Question

In Juiceland company Ltd, the relationship between output(Q) and number of hours of skilled labour (S) and unskilled labour(U) is :
Q=500S + 100U - 0.2S^2 - 0.3U^2
The hourly wage of skilled labour is $15, and the hourly wage of unskilled labour is $5. The firm can hire as much labour as it wants at these wage rates.
a) Juiceland COO recommends that the firm hire 400 hours of skilled labour and 100 hours of unskilled labour. Evaluate this recommendation.
b) If Juiceland decides to spend a total of $5000 on skilled and unskilled labour , how many hours of each type of labour should it hire?
c) If the price of a unit of output is $5 ( and does not vary with output), how many hours of unskilled labour should the company hire?

Answer 1

Answer on Question #79575, Economics / Macroeconomics

a) With the existing relationship

$$Q = 500S + 100U - 0.2S^2 - 0.3U^2$$

And provided hire 400 hours of skilled labour and 100 hours of unskilled labour

We can deduce the value of Q

$$Q = 500*400 + 100*100 - 0.2*400^2 - 0.3*100^2$$ $$Q = 200000 + 10000 - 32000 - 3000 = 175000$$

And the labor costs for such a release:

$$400 * 15 = 6000$$ $$100 * 5 = 500$$

**Conclusion**: for a given number of hours, we get output $Q = 175000$ for 1 hour and labor costs of $6,500$.

b) With labor costs of $5,000, we get a relationship:

$$15 * S + 5 * U = 5000,$$

Or

$$3S + U = 1000$$ $$U = 1000 - 3S$$

Provided that the number of hours of labor can not be negative, we get the following answers:

[1; 997] - 1 hour of skilled labor and 997 unskilled labor;

[2; 994]

[333; 1] - 333 hours of skilled labor and hour of unskilled labor.

But with the indicators of skilled labor less than 122 hours, index of $Q$ will be negative. So, we get the following answers: [122; 634], [123; 631], [124; 628]...[333;1].

Total 212 options for the use of skilled and unskilled labor at a total cost of $5,000.

At the same time - with the growth of hours of skilled labor production ($Q$) growth is also observed: from $\sim 836$ ($S = 122$, $U = 634$) to $\sim 144421$ ($S = 333$, $U = 1$).

**Conclusion** - with labor costs of $5,000$, the maximum production figure ($Q$) corresponds to the maximum use of skilled labor and the minimum use of unskilled labor and is $\sim 144421*$.

c) The relationship between production and hours of labor characterizes the indicator of labor productivity. In this pattern, the price indicator does not affect the value of others.

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* These indicators can be output through the graph of the constructed function or manually.