All Questions
Tagged with production-function microeconomics
112 questions
0
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36
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Labour saving technological change VS Capital saving technological change
I have been learning about technological change in relation with production functions(cobb Douglas) and isoquants and I'm very confused as the explanations can be somewhat vague and not clear enough.
...
- 95
1
vote
1
answer
101
views
Understanding Second-Order Approximations in Translog Production Functions
Consider a model with the following production technology:
\begin{equation}
Q_i=F(\Omega_i,K_i,S_i,N_i) = \Omega_i\Big(\nu N_i^\sigma+(1-\nu)(\tau K_i^\rho+(1-\tau)S_i^\rho)^{\frac\sigma\rho}\Big)^{\...
- 143
2
votes
1
answer
238
views
Is increasing Average Product(AP) always implying increasing Marginal Product(MP) in microeconomics?
I'm studying microeconomics and came across a statement that I'm not sure is correct:
"If average product ($AP = F(X)/X$) always increases from $X=0$, then marginal product ($MP = F'(X)$) also ...
- 23
0
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0
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35
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Calculating Production Functions using Input Prices
I am trying to estimate production functions and want to correct the endogeneity problem from direct estimation.
I found from these notes that it is possible to estimate production functions using ...
2
votes
1
answer
101
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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 ...
- 161
0
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2
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153
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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 ...
- 440
0
votes
1
answer
104
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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 ...
- 43
1
vote
1
answer
254
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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 ...
- 11
1
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0
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57
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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 ...
- 45
2
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1
answer
1k
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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 ...
- 45
1
vote
1
answer
232
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) ...
- 11
2
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0
answers
275
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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 ...
- 277
1
vote
1
answer
263
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 ...
- 57
2
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1
answer
186
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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 $f(x_1, x_1) = x_1^{0.5}x_2^{0.5}$, ...
2
votes
2
answers
144
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:
$$\max_{L_{t}}\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\alpha}...
- 175
1
vote
1
answer
129
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Supply function of a price-taking firm with a quadratic production function
For a firm with the production function
$$Q = 40L-L^2$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.
Following ...
- 13
1
vote
1
answer
127
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Solving Lagrangian FOCs: a few difficulties
I have an optimization problem from microeconomics that yields me the following first-order conditions based on a Lagrangian:
$ p_1 = \lambda \qquad(1)$
$ p_2 - \lambda (x_2^2+x_3^2)^{-1/3}x_2=0 \...
3
votes
1
answer
508
views
CES production function: How to show that $\sigma < 1$ implies essentialness?
Consider the CES production function:
$$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
2
answers
156
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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, ...
- 109
1
vote
1
answer
582
views
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. ...
- 23
0
votes
1
answer
492
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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 ...
- 119
2
votes
2
answers
742
views
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, ...
- 2,351
1
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0
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143
views
Total factor productivity (TFP) estimation in R via estprod package
good morning
I am trying to calculate the total factor productivity (TFP) for companies in the manufacturing industry through the Levinsohn-Petrin model. To do so, I use the prodest package in R. ...
0
votes
0
answers
86
views
Elasticity of substitution for 3 and more goods (interpretation)
Elasticity of substitution for 3 and more goods (interpretation)
Hello everyone,
I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function ...
- 834
0
votes
1
answer
43
views
How is production managed with respect to the long run vs the short run?
Assuming perfect competition, I think that firms are price takers in the labor/capital markets as well (in the short and long run), correct?
And I know that the Long-run total cost curve is derived by ...
- 61
3
votes
2
answers
487
views
How to find the price ratio $\frac{p_x}{p_y}$?
(Question) Suppose two individuals form an economy where each devotes $10$ hours of labour to produce goods $x$ and $y$. The utilities of the agents $S$ and $J$ are $U_S(x,y) = x^{0.3}y^{0.7}$ and $...
- 31
2
votes
0
answers
218
views
Find cost function for given production function
I have the following production function
$$f(x_1,x_2,x_3,x_4)=max\{\min\{x_1, x_2), x_3+2x_4\}\}\ge q$$
And I want to find the cost function.
What I think
(1) $P_1+P_2 <P_3$ and $P_3/P_4<1/2$
...
- 192
1
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1
answer
134
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Please check those two production functions, they also seems to be quasiconvex?
Those two are generic production functions that we usually see, and I check the definition of quasiconvex and quasiconcave on wikipedia, and it seems those two graphs satisfy both definitions, and ...
- 155
0
votes
1
answer
662
views
How to calculate Returns to Scale for Translog production function with two inputs?
I have a double-log (both inputs and output in logarithmic form) translog production function with 2 inputs [with Labour and Capital]. There are two squared terms, one for each of the inputs and there ...
0
votes
1
answer
523
views
isoquant of a leontief production function
Consider a firm that can produce q units of good G using two technologies and two production factors, $z_1$ and $z_2$. There are two ways how a firm
can produce the good G: It can use 2 units of $z_1$ ...
- 1,048
5
votes
3
answers
2k
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Cobb-Douglas Production Function - Finding units of labour to maximise production
Given production function $f(L,K)=16L^\frac{1}{4}K^\frac{3}{4}$, where each unit of labour costs £50 and each unit of capital costs £100 and you have a budget of £500,000. Find the number of units of ...
