The Theory of the Firm Under Perfect Competition — NCERT Solutions

A unit tax raises the firm's cost by the same amount per unit of output. So both LRMC and LRAC increase by the tax amount. As a result, the firm's long run supply curve shifts to the left.

So at any given market price, the firm now supplies fewer units.

An increase in input prices raises the cost of production. This usually raises the firm's marginal cost at each output level, so the MC curve shifts upward/leftward.

Therefore, the firm's supply curve shifts to the left: at any given market price, it now supplies less output.

If the number of firms increases, the market supply curve shifts to the right because more firms are supplying output in the market. If the number of firms decreases, the market supply curve shifts to the left.

The price elasticity of supply measures how responsive quantity supplied is to a change in the good's price.

It is defined as

$$e_s = \frac{\text{Percentage change in quantity supplied}}{\text{Percentage change in price}}$$

or

$$e_s = \frac{\Delta Q}{Q} \times \frac{P}{\Delta P}$$

where $\Delta Q$ is the change in quantity supplied and $\Delta P$ is the change in price.

Using market price = Rs 10 and $TR=p\times q$:

• For $q=0$, $TR=10\times 0=0$
• For $q=1$, $TR=10\times 1=10$
• For $q=2$, $TR=20$
• For $q=3$, $TR=30$
• For $q=4$, $TR=40$
• For $q=5$, $TR=50$
• For $q=6$, $TR=60$

Marginal revenue is the change in TR for a one-unit change in output. Since the price is constant at Rs 10, $MR=10$ at each additional unit. For $q=0$, MR is not defined in the usual table, so it is shown as a dash.

Average revenue is $AR=TR/q=p=10$ for every positive output level.

So the schedule is:

Profit is defined as TR − TC.

Compute for each output level:

• $q=0$: $0-5=-5$
• $q=1$: $5-7=-2$
• $q=2$: $10-10=0$
• $q=3$: $15-12=3$
• $q=4$: $20-15=5$
• $q=5$: $25-23=2$
• $q=6$: $30-33=-3$
• $q=7$: $35-40=-5$

The market price can be found from TR at one unit: $TR=p\times q$, so $p=5/1=Rs\ 5$.

So the completed table is:

Market price = Rs 5.

Profit is $\pi = TR - TC$.

Given price $p=Rs\ 10$, so for each output level:

• $q=0$: $TR=10\times 0=0$, profit $=0-5=-5$
• $q=1$: $TR=10$, profit $=10-15=-5$
• $q=2$: $TR=20$, profit $=20-22=-2$
• $q=3$: $TR=30$, profit $=30-27=3$
• $q=4$: $TR=40$, profit $=40-31=9$
• $q=5$: $TR=50$, profit $=50-38=12$
• $q=6$: $TR=60$, profit $=60-49=11$
• $q=7$: $TR=70$, profit $=70-63=7$
• $q=8$: $TR=80$, profit $=80-81=-1$
• $q=9$: $TR=90$, profit $=90-101=-11$
• $q=10$: $TR=100$, profit $=100-123=-23$

The maximum profit is 12 at output 5.

So the profit-maximising level of output is 5 units.

The market supply is the horizontal summation of the two firms' supplies.

Add firm 1 and firm 2 at each price:

• Price 0: $0+0=0$
• Price 1: $0+0=0$
• Price 2: $0+0=0$
• Price 3: $1+1=2$
• Price 4: $2+2=4$
• Price 5: $3+3=6$
• Price 6: $4+4=8$

So the market supply schedule is:

The market supply is the horizontal summation of the two firms' supplies.

Add the two columns price by price:

• Price 0: $0+0=0$
• Price 1: $0+0=0$
• Price 2: $0+0=0$
• Price 3: $1+0=1$
• Price 4: $2+0.5=2.5$
• Price 5: $3+1=4$
• Price 6: $4+1.5=5.5$
• Price 7: $5+2=7$
• Price 8: $6+2.5=8.5$

So the market supply schedule is:

There are three identical firms, so market supply at each price is 3 times firm 1's supply.

Multiply each quantity by 3:

• Price 0: $3\times 0=0$
• Price 1: $3\times 0=0$
• Price 2: $3\times 2=6$
• Price 3: $3\times 4=12$
• Price 4: $3\times 6=18$
• Price 5: $3\times 8=24$
• Price 6: $3\times 10=30$
• Price 7: $3\times 12=36$
• Price 8: $3\times 14=42$

So the market supply schedule is:

Use the formula for price elasticity of supply:

$$e_s = \frac{\text{Percentage change in quantity supplied}}{\text{Percentage change in price}}$$

Initial revenue at price Rs 10 is Rs 50, so initial quantity is

$$Q_1 = \frac{50}{10} = 5$$

New revenue at price Rs 15 is Rs 150, so new quantity is

$$Q_2 = \frac{150}{15} = 10$$

Now calculate percentage changes:

$$\Delta Q = 10 - 5 = 5$$

$$\%\Delta Q = \frac{5}{5} \times 100 = 100$$

$$\Delta P = 15 - 10 = 5$$

$$\%\Delta P = \frac{5}{10} \times 100 = 50$$

Therefore,

$$e_s = \frac{100}{50} = 2$$

Use the price elasticity of supply formula:

$$e_s=\frac{\%\text{ change in quantity supplied}}{\%\text{ change in price}}$$

Given:

• Old price $P_1=5$
• New price $P_2=20$
• Increase in quantity supplied $=15$
• $e_s=0.5$

First find the percentage change in price:

$$\%\Delta P=\frac{20-5}{5}\times 100=\frac{15}{5}\times 100=300\%$$

Now use elasticity:

$$0.5=\frac{\%\Delta Q}{300}$$

So,

$$\%\Delta Q=0.5\times 300=150\%$$

Let the initial output be $Q_1$. Then the increase of 15 units is 150% of $Q_1$:

$$15=150\%\text{ of }Q_1=\frac{150}{100}Q_1=1.5Q_1$$

$$Q_1=\frac{15}{1.5}=10$$

Final output:

$$Q_2=Q_1+15=10+15=25$$

So the initial and final output levels are $10$ units and $25$ units.

If the question expects the worked values from the standard school-method calculation, the answer is the pair of output levels just found; the book's exercise statement appears to involve a numerical inconsistency, so the computed result is what should be reported.

Use the price elasticity of supply:

$$e_s=\frac{\%\Delta Q}{\%\Delta P}$$

Given:

• Old price $P_1=10$
• New price $P_2=30$
• Old quantity $Q_1=4$
• $e_s=1.25$

Compute percentage change in price:

$$\%\Delta P=\frac{30-10}{10}\times 100=\frac{20}{10}\times 100=200\%$$

Now find percentage change in quantity:

$$\%\Delta Q=e_s\times \%\Delta P=1.25\times 200=250\%$$

So the increase in quantity is $250\%$ of the initial quantity:

$$Q_2-Q_1=250\%\text{ of }4=\frac{250}{100}\times 4=10$$

Therefore,

$$Q_2=Q_1+10=4+10=14$$

So the firm will supply 14 units at the new price.