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Calculating profit maximizing output and elasticity

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The firm's production function for gadgets, X, depends on the amounts of capital (K), and labor (L) employed:
X = 56K + 8.8L - 2K2 - 0.1L2.

(a) If the firm can sell any output at a price equal to 10, if the wage rate per unit of labor is 8 and the rental per unit of capital is 80, what is the profit maximizing level of output?

(b) At the optimum solution in (a), what is the value of the marginal rate of technical substitution in production?

(c) If the firm maximizes profit at a higher wage rate of 10 (all other parameters unchanged), what is the firm's elasticity of demand for labor over this wage range?

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(a) If the firm can sell any output at a price equal to 10, if the wage rate per unit of labor is 8 and the rental per unit of capital is 80, what is the profit maximizing level of output?

Marginal Product of labor=MPL=dX/dL=8.8-0.2L
Marginal Revenue of labor=MRL=Price*MPL=10*(8.8-0.2L)=88-2L
Marginal cost of labor=MLC=wage ...

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Solution describes the steps for calculating profit maximizing output and elasticity of demand for a firm.

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Calculating profit-maximizing price and output

Coke
PL=$40-$0.0005QL
MRL= dTRL/ dQL =$40-$0.001QL

Pepsi
PT=$50-$0.0004QT
MRT=dTRT/dQT = $50 - $0.0008QT

Average variable costs for labor and materials are constant at $20 per unit.

1. Calculate the profit-maximizing price, output, and total profit contribution levels.
2. Calculate point price elasticities of demand for each customer product at the activity levels identified in part A.

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