- 53
4
votes
2
answers
538
views
What kind of production function would give a cubic-shape cost function?
I would like a production function that gives a cost function with the following shape:
The figure was taken from "Microeconomic Theory: Basic Principles and Extensions, 12th edition", on ...
- 480
1
vote
1
answer
158
views
What is the elasticity of Substitution of the function X = (K+alpha)(L+beta)
For this function, the marginal rate of technical substitution is given by (K+alpha)/(L+beta). Generally we solve for K/L in terms of MRTS of two factors. Then differentiate to solve for elasticity of ...
- 11
2
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1
answer
90
views
If I have a production function f where the marginal product of all the input is constant, can f exhibit decreasing returns to scale? [duplicate]
Marginal product of input xi=
\begin{equation}
\frac{\partial f }{\partial x _{i}}
\end{equation}
Decreasing return to scale:
f(tx,ty) < t f(x,y) for t>1
- 21
2
votes
1
answer
283
views
Example of production function with negative returns with respect one input
Are there examples of production functions where increasing the input of one factor and keeping the other factor constant leads to reductions in total production?
- 1,938
3
votes
1
answer
62
views
Linearization and the effect of a change
There is the following system of four equations and four endogenous variables $(K,L,w,q)$. Assume $F$ is a concave function.
$\partial F(K,L)/\partial K = r + (1-p)$
$\partial F(K,L)/\partial L = w$
$...
- 195
1
vote
1
answer
412
views
Production Set: Not satisfying Free Disposal Assumption
I saw the figure which satisfies the free disposal assumption in Mas-Colell, Whinston and Green (1995), but wondering if there is a figure that DOES NOT satisfy the free disposal assumption? Any leads ...
- 21
1
vote
0
answers
246
views
Finding long run total cost function
I am trying to find the long run total cost function, given the firm's production function $y=L^α K^β$ where $α,β>0$ and two inputs $L$ and $K$ where $ L,K∈R_+^2$, with factor prices $w$ and $r$ ...
1
vote
1
answer
88
views
I need to prove how an increase in output p increases profit-max. Can someone help to understand why IFT implies that z is a unique maximizing point? [closed]
MWG 5C6 asks: "Suppose a concave prod function f(z) with inputs $(z_1,...,z_L-1)$ and also that $\partial f(z))/\partial z_l \geqslant 0$ for all l and $z\geqslant0$ and that $D^2f(z)$ is ...
3
votes
2
answers
843
views
Under what condition is a cost function strictly concave in prices?
Define the unit cost function as
$$
c(w) = \min_{z\geq 0} w\cdot z
$$
subject to $f(z)\geq 1$. Where $w$ is a vector of input prices, $z$ is the vector of inputs and $f$ is a production function. We ...
- 168
5
votes
1
answer
1k
views
CobbDouglas: Constant marginal costs and constant returns to scale
A company has a production function:
$$y=x_1^{\alpha}x_2^{1-\alpha}$$
where $0<\alpha<1$. Factor input 1 costs $w_1> 0$ and factor input 2 costs $w_2> 0$. The company wants to minimize its ...
- 175
1
vote
1
answer
1k
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How to find the cost function for perfect complements [closed]
Imagine I got a production function like it :
$$
\min\{x_1, x_2\}
$$
How can I find the cost function?
- 61
5
votes
2
answers
4k
views
What is the returns to scale of the production function q = min {K, L^(1/2)}?
I learned that when there is decreasing returns to scale, the average cost is always increasing.
But the professor told us today that the other way around might not always be true. So if average cost ...
- 305
1
vote
0
answers
73
views
How to explain the flattening of the SRAC curve?
I discovered that there is a way that Short-run average cost curve could become 'flatter' instead of shifting. Yet I cannot find an explanation of why and how it can become flatter.
For example, in ...
- 111
2
votes
1
answer
2k
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Finding the conditional input demand function
Find the conditional input demand function and cost function for the given production function $$f(a,b,c,d)=\min\{ a,2b\} + \max\{3c,4d\} $$
In The solution, The production function is defined as $f(x,...
- 192
3
votes
1
answer
1k
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If production function is concave, then demonstrate that profit function will also be concave
Show that concavity of firm's production function implies concavity of its profit function.
(Hint: For a concave function, first order conditions gives the vector that maximizes the function)
...
- 41
0
votes
1
answer
9k
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Calculate supply function based on production or cost function
Q1: A company has the following production function:
$$f(x_1,x_2) = 2x_1 + x_2$$.
The factor prices are $w_1=4$ and $w_2=3$. Calculate the company's supply function.
Q2: A company's cost function is
$$...
- 13
0
votes
0
answers
220
views
Prodest package in R for TFP estimation, gives different results
I have the following issue: each time I run the estimation of the TFP by using prodest package in R 4.0.3, I obtain different coefficients before the variables as well as omega variable is different:
...
- 109
3
votes
0
answers
55
views
Why is my elasticity of substitution wrong?
I am calculating elasticity of substitution for the following production function:
$$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$
where $A, a, b, \mu, \gamma$ are constants. $A, a, b, > 0$,...
- 1,935
2
votes
2
answers
144
views
Are prices part of total factor productivity?
I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology:
$$Y=AK^{\alpha}L^{\beta}$$
...
- 145