Question

Joe’s coffee house operates under the production function Q(L,K) = ln(L^2 ) + K^1/2 , 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.

Answer 1

Solution: -

a).Consider the given problem here the production function is given by, “Q=ln(L^2) + K^1/2.

=> dQ = (1/L^2)*2*L*dL + (1/2)*K^(-1/2)*dK, => (2/L)*dL + (1/2)*K^(-1/2)*dK = 0.

=> (2/L)*dL = (-1/2)*K^(-1/2)*dK, => dK/dL = (4/L) / [(-1)*K^(-1/2)] = (-4/L)*K^1/2.

=> dK/dL = (-4*K^1/2)/L < 0, => MRTS = (4*K^1/2)/L = 4*(K^1/2/L).

dMRTS/dL = 4*[(1/2)*K^(-1/2)*dK/dL*L – K^1/2]/L^2, where “dK/dL” is negative, => “-dK/dL” is positive.

=> dMRTS/dL = (-4)*[(1/2)*K^(-1/2)*(-dK/dL)*L + K^1/2]/L^2 < 0.

=> So, we can see that “d(MRTS)/dL” is negative, => as “Joe” substitutes labor for capital, => MRTS decreases.

Here “dK/dL < 0”, => the isoquant is negatively sloped and the MRTS is decreasing as “L” increases, => the isoquant is negatively sloped and convex to the origin.

b).Here the production function is given by, “Q= In(L^2) + K^1/2”. So, the “marginal product of “L” is given by, “MPL = ?Q/?L = (1/L^2)*2L = 2/L.

Now, ?MPL/?L = (-2)/L^2 < 0, => as “L” increases implied “MPL” decreases. So, because the “MPL” is downward sloping, => the short run production function is concave to the origin, => as “L” increases, => “Q” also increases but at the decreasing rate.

c).So, “MP” be the additional production form employing additional input to the production and the MRTS be the rate at which one factor decreases as the other factor increases to maintain the same level of production.

So, here the diminishing MP implied as the additional factor is employed in the production => the production also increases but at the decreasing rate. Now, the diminishing MRTS => as we employ more and more factor into the production we have reduce the other factor less, => MRTS decreses.