Joe’s coffee house operates under the production function Q(L,K) = ln(L2) + K1/2, where L is...

Joe’s coffee house operates under the production function Q(L,K) = ln(L2) + K1/2, where L is the number of worker hours and K is the number of coffee machine hours.

What happens to the marginal rate of technical substitution as Joe substitutes labor for capital, holding output constant? What does this imply about the shape of the corresponding isoquants? Justify.

What happens to the marginal product of labor as Joe uses more labor, holding capital constant? What does this imply about the shape of the short-run total product curve, where capital is fixed? Justify.

Explain the difference between diminishing marginal product and diminishing marginal rate of technical substitution.

Expert Solution

MRTS_L,K=-MP_L/MP_K

MPL=derivative of output with respect to L= 1/L²*2L= 2/L

MPK=derivative of output with respect to K= 1/2*K^-1/2

MRTS=-2/L*1/2*K^-1/2 =K^-1/2/L=-1/LK^1/2

The MRTS explains the slope of the isoquants. It explains the rate at which labour is substituted for capital. The negative value of MRTS means that the isoquants are negatively sloped or downward sloping to the right.

MP of labour is calculated as the change in the output while keeping the capital constant. It can be solved by finding the derivative of output with respect to the labour. As calculated above, MP of labour is 2/L. The short run product curve plots the output with respect to labour being on horizontal axis. The marginal product of labour would reduce as we increase the value of the labour used, so we can say that the firm is either operating in II stage wherein it is having the diminishing returns to the factor.

Diminishing marginal product means that the total returns to the output or the total change in the output is declining when we increase the amount of one factor while keeping the other factor constant.

Whereas diminishing the marginal rate of technical substitution means the rate at which one factor is decreasing while increasing the amount of another factor by one unit while keeping the total output constant.

Rahul Sunny answered 2 years ago

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