INTEGRATION OF MARGINAL FUNCTIONS TO TOTAL FUNCTIONS

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When economists want to derive "total function" from "marginal function",  they essentially integrating.

From the above statement, it follows that:

1. The antiderivative of the marginal cost function is the total cost function.

2. The anti-derivative of the marginal revenue function is the total revenue function.

3. The anti-derivative of the marginal propensity to consume is the consumption function.

4. The anti-derivative of the marginal propensity to save is the savings function.

Example 1

A firm's marginal cost function is $MC=Q^2+2Q+40$. Find the total cost function if the fixed costs are N50. Hence find

1. The total cost of producing 3 units

2. The average cost of producing 3 units

3. The average variable cost of producing the 3 units

Solution:

Recalled that total cost is the anti-derivative of the marginal cost function.

$TC=\int Q^2+2Q+40$

$TC=\frac{Q^3}{3}+\frac{2Q^2}{2}+\frac{40Q}{1}+c$

$TC=\frac{Q^3}{3}+Q^2+40Q+c$

Recalled that the fixed cost (which is constant) is 50. Hence

$$TC=\frac{Q^3}{3}+Q^2+40Q+50$$

1. When $Q=3$

$TC= \frac{3^3}{3}+3^2+40(3)+50$

$TC= 9+9+40(3)+50=188$

2. Recalled that average cost is total cost divided by the quantity.

$$AC=\frac{TC}{Q}$$

$AC=\frac{188}{3}=62.67$

3. Average variable cost is variable cosy divided by quantity.

$$AVC=\frac{VC}{Q}$$

But, remember that variable cost is the difference between the total cost and fixed cost.

$VC=TC-FC$

$VC=188-50=138$

Hence, AVC is computed as:

$AVC=\frac{138}{3}=46$

Example 2

If the marginal revenue function of a firm is $MR=100-5Q$. Obtain

1. The total revenue function

2. The total revenue of the 5th unit

3. The average revenue of the 5th unit.

Solution:

1. You may recall total revenue function is the anti-derivative of the marginal revenue.

$TR=\int 100-8Q$

$TR=\frac{100Q}{1}-\frac{8Q^2}{2}+c$

$TR=100Q-4Q^2+c$

We do not know the value of C. But, we know that TR will be zero when the firm does not produce anything.

Hence,

$0=100(0)-4(0)^2+c$

$c=0$

Therefore, the value of the constant is zero, Hence, the revenue function is 

$TR=100Q-4Q^2$

2. The revenue of the 5th unit is when Q is 5.

$TR=100(5)-4(5^2)$

$TR=500-100=400$

3. In our analysis of revenue, I noted that average revenue is calculated as total revenue divided by the quantity

$$AR=\frac{TR}{Q}$$

$AR=\frac{400}{5}=80$

It should be noted that average revenue is equal to the price for all firms. 

Example 3

Given that the marginal propensity to consume is expressed as $MPS=0.5-0.1Y^{\frac{-1}{2}}$ and savings is 15 when income is 100. Obtain the

1. Savings function

2. Marginal propensity to consume

3. Consumption function assuming consumption is 85 when savings is 100

Solution:

1. The savings function is the anti-derivative of the marginal propensity to save

$S=\int 0.5-0.1Y^{\frac{-1}{2}}$

$S=0.5Y-\frac{0.1Y^{\frac{-1}{2}+1}}{\frac{1}{2}}+c$

$S=0.5Y-\frac{0.1Y^{\frac{1}{2}}}{0.5}+c$

$S=0.5Y-0.2\sqrt{Y}+c$

$$S=0.5Y-0.2\sqrt{Y}+c$$

We do not know the value of the constant, but we know that savings are 15 when the national income is 100. 

$15=0.5(100)-0.2\sqrt{100}+c$

$15=50-2+c$

$c=-33$

Substituting $-33$ for c in the savings functions, it will be

$$S=0.5Y-0.2\sqrt{Y}-33$$

2. Recall that $MPC+MPS=1$

$MPC=1-MPS$

$MPC=1-(0.5-0.1Y^{\frac{-1}{2}})$ 

$MPC=0.5+0.1Y^{\frac{-1}{2}}$ 

3. The consumption function is the integral of the marginal propensity to consume

$C=\int 0.5+0.1Y^{\frac{-1}{2}}$

$C=0.5Y+\frac{0.1Y^{\frac{-1}{2}+1}}{\frac{1}{2}}+c$

$C=0.5Y+\frac{0.1Y^{\frac{1}{2}}}{0.5}+c$

$C=0.5Y+0.2\sqrt{Y}+c$

$$C=0.5Y+0.2\sqrt{Y}+c$$

We do not know the value of the constant, but we know that consumption is 85 when the national income is 100. 

$85=0.5(100)+0.2\sqrt{100}+c$

$85=50+2+c$

$c=33$

Substituting 33 for c in the consumption function.

$$C=0.5Y+0.2\sqrt{Y}+33$$

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