5
votes
Accepted
Is increasing Average Product(AP) always implying increasing Marginal Product(MP) in microeconomics?
Consider the following production function:
$f(x)=\begin{cases} x^2 & x < 1 \\ x^{\frac{3}{2}} & x \geq 1\end{cases}$
In this case, the average product is
$\text{AP}(x)=\dfrac{f(x)}{x}=\...
- 9,431
4
votes
How was the CES production function derived?
As a follow up to Ben's answer. Here's the derivation to get to the solution of the SODE.
For notational convenience, let $y = f(k)$ and $y' = f'(k)$. We have:
$$
\frac{y'(ky' - y)}{k y y''} = s.
$$
...
- 12.5k
3
votes
Accepted
Could you give an example of production function such that involves sunk costs?
Concepts:
Production function
Measured in units of output, depends on inputs. It has no information on prices or costs, so it cannot directly include fixed/sunk costs. The only way it can imply the ...
- 29.6k
3
votes
A problem with "Returns to Scale"
Assume that $Y$ has constant returns to scale, which means that if for all $t > 0$
$$
(-z_1, -z_2, q) \in Y \to (-tz_1, -tz_2, tq) \in Y
$$
We want to show that $\overline{Y}$ has non-increasing ...
- 12.5k
2
votes
What could a negative output elasticity of an input imply?
What could a negative output elasticity of an input imply?
If this would be actual rigorously and properly estimated result it would imply that when you increase the said input, you will get less ...
- 57.7k
2
votes
any contingent labor and capital demand functions shortcuts for cobb-douglas functions?
Let $q= f(x,y)$ be the prodution function. the equivalence of utility maximization for production is the output maximisation problem:
$$
q(p_x, p_y, c) = \max f(x,y) \text{ s.t. } p_x x + p_y y = c.
$$...
- 12.5k
2
votes
Is CES production representing the average of inputs?
Yes, all this does mean that the production function represents the average of two inputs $L$ and $K$ for different values of $\alpha$, given that $0<\gamma<1$. The key thing to consider here is ...
- 161
2
votes
Derive the input requirement set from production set
The input requirement set is defined as the set of all inputs required to produce at least the quantity $y$, or in other words, with a production function given by $y=f(x),$
$$ V(y)= \{x: f(x) \geq y \...
- 3,481
2
votes
Understanding Second-Order Approximations in Translog Production Functions
To address your first question - let's break it down by showing both sides of the equivalent terms.
For a general function $ f(s_i, n_i, k_i) $, the second-order Taylor expansion is:
\begin{align*}
f(...
- 935
1
vote
Costs and Increasing returns to scale
First, let's start with a definition of increasing returns to scale (IRS). A firm is said to have IRS if multiplying the quantity of each of its inputs by a common factor, $a$, causes its output to ...
- 17k
1
vote
Accepted
Solow model with three input factors
With the production function you specified you won’t get ‘clean’ per capita production function.
However, to get it in per capita terms you just literally have to divide both sides of equation by $L$ ...
- 57.7k
1
vote
Marginal and Average costs for constant returns to scale production function being constant
Constant returns to scale (CRS) means that for any positive real number $a$
$$
f(a \cdot k,a \cdot l) = a \cdot f(k,l).
$$
Let us assume that industry wants to produce one unit of output. Then they ...
- 29.6k
1
vote
When is the PPF convex to the origin?
Consider the following 2 goods (X and Y), 2 inputs model (L and K):
Production functions (with IRS): $x=(l_x+k_x)^2$, $y=l_y^2$
Input constraint: $l_x+l_y=1$, $k_x=1$
In this case, production ...
- 9,431
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