1. Differentiation in Economics
Dr. Ananda Sabil Hussein

2. Differentiation 𝑦= 𝑥 𝑛 𝑑𝑦 𝑑𝑥 =𝑛 𝑥 𝑛−1 Example: 𝑦= 2𝑥 4 𝑦 ′ =2.4 𝑥 3
𝑦 ′ =8 𝑥 3

3. Differentiate each function separately and add.
ℎ 𝑥 =𝑓 𝑥 +𝑔 𝑥
ℎ ′ 𝑥 = 𝑓 ′ 𝑥 + 𝑔 ′ 𝑥
Example:
𝑦= 𝑥 2 + 𝑥 50
𝑦 ′ =2𝑥+50 𝑥 49

4. Revenue and Cost TR = P x Q MR = dTR/dQ 𝑃=100−2𝑄 TR = (100 – 2Q)Q
MR = 100 – 4Q

5. TC = ACxQ
MC = dTC/dQ
𝐴𝐶=2𝑄 𝑄
TC = (2Q+6+13/Q)xQ
TC = 2Q2+6Q+13
MC = 4Q + 6

6. Production, Consumption and Saving
𝑀𝑃 𝑙 = 𝑑𝑄 𝑑𝐿
𝑀𝑃𝐶= 𝑑𝐶 𝑑𝑌
𝑀𝑃𝑆= 𝑑𝑆 𝑑𝑌
MPC + MPS = 1

7. If the consumption function is
𝐶=0.01 𝑌 𝑌+50
Calculate MPC and MPS when Y = 30
MPC = dC/dY
MPC = 0,02Y + 0,2
MPC (30) = 0,02(30) + 0,2
MPC = 0,8
MPS + MPC = 1
MPS + 0,8 = 1
MPS = 1-0,8 = 0,2

8. Elasticity 𝐸=− 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒
𝐸=− 𝑃 𝑄 𝑥 ∆𝑄 ∆𝑃
𝐸=− 𝑑𝑄 𝑑𝑃 𝑥 𝑃 𝑄
Demand is said to be
Inelastic if E<1
Unit elastic if E=1
Elastic if E>1

9. The price elasticity of supply is defined in an anlogous way to that demand. An increase in price leads to an increase in supply, so E is automatically positive.
𝐸= 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑠𝑢𝑝𝑝𝑙𝑦 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒
𝐸= 𝑃 𝑄 𝑥 ∆𝑄 ∆𝑃
𝐸= 𝑑𝑄 𝑑𝑃 𝑥 𝑃 𝑄

10. Given the demand function 𝑃=200− 𝑄 2
Calculate the elasticity as P falls from 136 to 119.
Solution:
By substituting P = 136 and P = 119 respectively and solving for Q, then
Q1 = 8
Q2 = 9
𝐸= 𝑥 1 −17 =−0.88

11. Practice
The fixed costs of producing a good are 100 and the variable costs are 2+ 𝑄 10 per unit.
Find the expression for TC and MC
Evaluate MC at Q = 30 and estimate the change in TC brought about by a two unit increase in output from a current level of 30 units.
At what level of output does MC = 22