Question

3. the production function is f(L, M)=4*(L^1/2) (M^1/2), where L is the number of units of labor and M is the number of machines used. If the cost of labor is $49 per unit and the cost of machines is $25 per unit, then how much will be the total cost of producing 7 units of output ?

Answer 1

As per the question the production function Q=f(L,M)=4L^1/2M^1/2

Where L=labour and M=Machine

Cost of labour or price of labour=$49

Cost of machine or price of machine=$25

Marginal Product (MP) of Labour =dQ/dL = 2 M^1/2L^-1/2=2 M^1/2/ L^1/2

Marginal Product (MP) of Machine =dQ/dM = 2 L^1/2M^-1/2=2 L^1/2/ M^1/2

Marginal rate of technical substitution (MRTS) = MP of Labour / MP of Machine

Marginal rate of technical substitution (MRTS) = (2 M^1/2/ L^1/2) / (2 L^1/2/ M^1/2) =M/L

At equilibrium level of output = MRTS = Price of labour / Price of Machine

At equilibrium level of output = M/L = 49 / 25   So M=49/25L and L=25/49M

As production function Q=f(L,M)=4L^1/2M^1/2

At 7 unit of output or Q=7 then

7= 4L^1/2M^1/2    (replacing the value of M=49/25L)

7= 4L^1/2(49/25L)^1/2   

7= 4(7/5)L^1/2L^1/2   

7=28/5L                so L=35/28 =1.25  

units of Labour at 7^th unit of output L= 1.25

Similarly

7= 4L^1/2M^1/2    (replacing the value of L=25/49M)

7= 4(25/49M)^1/2M^1/2      

7= 4(5/7)M^1/2M^1/2      

7=20/7M             so M=49/20=2.45  

Units of machine at 7^th unit of output M= 2.45

Total cost = units of labour x price of labour + units of machine x price of machine

Total cost = (1.25 x 49) + (2.45 x 25) = 61.25 + 61.25 = $122.5

The total cost of producing 7 units of output is $122.5