Catherine plan to start a house-made lunchbox business that sells healthy lunchbox during the COVID pandemic. The business has a production function

Q = 10K^0.3L^0.7

where K denotes the unit of capital inputs per hour and L denotes the unit of labor inputs per hour used in producing Q number of lunch box. The capital (kitchen, all the kitchen wares, and raw materials) rents for $20 per hour, labor can be hired for $30 per hour.

Catherine wants to applies her economics knowledge to make the optimal decision to minimize the cost of the production.

Question 1 a. Provide reason for whether Catherine's house-made lunchbox business has increasing, decreasing, or constant returns to scale.

Question 1 b. What is the Marginal Productivity of Labor (MPL) and Marginal Productivity of Capital (MPK) for Catherine's business? Hint: aX^bY^c/Y = acX^bY^c-1.

Question 1 c. Catherine is making decision in the short run (for one day): when the unit of capital are fix at 5 (kitchen and all the kitchen wares already purchased) and she can only change the labor input. Fill in the blanks in the first row except for AC, AFC and AVC of the table below (round to the closest integer)

Q K L MPL MPK TC TFC TVC AC AFC AVC

5 1 ? ? ? ? ? ? ? ?

55? ? ? ? ? ? ? ?

5 1 ? ? ? ? ? ? ? ?

55? ? ? ? ? ? ? ?

Question 1 d. Catherine now is making long run decision: both capital and labor inputs are variable now. Determine Catherine's optimal capital-labor ratio.

Question 1 e. In long run, Catherine has a budget of $10,000 (to spend on the inputs), find out the exact optimal labor input, capital input and total output levels.

question 2

Bonus 1.

Solve the Q1c and Q1d by using Lagrangian

a. Write down the Cost Minimization Problem.

b. Write down the corresponding Lagrangian.

c. Find the optimal capital-labor ratio.

d. Given $10,000 budget find the exact optimal amount of labor input, capital input and total output level.

Bonus 2.

Fill the rest of the blanks in Q1c. And draw a sketch of AC, AFC and AVC in the same graph (no need to be extremely precise, just with the numbers calculated from the table and the correct curvature).