The Production Function, Finding the Wage Rate, Rental Rate, and Labor's Share of Income

In this problem we're given a simple production function, a partially parameterized Cobb-Douglas Production Function. We derive output/production, then find the real wage rate (finding the marginal product of labor) and the rental rate of capital (finding the marginal product of capital along the way... actually the marginal product of land, but it's the same steps to find MPK). Lastly, finding the labor share of income, and the capital share of income. Plus a "trick" to find the factor share of income by just looking at the Cobb-Douglas production function.

Part 1 of 3 - This video solves parts a and b below.

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Suppose the production function in medieval Europe is Y=K^0.5 L^0.5, where K is the amount of land and L is the amount of labor. The economy begins with 100 units of land and 100 units of labor. Use a calculator and equations in the chapter to find a numerical answer to each of the following questions.
a. How much output does the economy produce?
b. What are the wage rate (skip to 5:33) and the rental price of land (skip to 18:00)?
c. What share of output does labor receive?
d. If a plague kills half the population, what is the new level of output?
e. What is the new wage and rental price of land?
f. What share of output does labor receive now?

From Mankiw "Macroeconomics" (Intermediate level), 8th edition, Chapter 3 (The Open Economy) - Problem 2
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[Leonar]

with MPL, it should be 0.5-1 when you derive, not 1-0.5

just thought I'd let you know :)

these videos are saving my life right now, thank you so much

[economicurtis]

at 18:25
Thank you for the comment - you are correct! (and youtube is preventing me from ansing your question with mathematical notation!)
d/dK(K^0.5) = K^(0.5-1) = K^(-0.5) !
(I wrote in error K^(1-0.5)... sorry, and thank you.)

However, the final answer is correct.
Since K^(-0.5) = (1/K)^0.5

[victormireles]

Dammit good video, you save me from a bad homework

[Nobody-kgmd]

Don’t tell my professor that you explain setting up equations better than he does

[economicurtis]

Thanks Darell! I appreciate the feedback to help ensure I'm clear in communicating the steps I take.

(And I'm not sure how your comment got marked as Spam before I could see it. That's kinda weird)

[economicurtis]

Word. Microsoft Word. I think from 2007 forward, they introduced an equation editor that's easier to use than LaTeX (you can type WYSIWYG, you can create your own short-cuts to math symbols you often use -- however you're stuck with their math font). Then I just zoom in for the video
And then at the end of the video, I have nice set of notes to use for then I go over the material in lecture!

[wrjazziel]

00:32 I thought I needed better glasses really bad LOL... JK... thanks for this video!

[XxLiinaaxX]

So P in real wage is 1? or is it the value of the factor of production L or K

[johannesvaasalainen]

can someone tell me what is the software he used for typing here?

[AnkitRajput-sgfd]

hey, can i get a vedio like this on chapter 17 investment function frrom mankiew 8th edition
by the way, its really helpful :)

[sojournsojourntraveler]

This over complicates simple realities in life and tries to quantify an argument to whoever.