Question

6. Consider the following Cobb - Douglas utility function: U = xayBzY *Note, it should be assumed that a, B.y > 0 Show that this production function can exhibit increasing returns to scale globally while maintaining diminishing returns for each individual input.

Answer 1

6.

Ans:

To check returns to scale, we multiply each output by t:

$(tX)^{\alpha }(tY)^{\beta }(tZ)^{\gamma }=t^{\alpha +\beta +\gamma }(X)^{\alpha }(Y)^{\beta }(Z)^{\gamma }=t^{\alpha +\beta +\gamma }U$

If alpha + beta + gamma > 1 then we have increasing returns to scale.

Condition required for increasing returns to scale:

$\alpha +\beta +\gamma >1$

To check diminishing returns, we need partial derivatives:

$\frac{dU}{dX}=\alpha X^{\alpha -1}Y^{\beta }Z^{\gamma }$

To have diminishing returns, we need:

$\alpha <1$

$\frac{dU}{dY}=\beta X^{\alpha}Y^{\beta-1 }Z^{\gamma }$

To have diminishing returns, we need:

$\beta <1$

$\frac{dU}{dZ}=\gamma X^{\alpha}Y^{\beta }Z^{\gamma-1 }$

To have diminishing returns, we need:

$\gamma <1$

Conditions required for increasing returns to scale globally and diminishing returns to individual inputs:

$\alpha &plus;\beta &plus;\gamma >1$

$\alpha ,\beta ,\gamma <1$

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