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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CES aggregator for intermediates with heterogeneous productivity

I'm following Atkeson, Burstein (2019 JPE), and cannot understand the aggregation result. There is measure $M(z)$ of firms with productivity $z$, with production function $$ y(z) = z k(z)^\alpha l(z)^...
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55 views

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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61 views

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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comparison of micro production functions

There are many different production function estimation methods, relevant for micro and firm data. For example Olley-Pakes, Levinsohn-Petrin, Ackerberg et al., Wooldridge etc. But does anyone know of ...
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46 views

Can technical change have a negative growth rate?

Using the following nested CES function I have backed out augmentation indices $A_L$ and $A_E$. Where $A_L$ is a labour-augmenting technical change index and $A_E$ is an energy-augmenting technical ...
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2k views

Derive the cost function and supply function from production function

I didn't study economics, but am quite interested in the topic. I came to the question whether I could derive the supply curve / marginal cost function from the production function and I actually ...
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131 views

Wages, capital: Substitution and Output Effects

Consider a CES production function $Y=f(K,L)$ with elasticity of factor substitution $\sigma>0$. The substitution effect of higher real wages naturally implies a shift along the isoquant to more $...
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89 views

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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1answer
64 views

The Household Production Function

I was wondering whether someone can explain the household production function (HPF). Specifically, the variant presented in Patanayak et al (2005). The paper uses the HPF to determine household ...
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60 views

Relationship between Elasticity of substitution of sectoral outputs and elasticity of substitution of inputs

There are two sectors Y1 and Y2. Composite output is given by CES form - Each sector employs Capital and Labor in combination through Cobb-Douglas Production Technology. The paper mentions that ...
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164 views

When is the PPF convex to the origin?

Given a 2X2 model (2 goods, 2 inputs), if the factor intensities (capital/labour ratio) of the two goods along the Pareto set are unequal, then we get a concave PPF. Can we get a convex PPF in some ...
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195 views

How to prove if production set Y satisfies free-disposal and CR of scale, then Y is convex, when there are only 2 commodities

Free-disposal and CR of scale can not imply convex production set. But it is true for single-input and single-output model. Therefore, I am wondering how to prove if production set Y satisfies free-...
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104 views

Linearization of VES production funtions

I know that the linearization of a CES (constant elasticity of substitution) funtion is a bit complicated. There is even an R package dedicated just for that - the econometric estimation and ...
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27 views

modelling Inconsistent production functions

last year I asked How do we estimate production functions?. That answer provided was insightful from an econometric perspective and has helped me in applying such an understanding to the workplace. ...
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1k views

How to find the “cost function” given the production function *as well as* the cost per unit produced and the fixed costs?

I'm working on the following homework problem, transcribed verbatim: A firm has a production function defined as $y = 8L^{1/4}K^{3/4}$. The firm faces costs of \$20 wage, \$60 rental rate of ...
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277 views

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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28 views

Parameter value for a CES production function

Consider a firm with the following CES production function, which utilizes only two production factors (capital and labor) whose prices are, respectively, $r > 0$ and $w > 0$: $$ y = \gamma \...
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35 views

Negative elasticity of substitution in a CES production function

I have empirically estimated the elasticity of substitution parameter in the following model: $$Y_t=[(A_1L_tK_{t})^{\rho} +(A_2M_{t})^{\rho}]^\frac{1}{\rho} $$ here, $Y_t$ is output, $A_i$ is a ...
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35 views

Optimization problem of a Cobb-Douglas function with 3 inputs

A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function: $$ Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3} $$ ...
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34 views

Kuhn-Tucker conditions in linear cost minimization

Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by $$ f(x,y) = ax + by $$ and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} ...
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If the relative share of inputs is constant in a production process, does that mean that the inputs are used in fixed proportion?

One of the assumptions in Euler's adding up theorem is: Fourth, the relative shares of the factors are constant and independent of the level of the product. Does this mean that the factors are ...
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273 views

What is the difference between imperfect substitutes and complements in a production function?

According to the following definition: ...
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73 views

Demand Elasticity, Factor Substitution: Independent?

