industry production function still be the sum of individual production functions. How ...
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Decreasing Costs, Increasing Returns to Scale, & C''(q)

Given a profit-maximizing firm with production function $f(x_1,x_2)$, I understand that we can formulate a firm's cost function $C(q)$ by using the contingent demand functions $x_1^c$ and $x_2^c$. We ...
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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 ...
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Monotonic Transformation

How does positive monotonic transformation of production function effect the resulting profit function? For example if we had production function $f(x) $ and that gave profit function $\pi(p,w)$. Now ...
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Homothetic Production Technologies

Can someone suggest a good resource on homothetic technologies and what properties they imply about cost function, profit function, input demands, output supply etc? Also is it possible to have ...
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651 views

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 ...
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Examples of how economists come up with a production function for a firm?

I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes ...
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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 ...
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Concave production function implies convex cost function

Let's assume we have an increasing production function $f:\mathbb{R^+} \to \mathbb{R^+}$ Now, assume this production function is concave and that the price of input z is fixed (this is a single-input ...
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Production function and isoquant slope

Given the company's production function $f(L,K)=L^{1/3}K^{3/4}$, find slope of the isoquant passing through $(L,K)=(20,40)$ is equal to $-4/5$ (K is on the vertical axis). I need to state whether ...
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What does it mean by 'intensive form'?

I encountered this phrase while reading up on growth models - that is to work with a function in intensive form. What does it mean when a function is in intensive form? Thanks!
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Model for simple production chain economy

Given a simplistic economical model with a finite number of possible resources, and a set of machines that can be produced (with a cost) that that take some or none input, and give some output per ...
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What exactly is L in a Cobb-Douglas production function?

Cobb-Douglas is $Y = AK^{1-\alpha}L^{\alpha}$. What exactly is $L$ in Cobb-Douglas? Is $L$ the number of workers available in a single year? Working hours?
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Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?

Marginal costs MC is defined as $MC=\frac{dC}{dq}$. Taking into account that $C=wL+rK$, $$MC=\frac{dC}{dq}=w\frac{dL}{dq}+r\frac{dK}{dq}$$ Recall that marginal product of labor $MP_{L}=\frac{\...
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How to derive cubic cost function from a problem of constrained optimization?

The cubic total cost function usually take the form $TC(q)=a+bq+cq^{2}+dq^{3} \qquad a,b,d>0, c<0$ and $c^{2}<4bd$ I know that from a constraint maximization problem $min\quad wL+vK$ ...
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417 views

Find Change in output from marginal products of labor/capital

A firm produces 231 doohickeys with 8.4 units of labour and 22.1 units of capital. the marginal product of labour is 18, the marginal product of capital is 20. Approximately how many doohickeys will ...
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112 views

Minimization of costs combination of factors of production

Given the production function $D(x,y,z)=\min\{2x, \left( 2y+4z\right)\}$, and knowing that P-price $P_x=30,P_y=20,P_z=5$ and volume of production $D=200$. How much of x,y and z is needed to minimize ...