All Questions
Tagged with production-function microeconomics
112 questions
10
votes
3
answers
4k
views
CES Production Function with $\rho>1$
In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the ...
- 616
8
votes
2
answers
2k
views
Deriving the translog production function
Ive been having difficulty deriving the translog production function defined as:
$$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$
I know ...
- 8,847
6
votes
2
answers
488
views
Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?
Marginal costs MC is defined as $MC=\frac{dC}{dq}$. Taking into account that $C=wL+rK$,
$$MC=\frac{dC}{dq}=w\frac{dL}{dq}+r\frac{dK}{dq}$$
Recall that marginal product of labor $MP_{L}=\frac{\...
- 61
5
votes
3
answers
2k
views
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
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
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
5
votes
3
answers
890
views
Why labour, capital, and output levels cannot be pinned down in perfect competition?
Consider a firm producing with the following technology:
\begin{equation}
Y = AL^{\alpha}K^{\beta}
\end{equation}
Assuming that factors are paid their marginal contribution to output, it can be ...
- 8,631
5
votes
2
answers
4k
views
Transformation Function
In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
- 195
5
votes
3
answers
2k
views
Examples of how economists come up with a production function for a firm?
I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes ...
- 51
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
4
votes
2
answers
370
views
Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor
Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form $y=f(L)$, for $L≥...
- 175
4
votes
1
answer
1k
views
Interpretation of the Cross Partials of the Cobb-Douglas
Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$
This has the cross-partial:
$$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$
Is the ...
- 69
4
votes
1
answer
15k
views
Why does marginal cost (derivative of total cost) differ from variable cost at each level?
Why does the marginal cost equation (as the derivative of total cost equation) make predictions of variable costs that are very different from costs calculated using the Total Cost equation?
Marginal ...
- 141
4
votes
0
answers
51
views
comparison of micro production functions
There are many different production function estimation methods, relevant for micro and firm data. For example Olley-Pakes, Levinsohn-Petrin, Ackerberg et al., Wooldridge etc.
But does anyone know of ...
- 41
4
votes
1
answer
602
views
Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8
Letting $f_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x_i$ is a scalar, commodity $i$ in the bundle) show that,
$\sigma_{ij}(\mathbf{x})\...
- 153
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
3
votes
3
answers
1k
views
How was the Cobb Douglas function derived?
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the ...
- 133
3
votes
5
answers
20k
views
Real world production-possibility frontier example?
I'm reading an economics textbook and trying to make sense of PPF (or PPC) concept. All examples I could find are like producing computers vs food. I can't understand how it can be evaluated and ...
- 155
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 ...
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
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.
$\partial F(K,L)/\partial K = r + (1-p)$
$\partial F(K,L)/\partial L = w$
$...
- 195
3
votes
1
answer
406
views
Kimball (1995) Specification of Final Good Production
Kimball (1995) defines production of the final good ($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 $...
- 10.8k
3
votes
3
answers
5k
views
Total cost function and (dis)economies of scale
I'm really confused about how I'd tackle this.
I have an equation, $TC(Q) = 100Q + 20Q^2 + 3Q^3$.
And I'm trying to find where the economies of scale, diseconomies of scale and constant return to ...
- 153
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
3
votes
0
answers
121
views
Is it possible to derive the marginal product of an input using a transformation function?
I'm using a transformation function $F(\cdot)$ to describe a production set $y = (x, z, L, K)$, where $x$ and $z$ are private goods denoted by positive numbers, $L$ is labour input, $K$ is capital ...
- 31
3
votes
1
answer
542
views
Find Change in output from marginal products of labor/capital
A firm produces 231 doohickeys with 8.4 units of labour and 22.1 units of capital. the marginal product of labour is 18, the marginal product of capital is 20. Approximately how many doohickeys will ...
- 33
3
votes
1
answer
1k
views
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
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
2
votes
2
answers
720
views
Returns to scale - Constant Function
Suppose we have a production function $f(z)=2$.
I am asked to determine whether the function exhibits increasing, decreasing, constant or no returns to scale.
For $t>0$, $f(tz)=2$.
I'm not sure ...
- 448
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
2
votes
1
answer
2k
views
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
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
2
votes
2
answers
2k
views
How to derive cubic cost function from a problem of constrained optimization?
The cubic total cost function usually take the form
$TC(q)=a+bq+cq^{2}+dq^{3} \qquad a,b,d>0, c<0$ and $c^{2}<4bd$
I know that from a constraint maximization problem
$min\quad wL+vK$
...
- 162
2
votes
1
answer
186
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
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
2
votes
1
answer
1k
views
Production Possibility Frontier for 3 goods
Is it possible to construct a Production Possibility Frontier (PPF) for 3 goods? Would this require a 3D graph and a part of a sphere, in one quadrant (as opposed to the usual 2D plot, with a part of ...
- 123
2
votes
1
answer
246
views
Why does a homothetic function have constant ratio of marginal products along rays?
A homothetic ordering is defined as
$x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda >0$
where $x,y \in \mathbb{R}^n$
Then, any differentiable function representing ...
- 880
2
votes
1
answer
3k
views
Decreasing Costs, Increasing Returns to Scale, & C''(q)
Given a profit-maximizing firm with production function $f(x_1,x_2)$, I understand that we can formulate a firm's cost function $C(q)$ by using the contingent demand functions $x_1^c$ and $x_2^c$. We ...
- 530
2
votes
1
answer
101
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}}$$
where $\gamma$ is ...
- 161
2
votes
1
answer
90
views
If I have a production function f where the marginal product of all the input is constant, can f exhibit decreasing returns to scale? [duplicate]
Marginal product of input xi=
\begin{equation}
\frac{\partial f }{\partial x _{i}}
\end{equation}
Decreasing return to scale:
f(tx,ty) < t f(x,y) for t>1
- 21
2
votes
2
answers
144
views
Are prices part of total factor productivity?
I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology:
$$Y=AK^{\alpha}L^{\beta}$$
...
- 145
2
votes
1
answer
1k
views
How to derive the input demand functions from a perfect substitutes production function
I am struggling to derive the input demand functions from a production function with inputs that are perfect substitutes.
The production function is as follows:
$f(x_1,x_2) = (x_1+x_2)^\frac{1}{2}$
I ...
- 45
2
votes
1
answer
1k
views
Nested CES Production Function
If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a,...
- 33
2
votes
0
answers
275
views
Solow Model - speed of convergence
This is a question also for those with a good expertise in micro. For micro guys who wanna go streight to the question, just jump to equation $(1)$
I'm studying the Solow growth model.
Let's write the ...
- 277
2
votes
0
answers
218
views
Find cost function for given production function
I have the following production function
$$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
2
votes
0
answers
1k
views
How to find the "cost function" given the production function *as well as* the cost per unit produced and the fixed costs?
I'm working on the following homework problem, transcribed verbatim:
A firm has a production function defined as $y = 8L^{1/4}K^{3/4}$. The firm faces costs of \$20 wage, \$60 rental rate of ...
- 191
2
votes
0
answers
109
views
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 ...
- 41
1
vote
2
answers
5k
views
Example production function with increasing returns to scale but diminishing marginal product [duplicate]
I know that diminishing marginal returns even to all factors of production doesn't imply decreasing returns to scale. But could you please give me just an example of such production function?
- 37
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
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