Lecture 17. It's on math for business. Jskwk

It's on the lecture 17

1. Application of Differentiation

2. Production Function M arginal product of labour is the derivative of output with respect to labour Output F unction : Q = − 4 L Find the Marginal product of labor when the workforce size is 9, 100, and 2500.  

3. Solution − 4 L = 300 * ½ - 4 = - 4 Substituting L = 9, 100 and 2500 in turn into the formula for MP L gives When L = 9; MP L = - 4 = 46 When L = 100 ; MP L = - 4 = 11 When L = 2500; MP L = - 4 = -1  

4. Interpretation Notice that the values of MP L decline with increasing L . In part (a) we see that to increase the number of workers from 9 to 10 would result in about 46 additional units of output. In part (b) we see that a 1 unit increase in labor from a level of 100 increases output by only 11. In part (c) the situation is even worse. This indicates that to increase staff actually reduces output! This production function illustrates the law of diminishing marginal productivity (sometimes called the law of diminishing returns ). It states that the increase in output due to a 1 unit increase in labor will eventually decline.

5. Elasticity D emand is elastic if the demand is relatively sensitive to changes in price. Similarly, demand is said to be inelastic if demand is relatively insensitive to price changes. Demand is inelastic if Unit elastic if Elastic if Arc Elasticity, E = x  

6. Arc Elasticity Demand Function: P = 200 − For P1, 136 = 200 - Q1 = 8 For P2, 119 = 200- Δ P = 119 − 136 = − 17 Δ Q = 9 − 8 = 1  

7. P = ½ (136 + 119) = 127.5 Similarly, averaging the Q values gives Q = ½ (8 + 9) = 8.5 E = x  

8. Point Elasticity Formula: E = x Practice: Given the demand function P = 50 − 2 Q find the elasticity when the price is 30. Is demand inelastic, unit elastic or elastic at this price?  

9. Solution P = 50 − 2 Q P + 2Q = 50 Q = 25 – ½ P = - ½ We are given that P = 30 so, at this price, demand is Q = 25 − ½ (30) = 10 E = x = x = -1.5 Since  

10. Practice Given the demand function P = calculate the magnitude of the price elasticity of demand when the price is (a) 9 (b) 45 (c) 90 Is the demand inelastic, unit elastic or elastic at these prices?  

11. Elasticity with Quadratic Function Given the demand function P = − − 4Q + 96 find the price elasticity of demand when P = 51. If this price rises by 2%, calculate the corresponding percentage change in demand.  

12. Solution When P = 51 51 = − − 4Q + 96 − − 4Q + 45 = 0 Q = Q = Q = - 9 and 5 Negative output can be rejected; Q = 5  

13. = = ? P = − − 4Q + 96 = -2Q -4 = = = = -  

14. Price elasticity of demand: E = x = x (- = - 0.73 0.73 = Demand changes by = -1.46% A 2% rise in price therefore leads to a fall in demand of 1.46%.  

15. Practice A supply function is given by Q = 40 + 0.1 Estimate the percentage change in supply when the price increases by 5% from its current level of 17.