A production process using two inputs, labor and capital, can be written as:
Q = 5LK MP_K = 5L MP_L = 5K
where Q represents output per day (tons). The unit costs of inputs are $150 for labor (L) and $1,000 for capital (K). Determine the least cost combination of L and K when output is produced at the rate of 1,000 tons per day. Determine the required outlay for 1,000 tons per day.

The least cost combination of inputs occurs where the ratio of prices of inputs equals the marginal rate of technical substitution of one input for another.
The price ratio is P_L/P_K = 150/1,000 = 0.15.
Now find the combination of L and K that will make MRTS equal to 0.15.
MRTS = = = 0.15
K = 0.15L
The output rate is 1000 = Q, thus
1000 = 5LK = 5L(0.15L) = 0.75L²
L = = 36.51 units.
K = 0.15(36.51) = 5.48 units.
The total outlay needed to purchase inputs to satisfy this production rate is:
I = P_LL + P_KK
I = 150(36.51) + 1,000(5.48)
= $5,476.50 + 5,480
I = $10,956.50 total outlay per day.