Q:

The automobile assembly plant you manage has a Cobb-Douglas production function given by p = 10x^0.2 y^0.8where P is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). Assume that you maintain a constant work force of 130 workers and wish to increase production in order to meet a demand that is increasing by 100 automobiles per year. The current demand is 1200 automobiles per year. How fast should your daily operating budget be increasing? HINT [See Example 4.] (Round your answer to the nearest cent.)= ? $ per yr

Accepted Solution

A:
Answer:The variation needed for the daily buget to follow the increase in production for the first year is 12.38 $/year.This value of Δy is not constant for a constant increase in production.Step-by-step explanation:We know that the production function is [tex]p = 10x^{0.2} y^{0.8}[/tex], and in the current situation [tex]p=1200[/tex] and [tex]x=130[/tex].With this information we can calculate the actual budget level:[tex]p_0 = 10x^{0.2} y^{0.8}\\\\1200=10*130^{0.2} y^{0.8}\\\\1200=26.47*y^{0.8}\\\\y=(1200/26.47)^{1/0.8}=45.33^{1.25}=117.62[/tex]The next year, with an increase in demand of 100 more automobiles, the production will be [tex]p_1=1300[/tex].If we calculate y for this new situation, we have:[tex]y_1=(\frac{p_1}{10x^{0.2}} )^{1.25}=(\frac{1300}{26.47} )^{1.25}=49.10^{1.25}=130[/tex]The budget for the following year is 130.The variation needed for the daily buget to follow the increase in production for the first year is 12.38 $/year.[tex]\Delta y=y_1-y_0=130.00-117.62=12.38[/tex]This value of Δy is not constant for a constant increase in production.