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25

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 function was introduced, Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The Review ...

15

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 characterized by constant per capita magnitudes. Verbally, the term "steady-state" seemed appropriate to describe such a situation. Then Romer and endogenous growth ...

14

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 $Q=[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$ and $$Q^{-\rho}=[a K^{-\rho} +(1-a) L^{-\rho} ]$$ By The total derivative via differentials we will have $$-\rho Q^{-\rho-1}dQ=- a\rho K^{-\rho-1}dK -(... 13 "I am not given wealth w although I suppose I could assume any firm who is purchasing has some budget." No. This is exactly where the fundamental microeconomic theory of the firm differs from the microeconomic Consumer Theory: the firm is not constrained by a budget. The reason is that this fundamental theory deals most and foremost with the "long-term" ... 9 The production function is:$$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} \cdot \rho\cdot l^{\rho-1}q_k = \frac{\partial q}{\partial k} = \frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot k^{\rho-1}$$... 8 Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will ... 7 Given the fixed input price w, the cost function can be written as$$ C(q)=f^{-1}(q)\times w $$where f^{-1} is the inverse of the production function f. From the discussion here, one can conclude that the inverse of a concave strictly increasing function is convex. Thus, C(q) is convex as well. Going back to your approach, you might like to have ... 7 Such a function is called homogeneous of degree 1. 7 The following is the basic idea if we are to estimate the parameters by linear regression. Take the natural log of the production function F(L,K)=L^aK^b, you will then get$$\ln(F)=a\ln(L)+b\ln(K).$$For each entity (e.g., firm) i, collect data on the production level F_i, the amount of labour L_i, and the amount of capital K_i. Note that ... 7 Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, L^*, K^*, or Y^*. I don't think this is true. You combined two equations into one, and thereby lost information. The optimization problem is \... 7 They both relate in one way or the other to extensions of the same mathematical object, namely the Cauchy-Euler differential equation, that has the form$$a_{n} x^n f^{(n)}(x) + a_{n-1} x^{n-1} f^{(n-1)}(x) + \cdots + a_0 f(x) = 0 \tag{1}$$where f^{(n)} denotes the n-th derivative of the function f. Its 1st-order version is$$a_{1} x f^{(1)}(x) + ...

7

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 Hessian of a linear homogeneous function is singular. So if you want to end up with a unit cost function that is strictly concave, you have to drop at the same ...

6

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. So they were both able to exert a lot of government influence over their development programs. In this respect, they are not similar to Caribbean countries. ...

6

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, single-valued function, then by the implicit function theorem applied on the implied equation $H = f(L,K)-q=0$, what we can say is only that $$\frac {\partial L}{\... 6 It can be both. The number of working hours are of course easier to interpret. You can also make the assumption that every employee works some fixed number of hours, say 8 per day (or 220 per month, or 38 a week, etc). In this case if you denote by n the number of workers and L the number of hours worked you have of course$$ n \cdot 8 = L $$and then$$ ...

6

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 function. Suppose we start with a homothetic production function $Q = f^*(K, L)$ and we rewrite this in intensive form as: $$\begin{matrix} q = f(k) & &... 6 The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as$$ Y = A[\alpha K^\gamma + (1 - \alpha)L^\gamma]^{1/\gamma} \tag{1} $$in this case X_1 = K, X_2 = L. Now we expand \ln Y around \gamma = 0 (recall the CES ... 5 It is valid but only in the short run with the assumption that the capital is fixed. By assuming that output depends on labor and capital you can write$$q=q(L,K)$$Now taking the total derivative$$dq=\frac{\partial q}{\partial L}dL+\frac{\partial q}{\partial K}dK$$In the short run capital is fixed such as dK=0$$\longrightarrow\quad dq=\frac{\...

5

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 through being a net importer), they'll have a capital account surplus (i.e., foreigners will on net buy more domestic assets or domestic owners will be net sellers ...

5

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, $Y = F(K,AL)$. Assuming constant returns to scale, i.e. $F(cK,cAL) = cF(K,AL)$, we can say $F(K,AL) = ALF(K/AL, 1)$. The intensive form of the production function, f, takes ...

5

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 example here, figure 2) is not a phase diagram, since reasonably we call "phase diagrams" those that contain zero-change loci, identify the crossing points of ...

5

So we want the Hessian to be NSD, so we need the PMs to alternate weakly. $H=\begin{bmatrix} F_{kk} & F_{kl} \\ F_{kl} & F_{ll} \end{bmatrix}~~NSD \iff~~~F_{kk},F_{ll}\leq0~~~\&~~F_{kk}F_{ll}-F_{kl}^2\geq0$ We are given that $F_{kk},F_{ll}<0$ so we need to figure out the cross partials. Constant returns to scale implies that we have a ...

5

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.

5

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 AI? The stock of labour is easy to calculate. The stock of capital is more complex, but at least all capital goods (even robots) have a market price. As such, ...

5

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 a competitive labor market you have $$w_t = \frac{\partial Q_t}{\partial N_t}.$$ If you assume the production function above then you indeed have  \frac{\...

5

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 $\Delta Q = \sum_{i=1}^{\ k}\frac{\partial Q}{\partial x_i}\Delta x_i$ Now we are interested in returns to scale, which is the change in output due to the multiplication of every input with a constant. We ...

5

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.g., $f(x_1,x_2) = x_1^2x_2^2$ does not. There are many non Cobb-Douglas functions that satisfy the Inada conditions, albeit they may not have nice compact ...

5

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 appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production ...

5

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/ goods market clearing. The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy. ...

5

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 $f:\mathbb{R}^n_+\to\mathbb{R}_+$ be a production function. We let $\Delta^n_{++}$ be the set of all points in $\mathbb{R}^n$ with all coordinates being strictly positive and summing to one. ...

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