1. **Stating the problem:** WX has a product priced at 25 per unit with annual demand 150,000 units. For every 1 unit increase in price, demand increases by 25,000 units. We have costs at three production levels and want to: a. Calculate total variable cost per unit. b. Calculate the selling price to maximize profits. --- 2. **Find variable costs per unit:** Given annual production and total cost components: | Units | Direct materials (P'000) | Direct labour (P'000) | Overhead (P'000) | |-------|-------------------------|----------------------|-----------------| | 100,000 | 200 | 320 | 400 | | 160,000 | 600 | 960 | 1200 | | 200,000 | 880 | 1228 | 1460 | Variable costs change with units. Fixed costs remain constant. Calculate variable cost for each component between levels: - Direct materials increase from 200 to 600 (between 100k to 160k units): increase = 600 - 200 = 400 P'000 for 60,000 units Variable cost per unit for direct materials = 400/60,000 = 6.67 - Direct labour increase from 320 to 960: increase = 640 P'000 over 60,000 units Variable cost per unit for direct labour = 640/60,000 = 10.67 - Overhead increase from 400 to 1200: increase = 800 P'000 over 60,000 units Variable cost per unit for overhead = 800/60,000 = 13.33 Total variable cost per unit = 6.67 + 10.67 + 13.33 = 30.67 3. **Deriving demand and revenue functions:** Let: - Price $P = a - b x$ where $x$ is quantity demanded in units. Given at $P = 25$, demand $x=150,000$, and for every 1 increase in price, demand increases by 25,000 units. Since demand increases as price increases, $b$ is negative (but problem states demand increases with price). This is unusual but we keep the given relation. Assuming linear inverse demand: $P = a - b x$ Given: Demand change per price increase: $\frac{\Delta x}{\Delta P} = 25,000/1 = 25,000$ So slope $b = \frac{-1}{25,000} = -0.00004$ (usually demand falls when price rises, here demand rises) Find $a$ using point $(x,P) = (150,000, 25)$: $$25 = a - b \times 150,000 \Rightarrow a = 25 + b \times 150,000 = 25 -0.00004 \times 150,000 = 25 - 6 = 19$$ Note $a=19$ is less than 25, so sign assumptions off. Alternatively, since demand increases by 25,000 when price increases by 1, demand function is: $$x = m P + c$$ Slope $m = 25,000$ At $P=25, x=150,000$, so: $$150,000 = 25,000 \times 25 + c = 625,000 + c \Rightarrow c = 150,000 - 625,000 = -475,000$$ So demand function: $$x = 25,000 P - 475,000$$ Invert to get price as function of quantity: $$P = \frac{x + 475,000}{25,000}$$ Rewrite as: $$P = a - b x$$ So: $$a = \frac{475,000}{25,000} = 19$$ $$b = -\frac{1}{25,000} = -0.00004$$ (Note price increases as quantity decreases: usual law.) 4. **Marginal revenue and profit maximization:** Given: Selling price: $P = a - b x$ Marginal revenue (MR): $MR = a - 2 b x$ Variable cost per unit: $VC = 30.67$ Profit maximization where MR = MC (marginal cost). Since VC per unit approximates MC, $$MR = MC \Rightarrow a - 2 b x = VC$$ Plug values: $$19 - 2 \times (-0.00004) x = 30.67$$ Simplify: $$19 + 0.00008 x = 30.67$$ $$0.00008 x = 30.67 - 19 = 11.67$$ $$x = \frac{11.67}{0.00008} = 145,875$$ Find corresponding price: $$P = 19 - (-0.00004) \times 145,875 = 19 + 5.835 = 24.835\approx 24.84$$ 5. **Final answers:** a. Total variable cost per unit = 30.67 b. Price maximizing profit = 24.84 ---