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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 ...
Ben's user avatar
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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} ...
BKay's user avatar
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8 votes
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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).$...
Elias's user avatar
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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 ...
Mark Machina's user avatar
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 ...
tdm's user avatar
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7 votes
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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 ...
Giskard's user avatar
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7 votes
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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-...
Alecos Papadopoulos's user avatar
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 ...
Bertrand's user avatar
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6 votes
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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}...
VCG's user avatar
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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 = ...
caverac's user avatar
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6 votes
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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 ...
Amit's user avatar
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5 votes
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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 ...
luchonacho's user avatar
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5 votes
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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 ...
luchonacho's user avatar
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5 votes
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Transformation Function

I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was ...
Julia B's user avatar
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5 votes
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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.
Kitsune Cavalry's user avatar
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5 votes
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The Econometrics of the Stone Geary Production function

Stone-Geary production functions attempt to reflect the real-world observation that for production to be feasible, there exist minimum thresholds in the quantities of inputs employed. This is not ...
Alecos Papadopoulos's user avatar
5 votes
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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 ...
Giskard's user avatar
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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 ...
Bensstats's user avatar
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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}{\...
user18214's user avatar
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5 votes
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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....
Giskard's user avatar
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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 ...
1muflon1's user avatar
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5 votes
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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 ...
Jesper Hybel's user avatar
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5 votes
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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/...
Albert Zevelev's user avatar
5 votes
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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}...
Michael Greinecker's user avatar
5 votes
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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 ...
Jesper Hybel's user avatar
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5 votes
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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 ...
Macosso's user avatar
  • 366
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
tdm's user avatar
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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)...
bomadsen's user avatar
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5 votes
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What does omitting capital from production function assume about capital?

One way to think about it is the following: Given a production function $F(L, K)$ of two inputs $L$ and $K$, if $K$ is fixed at $\overline{K}$ in the short run, then we can define the short run ...
Amit's user avatar
  • 9,196
5 votes
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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}=\...
Amit's user avatar
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