Suppose a competitive firm’s production function is q = L 0.5K0.5 and the wage rate is w = $5 and the rental rate is r = $0.8

(a) If K is fixed at 25 units in the short-run, what is the short-run production function?

(b) What is the equation that determines how much labor the firm should hire to produce some quantity, q?

(c) Use your answer to (b) to derive the short-run total cost curve.

(d) What is the firm’s marginal cost curve? What is the firm’s average total cost curve?

(e) If the price of the firm’s product is $6, how many units should the firm choose to sell?

(f) What are its short-run profits?

Thanks for your help!

A flat furnace wall is constructed of a 4.5 in (114 mm) layer of Silo-cel brick, with a thermal conductivity of 0.08 Btu/ft·h·˚F (0.138 W/m·˚C), backed by a 9-in. (229-mm) layer of common brick, of conductivity 0.8 Btu/ft·h·˚F(1.38 W/m.˚C). The temperature of the inner face of the wall is 1400˚F (760˚C), and that of the outer face is 170˚F(76.6˚C).

(a) What is the heat loss through the wall?

(b) What is the temperature of the interface between the refractory brick and the common brick?

(c) Supposing that the contact between the two brick layers is poor and that a “contact resistance” of 0.50˚F·ft2.h/Btu (0.088˚Cm2/W) is present, what would be the heat loss?

Suppose your firm has decided to use a divisional WACC approach to analyze projects. The firm currently has four divisions, A through D, with average betas for each division of 0.8, 1.1, 1.5, and 1.7, respectively. Assume all current and future projects will be financed with 35 percent debt and 65 percent equity, the current cost of equity (based on an average firm beta of 1.3 and a current risk-free rate of 6 percent) is 15 percent and the after-tax yield on the company’s bonds is 11 percent. What will the WACCs be for each division? (Do not round intermediate calculations and round your final answers to 2 decimal places.) WACCs

Division A %

Division B %

Division C %

Division D %

A steam power plant operates on a regenerative Rankine cycle with two feedwater heaters, one closed and one open. Steam enters the turbine at 12.5 MPa and 550ºC and exhausts to the condenser at 10 kPa. Steam is extracted from the turbine at 1 MPa for the closed feedwater heater and 0.8 MPa for the open one. The feedwater is heated to the condensation temperature of the extracted steam in the closed feedwater heater. The extracted steam leaves the closed feedwater heater as a saturated liquid, which is subsequently throttled to the open feedwater heater. Show the cycle on a T-s diagram with respect to saturation lines, and determine(a) the mass flow rate of steam through the boiler for a net power output of 350 MW and (b) the thermal efficiency of the cycle.

1. The U.S. Census Bureau computes quarterly vacancy and homeownership rates by state and metropolitan statistical areas. The following data are the rental vacancy rates in percentage (%) grouped by region for the last quarter of 2018 using a sample of 4 statistical metropolitan areas.

                                     Vacancy rates (%)

1. You are hired as statistician to test whether there are any differences among the regions regarding vacancy rates and you should allow for 10% error.
2. Do you think that the mean vacancy rate is the same for these regions? Why

Patients with chronic kidney failure may be treated by dialysis, in which a machine removes toxic wastes from the blood, a function normally performed by the kidneys. Kidney failure and dialysis can cause other changes, such as retention of phosphorus, that must be corrected by changes in diet. A study of the nutrition of dialysis patients measured the level of phosphorus in the blood of several patients on six occasions. Here are the data for one patient (in milligrams of phosphorus per deciliter of blood).

The measurements are separated in time and can be considered an SRS of the patient's blood phosphorus level. Assume that this level varies Normally with

σ = 0.8 mg/dl.

(a) Give a 95% confidence interval for the mean blood phosphorus level.

Thorpe and Company is currently an​ all-equity firm. It has three million shares selling for ​$32 per share. Its beta is 0.8​, and the current​ risk-free rate is 2.1​%. The expected return on the market for the coming year is 9.1​%. Thorpe will sell corporate bonds for ​$32,000,000 and retire common stock with the proceeds. The bonds are​ twenty-year semiannual bonds with a 9.4​%coupon rate and ​$1,000 par value. The bonds are currently selling for ​$893.61 per bond. When the bonds are​ sold, the beta of the company will increase to 1.3 What was the WACC of Thorpe and Company before the bond​ sale? What is the adjusted WACC of Thorpe and Company after the bond sale if the corporate tax rate is 15%​? Hint​: The weight of equity before selling the bond is​ 100%.

Company Q’s current return on equity (ROE) is 14%. It pays out one-half of earnings as cash dividends (payout ratio=0.5). The current book value per share is $50. Book value per share will grow as Q reinvests earnings. Assume that the ROE and payout ratio stays constant for the next four years. After that, competition forces ROE down to 11.5% and the payout ratio increases to 0.8. The cost of capital is 11.5%.

1. What are Q’s EPS and dividends next year? How will EPS and dividends grow in years 2, 3, 4, 5, and subsequent years?
2. What is Q’s stock worth per share? How does that value depend on the payout ratio and growth rate after year 4?

3. Suppose that a closed economy could be described by the following set of equations: Production Function: Y = 120K1/3 L2/3

̅̅ Factors of production: K=125; L=1000

̅̅ Government Behavior: G= 3000; T = 2500

Consumption Behavior: C = 400 + 0.8(Y –T)
Investment Behavior: I = 16500 – 1000r
a. Calculate the equilibrium level of output (Y)
b. Prove that the production function has the property of constant returns to scale.

c. Calculate the marginal product of labor (MPL)

d) Calculate the equilibrium levels of the following variables

(i) Consumption (C)

(ii) Private Savings

(iii) Public Savings

(iv) National Savings (S)

(v) real interest rate (r)