derivation of an individual firm’s demand for labour using marginal revenue product

Labour Market Forces and Government Intervention

1️⃣ Individual Firm’s Demand for Labour

Think of a firm as a chef running a restaurant 🍽️. The chef wants to decide how many cooks (workers) to hire to maximise profit. The key idea is the Marginal Revenue Product of Labour (MRPL).

  1. Production Function – The chef can produce a certain amount of food Q depending on the number of cooks L:

    2. Q = f(L)

  2. Marginal Product of Labour (MPL) – The extra output from hiring one more cook:

    3. MP_L = \frac{dQ}{dL}

  3. Marginal Revenue (MR) – In a competitive market, the price of the product is constant. So:

    4. MR = P

  4. Marginal Revenue Product of Labour (MRPL) – The extra revenue from hiring one more cook:

    5. MRPL = MPL \times MR = MPL \times P

  5. Hiring Decision – The chef hires cooks until the extra revenue equals the extra cost (wage W):

    6. MRPL = W

Putting it all together:

StepFormula
ProductionQ = f(L)
MPLMP_L = \frac{dQ}{dL}
MRPLMRPL = MP_L \times P
Hiring RuleMRPL = W

🔍 Example: Suppose the firm’s production function is Q = 10\sqrt{L} and the price of the product is P = \\$20. The wage is W = \\$10.

  1. MPL = \frac{d}{dL}(10\sqrt{L}) = \frac{5}{\sqrt{L}}

  2. MRPL = MP_L \times P = \frac{5}{\sqrt{L}} \times 20 = \frac{100}{\sqrt{L}}

  3. Set MRPL = W: \frac{100}{\sqrt{L}} = 10

  4. Solve: \sqrt{L} = 10 \;\Rightarrow\; L = 100

So the firm should hire 100 cooks to maximise profit. 🎉

2️⃣ Government Intervention in the Labour Market

Governments sometimes step in to influence hiring decisions. Here are the most common tools:

  • Minimum Wage – Sets a floor on wages. If the market wage W is below the minimum, firms must pay the higher rate, which can reduce hiring.

  • Subsidies – The government pays part of the wage, effectively lowering W for the firm. This can increase hiring.

  • Training Grants – Reduce the cost of increasing MP_L by improving worker skills.

  • Taxation – Labour taxes raise the effective wage cost, shifting the demand curve left.

📈 Visualising the Effect – Imagine the demand curve for labour is a downward‑sloping line. A subsidy shifts it right (more hiring), while a tax or minimum wage shifts it left (less hiring).

3️⃣ Key Takeaways for A‑Level Economics

  1. The MRPL is the product of the marginal product of labour and the marginal revenue from selling output.

  2. A firm hires workers until MRPL equals the wage.

  3. Government policies can shift the demand for labour by changing either the wage or the marginal product.

  4. Understanding these concepts helps explain real‑world phenomena like unemployment, wage gaps, and the impact of policy changes.

💡 Remember: Think of the firm as a chef, the workers as cooks, and the government as the restaurant owner who can set rules (like minimum wage) or give incentives (like subsidies) to influence how many cooks the chef hires.