Differentiation in Economics Dr. Ananda Sabil Hussein

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

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

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

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

If the consumption function is 𝐶=0.01 𝑌 2 +0.2𝑌+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

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

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

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= 136 8 𝑥 1 −17 =−0.88

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