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Lagrangian Multiplier

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Please show steps to resolve these problems please use Lagrangian Multiplier if possible.

1. A manufacturer has the following production function:

Q=100 K^. 2 L^. 9

If the price per unit of labor is $20 and the price per unit of capital is $10,
a) What is the optimal combination of labor and capital to use in order to maximize output for a total cost of $2,200?
b) How much output will be produced?
c) What is the equation for the expansion path?
d) What is the meaning and value of λ (lambda)?
e) What is the value of the marginal rate of technical substitution for labor and capital at the optimal point?
f) Is there diminishing marginal productivity for labor and capital? Explain
g) Are there increasing returns to scale? Explain

2. Given the following short-run production function, Q = 1500L + 60L^2 - L^3
where Q is output and L is variable input, find
a. The point of diminishing marginal returns.
b. The point where the elasticity of production is equal to one.
c. The boundary between stage II and stage III.
d. Show that marginal product is equal to average product when average product is at its maximum.

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Lagrangian Multiplier is utilized in the problems.

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