Question

Paco's Painting Service does interior and exterior house painting. They have a production function that uses three factors: brushes (B), paint (P), and labor (L). In the short run, Paco's has already committed to orders of brushes and paint, but can adjust the number of hours of labor they use from week to week. Therefore, their short-run production function depends only on labor and is given by ?(?)=√L/2 , which implies that Paco's marginal product of labor is ???=1/4√L

In the questions below, give all numerical answers to two decimal places.

a) Suppose that Paco charges $4000 for a paint job and that the current wage he must pay his workers is $28 per hour.

How many worker hours should Paco use to maximize his profit? ________

How many painting jobs will Paco complete using this number of worker hours? ________

In addition to hiring workers, Paco spends $150 on brushes (50 brushes at $3 each) and $3000 on paint (100 gallons at $30 per gallon). What is Paco's profit using these values for his three factors (labor, brushes, and paint)? $_______

b) Now suppose that Paco is able to negotiate a lower price for paint and can buy the 100 gallons he needs for $27 instead of $30. Due to this cost savings, Paco's profit-maximizing choice of labor will __________(increase/decrease/stay the same) and he will use ________ worker hours. He will complete ________ projects (houses painted) and earn a profit of $_________

Answer 1

Solution :-

(A) :-

short-run production function depends only on labor and is given by :-

?(?)=√L/2

Paco's marginal product of labor is :-

???=1/4√L

Supposing that Paco charges = $4000 for a paint job .

The current wage he must pay his workers is = $28 per hour.

Optimal number of worker hires at following point :-

MPL = w/p

Where, w = wage paid to workers

p = price charges for print job

MPL = w/p

1/4 √L = 28/4000

4000 = 28 x 4√L

√L = 4000/( 28 x 4)

√L = 1000/28

L = ( 1000/28)^2

[ L = 1275.51 ]

Worker hours should Paco use to maximize his profit is = 1275.51

q = √L/2

q = ( 1000/28)/2

q = 1000/(28 x 2)

q = 500/ 28

[q = 17.85 ]

Painting jobs Paco complete using number of worker hours is 17.85

In addition to hiring workers,

Paco spends $150 on brushes (50 brushes at $3 each) and

$3000 on paint (100 gallons at $30 per gallon).

Total Cost (TC) = labour cost + brushes + paint

= wL + 150 + 3000

= 28 x 1275.51 + 150 + 3000

= 35714.28 + 150 + 3000

[TC = 38864.28 ]

Total Revenue (TR ) = Price x quantity

= 4000 x 17.85

[ TR = 71400 ]

Profit = TR - TC

= 71400 - 38864.28

[ Profit = 32535.72 ]

So, Paco's profit is = $32535.72

(B) :-

Now suppose that Paco is able to negotiate a lower price for paint and can buy the 100 gallons he needs for $27 instead of $30.

Optimal number of worker hires at following point :-

MPL = w/p

1/4 √L = 28/4000

L = (1000/28)^2

[ L = 1275.51]

q = √L/2

= 1000/(28 x 2)

[q = 17.85 ]

Now, paint price gets lower from $30 to $27

Paco can buy paint = 100 gallons

Paint cost = 27 x 100

= 2700

TC = labour cost + brushes + paint

= wL + 150 + 2700

= 28 x 1275.51 + 150 + 2700

= 35714.28 + 150 + 2700

[TC = 38564.28 ]

TR = Price x quantity

= 4000 x 17.85

[TR = 71400 ]

Profit = TR - TC

= 71400 - 38564.28

[Profit = 32835.72 ]

Due to cost savings, Paco's profit-maximizing choice of labor will stay the same and

he will use 1275.51 worker hours.

He will complete 17.85 projects (houses painted) and earn a profit of $32835.72.