Law of Diminishing Returns Principle that, as the use of one input ...
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 = MPL/ MP K
Q = 2K + L, MPL = 1, MPK = 2, therefore MRTS = MPL/MPK = ½ 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 Q = ALCKD, MPL = cALC-1KD MPK = dALCKD-1 MRTS = MPL/ MPK = cALC-1KD/ dALCKD-1 = c/d x K/L Note: Output here is Cardinal, so Q = ALCKD is NOT equivalent to Q = LCKD Diminishing MRTS? L goes up, K down. MRTS = c/d x K/L Yes decreasing 2) Linear (Perfect Substitutes) Q = aL + bK MPL = a MPK = b MRTS = MPL/MPK = a/b 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 to Scale - Output increases by the same percentage as inputs
Production Functions and Returns to Scale Let L’ = xL
K’ = xK
where x is proportion of increase
1) Perfect Substitutes Q = aL + bK Q’ = aL’ + bK’ Q’ = a(xL) + b(xK) Q’ = x(aL + bK) Thus Q’ = xQ Constant 2) Perfect Compliments Q = min {aL, bK} Q’ = min {axL, bxK} Q’ = x . min{aL, bK} Q’ = xQ Constant 3) Cobb – Douglas Q = ALCKD Q’ = A(L’)C(K’)D Q’ = A(xL)C(xK)D Q’ = XC+DQ Increasing: Q’ > xQ XC+DQ > xQ
c+d >1
Decreasing Q’ < xQ XC+DQ < xQ
c+d 1, therefore increasing Q = 10L1/4K1/3, c+d = 7/12 < 1, therefore decreasing Q = 1/2L1/4K3/4, c+d = 1 = 1, therefore constant Diminishing MRTS vs Returns to Scale Diminishing MRTS: fall in MRTS moving down the isoquant (L increasing, K decreasing) Returns to Scale: Change in Q from L and K increasing Production function can have diminishing MRTS and increasing returns of scale
Short-run and Long-run Long-run – Firms can change all inputs Short-run – At least one input is fixed, that input will typically be K, at K-bar -
Even if firm produces nothing (Q = 0) it still has to pay for capital
Explicit Cost – Costs involving a cash outlay Implicit Cost – Costs not involving a cash outlay Opportunity Cost – Value of the next best alternative Economic Cost – Sum of a firms explicit and implicit costs Accounting Cost – Total of explicit costs Economic Profit – Profit where costs include both explicit and implicit costs Accounting Profit – Profit where costs include only explicit costs Sunk Cost – Cost already been incurred, and therefore cannot be avoided Non-Sunk Cost – Costs that can be avoided
Cost Minimization Problem We are given output, then solve for cost Wage (w) – cost per unit of labour Rental Rate (r) – cost per unit of capital Isocost Line (similar to budget line) – a set of (L,K) combinations that have the same total cost (TC) TC = wL + rK K = -w/rL + TC/r Slope = -w/r Example: W = 5, r= 10, tc = 40 40 = 5L +10K K = 40/10 = 4
L = 40/5 = 8
Y-int(K) = TC/r X-int(L) = TC/w
Firms Problem (Long-run) -
Minimize (L,K) TC = wL,rK Subject to Q* = F(L,K) The firm wants to find the cheapest way ie (L,K) bundle of producing Q*
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 = MPL/ MP K
Q = 2K + L, MPL = 1, MPK = 2, therefore MRTS = MPL/MPK = ½ 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 Q = ALCKD, MPL = cALC-1KD MPK = dALCKD-1 MRTS = MPL/ MPK = cALC-1KD/ dALCKD-1 = c/d x K/L Note: Output here is Cardinal, so Q = ALCKD is NOT equivalent to Q = LCKD Diminishing MRTS? L goes up, K down. MRTS = c/d x K/L Yes decreasing 2) Linear (Perfect Substitutes) Q = aL + bK MPL = a MPK = b MRTS = MPL/MPK = a/b 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 to Scale - Output increases by the same percentage as inputs
Production Functions and Returns to Scale Let L’ = xL
K’ = xK
where x is proportion of increase
1) Perfect Substitutes Q = aL + bK Q’ = aL’ + bK’ Q’ = a(xL) + b(xK) Q’ = x(aL + bK) Thus Q’ = xQ Constant 2) Perfect Compliments Q = min {aL, bK} Q’ = min {axL, bxK} Q’ = x . min{aL, bK} Q’ = xQ Constant 3) Cobb – Douglas Q = ALCKD Q’ = A(L’)C(K’)D Q’ = A(xL)C(xK)D Q’ = XC+DQ Increasing: Q’ > xQ XC+DQ > xQ
c+d >1
Decreasing Q’ < xQ XC+DQ < xQ
c+d 1, therefore increasing Q = 10L1/4K1/3, c+d = 7/12 < 1, therefore decreasing Q = 1/2L1/4K3/4, c+d = 1 = 1, therefore constant Diminishing MRTS vs Returns to Scale Diminishing MRTS: fall in MRTS moving down the isoquant (L increasing, K decreasing) Returns to Scale: Change in Q from L and K increasing Production function can have diminishing MRTS and increasing returns of scale
Short-run and Long-run Long-run – Firms can change all inputs Short-run – At least one input is fixed, that input will typically be K, at K-bar -
Even if firm produces nothing (Q = 0) it still has to pay for capital
Explicit Cost – Costs involving a cash outlay Implicit Cost – Costs not involving a cash outlay Opportunity Cost – Value of the next best alternative Economic Cost – Sum of a firms explicit and implicit costs Accounting Cost – Total of explicit costs Economic Profit – Profit where costs include both explicit and implicit costs Accounting Profit – Profit where costs include only explicit costs Sunk Cost – Cost already been incurred, and therefore cannot be avoided Non-Sunk Cost – Costs that can be avoided
Cost Minimization Problem We are given output, then solve for cost Wage (w) – cost per unit of labour Rental Rate (r) – cost per unit of capital Isocost Line (similar to budget line) – a set of (L,K) combinations that have the same total cost (TC) TC = wL + rK K = -w/rL + TC/r Slope = -w/r Example: W = 5, r= 10, tc = 40 40 = 5L +10K K = 40/10 = 4
L = 40/5 = 8
Y-int(K) = TC/r X-int(L) = TC/w
Firms Problem (Long-run) -
Minimize (L,K) TC = wL,rK Subject to Q* = F(L,K) The firm wants to find the cheapest way ie (L,K) bundle of producing Q*
