# 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, ...
1 vote
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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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### 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
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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 ...
1 vote
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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, ...
1 vote
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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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### 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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### 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
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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 ...
1 vote
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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 ...
1 vote
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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 ...
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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### 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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### 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. ...
396 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 $A$ ...
79 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 ...
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### 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. ...
274 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 ...
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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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### 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 $f(x_1, x_1) = x_1^{0.5}x_2^{0.5}$, ...
I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem: $$\max_{L_{t}}\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\alpha}... 1 vote 1 answer 89 views ### Are homothetic additively separable preferences always equivalent to CES? Are homothetic additively separable preferences always a monotonic transformation of CES preferences? In technical language, the question is the following: Let n>1, and let f:\mathbb{R}^n_{\ge 0}... 2 votes 0 answers 68 views ### Elasticity of substitution between capital and effective labour While going through the derivation of elasticity of substitution between capital and effective labour in economic materials for a Slow growth model, I found the following step there: \frac{\partial ... 1 vote 1 answer 109 views ### Supply function of a price-taking firm with a quadratic production function For a firm with the production function$$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 ...
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 \...