Question

a. Suppose the production function of a restaurant is as follows: Q- Assume it is the long run. (ao points) a) Carefully draw isoquants for Q4o, Q-so, and Qbo. Show all your work b) Suppose the price of labor is sio and the price of capital is sto. Draw the isocost lines for TC-8o, 100, and 120. Briefly explain how you drew the lines How much capital and how much labor will be used to produce at each output? Label the cost-minimizing combinations for each output What kind of returns to scale it is? c) d)

Answer 1

A) production function is Q = 10(KL)^1/2

For Q= 40, product of K and L should be 16, which is possible from following combination: (1,16) ,(2,8), (4,4), (8,2) and (16,1)

For Q=50, KL should be 25, which is possible from following combination: (1,25), (5,5) and (25,1)

For Q= 60, KL should be 36, which is possible from following combination (1,36), (2,18), (3,12), (4,9), (6,6), (9,4), (12,3) and (36,1)

This will be isoquant lines for all 3

40 35 0 050 433 05050 211

B) Price of 1 labor unit is 10 and price of 1 capital unit is 10

So for TC = 80,100 and 120 restaurant can get in total 8, 10 and 12 units of labor and capital in combination like 1,7 or 1,9 1,11 or similar to that with sum of both units being 8,10 and 12. Following will be isocost lines:

2 0 9876543210 8 6 3 2 876543210 0123

TC-120 12 10 6 0 6 8 10 12 0 2 4. Seriesi
Following are the outputs for each combination. You can observe that for the given production function, output will be maximum if equal number of capital and labour units are purchased. Also these points are tangents to isoquant lines of 40,50 and 60

5542 4 15971 9614 5 345 5 5 0 1 2 3 4 5 6 78901 1 008808800 345959543 2 2 4 4 987654321 123456789 719 t542 u467 p648 t233 1230321 5084805 917 245 764 332 7654 3 2 1 1 2.3 4.5 6 7

D) We can see that it is constant rate of return to scale as proportion of units of capital and labour required to increase output is constant:

4K and 4L gives 40 units, 5k and 5L gives 50 units and 6k and 6L gives 60 units. You can check further that 7K and 7L will give 70 units. With every 1 unit increment in K and L together, output increases by 10 units.

Thanks