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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Derivation from Solow-type neoclassical growth equation?

In Chapter 6 of the 12th edition of "Economic Development" by Michael P. Todaro & Stephen C. Smith, an equation is introduced to illustrate the consequences of rapid population growth, ...
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Introducing productive sector into an exchange economy where only one agent is endowed with input

I'm trying to find a competitive equilibrium for an economy with consumers and some outside productive sector. Consider an economy with two consumption goods $x_1, x_2$ and two individuals $A,B$ . ...
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Toy model equilibrium calculation: How a small system's firm determines wage and price?

I am trying to model a small system, with limited population and a single firm. Personally I have no economic background but only taken a simple microeconomics course and I am quite interested in ...
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Could you give an example of production function such that involves sunk costs?

I am looking for an example of a single output-single input production function such that involves sunk costs. I have in my mind that a drug - firm that is motivated to make a new drug, the drug has ...
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From Cobb-Douglas Production Function to Profit Function

A firm's output is given by the Cobb-Douglas production function $$Y_t=X_tK_t^{\alpha_K} L_t^{\alpha_L}$$ where $\alpha_K\approx\frac{1}{3}$ is the capital share and $\alpha_L$ the labor share. ...
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Slope of isoquants

Consider a production function $f(L,K)=\sqrt{KL}$. The |MRTS|=$K/L$, and $\frac{d|MRTS|}{dl}=\frac{-K}{L^2}$ However, if I use the expression given in Nicholson and Snyder (Microeconomic Theory, ...
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Derive the input requirement set from production set

This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like this question but my emphasize is different. Q: True or False? If V(y) is a convex set, then the ...
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Solow model with three input factors

My problem is that I want to construct a Solow model with three input factors; labour, capital and energy. But when trying to divid the equation by labour to get the per capita variables, it doesn’t ...
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A problem with "Returns to Scale"

Suppose that $Y\subset R^3$ is a production set satisfying the free disposal condition: if $y\in Y$ and $y'\leq y$ then $y'\in Y.$ Suppose the technology of production uses good 1 and good 2 as inputs ...
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Marginal and Average costs for constant returns to scale production function being constant

Suppose that we are dealing with a production function $q = f(k,l)$, of inputs capital and labor. If this function exhibits constant returns to scale then I know that both the marginal cost and ...
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The consideration of export in calculating national income

Is export/imports a part of factor income i.e returns from the act of production? If not, then here is my counter argument. Export is the equivalent value which is added to the goods when it is ...
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What could a negative output elasticity of an input imply?

Output elasticity of an input means (consider the non-calculus formulation) the percent change in output for a percent change in input (it is customary to substitute “change” with “increase”). Let’s ...
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Modelling the optimal mix of labour

I am trying to model the profit maximization decision of a firm that uses two types of labor, workers A and workers B. I started by drawing the marginal product and marginal cost curves (lines, for ...
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Where does the optimal point of production occur in the graph of short-run cost curves?

When MC, ATC, AVC, and AFC are in the same diagram, where does the optimal point of production occur? I know production is optimal when MR=MC but the question doesn't give you the price of the good. ...
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Help with Solow model

I need help with the following question, I would really appreciate any help. For the general case of any production function, the differential equation for k(superscript dot) looked as follows: k(...
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relation between elasticity of substituion and mrts

elasticity of substituion has been defined as the as the percentage change in capital labour ratio given change in mrts. but mrts itself tries to explain how a change in labour results in a change in ...
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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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Connection between roundabout production and decreasing returns to scale?

I've heard the statement that roundabout production as in Basu 1995 leads to decreasing returns to scale in production. What is the connection between the two?
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Leontief function nested in a cobb-douglas function for a computable general equilibrium

I am currently trying to build a CGE model, and I'm stuck with the specification of the agriculture sector. I'm trying to understand how to do nested production functions and also how to solve them. I ...
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Why is the Hicksian form of the CES demand used in CGE model forms rather than the Marshallian

I am curious to know why the Hicksian form of the CES is used in CGE models rather than the Marshallian form. I have a few hypotheses, but I am not sure which one is correct. If any? Hypothesis 1: In ...
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Stata command for Dynamic Panel Production Function estimation

Consider a production function to be estimated, $$(*) y_{it} = \beta_0 +\beta_k k_{it} +\beta_l l_{it} + a_i +\omega_{it} +\varepsilon_{it}$$ where $\omega_{it}=\rho\omega_{i,t-1}+\xi_{it}$. The ...
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How do I derive optimal tax on pollution causing intermediate products?

I am reading "The Environment and Directed Technical Change" by Acemoglu et al. (2012). I cannot understand how the optimal tax in \eqref{eqA10} is derived. $$\tau_t = \frac{\omega_{t+1} \xi}...
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Technology Parameter In Converted Minimisation Problem

Question: I want to understand what's going on with respect to the technology parameter $A$ when i convert this minimisation problem into a maximisation problem. The issue is only revealed when i use ...
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Global returns to scale

I have a production function of the form $f(x_1,x_2) = x_1^a x_2^b$ and I am trying to figure out what the global returns to scale would be given that $a,b \in (0,1)$. This production function is ...
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How to derive the input demand functions from a perfect substitutes production function

I am struggling to derive the input demand functions from a production function with inputs that are perfect substitutes. The production function is as follows: $f(x_1,x_2) = (x_1+x_2)^\frac{1}{2}$ I ...
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Hessian Matrix Test - When does it fail?

