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Law of Diminishing Marginal returns

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- Law of Diminishing Marginal returns
- As units of one input are added (with all other inputs held constant), a point will be reached where the resulting additions to output will begin to decrease (marginal product will decline).
- Here diminishing returns will occur for any workers beyond 40.

MP Increasing

Total Product Increasing

MP Decreasing

Decreasing

Optimal Use of an Input

When adding an additional unit of an input, firm will face additional benefit (increased output) and additional cost.

We know profit max is where MR = MC, or marginal profit = zero.

BUT, in production, we’re changing input amounts, not prices or quantity. Must find where marginal profit of adding another input = zero. (where marginal revenue product = MC).

Optimal Use of an Input

Marginal Benefit Side

Marginal Revenue Product of Labor (MRPL)

Extra revenue resulting from an increase in labor.

MRPL = (Price of output) x (marginal output / unit of labor)

MRPL = (MR)(MPL)

(how many Q’s will each additional unit of labor produce

times how much can we sell each for)

Example: When increasing labor from 20 to 30 workers, marginal product per worker is 4.5 (from table). We can sell each unit of output for $40.

MRPL = (40)(4.5)

= $180 per worker

Optimal Use of an Input

Marginal Cost Side

Marginal Cost of an input

Amount an additional unit of input adds to firm’s costs.

Marginal cost of labor is typically the wage.

Profit maximizing workforce is:

M L = MRPL – MCL

Labor should be increased until

MRPL = MCL

Optimal Use of an Input

- Example 1
- Production function: Q = 60L –L2
- Price of output = $2 per unit
- Price of labor = $16/hour

- How many workers should the firm hire?
- MPL = ∂Q/ ∂L = 60 – 2L
- MRPL = ($2)(60-2L) = 120- 4L
- Setting MRPL = MC, 120- 4L = 16
- L = 26

Optimal Use of an Input

- Example 2
- Production function: Q = 10L –.5L2
- Price of output = $10 per unit
- Price of labor = $40/hour

- How many workers should the firm hire?

- Example 3
- Production function: Q = 10L –.5L2 + 24K – K2
- Price of output = $10 per unit
- Price of labor = $40/hour
- Price of K = 80

- What is the optimal quantity of each input?

Long Run Production

IN LR, all inputs are variable.

With variable K, firms can make decisions about scale (size) of their production facility.

Returns to Scale

Measure of the percentage change in an output resulting from a given percentage change in inputs.

Returns to Scale

- Constant Returns to Scale
- When a given percentage change in inputs results in an equal percentage change in outputs.
- If inputs are doubled, output is doubled.
- 10% increase in inputs results in 10% increase in output.
- Firms inputs are equally productive whether smaller or larger levels of output are produced.

Returns to Scale

- Increasing Returns to Scale
- When a given percentage change in inputs results in greater percentage change in outputs.
- If inputs are doubled, output is more than doubled.
- 10% increase in inputs results in a greater than 10% increase in output.
- Firms inputs are more productive when producing larger levels of output.

Returns to Scale

- Decreasing Returns to Scale
- When a given percentage change in inputs results in a smaller percentage change in outputs.
- If inputs are doubled, output is less than doubled.
- 10% increase in inputs results in a less than 10% increase in output.
- Firms inputs are more productive when producing smaller levels of output.

Least-Cost Production

With both inputs variable, firm must decide how much L and how much K to use for minimizing cost of producing a given level of output.

Least cost production is where ratios of marginal products to input costs are equal across all inputs. (Extra output per dollar of input must be the same for all inputs).

MPL/PL = MPK/PK

Least-Cost Production

- Example 1
- Production function: Q = 10L –.5L2 + 24K – K2
- Price of labor = $40/hour
- Price of K = 80

- What is the optimal combination of K and L?

MPL/PL = MPK/PK

MPL = ∂Q/ ∂L = (10 – L)/40 MPK = ∂Q/ ∂K = (24 - 2K)/80

10 – L = 24 – 2K

40 80

L = K – 2

Least-Cost Production

- Example 2
- Production function: Q = 40L –L2 + 54K – 1.5K2
- Price of labor = $10/hour
- Price of K = $15

- What is the optimal combination of K and L?

MPL/PL = MPK/PK

MPL = ∂Q/ ∂L = (40 - 2L)/10 MPK = ∂Q/ ∂K = (54 – 3K)/15

40 – 2L = 54 – 3K

10 15

L = K + 2

Cobb-Douglas Production Function

Most common form of production function:

Q = cLαKβ

Where C is constant, and

α and β are between zero and 1.

- MPL decreases as L increases, but increases as K increases. (vice versa for K)
- Nature of returns to scale can be seen in sum of the exponents:
- Ifα+ β = 1Constant returns
- If α+ β< 1Decreasing returns
- If α+ β > 1Increasing returns

Cobb-Douglas Production Function

Example:

Q = L.5K.5

- Constant returns to scale
- Both inputs exhibit diminishing returns as quantity of the input is increased.

Q = L.7K.4

- Increasing returns to scale
- Both inputs exhibit diminishing returns as quantity of the input is increased.