Please help me with ALL parts of the questions You own a small furniture shop th
ID: 1208572 • Letter: P
Question
Please help me with ALL parts of the questions
You own a small furniture shop that makes custom wooden bed
frames. The market price of your bed frames if p=$5,000, you pay your workers w1=$100 an hour
(let x1=worker hours), and you rent furnituremaking equipment (x2) each month for w2=$1000 per
machine. It takes at least one month to install new machines so this month your equipment is fixed
at x2=16 machines. Your production function for bed frames in one month is y=x1^1/2 x2^1/4.
a) calculate cost function for furniture this month.
b1) Solve for optimal output this month
b2) how many worker hours will be used
c1) Using the cost function from part a), give equations for average cost, average variable cost, and marginal cost.
c2) at what quantity will average cost reach its minimum?
d) now consider the production decision for next month, when both number of worker hours and number of machines can be changed
d1) if you want to produce exact same amount as previous month, would you use the same ratio of worker hours to equipment? if not, what ratio would you use and how wourld your average cost compare to your average cost last month? provide calculation to support answer.
d2) does long-run average cost function decrease as output is increased, increase, or stay the same? explain.
Explanation / Answer
y = x11/2x21/4 = 2. x11/2 [Since x2 = 16]
(a)
Total cost, TC = w1x1 + w2x2 = 100x1 + 16,000
Since y = 2. x11/2,
x11/2 = y / 2
Squaring both sides,
x1 = y2 / 2
So, TC = 100. (y2 / 2) + 16,000 = 50. y2 + 16,000
(b-1) Output is optimal when MPx1 / MPx2 = w1 / w2
MPx1 = dy / dx1 = (1/2). x1 - 1/2.x21/4
MPx2 = dy / dx2 = (1/4). x11/2.x2 - 3/4
So, MPx1 / MPx2 = 2. (x2 / x1) = 100/1000 = 1 / 10
x2 / x1 = 1 / 20
x1 = 20x2 = 20 x 16 = 320
x2 = 16 (given)
y = 2 x 4 = 8
(b-2) x1 = 320 hours
(c-1)
AC = TC / y = 50y + (16,000 / y)
AVC = 50y
MC = dTC / dy = 100y
(c-2)
AC is minimum when dAC / dy = 0
50 - (16,000 / y2) = 0
16,000 / y2 = 50
y2 = 16,000 / 50 = 320
y = 17.89
Note: First 5 sub-parts are answered.