When does the hessian matrix test fail. I understand we are testing the definiteness of the Matrix, and i also understand that because it's a symmetric $n•n$ matrix, we have a principal minor ...
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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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Envelope Theorem and Factor Mix Intuition

Brief Summary of question: I'm frustrated with the intuition of the envelope theorem that when input costs change, our envelope theorem tells us we do not need to re-optimise demand levels. Context: I ...
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CRS, Homothetic Functions, and constant MRTS

Questions When our Isoquant map exhibits constant MRTS along a ray from the origin making. Why do we make specific reference to. Constant returns to scale Homothetic Functions I'm asking because it ...
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Conceptualising the effect of changes to the substitution parameter in a CES production function

I'm trying to have a conceptual understanding of what happens to the CES production function when the substitution parameter $\rho = \frac{\epsilon - 1}{\epsilon}$ changes, where $\epsilon$ is the ...
anson's user avatar
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Constant returns and (weak/strict) concavity

Suppose I have a constant returns production function $Q = f(X,Y,Z)$, where $X$, $Y$, and $Z$ are the inputs. Because of constant returns, the Hessian matrix of second-order partial derivatives (f_ij) ...
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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 ...
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Solow Model - speed of convergence

This is a question also for those with a good expertise in micro. For micro guys who wanna go streight to the question, just jump to equation $(1)$ I'm studying the Solow growth model. Let's write the ...
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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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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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Derive cost function from production function

proportions production function as follows: where the price of input is 1 and z2 is supposed to be a fixed factor of production. I've been having trouble finding the cost function because if z2 isn't ...
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Optimal production plan in monopol

Suppose a monopolist faces the following demand curve: $𝑃(𝑄) = 120 − 3𝑄$, where $𝑄 = 𝑞_1 + 𝑞_2$. The monopolist has two factories. Factory 1 and factory 2 have the following marginal costs: $$ 𝑀...
Fjeeds Arcade's user avatar
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Solve long run production function of a firm using technical rate of substitution

I don't understand the solution to a question which deals with the long run production function of a firm. The question is: Suppose a firm has a production function $f(x_1, x_1) = x_1^{0.5}x_2^{0.5}$, ...
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Maximization of CD production function

I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem: $$\max_{L_{t}}\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\alpha}...
Alessandro's user avatar
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Elasticity of substitution between capital and effective labour

While going through the derivation of elasticity of substitution between capital and effective labour in economic materials for a Slow growth model, I found the following step there: $\frac{\partial ...
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Supply function of a price-taking firm with a quadratic production function

For a firm with the production function $$Q = 40L-L^2$$ where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$. Following ...
honkhonk's user avatar
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Are homothetic additively separable preferences always equivalent to CES?

Are homothetic additively separable preferences always a monotonic transformation of CES preferences? In technical language, the question is the following: Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}...
cfp's user avatar
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Solving Lagrangian FOCs: a few difficulties

I have an optimization problem from microeconomics that yields me the following first-order conditions based on a Lagrangian: $ p_1 = \lambda \qquad(1)$ $ p_2 - \lambda (x_2^2+x_3^2)^{-1/3}x_2=0 \...
Econometric Novice's user avatar
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Total Factor Productivity and Leontief Production Function

Just a bit of a conceptual question regarding the Leontief production function and the concept of total factor productivity (TFP). In particular, I wondered if these two concepts are at odds? The ...
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CES production function: How to show that $\sigma < 1$ implies essentialness?

Consider the CES production function: $$Y = f(K, L) = (a \cdot K^\rho + (1 - a) \cdot L^\rho )^{1/\rho}$$ The elasticity of substitution is $\sigma = 1/(1 - \rho)$. I remember that, if the elasticity ...
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Naive Question About PPFs

I am a maths major, and am taking an introduction to microeconomics course this semester, and am confused by how we deduce the shape of PPF's. For example, I was given the following problem: Larry, ...
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Convexity of production sets and input requirement sets

The following question is from Microeconomic Analysis by Hal R Varian. True or false? If V(y) is a convex set, then the associated production set Y must be convex. The solution available says; False. ...
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Explain the definition of a primal shifter versus an input shifter parameters in the standard CES function

I have run into a CES function that seems to be very closer to standard but with a small disaggregation of the share parameter into two parameters (primal share) and (input shift). I am hoping someone ...
user42955's user avatar
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Effects of retirement and unemployment on PPF

Which of the following will not shift a country's production possibility frontier (PPF) ? An increase in the age at which people retire or a fall in unemployment ? To me, it is the increase in the age ...
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How are returns to scale of a non homogeneous production function defined?

Most of the production functions encountered in Intermediate Microeconomics are homogeneous (Cobb-Douglas, perfect substitutes, perfect complements). So their returns to scale are easy to get, ...
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