30 votes
Accepted

How can I obtain Leontief and Cobb-Douglas production function from CES function?

The proofs I will present are based on techniques relevant to the fact that the CES production function has the form of a generalized weighted mean. This was used in the original paper where the CES ...
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17 votes

How can I obtain Leontief and Cobb-Douglas production function from CES function?

The regular method of obtaining Cobb-Douglas and Leotief is L'Hôpital's rule. Another methods should be used too. Setting $ \gamma=1$ will be return $Q=[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\...
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  • 713
15 votes
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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 ...
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13 votes
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How to derive firm's cost function from production function?

"I am not given wealth $w$ although I suppose I could assume any firm who is purchasing has some budget." No. This is exactly where the fundamental microeconomic theory of the firm differs from the ...
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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} ...
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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 ...
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7 votes
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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 ...
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  • 967
7 votes
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Is a production function bilinear?

Such a function is called homogeneous of degree 1.
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  • 3,202
7 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).$...
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  • 923
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 ...
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  • 26.7k
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-...
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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 ...
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  • 2,692
7 votes
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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 ...
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  • 8,642
6 votes
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Lessons From Successfully small island economies

I like this question, in the sense that although it is broad, I think it can be answered concisely and factually. Singapore and China (from your previous question) are largely authoritarian countries....
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  • 558
6 votes

Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?

Question: is the following correct ? $$\frac{dL}{dq}=1/\frac{\partial q}{\partial L},\;\frac{dK}{dq}=1/\frac{\partial q}{\partial K}$$ In general, no. Since $q= f(L,K)$ is a multivariable, ...
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6 votes

What exactly is L in a Cobb-Douglas production function?

It can be both. The number of working hours are of course easier to interpret. You can also make the assumption that every employee works some fixed number of hours, say 8 per day (or 220 per month, ...
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6 votes
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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 ...
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  • 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 = ...
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  • 1,206
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 ...
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  • 5,267
5 votes

Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?

It is valid but only in the short run with the assumption that the capital is fixed. By assuming that output depends on labor and capital you can write $$q=q(L,K)$$ Now taking the total derivative $...
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  • 206
5 votes
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Financial investment in the composition of GDP

What you're describing is a change in the capital account, not in GDP. They're related through the balance of payments, in that if a country is running a current account deficit (usually arising ...
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5 votes
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What does it mean by 'intensive form'?

The intensive form of the production function is derived from the following. Let us assume a production function, F, with inputs of capital, K, and labor and technology, AL. Thus, output, $Y = F(K,AL)$...
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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 ...
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  • 8,477
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 ...
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5 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}...
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  • 756
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.
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  • 6,407
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 ...
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  • 8,477
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 ...
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  • 26.7k
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}{\...
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  • 753
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....
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