More Matter, Less Art. -Shakespeare (Hamlet, I think)
1. Suppose dentist estimates the production function for cleaning services to be :
lnQ = 7.86 +.2lnK+ .6lnD+ .1lnH
A unit of capital (K) represents a chair and a set of tools and costs $1000 per year, a unit of D represents one full-time dentist whose salary is $100,000 per year, and each unit of H represents one full time hygienist at a cost of $12.50 per hour or $25,000 per year.
Is this a realistic production function? Explain why or why not.
Yes, all coefficients are positive and less than one. This implies diminishing marginal returns. Also if any single input is zero, the output will be zero.
If the dental office has one dentist and two hygienists and three units of capital, what is the annual production? Is this the best combination of inputs for the level of output achieved? Thoroughly explain your answer.
Q= 2591(3).2(1).6(2).1 =3459 (annual production)
Optimal combination = MP/p for all inputs
MPk = 231; mpk/p = 231/1,000 =.23
Mpd =2075 ; mpd/p = 2075/100,000 =.02
Mph = 172; mph/p = 172/25000 =.0069
So, capital is the most productive per dollar. Get some more chairs, and get rid of a hygenist.
Given the choice of expanding the current location, or opening a second brand new location elsewhere, which would you reccomend? Explain your answer based on the production function.
Becausee we have diseconomies of scale (.2+.6+.1 <1), you are better off with multiple small offices, rather than one large office.
If the firm's short-run variable cost function for cleanings is estimated to be:
TVC = 5q +.0065q2
Does this short run function meet the same realism requirements as the production function?
Since mc = 5+.013q is increasing, we have diminishing returns so, yes.
What are the fixed costs in the short run? (I’m thinking the chair and the dentist (103,000)
Cleanings are sold in a competitive market at a price of $50. (Insurance companies also cover $50, so the dentist has no control over the price)
Based on your results from the production function, is the dentist maximizing profits? If not, How should resources be adjusted?
Max profit at MR = MC
50 = 5-.013q
45=.013Q
q = 3461 (that’s pretty close to the production function answer)
AVC= 5 + .0065(3461) = $27.50
What is the contribution to fixed costs and profit (P-AVC) on each cleaning if the dentist is maximizing profits? = 50 – 22.5 = $27.50
Assume that the profit margin on cleanings does not vary with output (why is that unrealistic? Because average costs increase, the margin decreases ) and the dentist can make $40 contribution to fixed costs and profit on each tooth pulled (also irrespective of output). Further, each tooth pulled requires 20 minutes dentist time, 20 minutes hygienist time, and 48 minutes in the chair. Cleanings require 6 minutes dentist time, 20 minutes hygienist time and 24 minutes in the chair. What's more, the dentist does not adjust resources from those available in question one. How many cleanings and pullings should the dentist do each 8 hour day?
So maximize: л = 27.5*C + 40*P
the constraints are
Dentist6C +20P < 480 (8 hours * 60 minutes* 1 dentist)
Hygenist20C +20P < 960 (8 *60 * 2 chairs)
Chair24C +40P < 1440 (8 * 60* 3 chairs )
Cleanings
80dentist
34*27.5 +13.8*40 =$1487 (max)
60$1320
Chair
48 $960
Hygenist
243648Pullings