16
votes
Accepted
Solow Model: Steady State v Balanced Growth Path
This is when the attempt at accuracy creates confusion and misunderstanding.
Back in the day, growth models were not incorporating technological progress, and led to a long-run equilibrium ...
- 33.4k
9
votes
CES: Production function: Elasticity of substitution $\sigma = 1/(1 + \rho)$
The production function is:
$$q = (l^\rho + k^\rho)^\frac{1}{\rho}$$
The MPL and MPK are respectively:
$$q_l = \frac{\partial q}{\partial l} = \frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} ...
- 16.3k
8
votes
Accepted
How do we estimate production functions?
The following is the basic idea if we are to estimate the parameters by linear regression.
Take the natural log of the production function $F(L,K)=L^aK^b$, you will then get $$\ln(F)=a\ln(L)+b\ln(K).$...
- 973
8
votes
CobbDouglas: Constant marginal costs and constant returns to scale
Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is ...
- 81
8
votes
Confusing on the CRS Property of CES Function
Consider the function $f(K,L) = [\alpha K^\rho + \beta L^\rho]^{1/\rho}$. We want to evaluate the limit of $f$ when $\rho \to 0$.
$$
\lim_{\rho \to 0} f(K,L) = \lim_{\rho \to 0} [\alpha K^\rho + \beta ...
- 10.3k
7
votes
Accepted
7
votes
Accepted
Concave production function implies convex cost function
Given the fixed input price $w$, the cost function can be written as
$$
C(q)=f^{-1}(q)\times w
$$
where $f^{-1}$ is the inverse of the production function $f$. From the discussion here, one can ...
- 967
7
votes
Accepted
Why labour, capital, and output levels cannot be pinned down in perfect competition?
Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and ...
- 28.7k
7
votes
Accepted
Euler's Theorem
They both relate in one way or the other to extensions of the same mathematical object, namely the Cauchy-Euler differential equation, that has the form
$$a_{n} x^n f^{(n)}(x) + a_{n-1} x^{n-1} f^{(n-...
- 33.4k
7
votes
Under what condition is a cost function strictly concave in prices?
No function that is homogeneous of degree one, is at the same time strictly concave in its arguments. If the function is differentiable (or non-differentiable at a finite number of points), then the ...
- 3,144
6
votes
Accepted
How to show the production function is concave in K and L but not strictly so?
So we want the Hessian to be NSD, so we need the PMs to alternate weakly.
$H=\begin{bmatrix} F_{kk} & F_{kl} \\
F_{kl} & F_{ll}
\end{bmatrix}~~NSD \iff~~~F_{kk},F_{ll}\leq0~~~\&~~F_{kk}...
- 766
6
votes
Accepted
How was the CES production function derived?
The CES function can be derived directly from the condition of constant elasticity of substitution. There are various ways to do this, but the simplest derivation occurs for a homothetic production ...
- 235
6
votes
Deriving the translog production function
The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as
$$
Y = ...
- 1,216
6
votes
Accepted
Aggregate production function and returns to scale
I am assuming that you are interested in finding the aggregate production function when you have two plants and they use same inputs. So if you have $k$ units of capital and $l$ units of labor in ...
- 8,206
5
votes
Solow Model: Steady State v Balanced Growth Path
Following the conversation with user @denesp at the comments of my previous answer, I have to clarify the following: the usual graphical device we use related to the basic Solow growth model (see for ...
- 33.4k
5
votes
Accepted
CES Production Function with $\rho>1$
The problem with $\rho>1$ is that it means the marginal product of factors is not decreasing ($\rho<1$) or constant ($\rho=1$) but increasing, which is an odd assumption. Such functions yield ...
- 8,581
5
votes
Accepted
Returns to scale - Constant Function
You want to find a relation between $tF(z)$ and $F(tz)$ for all $t>1$ (or $0$ for CRS).
So since $2t=tF(z)>F(tz)=2$ for all $t>1$, we see decreasing returns to scale.
- 6,578
5
votes
Accepted
Is Artificial Intelligence a completely new (and underestimated) production factor?
My guess is that it is not a new factor of production, but simply a type of total factor productivity. This is because:
factors of production are a stock, which produce services. What is the stock of ...
- 8,581
5
votes
Accepted
Notation of a Cobb-Douglas function printed in 1989
Seems to be the second one, so $$ Q_t = A_t*(K_t^\alpha N^\beta_t T_t^\rho). $$
Two clues:
This is the usual specification.
On the top of page 14 it is written that
$$
w_t = \beta Q_t/N_t.
$$
Given ...
- 28.7k
5
votes
Accepted
Why is the Cobb-Douglas production function so popular?
The reason why Cobb Douglas production functions are so popular stem from the fact that the following assumptions are satisfied while remaining statistically rigorous1:
Recall the Cobb- Douglas ...
- 869
5
votes
What's the relationship between Output elasticity and Returns to scale?
Define the production function $Q=f(x_1,..,x_k)$, where $x_i$ denotes the ith input.
Next recall that the total differential of output can be written as
$\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\...
- 805
5
votes
Accepted
List of production functions that satisfy the Inada conditions
The Wikipedia statement should come with a citation or proof, as it is not entirely accurate: Cobb-Douglas functions with increasing returns to scale do not necessarily satisfy the Inada conditions, e....
- 28.7k
5
votes
Accepted
How was the Cobb Douglas function derived?
What is the proof of this formula?
There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most ...
- 54.4k
5
votes
Accepted
Finding the conditional input demand function
Output $z$ is given as $z = x + y$ where $x=min(a,2b)$ and $y = max(3c,4d)$.
So assume that you want $x=12$ then $a=12$ AND $b=6$. Since this part of the production delivers only the minimum of ...
- 3,286
5
votes
Accepted
Walras Law in a production economy with fixed costs
Partial answer:
for simplicity let $P_c =1$.
The budget constraint: $c= wn + \Pi$
Simplify (plug in $\Pi$): $c= F(n)- fc$
Goods clearing: $c = F(n)$
The household's budget constraint is inconsistent w/...
- 340
5
votes
Accepted
Under what condition is a cost function strictly concave in prices?
As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems.
So let $f:\mathbb{R}^n_+\to\mathbb{R}...
- 11.7k
5
votes
Accepted
How to get this Production Function in Growth Rates
Given
$$Y = Af(K,L,Z)$$
it follows that
$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$
where dotted ...
- 3,286
5
votes
Accepted
Cobb-Douglas Production Function - Finding units of labour to maximise production
If your aim is to maximize the production then your approach is correct.
But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with ...
- 356
5
votes
Cobb Douglas Production: Identification issues for technical change
Maybe the following is useful:
Sato (1970), The Estimation of Biased Technical Progress and the Production Function, International Economic Review, 11, 179-208
JSTOR link
- 10.3k
5
votes
Prove that if a production function is such that f'>0 and f''<0, then f'<Average Product
Assuming $f''<0$ implies strict concavity and hence
$$tf(y) + (1-t)f(x)<f(ty+(1-t)x) \Leftrightarrow f(y) - f(x)<\frac{f(ty+(1-t)x)-f(x)}{t}$$
$$f(y) - f(x)<\frac{f(x + t(y-x))-f(x)}{t(y-x)...
- 313
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