Question

George produces computer software (user friendly). His firm's production function is Q = 1K + 2L, where Q is the programs, K is capital employed, and L is the labour used.

If George faces factor prices of Pk=6 and Pl =6, the cheapest way to produce Q = 40 is:

Part 1: By using how many units of capital?  

Part 2: By using how many units of labour?  

If George faces factor prices of Pk=7 and Pl=21, the cheapest way to produce Q = 40 is

Part 3: By using how many units of capital?  

Part 4: By using how many units of labour?  

Answer 1

As per the question the George produces computer software has the production function Q=1K+2L

Marginal Product (MP) of Labour =dQ/dL = K+2

Marginal Product (MP) of Capital =dQ/dK = 1+2L

Marginal rate of technical substitution (MRTS) = MP of Labour / MP of Capital = K+2/1+2L  

(A) if price of capital (r) = 6 and price of labour (w) =6, output (Q) =40

At equilibrium level of output = MRTS = Price of labour (w) / Price of capital (r)

ÞK+2/1+2L   = 6/6

ÞK+2/1+2L   = 1

ÞK+2   = 1+2L

ÞK =2L-1

As per production function Q=1K+2L

For producing 40 units of output the production function will be

Þ 40 = 1K+2L     (replacing the value of K =2L-1)

Þ 2L-1 +2L = 40  

Þ 4L = 41           Þ L = 10.25

For producing 40 units of output the production function will be

Þ 40 = 1K+2L     (replacing the value of L =10.25)

Þ 40 = 1K+2(10.25)         Þ K = 40 – 20.5   Þ K = 19.5

Part 1.Equilibrium units of labour (K) =19.5

Part 2. Equilibrium units of labour (L) =10.25

(B) if price of capital (r) = 7 and price of labour (w) =21, output =40

At equilibrium level of output = MRTS = Price of labour (w) / Price of capital (r)

ÞK+2/1+2L   = 21/7

ÞK+2/1+2L   = 3

ÞK+2   = 3+6L

ÞK = 1+6L

As per production function Q=1K+2L

For producing 40 units of output the production function will be

Þ 40 = 1K+2L     (replacing the value of K =1+6L)

Þ 1+6L +2L = 40  

Þ 8L = 39           Þ L = 4.875

For producing 40 units of output the production function will be

Þ 40 = 1K+2L     (replacing the value of L =4.875)

Þ 40 = 1K+2(4.875)         Þ K = 40 – 9.75   Þ K = 30.25

Part 3. Equilibrium units of Capital (K) =30.25

Part 4. Equilibrium units of labour (L) =4.875