Special emphasis on bonus questions please

Catherine plan to start a house-made lunchbox business that sells healthy

lunchbox during the COVID pandemic. The business has a production

function

Q = 10K0:3L0:7

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.

a. Provide reason for whether Catherine's house-made lunchbox busi-

ness has increasing, decreasing or constant returns to scale.

b. What is the Marginal Productivity of Labor (MPL) and Marginal

Productivity of Capital (MPK) for Catherine's business?

Hint: @aXbY c

@Y = acXbY c1

c. Catherine is making decision in the short run (for one day): when

the unit of capital are x at 5 (kitchen and all the kitchen wares

already purchased) and she can only change the labor input. Fill in

the blanks in the rst 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

5 5

5 7

5 10

1

d. Catherine now is making long run decision: both capital and labor

inputs are variable now. Determine Catherine's optimal capital-labor

ratio.

e. In long run, Catherine has a budget of $10,000 (to spend on the

inputs), nd out the exact optimal labor input, capital input and

total output levels.

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 nd 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).

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