Here we apply a variant of the Cobb-Douglas function to the modeling of research
ID: 2834254 • Letter: H
Question
Here we apply a variant of the Cobb-Douglas function to the modeling of research productivity. A mathematical model of research productivity at a particular physics laboratory is
P = 0.03x0.5y0.1z0.3
where P is the annual number of groundbreaking research papers produced by the staff, x is the number of physicists on the research team, y is the laboratory's annual research budget, and z is the annual National Science Foundation subsidy to the laboratory.
Find the rate of increase of research papers per government-subsidy dollar at a subsidy level of $1,000,000 per year and a staff level of 10 physicists if the annual budget is $100,000. (Give your answer to three significant figures.)
Explanation / Answer
To find the rate of increase with respect to a specific variable you take the derivative with respect to that variable while holding the others constant. This is called a partial derivative. In this case z is the government subsidy variable.
The thinking behind it is that you have a set number of researchers and budget, but the subsidy may change. So P is really like c*z^(0.3) where c is the constant product of all the constant terms in the expression. I.E., z is the only variable.
To differentiate you just use the power rule for a derivative of a variable to a constant exponent.
?P/?z = 0.03 x^(0.5) y^(0.1) (0.3)*z^(-0.7)
To find ?P/?z @ (x,y,z) = (10,100000,1000000) you just plug in...
?P/?z ? 5.67 * 10-6 papers/dollar