# Tag Info

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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 ...

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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 ...

12

"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" ...

11

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 $$-\rho Q^{-\rho-1}dQ=- a\rho K^{-\rho-1}dK -(... 9 The production function is:$$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}$$... 7 Such a function is called homogeneous of degree 1. 7 The following is the basic idea if we are to estimate the parameters by linear regression. Take the natural log of the production function F(L,K)=L^aK^b, you will then get$$\ln(F)=a\ln(L)+b\ln(K).$$For each entity (e.g., firm) i, collect data on the production level F_i, the amount of labour L_i, and the amount of capital K_i. Note that ... 7 Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, L^*, K^*, or Y^*. I don't think this is true. You combined two equations into one, and thereby lost information. The optimization problem is \... 6 I like this question, in the sense that although it is broad, I think it can be answered concisely and factually. Singapore and China (from your previous question) are largely authoritarian countries. So they were both able to exert a lot of government influence over their development programs. In this respect, they are not similar to Caribbean countries. ... 6 It can be both. The number of working hours are of course easier to interpret. You can also make the assumption that every employee works some fixed number of hours, say 8 per day (or 220 per month, or 38 a week, etc). In this case if you denote by n the number of workers and L the number of hours worked you have of course$$ n \cdot 8 = L $$and then$$ ...

6

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 ...

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: $$\begin{matrix} q = f(k) & &... 6 They both relate in one way or the other to extensions of the same mathematical object, namely the Cauchy-Euler differential equation, that has the form$$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) + ...

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

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 $$\frac {\partial L}{\... 5 The intensive form of the production function is derived from the following. Let us assume a production function, F, with inputs of capital, K, and labor and technology, AL. Thus, output, Y = F(K,AL). Assuming constant returns to scale, i.e. F(cK,cAL) = cF(K,AL), we can say F(K,AL) = ALF(K/AL, 1). The intensive form of the production function, f, takes ... 5 So we want the Hessian to be NSD, so we need the PMs to alternate weakly. H=\begin{bmatrix} F_{kk} & F_{kl} \\ F_{kl} & F_{ll} \end{bmatrix}~~NSD \iff~~~F_{kk},F_{ll}\leq0~~~\&~~F_{kk}F_{ll}-F_{kl}^2\geq0 We are given that F_{kk},F_{ll}<0 so we need to figure out the cross partials. Constant returns to scale implies that we have a ... 5 You want to find a relation between tF(z) and F(tz) for all t>1 (or 0 for CRS). So since 2t=tF(z)>F(tz)=2 for all t>1, we see decreasing returns to scale. 5 Seems to be the second one, so$$ 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

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 ...

4

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 $dK=0$ $$\longrightarrow\quad dq=\frac{\... 4 First, there are direct negative effects from inflation, known as the shoe leather cost and menu costs of inflation. These are direct costs which come from price changes: People will carry less cash, need to update their knowledge about prices, firms need to update their prices and wages, and similar. These are usually small, but typically higher given ... 4 Following the conversation with user @denesp at the comments of my previous answer, I have to clarify the following: the usual graphical device we use related to the basic Solow growth model (see for example here, figure 2) is not a phase diagram, since reasonably we call "phase diagrams" those that contain zero-change loci, identify the crossing points of ... 4 Constant returns of scale As observed above, in mathematical terms the function is homogeneous of degree 1. But in terms of economic theory, this is called a production function with constant returns of scale. It means that if you change the inputs in a determinate proportion, the output varies in the same proportion. 4 The purpose of the paper under consideration is to examine/show the "investment rule" that leads to "intergenerational equity", which with constant population translates into constant consumption. The investment rule under examination is (last line of p. 973) "invest all net returns from exhaustible resources in reproducible capital" (and consume the rest).... 4 My guess is that it is not a new factor of production, but simply a type of total factor productivity. This is because: factors of production are a stock, which produce services. What is the stock of AI? The stock of labour is easy to calculate. The stock of capital is more complex, but at least all capital goods (even robots) have a market price. As such, ... 4 Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at \bar{K}. Then, the optimal choice of capital and labour is given by:$$\frac{L^*}{a}=\frac{\bar{K}}{b}$$Therefore, optimal labour is:$$L^* = \frac{a}{b}\bar{K} The marginal product of labour depends on how actual labour relates to optimal labour: ...

4

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 about initial investment but that one cannot use miniscule levels of inputs and obtain "very little" output -for production to even begin, one needs input amounts ...

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Textbooks: Mas-Colell, Whinston, and Green (1995). Microeconomic Theory. Chapter 5 Geoffrey Reny. Advanced Microeconomic Theory, 3rd edition. Chapter 3 Kreps (1990). A Course in Microeconomic Theory. Chapter 7 Varian (1992). Microeconomic Analysis, 3rd edition. Chapter 5 Lecture notes: I think undergraduate notes are more useful on this respect. For ...

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The question is broad, but I believe there is plenty of literature that defines this concept in similarly broad terms. The following is adapted from the Wikipedia on Pareto Efficiency, which is the mathematical basis of the Production Possibilities Frontier. There may be better definitions out there, but this one should probably work in a lot of cases: The ...

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