Charlie has a repair shop that uses three inputs: Unskilled Labor (quantities of which will be denoted by L), Skilled Labor (H) and Machinery (K). The inputs (L,H,K) produce: Y=L1/6 H1/3K1/2 units of repairs. Charlie is a price-taker in all markets. The wages of unskilled and skilled labor are wL and wH respectively, and the running
Charlie has a repair shop that uses three inputs: Unskilled Labor (quantities of which
will be denoted by L), Skilled Labor (H) and Machinery (K). The inputs (L,H,K)
produce:
Y=L1/6 H1/3K1/2
units of repairs. Charlie is a price-taker in all markets. The wages of unskilled and
skilled labor are wL and wH respectively, and the running cost of K units of machinery
is wK. Charlieís repair shop is equipped with K = 1 unit of machinery, which cannot
be adjusted. To produce any positive output all the machinery in the repair shop must
be turned on, but if no output is produced he can choose not to turn on the machinery.
There is no upper limit on the output of Charlieís repair shop. He is a profit maximizer, and hence for any level of output y he will adjust (L, H) to minimize cost. Input prices
are (wL,wH,wK) = (1,2,27):
(a) Derive Charlieís short-run cost function
sc(wL,wH,wK, y,K(complement)), and then specialize it to the given input prices. Determine the cost-minimizing ratio:
L(wL,wH,wK,y,K(complement))
H(wL,wH,wK,y; K(complement))
of unskilled to skilled labor.
(b) Let the output price be p. What is Charlieís optimal output and profit as a function of p?
(c) Consider the following model of long-run competition. Repair shops of the kind described above can freely enter and exit the market, each furnished with K = 1; but there are no repair shops with higher or lower levels of K. What is the price p in the long-run equilibrium? How much does each repair shop produce? Find
how many repair shops will be operating in the long-run equilibrium if the market
demand curve is:
p + Y = 1050
![The image shows a mathematical expression involving two functions, \( L \) and \( H \). It is presented as a fraction:
\[
\frac{L(w_L, w_H, w_K, y, \overline{K})}{H(w_L, w_H, w_K, y, \overline{K})}
\]
### Explanation:
- **Numerator: \( L(w_L, w_H, w_K, y, \overline{K}) \)**
This represents a function \( L \) that takes five variables as inputs:
- \( w_L \)
- \( w_H \)
- \( w_K \)
- \( y \)
- \(\overline{K}\)
- **Denominator: \( H(w_L, w_H, w_K, y, \overline{K}) \)**
Similarly, this represents another function \( H \) with the same set of inputs:
- \( w_L \)
- \( w_H \)
- \( w_K \)
- \( y \)
- \(\overline{K}\)
These types of expressions are often used in mathematical modeling, economics, or statistical analysis to compare or relate two different functions or processes using a ratio.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7f4e788-ef76-42cd-ba71-0d980b205d9f%2F4315fd86-3b49-457a-aa5d-4059da793f03%2Fgk4fu7e_processed.png&w=3840&q=75)
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