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# Technical Rate of Substitution and Optimal Output

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Consider a firm that uses two inputs. The quantity used of input 1 is denoted by x_1 and the quantity used of input 2 is denoted by x_2. The firm produces and sells one good using the production function f(x_1,x_2)=4x_1^0.5+3x_2^0.5. The final good is sold at price P=\$10. The prices of inputs 1 and 2 are w_1=\$2 and w_2=\$3, respectively. The markets for the final good and both input goods are treated as competitive markets by the firm, that is, it takes prices as given.

1) Find the technical rate of substitution. Does the technology show diminishing technical rate of substitution? Explain.

Assume in the short run that x_2 is fixed at (x_2 ) ?=100.

2) Write down the firm's profit function and the firm's short run profit maximization problem. Find the firm's optimal use of input 1, the associated optimal quantity of the output, and the firm's profit level.