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

To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on constant elasticity of substitution. In particular, The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969). Then, the CES utility function was popularized by Dixit and Stiglitz (1977) ...


6

The C.E.S functional has been introduced in Economics in the context of production theory, by Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The review of Economics and Statistics, 225-250. There you can find a discussion of how it was derived. A more pedagogic and detailed ...


5

A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning. It can be easily verified that the solution to \begin{equation} \max_{x_1,x_2}\;x_1^{a_1}x_2^{a_2}\qquad\text{s.t.}\; \pi_1x_1+\pi_2x_2\le w \end{equation} is \begin{equation} x_1^*=\frac{a_1}{a_1+a_2}\frac{w}{\pi_1}\quad\text{and}\quad x_2^*=...


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 ...


4

If one reads the original article by Cobb and Douglas (1928), https://www.aeaweb.org/aer/top20/18.1.139-165.pdf , one will find at the end of page 152 that the authors stress that they took into account two properties that had been theoretically discussed in the past: That production exhibits constant returns to scale, meaning that doubling all inputs will ...


4

Demand is positive, so $A>0$. If $p_1$ goes to $\infty$, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\alpha < 0$. If $p_2$ goes to 0, $x_1$ cannot go to $\infty$, since $p_1x_1$ is bounded by $M$. Thus $\beta\ge 0$. If $M$ goes to 0, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\gamma > 0$. If both prices and ...


4

Simply multiply and divide one $\left(a_{F} \boldsymbol{F}\right)^{\frac{(\sigma-1)}{\sigma}}$ in the bracket and then take one outside the bracket. And by the way your FOC is incorrect in that $\frac{\sigma}{(\sigma-1)}$ should be cancelled out.


3

You question can be answered using a revealed preference argument. Let $B = \{q \in \mathbb{R}^n_+| p' q \le m\}$ be some budget set of a consumer (i.e. $B$ gives all possible bundles that the consumer can choose). Let $q^\ast$ be the optimal choice from $B$, i.e. the bundle that optimizes the utility. Then for any other bundle $q \in B$, it must be that $u(...


3

The Cobb Douglas production function with constants returns to scale $$y = \prod_i x_i^{\alpha_i} = A \prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i} ,$$ where $A:= \prod_i \alpha_i^{\alpha_i}$ annoying constant. Cost minimization with perfect competition $$\min_x \ \ p^\top x\ \lvert \ y = \prod \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i},$$ implies ...


2

Note that $$ln(x_1)+2*ln(x_2) = ln(x_1)+ln(x_2^2) = ln (x_1 * x_2^2),$$ and note that $$MRS_v = \frac{g'(x_1 * x_2^2) x_1 * 2 x_2}{g'(x_1 * x_2^2) x_2^2}$$ such that the derivative of $g$ cancels out. For more intuition, see here.


1

The "a fixed fraction of her income" means a fixed share of the total expenditure, not the share of total goods purchased. This is a property of a Cobb–Douglas utility function but not a property of a general CES utility function. This is because that with a CD form, you have the elasticity of substitution to be 1. You can easily find this in a two ...


1

The first order conditions equate marginal revenue per factor to the price of that factor: \begin{align} p\cdot\alpha\frac{y}{x_1} &= w_1\\ p\cdot\beta\frac{y}{x_2} &= w_2, \end{align} Where I used the property of power function $(x^n)'_n = n \frac{x^n}{x}$. Divide the second FOC by the first to get the relation between the relative prices and the ...


1

The intuition you have is correct. Mathematically you can show it by first deriving the optimal choices with the lump sum income tax. So you will set up the following lagrangian: $$\mathcal{L} = x^{1/2}_1 x^{1/2}_2 - \lambda [x_1p_1+x_2p_2 - m + T] $$ This gives you 3 FOC's the budget constraint and: $$ 0.5x_1^{-0.5} x_2^{0.5} = \lambda p_1 \\ 0.5x_2^{-0.5} ...


1

If you want to control for some deterministic trend you could add a trend term to your equation. For example, if you think there is a linear trend you can add $\gamma t$: $$\Delta \ln Y_t= \mu+ \gamma t + \alpha \Delta \ln K_t+ \beta \Delta \ln L_t+\epsilon_t.$$ However, this being said since the equation is already estimated in first differences you have to ...


1

In micro-econometric work, prices may or may not be part of TFP. The literature recognizes two versions: TFPQ (Quantity-based) TFPR (Revenue-based) Clearly, TFPR must include prices (Revenue = quantity*prices). Many scholars argue that TFPQ is the purer and most correct measure of TFP. In the sense that higher productivity means producing more output (...


1

No actually by default when you use Cobb-Douglas function the output is not even measured in monetary units but rather as output per unit of time. This output per time can be still called income without assigning it any 'monetary' value. For example, in Robinson Crusoe economy if you catch 5 fish then those 5 fish are your income from economic perspective. ...


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