Given $ Y=f(K,L;\sigma) $, the effect on labor from a change in the price of capital can be gauged through a substitution effect and a scale effect: \begin{align*} \frac{\partial L}{\partial r} & ...
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143 views

Estimating Endogenous vs Exogenous growth models

Exogenous economic growth models (mostly those that take technological progress, or change in TFP as exogenous and random) are rather easy to estimate econometrically. But how about endogenous ...
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42 views

How change in technology affects the price of labour and capital?

If the technology improves how does the price of labour and capital change. We take the neo-classical assumption that demand for each factor depends on it's marginal productivity, and demand and ...
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73 views

Sources of Growth and co-integration: production function approach

I am experimenting with time series data to gauge the importance of factors of production i.e. labour force, capital stock, energy, land, etc. in output growth. One venue I am looking into is the ...
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72 views

Technology, Prices, and the Derived Demand for Energy

I was reading the paper by Berndt and Wood (1975), "Technology, Prices, and the Derived Demand for Energy". It was an interesting paper to read but there has not been anything done on this in nearly ...
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162 views

Relationship Input Distance Function and Output Distance Function

I was wondering if anybody knows how input distance functions (IDF) and output distance functions (ODF) relate to each other. One of the advantages of distance functions over cost and revenue ...
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277 views

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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646 views

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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68 views

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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303 views

Given Cobb-Douglas production functions for 2 factories (same owner), how will the owner produce $y$?

So my question is this: A company owns two factories, A and B, each with the following production functions: $f_A(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}$ $f_B(x_1,x_2)=x_1^{\beta}x_2^{1-\beta}$ Now ...
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35 views

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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37 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}}...
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Capital in terms of labor

I have a question that asks to find $\frac{\partial K}{\partial L} $ from $Q=cL^aK^b$, when $Q$ and $c$ are constants. It lists 4 answer choices but I’m just not sure how to approach it. Implicit ...
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22 views

The shift of Marginal Cost Curve

In standard microeconomics textbooks they usually assume that the cost curve consists of just two variables which are Capital and Labor ( I'm talking about this equation: TC = rK + wL) So when we ...
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1answer
19 views

Percentage of workforce change, with two PPFs

A hypothetical economy produces two goods, X and Y. The performance (yield) for every worker is steady, and every worker for Y can produce 10 units of product. If $L_x+L_Y=200 $ (meaning that the ...
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When do we have diminishing marginal returns to labor?

When we have a production function exhibiting constant returns to scale, with only labor and technology, why don't we have any diminishing marginal returns to labor?
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21 views

Production for multiplant firm?

A multi-plant firm will never produce at a plant with an increasing marginal cost when they own another plant with a decreasing marginal cost. Is this true? My reasoning behind this is that firms ...
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1answer
132 views

Decreasing and increasing returns to scale question

Hi, I have deduced that this function exhibit increasing returns to scale but I am not sure how to verify part d. My answer doesn't show that there is decreasing returns to scale but I can't be sure d ...
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267 views

Question about an economy with 3 components household, firm and government with functions given

I've spent considerable amount of time on this question, in vain. It is from one of the competitive exams for admission to a grad econ program. Help would be tremendously appreciated. Thanks. An ...
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59 views

Economic interpretation of a CES production function

I am following a paper where a production function of this type is used. $$Y=\left [\beta K^{- \rho}+\alpha \eta \left (\frac{K}{L} \right )^{-c(1+\rho)}L^{- \rho} \right ]^{-1/\rho}$$ It is a ...
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2answers
1k views

Perfect complement graph and isoquant

$f(x_1,x_2) = min \{x_1,x_2\} + x_2$ if that was the production, what would the isoquant be? Would it simply follow $x_1 = x_2$? I'm not entirely sure what it the graph would look like.
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418 views

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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1answer
103 views

Anything written on the process behind “multiplicative” production function?

Intuitively, one might make a naive first guess that the production function of the economy, or of a firm, should be a "leontief" production function: "for example, you need both a factory and a ...
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1answer
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Quick Question on Production Possibilities Frontier Curve

Can someone please tell me how or why the curve shifts outward. In the textbook, I was given that: "But if we cut production of mobile phones to 3 million this year, we can produce 2 mobile phone ...
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2answers
203 views

Production function question

Some help with this would be appreciated.