Production and Costs — NCERT Solutions

The average fixed cost (AFC) curve is downward sloping and is a rectangular hyperbola. It slopes downward because TFC is constant in the short run, while output increases. Since $AFC = \frac{TFC}{q}$, as $q$ rises, AFC falls. As output approaches zero, AFC becomes very large; as output becomes very large, AFC approaches zero.

The SMC, AVC, and SAC curves are all U-shaped.

• SMC first falls and then rises.
• AVC also first falls and then rises.
• SAC is also U-shaped and lies above AVC.

Also, the SMC curve cuts the AVC curve from below at the minimum point of AVC, and it cuts the SAC curve from below at the minimum point of SAC.

The SMC curve cuts the AVC curve from below at the minimum point of the AVC curve because as long as SMC is less than AVC, the AVC keeps falling. When SMC becomes greater than AVC, the AVC starts rising. Therefore, the point where SMC equals AVC and changes from below to above is the minimum point of AVC.

The SMC curve cuts the SAC curve from below at the minimum point of the SAC curve. The reason is the same as for AVC: when SMC is less than SAC, SAC falls; when SMC becomes greater than SAC, SAC rises. Hence, the intersection from below occurs at the minimum point of SAC.

The short run marginal cost curve is U-shaped because of the law of variable proportions. Initially, as output increases, the marginal product of the variable input rises, so producing one extra unit of output requires less and less input, causing SMC to fall. After a certain point, marginal product falls, so producing extra output requires more and more input, causing SMC to rise.

In the long run, both LRAC and LRMC are U-shaped.

• LRAC first falls due to increasing returns to scale, then becomes constant under constant returns to scale, and finally rises due to decreasing returns to scale.
• LRMC is also U-shaped and cuts the LRAC curve from below at its minimum point.

Using average product and marginal product:

• $AP_L = \dfrac{TP_L}{L}$
• $MP_L = TP_L - TP_{L-1}$

Given TP schedule:

So the AP schedule is $15, 17.5, 16.67, 10, 9.6$ and the MP schedule is $15, 20, 15, -10, 8$.

Given $AP_L = \dfrac{TP_L}{L}$, so $TP_L = AP_L \times L$.

Compute TP first:

• At $L=1$, $TP=2\times 1=2$
• At $L=2$, $TP=3\times 2=6$
• At $L=3$, $TP=4\times 3=12$
• At $L=4$, $TP=4.25\times 4=17$
• At $L=5$, $TP=4\times 5=20$
• At $L=6$, $TP=3.5\times 6=21$

Now marginal product:

• $MP_1 = 2-0 = 2$
• $MP_2 = 6-2 = 4$
• $MP_3 = 12-6 = 6$
• $MP_4 = 17-12 = 5$
• $MP_5 = 20-17 = 3$
• $MP_6 = 21-20 = 1$

So the schedules are:

Since $MP_L = TP_L - TP_{L-1}$ and $TP_0=0$, add successive marginal products to get total product.

• $TP_1 = 0+3 = 3$
• $TP_2 = 3+5 = 8$
• $TP_3 = 8+7 = 15$
• $TP_4 = 15+5 = 20$
• $TP_5 = 20+3 = 23$
• $TP_6 = 23+1 = 24$

Now average product is $AP_L = \dfrac{TP_L}{L}$.

• $AP_1 = 3/1 = 3$
• $AP_2 = 8/2 = 4$
• $AP_3 = 15/3 = 5$
• $AP_4 = 20/4 = 5$
• $AP_5 = 23/5 = 4.6$
• $AP_6 = 24/6 = 4$

So the schedules are:

The total fixed cost (TFC) is the cost at zero output and remains constant.

From the table, at $Q=0$, $TC=10$, so

• TFC = 10 at every level of output.
• TVC = TC - TFC
• AFC = TFC/Q for $Q>0$
• AVC = TVC/Q
• SAC = TC/Q
• SMC = change in TC since output rises by 1 unit each time

Calculations:

So the TFC schedule is constant at Rs 10 and the other schedules are as above.

Given $AFC = \dfrac{TFC}{Q}$ and at $Q=4$, AFC = 5.

So,
$$TFC = AFC \times Q = 5 \times 4 = 20.$$
Thus TFC = Rs 20 and remains constant.

Now compute:

• $TVC = TC - TFC$
• $AFC = 20/Q$
• $AVC = TVC/Q$
• $SAC = TC/Q$
• $SMC = \Delta TC$ (since output rises by 1 unit)

So the required schedules are obtained from these calculations.

The table is for SMC and TFC is given as Rs 100.

Since $SMC = \Delta TC$ for each extra unit of output, and $TC = TFC + TVC$, we can get TVC by cumulatively adding SMCs from zero output.

At $Q=0$, $TVC=0$ and $TC = TFC = 100$.

Now add marginal cost values:

• At $Q=1$, TVC = $0 + 500 = 500$, so TC = $100 + 500 = 600$
• At $Q=2$, TVC = $500 + 300 = 800$, so TC = $900$
• At $Q=3$, TVC = $800 + 200 = 1000$, so TC = $1100$
• At $Q=4$, TVC = $1000 + 300 = 1300$, so TC = $1400$
• At $Q=5$, TVC = $1300 + 500 = 1800$, so TC = $1900$
• At $Q=6$, TVC = $1800 + 800 = 2600$, so TC = $2700$

Now compute:

• $AVC = TVC/Q$
• $SAC = TC/Q$

So the schedules are as above.

Use the production function:
$$Q = 5L^{1/2}K^{1/2}$$
Substitute $L=100$ and $K=100$:
$$Q = 5\times \sqrt{100}\times \sqrt{100} = 5\times 10 \times 10 = 500$$
So the maximum possible output is 500 units.

Given
$$Q = 2L^2K^2$$

For $L=5$ and $K=2$:
$$Q = 2\times 5^2 \times 2^2 = 2\times 25 \times 4 = 200$$

For $L=0$ and $K=10$:
$$Q = 2\times 0^2 \times 10^2 = 0$$

So the maximum outputs are 200 and 0.

Given
$$Q = 5L + 2K$$
With $L=0$ and $K=10$:
$$Q = 5(0) + 2(10) = 20$$
So the maximum possible output is 20.