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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53 views

Homogenous production function

Could one define a production function which is homogenous as having constant elasticity of substitution. Just want clarification . Thanks
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Question on returns of scale in isoquants

What does the dotted line in the diagram show and what are they called? I am also wondering why the steepness of the isoquants changes at Q=3 for the third diagram. Thanks
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530 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(...
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308 views

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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455 views

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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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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217 views

Finding economy's factor price for perfect complement production function

Q.Consider an economy producing a single good by a production function $$Y = min [K, L]$$ where Y is the output of the final good. K and L are input use of capital and labour respectively. Suppose ...
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74 views

Quasiconcavity and homogeneity

How to prove that if $f$ is strictly quasi-concave and homogeneous of degree 1, then $f$ is concave? It was left as an exercise by Silberberg & Suen (2001), p.140. I simply could not elaborate ...
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2 firms production decision for one agent

I'm trying to solve the optimal production $\{x,y\}$ for a risk neutral agent with weight $w$ in firm $X$ and weight $1-w$ in firm $Y$. Each firm has marginal cost $c^X$ and $c^Y$ respectively. The ...
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General Equilibrium with Linear Production

I don't think I understand how optimization problems with a linear function work as of now. If you have a production economy with two agents, two goods and Cobb-Douglas utility representation, and you ...
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373 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,...
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Estimating production functions with time series data

A while back I asked How do we estimate production functions? The answers given address cases when dealing with cross-sectional data, However most of the data I've been seeing is given by a time ...
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What is the use of knowing elasticity of average product?

Firstly they have given output elasticity of a factor, use of which is clear to me. It says to what degree my total output will vary due to a change in the quantity of a factor. But what else do we ...
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Substitutive or supportive factors of production

Given the function: $y=A K^\alpha L^\beta, A,\alpha,\beta >0$ $y=aK+bL^{0.5}, a,b>0$ Decide whether capital and labour are substitute or supportive factors of production. How to solve this ...
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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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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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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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Log differentiation of aggregate production function [closed]

Suppose we have an aggregate production function $Y = A F(K, L)$. I'm following some slides which then states that we if log differentiate we get the following: $\frac{\dot{Y}}{Y} = \frac{\dot{A}}{A} +...
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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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What is the difference between imperfect substitutes and complements in a production function?

According to the following definition: ...
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262 views

Derive the cost function for a Homothetic production function

I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form $C(w, q) = a(w)b(q)$. Since the production function is ...
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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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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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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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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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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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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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263 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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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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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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301 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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549 views

Marginal product versus marginal productivity in a Cobb-Douglas production function [closed]

CFA Kaplan Schweser claim that Cobb-Douglas function exhibits constant marginal product of capital but diminishing marginal productivity of capital. I think this statement is not right. My view ...
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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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Too much of a good thing - examples of production (or utility) functions that are initially increase in a factor and then decrease

I'm attempting to build a spatial model that estimates labour distributions after climate change impacts agricultural production. The key point to note is that some rainfall is good, and some ...
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Graduate level texts/notes that cover the Short Run and Long Run Costs of firms' production

In undergrad econ we learned that it would cost a firm more money to increase its production in short term than in long term. MWG does not seem to cover this topic. I need a Graduate level texts/...
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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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Deriving long-run cost functions from production function

Suppose that I have a production function $(aK + bL)^3$ in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function ...
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In a perfectly competitive industry, why is apple considered best example of commodity? [closed]

Out of these options, shampoo apples ice cream hotels Why is apple considered to be the best example of commodity in a perfectly competitive industry? And Why not others?
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How does hedging using futures work?

I can't understand how it makes sense. From what I've read, corn producers (for example) sell their corn at current price but deliver the corn later, in order to protect themselves from decline in ...
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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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How do you find marginal revenue if you don't have a production function?

I was given a word problem. No formulas. So I set up the following equations: Demand Function \begin{equation} D(p) = a - p \end{equation} Cost Function: \begin{equation} c(q) = 9 + 10q \end{...
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In a production function, the technology can vary across the firms and times. But the “power” must be same for the same industry?

Let $F_t=A_tK_t^\alpha L_t^{1-\alpha}$ be the production function with two parameters. In regression, we know the firm level $F_t$, $K_t$, and $L_t$. We want to estimate $A_t$ and $\alpha$. I've ...
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How to prove that profit function is convex in price (with smaller price)?

According to this site, if output price increases from $p*$ to $p'$ and factor prices remain constant, then a new production bundle chosen must yield at least the same amount of profits as the old ...
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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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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 ...
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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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Price Derivation in a Production Function

Here is a composite goods production function: And here is the price ratio of Ys and Yg, derived from their marginal products: Then the author normalized the price of final goods Y to 1 and somehow ...
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intuitive interpretation of the marginal return/productivity of capital being less than one

Let's assume you have a production function, f, and you want to know how the output changes with respect to capital, everything else constant (ceteris paribus), so you want to know the marginal ...
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480 views

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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960 views

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 ...