Consider a firm with a production function of the following form:F (L, K) = L(1/4)K(1/2)

20

the corresponding marginal products are
MPL = 1 L(−3/4)K(1/2)

4 20MPK = 1 L(1/4)K(−1/2)

2 20
The cost of a unit of labour is the wage rate w and the cost of a unit of capital is the

rental rate r. The firm is free to adjust all factors of production. (19 points)
a. Does this production function exhibit decreasing returns, increasing returns, or

constant returns to scale?

b. Set up the cost minimization problem and derive the conditional input demand functions for the firm.

c. Provide an expression for the total cost curve for a given level of output q (the cost function should be a function of only the exogenous variables).

In many agricultural and biological experiments, one may use a two‑way model with only one observation per cell. When one of the factors is related to the grouping of experimental units into more uniform groups, the design may be called a randomized complete block design (RCBD). The analysis is similar to a two‑way analysis of variance (question B) except that the model does not include an interaction term.

            The specific leaf areas (area per unit mass) of three types of citrus each treated with one of three levels of shading are stored in Table C. The first column contains the code for the shading treatment, the second column contains the code for the citrus species, and the third column contains the specific leaf area. Assume that there is no interaction between citrus species and shading. Carry out a two‑way analysis of this data.

            The shading treatment and citrus species are coded as follows:

            Treatment        Code                Species                        Code

            Full sun           1                      Shamouti orange         1

            Half shade       2                      Marsh grapefruit          2

            Full shade        3                      Clementine mandarin 3

nCopy the treatment code, the species code, and the specific leaf area into the EXCEL worksheet, label the columns and look at the data.

                                                                                                                                       {Example 1}

nPerform a two‑way (without interaction) analysis of this data and answer the following questions. Use a 5% significance level.

{Example 26}

            Recall that the confidence interval for a difference between two means is based on a calculation of the margin of error of the estimated difference. With a common variance (Error MS) and the same number of observations in all shading treatments, the margin of error of an estimated difference will be the same whether we calculate it for treatments 1 and 2, 1 and 3, or 2 and 3. This margin of error of the difference between two means is sometimes referred as the least significant difference (LSD).

nCalculate the LSD for comparing shading treatments in this experiment.

LSD = critical tvalue ´standard error of difference.

Use the critical t value with 4 degrees of freedom is t _0.025,4= 2.776.

n is the number of times of times each treatment was tested (in this case n = 3 for the 3 species).

Consider the following reactions:

Zn^2+ + 4H3 <----> Zn(NH3)4^2+ beta 4 = 5.01 x 10^8

Zn^2+ + 2e^- <-----> Zn(s) E degree = -0.762 V

Assuming there is negligible current and nearly all Zn^2+ is in the form Zn(NH3)4^2+. what cathode potential (vs. S.H.E.) would be required to reduce 99.99% of the Zn^2+ from a solution containing 0.20 M Zn^2+ in 1.4 M ammonia? (Assume T= 298 K)

________V

2. Given A = | 2 1 0 1 2 0 1 1 1 |.

(a) Compute eigenvalues of A.

(b) Find a basis for the eigenspace of A corresponding to each of the eigenvalues found in part (a).

(c) Compute algebraic multiplicity and geometric multiplicity of each eigenvalue found in part (a).

(d) Is the matrix A diagonalizable? Justify your answer

Suppose gg is a function which has continuous derivatives, and that g(1)=1,g′(1)=5, g″(1)=4, g‴(1)=2

(a) What is the Taylor polynomial of degree 2 for g near 1?
P2(x)=

(b) What is the Taylor polynomial of degree 3 for g near 1?
P3(x)=

(c) Use the two polynomials that you found in parts (a) and (b) to approximate g(1.1).
With P2, g(1.1)≈
With P3, g(1.1)≈

Question2.

Let A = [2 1 1

1 2 1

1 1 2 ].

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

Let A = 2 1 1

1 2 1

1 1 2

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

Problem 13-4 Various liabilities [LO13-1, 13-2, 13-3, 13-4]

The unadjusted trial balance of the Manufacturing Equitable at December 31, 2018, the end of its fiscal year, included the following account balances. Manufacturing’s 2018 financial statements were issued on April 1, 2019.

Other information:

1. The bank notes, issued August 1, 2018, are due on July 31, 2019, and pay interest at a rate of 9%, payable at maturity.
2. The mortgage note is due on March 1, 2019. Interest at 8% has been paid up to December 31 (assume 8% is a realistic rate). Manufacturing intended at December 31, 2018, to refinance the note on its due date with a new 10-year mortgage note. In fact, on March 1, Manufacturing paid $492,500 in cash on the principal balance and refinanced the remaining $952,500.
3. Included in the accounts receivable balance at December 31, 2018, were two subsidiary accounts that had been overpaid and had credit balances totaling $20,700. The accounts were of two major customers who were expected to order more merchandise from Manufacturing and apply the overpayments to those future purchases.
4. On November 1, 2018, Manufacturing rented a portion of its factory to a tenant for $31,200 per year, payable in advance. The payment for the 12 months ended October 31, 2019, was received as required and was credited to rent revenue.

Required:
1. Prepare any necessary adjusting journal entries at December 31, 2018, pertaining to each item of other information (a–d).
2. Prepare the current and long-term liability sections of the December 31, 2018, balance sheet.

Problem 13-4 Various liabilities [LO13-1, 13-2, 13-3, 13-4]

The unadjusted trial balance of the Manufacturing Equitable at December 31, 2018, the end of its fiscal year, included the following account balances. Manufacturing’s 2018 financial statements were issued on April 1, 2019.
   

   
Other information:

1. The bank notes, issued August 1, 2018, are due on July 31, 2019, and pay interest at a rate of 10%, payable at maturity.
2. The mortgage note is due on March 1, 2019. Interest at 9% has been paid up to December 31 (assume 9% is a realistic rate). Manufacturing intended at December 31, 2018, to refinance the note on its due date with a new 10-year mortgage note. In fact, on March 1, Manufacturing paid $250,000 in cash on the principal balance and refinanced the remaining $950,000.
3. Included in the accounts receivable balance at December 31, 2018, were two subsidiary accounts that had been overpaid and had credit balances totaling $18,000. The accounts were of two major customers who were expected to order more merchandise from Manufacturing and apply the overpayments to those future purchases.
4. On November 1, 2018, Manufacturing rented a portion of its factory to a tenant for $30,000 per year, payable in advance. The payment for the 12 months ended October 31, 2019, was received as required and was credited to rent revenue.

   
Required:
1. Prepare any necessary adjusting journal entries at December 31, 2018, pertaining to each item of other information (a–d).
2. Prepare the current and long-term liability sections of the December 31, 2018, balance sheet.