Homework Assignment 4

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

Only two randomly-chosen questions will be graded.

Question 1:

Consider the following production function that depends only on labor: $Q=4L^{1/3}$

  1. Write a combination of input and output that are technically efficient. In other words, for what level of $L$ and $Q$, is the technology efficient?

  2. Write a combination of input and output that are technically inefficient. In other words, for what level of $L$ and $Q$, is the technology inefficient?

  3. Write a combination of input and output that are technically unattainable. In other words, for what level of $L$ and $Q$, is the technology unattainable?

Solutions 1:

  1. Anything such that $Q=4L^{1/3}$. An example would be $\left(L=8,Q=8\right)$.

  2. Anything such that $Q<4L^{1/3}$. An example would be $\left(L=8,Q=5\right)$.

  3. Anything such that $Q>4L^{1/3}$. An example would be $\left(L=8,Q=10\right)$.

Question 2:

Consider the following production function that depends only on labor: $Q=4L+12L^{2}-6L^{3}$

  1. Compute the $APL$ (average product of labor).

  2. Compute the $MPL$ (marginal product of labor).

  3. What is the value of $L^{*}$ at which $APL$ is the highest?

  4. For $L > L^{*}$, which one is bigger, $APL$ or $MPL$? How about when $L < L^{*}$ and $L = L^{*}$?

  5. Draw $APL$ and $MPL$ on the y-axis as a function of $L$ on the x-axis. Label the point of the intersection of $APL$ and $MPL$.

Solutions 2:

  1. $APL=\frac{Q}{L}=4+12L-6L^{2}$

  2. $MPL=\frac{\partial Q}{\partial L}=4+24L-18L^{2}$

  3. $\frac{\partial APL}{\partial L} = 12 - 12L$ By setting it equal to zero, we get $L^{*} = 1$.

  4. $MPL < APL$ for $L>1$. As the marginal product is lower than the average product, $APL$ decreases as $L$ increases. $MPL > APL$ for $L<1$. As the marginal product is higher than the average product, $APL$ increases as $L$ increases. $APL = MPL$ when $L=1$.

  5. See the graph below or here.

Question 3:

Consider the following production function: $Q=AL^{a}K^{b}$. Assume $A>0$. Further assume $0<a<1$, and $0<b<1$.

  1. What is the Marginal Product of Labor $(MP_{L})$? Is it diminishing as $L$ increases? What is the Marginal Product of Capital $(MP_{K})$? Is it diminishing as $K$ increases?

  2. What is the Average Product of Labor $(AP_{L})$? What is the Average Product of Capital $(AP_{K})$?

  3. What is the $TRS_{L,K}$ ? Is the absolute value of $TRS_{L,K}$ diminishing in $K$? Is it diminishing in $L$?

  4. Are there constant, decreasing, or increasing returns to scale? How does this depend on the parameters?

Solutions 3:

  1. $$ \begin{align} MP_{L}& = \frac{\partial Q}{\partial L}=AaL^{a-1}K^{b} \\ \frac{\partial MP_{L}}{\partial L} &= Aa\left(a-1\right)L^{a-2}K^{b}<0 \\ MP_{K} &= \frac{\partial Q}{\partial K}=AbL^{a}K^{b-1} \\ \frac{\partial MP_{K}}{\partial K} &= Ab\left(b-1\right)L^{a}K^{b-2}<0 \\ \end{align} $$ Since we know $0<a<1$ and $0<b<1$, we know that both marginal products are diminishing. This is because $a-1<0$ and $b-1<0$, and all other terms are positive.

  2. $AP_{L} = \frac{Q}{L}=AL^{a-1}K^{b}$ and $AP_{K} = \frac{Q}{K}=AL^{a}K^{b-1}$

  3. $$ \begin{align} |TRS_{L,K}| &= \frac{MP_{L}}{MP_{K}} \\ &= \frac{AaL^{a-1}K^{b}}{AbL^{a}K^{b-1}}=\frac{a}{b}\frac{K}{L}. \end{align} $$ This diminishes as we increase $L$ and decrease $K$.

  4. $$ \begin{align} Q \left(TL,TK\right) &= A\left(TL\right)^{a}\left(TK\right)^{b} \\ &= AT^{a}L^{a}T^{b}K^{b} \\ &= T^{a+b}AL^{a}K^{b} \\ &= T^{a+b}Q\left(L,K\right) \end{align} $$ If $a+b=1$, the technology exhibits constant returns to scale. If $ a+b<1$, the technology exhibits decreasing returns to scale. If $ a+b>1$, the technology exhibits increasing returns to scale.

Question 4:

Consider the following production function: $Q=\left(6L+3K\right)^{1/2}$

  1. What is the Marginal Product of Labor $(MP_{L})$? What is the Marginal Product of Capital $(MP_{K})$? Are they diminishing?

  2. What is the Average Product of Labor $(AP_{L})$? What is the Average Product of Capital $(AP_{K})$?

  3. What is the $TRS_{L,K}$ ? Is the absolute value of $TRS_{L,K}$ diminishing in $L$ or $K$?

  4. Are there constant, decreasing, or increasing returns to scale?

Solutions 4:

$MP_{L} = \frac{\partial Q}{\partial L}=3\frac{1}{\left(6L+3K\right)^{1/2}}$

$\frac{\partial MP_{L}}{\partial L} = -\frac{9}{\left(6L+3K\right)^{3/2}}<0$

$MP_{K}=\frac{\partial Q}{\partial K}=\frac{3}{2}\frac{1}{\left(6L+3K\right)^{1/2}}$

$\frac{\partial MP_{K}}{\partial K}=-\frac{9}{4}\frac{1}{\left(6L+3K\right)^{3/2}}<0$ .

$MP_L$ is diminishing in $L$ and $MP_K$ is diminishing in $K$

2. $AP_{L}=\frac{Q}{L}=\frac{\left(6L+3K\right)^{1/2}}{L}$

$AP_{K}=\frac{Q}{K}=\frac{\left(6L+3K\right)^{1/2}}{K}$

3. $TRS_{L,K}=-\frac{MP_{L}}{MP_{K}}=-\frac{3\frac{1}{\left(6L+3K\right)^{1/2}}}{\frac{3}{2}\frac{1}{\left(6L+3K\right)^{1/2}}}=-2$

No, it is constant.

4. $Q\left(TL,TK\right)=\left(6TL+3TK\right)^{1/2}$

$=\left[T\left(6L+3K\right)\right]^{1/2}$ $=T^{1/2}\left(6L+3K\right)^{1/2}$ $=T^{1/2}Q\left(L,K\right)$

The technology exhibits decreasing returns to scale.