1. Given production function F(L, K) = L^α K^β, where α + β...

1. Given production function $F(L, K) = L^\alpha K^\beta$, where $\alpha + \beta = 0.9$; we use $p$ for the price of production goods, $w$ for the wage and $\rho$ for the price of capital. In the short run, $K = \bar{K}$, capital is fixed. solve $L^*$ in terms of $(p, w, \rho, \bar{K})$.

Transcript

00:04 Now, in this question, basically, we ask to maximize the production p, which is a function given by b times a to the alpha, k to the 1 minus alpha.

00:14 Now, both b and alpha are just constants, and alpha, it's smaller than one, right? and b, it's a both b and alpha positive constants, alibi less less than one, right? and a or k are, you know, are valuable, right? and you need to maximize this p, subject to the following constraints.

00:34 Which is m -a -l plus m -k has to be equals p, little p is another constant, right? so basically use like a long -o method, right? so you need to construct the function.

00:44 I'm going to call the f, which is a function of k and a.

00:47 And that's given by p, the objective function minus lambda, which is like a larger multiple l, and times this constraint function, that's m -a -l plus m -kl, basically.

00:59 And then you take the partial derivative of this, which is back to k, and that, of course, first take the partial derivative of pay with respect to k, and that course, gives you b, l to the alpha and times 1 minus alpha times k to the minus alpha, right? and then multiply this m times n, which is the derivative, which is the derivative of this with the k, right? and you said this to be zero, of course.

01:26 And similarly, you can take the partial derivative of this respect to air, and you'll find this to be given by a b, alpha, air, alpha minus 1 times k 1 minus alpha and minus lambda m right equals 0.

01:43 So we can simplify these equations a little bit, basically.

01:49 For example, the first equation we can simplify like b1 minus alpha and times l k minus sorry not al k, right, it's a over k right? so this is alpha l divided by k, right, and equals lambda m, basically.

02:06 And this equation, of course, can be similarly simplified, written as b times alpha times, we can still write this, l over k, and this is alpha minus 1 equals lambda m, right? now, usually, we can actually easily find the, we can easily find the l over k, right, by taking the, deal of the ratio of these two equations...

Question

1. Given production function $F(L, K) = L^\alpha K^\beta$, where $\alpha + \beta = 0.9$; we use $p$ for the price of production goods, $w$ for the wage and $\rho$ for the price of capital. In the short run, $K = \bar{K}$, capital is fixed. solve $L^*$ in terms of $(p, w, \rho, \bar{K})$.

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