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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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1 answer
57 views

Is CES production representing the average of inputs?

I know that the Constant elasticity of substitution production function is given by: $$Q=\gamma\left[\delta L^{-\alpha}+\left(1-\delta\right)K^{-\alpha}\right]^{\frac{-1}{\alpha}}$$ where $\gamma$ is ...
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0 answers
22 views

Production and Cost [closed]

How to answer this question: "A production function following constant returns to scale can follow diminishing returns to a factor. Justify with logic".
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0 answers
19 views

Origin of terms Harrod neutral, Solow neutral and Hicks neutral

I just looked for the same question on the exchange but I didn't find anything, I even tried to look for the answer on Google but it seems it's not present. Could someone give me the origin of these ...
1 vote
1 answer
114 views

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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question about production optimization

the question is, if Q = AK^a(HL)^b and the parameters are: (A =100) (K = 10000$) (H = 1) (L = 100 person) (a = 0.5) (b = 0.5) P = 5 per unit, R = interest rate of 3 percent per year , W = 3 per ...
1 vote
1 answer
51 views

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, ...
1 vote
1 answer
35 views

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 $A,B$ . ...
3 votes
1 answer
91 views

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 ...
1 vote
0 answers
25 views

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 ...
0 votes
2 answers
132 views

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 ...
1 vote
1 answer
62 views

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, ...
1 vote
0 answers
69 views

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. ...
0 votes
1 answer
41 views

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 ...
1 vote
1 answer
68 views

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 ...
2 votes
1 answer
100 views

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 ...
1 vote
1 answer
29 views

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 ...
1 vote
2 answers
152 views

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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36 views

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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0 answers
77 views

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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0 answers
37 views

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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0 answers
31 views

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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89 views

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 ...
-1 votes
1 answer
1k 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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0 answers
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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?
2 votes
1 answer
334 views

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. ...
2 votes
2 answers
404 views

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 $A$ ...
2 votes
0 answers
80 views

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 ...
12 votes
5 answers
4k views

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. ...
3 votes
1 answer
309 views

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 ...
0 votes
1 answer
51 views

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}...
1 vote
0 answers
88 views

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 ...
1 vote
0 answers
76 views

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 ...
1 vote
0 answers
21 views

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 ...
1 vote
0 answers
55 views

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 ...
2 votes
1 answer
886 views

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 ...
0 votes
0 answers
73 views

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 ...
2 votes
1 answer
250 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 ...
2 votes
1 answer
261 views

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 ...
1 vote
1 answer
195 views

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) ...
1 vote
0 answers
51 views

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 ...
1 vote
1 answer
140 views

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 ...
2 votes
0 answers
207 views

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 ...
0 votes
1 answer
163 views

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(...
1 vote
1 answer
204 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 ...
4 votes
2 answers
713 views

Decision over "max" production function:

I've been presented with the following problem: $$y=3(x_3)^{\frac13}(\max\{x_1,8x_2\})^{\frac13}$$ And the objective is to both maximize profit and minimize cost. First of all, if the problems are ...
0 votes
2 answers
166 views

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...
0 votes
1 answer
209 views

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 ...
2 votes
1 answer
162 views

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 $f(x_1, x_1) = x_1^{0.5}x_2^{0.5}$, ...
2 votes
2 answers
141 views

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}...
1 vote
1 answer
101 views

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}...

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