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Production function and equilibrium output

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The technology of a firm making high end, solid gold bracelets in Soho (NYC) is described by the production function:

q = 6.0 L3/4K1/5

where q is the number of bracelets produced per year, L is the number of metallurgist employed by this firm and K is the number of capital units used, measured in square footage of factory floor space. Capital is available at a cost of $3.745 per square foot per year and this price is guaranteed regardless of the size of the facility the firm requires at any given time.

The product and labor market conditions facing this firm are described by:

Product Market: Labor Market:
Demand: P = 13,200 - 0.8Q w = 95,000 - 2.5 QL
Supply: P = 258.18 + 0.18Q w = 15,000 + 1.5 QL

where P is the price (rounded to the nearest quarter dollar) of bracelets similar to those produced by the firm in question on the national market, Q is the number of similarly styled bracelets produced by all firms in the national market, w is the annual salary, measured in dollars per year, paid to metallurgists who work in this industry, and QL is the number of metallurgists employed in this industry nationally.

The following relate only to the firm whose technology is given above:

1. Initially, the firm leases a facility with 248,832 square feet. With this size facility, determine the output level of the firm, the number of workers it employs and the profit of the firm.

2. Three years later, the firm renegotiates its capital lease so that it can employ exactly the amount of capital it feels necessary to maximize its profits. Determine the output level of the firm, the number of workers it employs and the profit of the firm.

3. Suppose the firm decides to set an output target so that it will be making a 50 per cent return on its capital (i.e., it wants to make a profit equal to 50 per cent of its capital cost). Determine the output level of the firm, the number of workers it employs and the profit of the firm.

4. Suppose the metallurgists working for the firm in question negotiate a 2.5 per cent wage hike over and above the current wage in the labor market. Determine many workers, if any, lose their jobs at the new wage. Then determine the substitution and output effects on labor of the wage

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Solution Summary

This posting is about a solid gold bracelet manufacturer. The production function for the manufacturer is given with inputs as number of metallurgist employed and number of capitals units used (factory floor space). The external environmental conditions for the manufacturer are characterized by product market conditions.

The solution provides step by step details to derive equilibrium output level and level of resources used. Scenarios like renegotiation of capital lease, target output level, wage hike are analyzed.

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Problem:
The technology of a firm making high end, solid gold bracelets in Soho (NYC) is described by the production function:
q = 6.0 L3/4K1/5 --------------------5
where q is the number of bracelets produced per year, L is the number of metallurgist employed by this firm and K is the number of capital units used, measured in square footage of factory floor space. Capital is available at a cost of $3.745 per square foot per year and this price is guaranteed regardless of the size of the facility the firm requires at any given time.

The product and labor market conditions facing this firm are described by:

Product Market: Labor Market:
Demand: P = 13,200 - 0.8Q ------------1
Supply: P = 258.18 + 0.18Q -------------2
w = 95,000 - 2.5 QL ----------------3
w = 15,000 + 1.5 QL --------------4

where P is the price (rounded to the nearest quarter dollar) of bracelets similar to those produced by the firm in question on the national market, Q is the number of similarly styled bracelets produced by all firms in the national market, w is the annual salary, measured in dollars per year, paid to metallurgists who work in this industry, and QL is the number of metallurgists employed in this industry nationally.

The following relate only to the firm whose technology is given above:

1. Initially, the firm leases a facility with 248,832 square feet. With this size facility, determine the output level of the firm, the number of workers it employs and the profit of the firm.

First calculate the level of price and wage rate that firm will be taking from the market. This will be determined from the market equilibrium in product market and labor market.

For equilibrium in product market
Equate Demand = Supply
Thus from equation 1 and 2, we get 13,200-0.8Q=258.18+0.18Q
0.98Q = ...

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