Basic Properties of Utility and Production Functions

I have attached the screenshot of the question so it's easier to understand because of formatting....

(3 points) Consider the utility function u(c) = −c−2 + γ, where γ > 0.

- (a) Is marginal utility decreasing for this utility function? Show your work to prove
- your answer.
- (b) Find the coefficient of relative risk aversion.
- (c) Does this utility satisfy the first and second Inada conditions?

(2 points) Prove whether or not the following production functions exhibit constant returns to scale:

(a) F(K,L) = AKαL1−α, where α is between 0 and 1 and A > 0. (b) F(K,L) = AKL where A > 0.

(5 points) Consider a firm that uses capital K and land M as inputs to produce a final good, Y . Suppose that the technology for this firm is given by:

11 F(K,M)=2K2 +3M3

The firm must pay rental rate μ(1 + τK ) for each unit of capital rented, where τK > 0 represents a capital tax. It also must pay γ for each unit of land rented. Set up the firm's profit maximization problem and take FOCs. Find the optimal capital-land ratio. What happens when τK increases? Give some intuition.

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