Applications and Linear Functions Example 1 – Production Levels

1. Applications and Linear Functions • Example 1 – Production Levels Suppose that a manufacturer uses 100 lb of material to produce products A and B, which require 4 lb and 2 lb of material per unit, respectively. Solution: If x and y denote the number of units produced of A and B, respectively, Solving for y gives

2. Demand and Supply Curves • Demand and supply curves have the following trends:

3. Demand Function • Relationship between demand amount of product and other influenced variables as product price, promotion, appetite/taste, quality and other variable. • Q = f(x1,x2,x3,……xn)

4. Demand Function D : Q = a –b P Q 22 20 18 16 14 12 10 P 100 200 300 400 500 600

5. Linear Demand function Q = a - b P Q : amount of product P : product price b : slope ( - ) a : value of Q if P = 0 Q 0 P

6. Property of Demand function • Value of q and p always positif or >= 0 • Function is twosome/two together, each value of Q have one the value of P, and each value of P have one the value of Q. • Function moving down from left to the right side monotonously

7. Supply function • Relationship between Supply amount of product and other influenced variables as product price, technology,promotion, quality and other variable. • Q = f(x1,x2,x3,……xn)

8. Supply Function S : Q = a +b P 22 20 18 16 14 12 10 100 200 300 400 500 600

9. Linear Function Supply Q = a + b P Q : Amount of product P : product orice b : slope ( + ) a : value of Q if P = 0 Q 0 P

10. Property of Supply Function • Value of q and p always positif or >= 0 • Function is twosome/two together, each value of Q have one the value of P, and each value of P have one the value of Q. • Function moving up from the left to the right side monotonously

11. The point of market equilibrium • Agreement between buyer and seller directly or indrectly to make the transaction of product with certain price and amount of quantity. • In mathematics the same like crossing between demand and supply function

12. Equilibrium • The point of equilibrium is where demand and supply curves intersect.

13. D: P = 15 - Q • S :P = 3 + 0.5Q • A. Determine equilibrium point • B. Graph D, S function

14. Exercise : Price - Demand At the beginning of the twenty-first century, the world demand for crude oil was about 75 million barrels per day and the price of a barrel fluctuated between $20 and $40. Suppose that the daily demand for crude oil is 76.1 million barrels when the price is $25.52 per barrel and this demand drops to 74.9 million barrels when the price rises to $33.68. Assuming a linear relationship between the demand x and the price p, find a linear function in the form p = ax + b that models the price – demand relationship for crude oil. Use this model to predict the demand if the price rises to $39.12 per barrel.

15. Exercise : Price - Demand Suppose that the daily supply for crude oil is 73.4 million barrels when the price is $23.84 per barrel and this supply rises to 77.4 million barrels when the price rises to $34.2. Assuming a linear relationship between the demand x and the price p, find a linear function in the form p = ax + b that models the price – demand relationship for crude oil. Use this model to predict the supply if the price drops to $20.98 per barrel. What’s equilibrium point and make a graph in the same coordinate axes

16. Example 1 – Tax Effect on Equilibrium Let be the supply equation for a manufacturer’s product, and suppose the demand equation is . a.If a tax of $1.50 per unit is to be imposed on the manufacturer, how will the original equilibrium price be affected if the demand remains the same? b. Determine the total revenue obtained by the manufacturer at the equilibrium point both before and after the tax.

17. Solution: a. By substitution, Before tax, and After new tax, and

18. Solution: • Total revenue given by • Before tax • After tax,

19. BREAK EVENT POINT • BEP is identifying the level of operation or level output that would result in a zero profit. The other way thatr the firm can’t get profit or don’t have loss • TC= FC + VC • TC : Total Cost • FC : Fixed Cost • VC : Variabel Cost • VC = Pp x Q = cost production per unit x • amount of product

20. TR = Pj x Q • Tr : Total Revenue • Pj : Selling Price • Q : Amount of product Profit = TR –TC BEP  TR=TC

21. TR $ TC profit C bep BEP FC loss Q 0 Q bep

22. Example 2 – Break-Even Point, Profit, and Loss A manufacturer sells a product at $8 per unit, selling all that is produced. Fixed cost is $5000 and variable cost per unit is 22/9 (dollars). a. Find the total output and revenue at the break-even point. b. Find the profit when 1800 units are produced. c. Find the loss when 450 units are produced. d. Find the output required to obtain a profit of $10,000.

23. Break-Even Points • Profit (or loss) = total revenue(TR) – total cost(TC) • Total cost = variable cost + fixed cost • The break-even point is where TR = TC.

24. Solution: a. We have At break-even point, and b. The profit is $5000.

25. BEP Exercise • A firm produce some products where the cost per unit is Rp 4.000,- and selling price per unit is Rp12.000,-.Management developed that fixed cost is Rp 2.000.000,-Determine the amount of product where the firm should sell amount of product so that the break event point achieved. • a. Find the total output and revenue at the break-even point. • b. Find the profit when 1600 units are produced. • c. Find the loss when 350 units are produced. • d. Find the output required to obtain a profit of Rp 7,000.