XV. Cost-minimisation
XV. Definitions
- Accounting costs (i.e., out-of-pocket expenses, historical costs, depreciation, &
other bookkeeping entries)
o L: wage payments
o K : historical cost of capital & some depreciation rule
- Economic costs
o Opportunity (= implicit) costs
Payment required to keep input in its present employment
Level of remuneration received by input in its next best alternative
employment
L: amount earned by worker in next best alternative employment
wage rate in labour market denoted by w
K : cost of capital sunk (≠ influencing present/future decisions) implicit
cost of capital = rental rate denoted by v
XV. Cost-minimisation problem
- Firm = cost-minimiser producing any given output level y ≥ 0 at smallest
possible total cost
- C(y) = firm’s smallest possible total cost for producing y units of output
- C(y) = firm’s total cost function
- Firm facing input prices ¿) total cost function = C (w , v , y )
- Time-frame affecting variability of factors (= labour & capital)
o Short-run: only labour variable, capital fixed
o Long-run: labour & capital variable
- Firm combining labour & capital for producing output y
o Production function: y=f (L , K)
o Output level y ≥ 0 as given
o Cost of an input bundle (L, K) = wL+ vK
- Cost-minimisation problem: finding bundle of labour & capital yielding smallest
possible total cost
min wL+ vK
L≥0 , K ≥ 0
subject to f ( L, K ) = y ≥ 0
o Note: first-order partial derivation of cost function with respect to each input
with Lagrangian set to 0
Marginal product of labour
Marginal product of capital
- Condition for minimisation of costs: hiring capital & labour until ratio of marginal
products = ratio of input/market prices technical substitution = market
substitution
XV. Definitions
- Accounting costs (i.e., out-of-pocket expenses, historical costs, depreciation, &
other bookkeeping entries)
o L: wage payments
o K : historical cost of capital & some depreciation rule
- Economic costs
o Opportunity (= implicit) costs
Payment required to keep input in its present employment
Level of remuneration received by input in its next best alternative
employment
L: amount earned by worker in next best alternative employment
wage rate in labour market denoted by w
K : cost of capital sunk (≠ influencing present/future decisions) implicit
cost of capital = rental rate denoted by v
XV. Cost-minimisation problem
- Firm = cost-minimiser producing any given output level y ≥ 0 at smallest
possible total cost
- C(y) = firm’s smallest possible total cost for producing y units of output
- C(y) = firm’s total cost function
- Firm facing input prices ¿) total cost function = C (w , v , y )
- Time-frame affecting variability of factors (= labour & capital)
o Short-run: only labour variable, capital fixed
o Long-run: labour & capital variable
- Firm combining labour & capital for producing output y
o Production function: y=f (L , K)
o Output level y ≥ 0 as given
o Cost of an input bundle (L, K) = wL+ vK
- Cost-minimisation problem: finding bundle of labour & capital yielding smallest
possible total cost
min wL+ vK
L≥0 , K ≥ 0
subject to f ( L, K ) = y ≥ 0
o Note: first-order partial derivation of cost function with respect to each input
with Lagrangian set to 0
Marginal product of labour
Marginal product of capital
- Condition for minimisation of costs: hiring capital & labour until ratio of marginal
products = ratio of input/market prices technical substitution = market
substitution
