Question

Assume a firm' production function is Q = 3K +L • In this case, inputs (K and L) are perfect substitutes. Can you give a real example where this production function works? Assume price of capital is r = 5, and price of labor is w = 1 How many units of capital and labor is need to produce Q=60 in cheapest way? O Show your logic using cost minimization condition, and Analyze it graphically

Answer 1

1)For a linear production function , where the inputs are perfect substitutes, it is difficult to find a real life example because generally production can't be carried out by employing just labour or capital. It requires a mix of two. We can think of it as factory workers being replaced by robots which have the same productivity.

2)Given r= 5 and w=1 , Q= 3K+L , Q=60

To find out the equilibrium level of input used in the production process we need to use the Optimality condition obtained from the profit maximization/ cost minimization problem :

Marginal rate of technical substitution= price ratio of inputs

Marginal product of labour/Marginal Product of capital = w/r

( dQ/ dL)/ ( dQ/ dK) = w/r

Here MRTS = 1/3 and w/r = 1/5

MRTS > price ratio , this means that Labour is one- third as productive as capital but costs one-fifth of it. Therefore it is cheaper to use only labour in the production process.

Hence to produce 60 units of output we need 60 units of labour as per the given production function and 0 units of capital.

Graphically :

X For Q = 26th MRTS: 13 For c= Lt sk wo Ś. e If only capital is used. ----06-3KFC - 1t5k ģ. Himal use A of sources lo 20 30 40 50

From the graph above it can be seen that the slope of production function is greater than the slope of cost line, hence by the above explanation it would be optimal for the producer to use only Labour in order to reach the highest isoquant at the minimum costs.