Consider the following two-period model of dynamically efficient extraction of a non-renewable natural resource. The constant social marginal cost of extraction is 40 in each period and the total stock of the resource is Q = 300 units. Moreover, the social marginal benefit is MB(Qt) = 200Qt, where Qt is the quantity of resource extracted in period t, for t = 0; 1. The discount factor is 0:8.

(a) What is the efficient quantity of resource extracted in each period? Provide a graphical representation of the solution.

(b) What is the marginal user cost (or scarcity rent) of the resource in each period?

(c) Suppose that there is a market to trade the resource. What is the equilibrium price corresponding to each period? Justify the answer.

(d) Suppose that it is now expected that because of an extraction technology improvement, while the worst period marginal cost of extraction will still remain MXC1 = 40, the second period one will now de crease to MXC2 = 20. Answer the previous questions (a)-(c) under this alternative premise.

(e) How will the answers to questions (a)-(c) change if, because of new discoveries, the known reserves of the natural resource become Q = 400

1: Find the matrix that diagonalizes A = 10 1.1: Find eigenvalues of the matrix A. (10 points) 1.2: Find eigenvectors for each eigenvalue of A. (10 points) 1.3: Diagonalize the matrix A . That is, find an invertible matrix S and a diagonal matrix D such that S" AS = D. (10 points) Solution 1.1:Question 1 Write a function that takes two matrices (say, A and B) as arguments and return the matrix multiplication of A and B. Recall that in matrix multiplication, let C = A X B, then Cij = 2k AikBkj. Notice that in matrix multiplication, the number of columns in A must be the same as the number of rows in B. So in your program, if the inputs do not satisfy the condition, then provide an error message. Notice that, you are not allowed to use the R internal matrix multiplication function for this question, although you can verify your solution with it. Part 2: Identify the type of chemical reaction for each blank space. If the reaction is an oxidation/reduction reaction, state the oxidizing agent and the reducing agent. [13 marks] (2) (C 16) R-CH2 -CH2-CH2-C-5-COM Palmitoyl-CoA acyl-CoA A FAD dehydrogenaseFADHa R- CH2 - C -C CS-COM NADIN + H trans-A Enoyl-CoA HADP enoyl-CoA H20 hydratase R-CH2 -C-CH2 -C-5-COA L-B-Hydroxy- acyl-CoA #-hydroxyacyl-CoA NAD+ dehydrogenase NADH + H* R-CH2-C-CH2-C-5-COA B-Ketoacyl-CoA NADPH + H' acyl-CoA COA-SH O acetyltransferase NADP (thiolase) ( 14) R-CH2-C-5-COM + CH3-C-S-COA CH,--06-2 Saturated acyl group, (C,4) Acyl-COA lengthened by two carbons (myristoyl-CoA) Acetyl -CoA