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Ch. 6 – Inputs and Production Functions
Q: How does a firm choose the combination of input to maximize output?
=maximum quantity of output that a firm can produce given the quanities of inputs that it
Q = f(K,L)
If the firm is producing at technically inefficient points like A and B, then the firm is not
maximizing output and producing efficiently.
Labour requirements function inverts the production function to show the min amount
of L required to produce a given level of output.
Caves and Barton found that the typical manufacturer produced only 63% of their
potential output given their stocks of L and K.
Firms that were more likely to be efficient faced more competition:
1. More competition from abroad
2. More competition domestically P a g e | 2
3. Had better transportation infrastructure
Total Product Functions
=Single input production functions. P a g e | 3
The TP function above shows output as a function of one input holding constant the quantity of
the other input. (This represents a Short-run scenario. Usually, we say that L is variable and K
is fixed in the SR. ) In long run, all inputs are variable.
Points to note about the graphs:
1. Marginal Product of Labour (MPL) = slope of TP curve
= dQ/dL (derivative of TP curve)
2. Average product of Labour (APL) = Q/L (output per unit of L)
Note: AP is the slope (Qo/Lo) of a ray from the origin to the TP curve
3. TP curve maximized where MPL=0.
4. The production function displays diminishing returns.
Law of diminishing returns = the marginal product of an input eventually
decreases as the input increases if other inputs are held constant.
5. Demonstrates 3 Properties of all MP and AP curves:
-AP increases when MP > AP
- AP decreases when MP λf(K,L)
Increasing inputs by the same percentage increases output by a bigger percentage.
Due to economies of scale (decreasing long run average costs LRAC).
Usually one firm in this market: utilities, oil pipelines, etc. P a g e | 19
2) Constant Returns
F(λK,λL) > λf(K,L)
Increasing inputs by the same amount will increase output by the same percentage.
3) Decreasing returns
F(λK,λL) < λf(K,L)
Due to diseconomies of scale (increasing LRAC).
Output increases less than the same percentage increase in all inputs.
To determine whether or not a production function displays IRS,DRS, or CRS, we
increase all inputs by some percentage λ and try to manipulate to function to see how λ
affects the original production function.
Example 2: What are the returns to scale for Q = aK + bL?
A(λK) +b(λL) = λ (aK + bL) = λQ.
Constant returns to scale if λ >0.
If we were to scale up all inputs by a factor (that is, replace K by K, and L by L), the
resulting output would equal Q. Therefore a linear production function has constant
returns to scale.
Returns to Scale differs from Marginal Returns (MP)
Returns to scale measures what happens to output when the firm increases ALL inputs
the same amount. Marginal returns measure what happens when you only increase one
input. P a g e | 20
Above: Holding K fixed at 10 units, there are diminishing returns to L as L
increases by increments of 10 units.
When fixed at K=10:
However, there are constant returns to scale if