Law of Diminishing Returns
Principle that, as the use of one input increases holding all other inputs constant, marginal product will
eventually decease for that input.
Isoquant – All the different combinations of inputs that produce the same level of output
- Example: Q = 2K + L, Q-bar = 10
Marginal Rate of Technical substitution (MRTS) – The rate at which the quantity of capital can be
reduced for every one unit increase in labour, holding output constant
- The negative of the slope of the Isoquant
- MRTS L,K= MP L MP K
Q = 2K + L, MP =L1, MP = K, therefore MRTS = MP /MP = ½L K
Diminishing MRTS – Feature of a production function whereas the quantity of labour increases while
staying on the same isoquant (K down), MRTS decreases, and isoquant is getting flatter. Isoquant is
bowed towards the origin.
Special Production Functions
1) Cobb- Douglas
C D
Q = AL K ,
C-1 D
MP L cAL K
MP K dAL K C D-1
C-1 D C D-1
MRTS = MP / LP = cKL K / dAL K = c/d x K/L
C D C D
Note: Output here is Cardinal, so Q = AL K is NOT equivalent to Q = L K
Diminishing MRTS? L goes up, K down. MRTS = c/d x K/L Yes decreasing
2) Linear (Perfect Substitutes)
Q = aL + bK
MP L a
MP K b
MRTS = MP /MP = a/b
L K
Example: Fast vs slow computer
3) Fixed Proportion (Perfect Compliments)
Q = min{aL,bK}
- Can't substitute one input for another and produce the same level or output
o No MRTS Word Problems – translating description into a production function
It takes 3 eggs and 2 pieces of toast to make 2 breakfasts (and you can't substitute eggs for toast)
Q = min{2E,3T} at E = 3, and T = 2, Q = 6, to high want Q = 2
Rescale the production function to get Q = 2
2 quantity we want
6 quantity we got
Q = min {1/3 x 2E, 1/3 3T}
Q = min {2/3 E, T}
Example 2:
Capital is twice as productive as labour, always. If the firm hires 2 labour and 2 capital, it makes Q = 12
Know: Q = aL + bK, and b/a = 2
Start with Q = L + 2K, at L = 2, K = 2, Q has to be 12
Plug in L = 2, and K = 2 and Q = 2 + 2(2) = 6. Not right, need to scale
12 quantity you want
6 quantity you got
Scale by 12/6 = 2
Q = 2L + 4K
Returns to Scale (RTS)
- How much output increases when all inputs increase by the same percentage
- Example, If both labour and capital double, how much does output increase?
Increasing Returns to Scale - Occurs when output increases by a larger percentage then input
Decreasing Returns to Scale – Output increases by a smaller percentage then input
Constant Returns t

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