Al Deardwarf’s cousin, Zwerg, makes plaster garden gnomes. The technology in the

Question

Al Deardwarf’s cousin, Zwerg, makes plaster garden gnomes.

The technology in the garden gnome business is as follows. You need a

gnome mold, plaster, and labor. A gnome mold is a piece of equipment

that costs $1,000 and will last exactly one year. After a year, a gnome

mold is completely worn out and has no scrap value. With a gnome

mold, you can make 500 gnomes per year. For every gnome that you

make, you also have to use a total of $7 worth of plaster and labor. The

total amounts of plaster and labor used are variable in the short run. If

you want to produce only 100 gnomes a year with a gnome mold, you

spend only $700 a year on plaster and labor, and so on. The number

of gnome molds in the industry cannot be changed in the short run. To

get a newly built one, you have to special-order it from the gnome-mold

factory. The gnome-mold factory only takes orders on January 1 of any

given year, and it takes one whole year from the time a gnome mold is

ordered until it is delivered on the next January 1. When a gnome mold

is installed in your plant, it is stuck there. To move it would destroy it.

Gnome molds are useless for anything other than making garden gnomes.

For many years, the demand function facing the garden-gnome industry

has been D(p) = 60, 0005, 000p, where D(p) is the total number

of garden gnomes sold per year and p is the price. Prices of inputs have

been constant for many years and the technology has not changed. Nobody

expects any changes in the future, and the industry is in long-run

equilibrium. The interest rate is 10%. When you buy a new gnome mold,

you have to pay for it when it is delivered. For simplicity of calculations,

we will assume that all of the gnomes that you build during the one-year

life of the gnome mold are sold at Christmas and that the employees and

plaster suppliers are paid only at Christmas for the work they have done

during the past year. Also for simplicity of calculations, let us approximate

the date of Christmas by December 31.

(a) If you invested $1,000 in the bank on January 1, how much money

could you expect to get out of the bank one year later? $1,100. If

you received delivery of a gnome mold on January 1 and paid for it at that

time, by how much would your revenue have to exceed the costs of plaster

and labor if it is to be worthwhile to buy the machine? (Remember that

the machine will be worn out and worthless at the end of the year.)

$1,100.

(b) Suppose that you have exactly one newly installed gnome mold in

your plant; what is your short-run marginal cost of production if you

produce up to 500 gnomes? $7. What is your average variable cost

for producing up to 500 gnomes? $7. If you have only one gnome

mold, is it possible in the short run to produce more than 500 gnomes?

No.

(c) If you have exactly one newly installed gnome mold, you would produce

500 gnomes if the price of gnomes is above 7 dollars. You

would produce no gnomes if the price of gnomes is below 7 dollars.

You would be indifferent between producing any number of gnomes

between 0 and 500 if the price of gnomes is 7 dollars.

Please explain in part b why the answer is No. Also, explain the steps to reach the correct answers for part d and e.

(d) If you could sell as many gnomes as you liked for $10 each and none at a higher price, what rate of return would you make on your $1,000 by investing in a gnome mold? 50% . Is this higher than the return from putting your money in the bank? Yes. What is the lowest price for gnomes such that the rate of return you get from ivesting $1000 in a gnome mold is as at least 10%? $9.20. Could the long-run equilibrium price of gnomes be lower than this? No . e) At the price you found in the last section, how many gnomes would be demanded each year? 14,000. How many molds would be purchased each year? 28. Is this a long-run equilibrium price? Yes.

Explanation / Answer

b) In the short run, only the variable factors of production can be changed and we cannot change the fixed factors of production. So in order to increase more than 500 gnomes,we can only change the variable factors like labor and plaster but we cannot change the fixed factor like gnome mold. Maximum capacity of production of gnomes by a gnome mold is 500 only and thus we cannot produce more than 500 gnomes in the short run.

d) Amount of money invested in making 500 gnomes = $1000 + 500*7

= 1000 + 3500

= $4500

Revenue earned by selling 500 gnomes = 500 * 10

= $5000

Profit = 5000 - 4500

= $500

So, Rate of Return = Profit / Investment * 100

= 500 / 1000 * 100

= 50%

This is a higher return than the return from putting money in the bank.

Rate of return by putting money in the bank = 1100 - 1000/ 1000*100

= 100/ 1000 * 100

= 10%

Thus 50% is higher than 10%.

Price for the gnome to earn a rate of return of 10%

Rate of Return = Profit / Investment * 100

10 = 500*x - 4500 / 1000 * 100

10 = 50000x - 450000 / 1000

-50000x = -450000 - 10000

x = -460000/ 50000

= 9.2

Thus, $9.2 is the lowest price charged for gnome to earn a rate of returns of 10%

e) Price = $9.20

Demand Equation : D(p) = 60000 - 5000p

= 60000 - 5000*9.2

= 60000 - 46000

= 14000

So Quantity Demanded at price $9.20 = 14000 gnomes

Each mold produces 500 gnomes

So, in order to produce 14000 gnomes

No. of molds required = 14000 / 500

= 28

28 molds are purchased to produce 14000 molds

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