Question

2. Determine the returns to scale of the following functions. Show your work. a) Q=4K + 3L b) Q= 2KL c) Q 10K/L

Answer 1

Returns to scale is with what proportion the output increases given the input increases.

If it increases with the same proportion ,we have constant returns to scale.

If it increases more then the increase in input, the three returns are increasing.

In the increase is less than the increase in inputs then the return is decreasing returns.

a) Q= 4K+3L

Suppose both inputs are increased by X. So K is increased to Kx and L to xL.

So Qnew= 4Kx+3Lx= x(4K+3L)= xQ

So Qnew = Qx ie the new output is x times the old one which means that the production function gives constant returns to scale.

b) Q= 2KL

As we did in the previous part , we would increase the inputs by x.

Qnew= 2(Kx)(Lx) = x² (2KL) = x² Q

This implies that when the inputs are increased by x , the output increases by x² . Hence gives increasing returns to scale.

c) Q= 10K/L

We increase both the inputs by x.

So Qnew = 10(Kx)/(Lx) = 10K/L = Q

Qnew = Q

This implies that even after increasing the inputs by X , the output does not increase, so it gives decreasing returns to scale.

We are given the fixed cost is $10 and the price is $8 (fixed)

Total product	AFC	AVC	ATC	TC	MC	TR	PROFIT
0				10		0	(-)10
1	10/1=10	20-10=10	20	20	10	8	(-)16
2	10/2= 5	14-5=9	28/2= 14	28	8	16	(-)12
3	10/3=3.34	11-3.34=7.66	33/3= 11	33	5	24	(-)9
4	10/4= 2.5	9-2.5=6.5	36/4=9	36	3	32	(-)4
5	10/5= 2	8-2=6	40/5=8	40	4	40	0
6	10/6= 1.67	8-1.67=6.33	48/6=8	48	8	48	0
7	10/7=1.43	8.28-1.43=6.85	58/7= ​​​​​8.28	58	10	56	(-)2
8	10/8= 1.25	9-1.25= 7.75	72/8= 9	72	14	64	(-)8
9	10/9= 1.11	10-1.11=8.89	90/9=10	90	18	72	(-)18
10	10/10= 1	11-1=10	110/10=11	110	20	80	(-)30

AFC is the average fixed cost given by TFC/number of units

We are given the TFC as $10 and so the column of AFC is filled according to this formula Given above.

Next we have the ATC which is Total cost / no of units

We are given the total cost in each case and by putting this formula we can find the ATC.

For AVC, we can either take the difference of ATC and AFC or we can find TVC and divide it by the number of units of goods produced at each level. Column of AVC is filled according to this.

MC is the marginal cost at each unit which is given by

MCn = TCn - TC(n-1) where n is the number of units produced , so TC(n-1) is the total cost at n-1 units.

MC column is filled accordingly.

TR is total revenue. Since we are given that price is fixed , we can simply multilply the price and units produced and the resultant would be TR . The TR column is filled accordingly.

Profit is given by = TR- TC

Note : Most of the answers already filled are correct except the MC at unit 6 which should be 8 and AVC at unit 6 which should be 6.33.

(You can comment for doubts)