Production and Cost
Christopher Makler
Stanford University Department of Economics
Econ 50 : Lecture 15
Unit I: The “Real Economy"
Labor
Fish
🐟
Capital
Coconuts
🥥
[GOODS]
⏳
⛏
[RESOURCES]
Utility
🤓
The story thus far...
Unit IIa: Consumers and Prices
Labor
Fish
🐟
Capital
Coconuts
🥥
[GOODS]
⏳
⛏
[RESOURCES]
🤓
p_1
p_2
m
Consumer
The story thus far...
Unit IIb: Theory of the Firm
Labor
Firm
🏭
Capital
⏳
⛏
Customers
🤓
p
w
r
q
L
K
This unit: analyze the firms
consumers buy things from
Unit IIb: Theory of the Firm
Firm
🏭
Costs
wL + rK
Customers
🤓
p
q
This unit: analyze the firms
consumers buy things from
Unit IIb: Theory of the Firm
Firm
🏭
Costs
wL + rK
Revenue
pq
From the firm's perspective, they get revenue and pay costs...
Unit IIb: Theory of the Firm
Costs
wL + rK
Revenue
pq
Profit
pq - (wL + rK)
Next week: Solve the optimization problem
finding the profit-maximizing quantity \(q^*\)
...which is what we call profits
Unit IIb: Theory of the Firm
Costs
c(q) = wL(q) + rK(q)
Revenue
Profit
pq
pq - c(q)
Today: Solve the cost minimization problem
and derive the cost function \(c(q)\)
Unit IIb: Theory of the Firm
Costs
c(q) = wL(q) + rK(q)
Revenue
r(q) = p(q) \times q
Profit
r(q) - c(q)
Today: Solve the cost minimization problem
and derive the cost function \(c(q)\)
Friday: Derive the revenue function \(r(q)\)
Unit IIb: Theory of the Firm
Costs
c(q) = wL(q) + rK(q)
Revenue
r(q) = p(q) \times q
Profit
r(q) - c(q)
Today: Solve the cost minimization problem
and derive the cost function \(c(q)\)
Friday: Derive the revenue function \(r(q)\)
Next Monday: Solve the profit maximization problem
Unit IIb: Theory of the Firm
Output Supply
q^*(p)
Input Demands
L^\star(w)
Next Wednesday: Analyze the comparative statics
of how the optimal choice changes with prices
for the special case of a price-taking firm
K^\star(r)
Today: Production and Costs
Review: Cost Minimization
Long-Run Conditional Demands and Cost Functions
Short-Run Conditional Demands and Cost Functions
Cost Minimization
Cost Minimization Subject to a Utility Constraint
Cost Minimization Subject to an Output Constraint
\min p_1x_1 + p_2x_2 \text{ s.t. } u(x_1,x_2) = U
\min wL + rK \text{ s.t. } f(L,K) = q
\text{solutions}: x_1^c(p_1,p_2,U), x_2^c(p_1,p_2,U)
\text{solutions}: L^c(w,r,q), K^c(w,r,q)
Hicksian Demand
Conditional Demand
pollev.com/chrismakler
If labor is shown on the horizontal axis and capital is shown on the vertical axis, what is the magnitude of the slope of the isocost line, and what are its units?
Cost Minimization: Lagrange Method
\mathcal{L}(L,K,\lambda)=
wL + rK +
(q - f(L,K))
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L
First Order Conditions
\frac{\partial \mathcal{L}}{\partial x_2} = r - \lambda MP_K
\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)
= 0 \Rightarrow \lambda = w \times {1 \over MP_L}
= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}
MRTS (slope of isoquant) is equal to the price ratio
\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}
\text{Set two expressions }
\text{for }\lambda \text{ equal:}
\frac{MP_L}{MP_K} = \frac{w}{r}
f(L,K) = \sqrt{LK}
Tangency condition: \(MRTS = w/r\)
Constraint: \(q = f(L,K)\)
MRTS = {MP_L \over MP_K} =
{{1 \over 2}L^{-{1 \over 2}}K^{1 \over 2} \over {1 \over 2}L^{1 \over 2}K^{-{1 \over 2}}}
= {K \over L}
Conditional demands for labor and capital:
Total cost of producing \(q\) units of output:
Expansion Path
A graph connecting the input combinations a firm would use as it expands production: i.e., the solution to the cost minimization problem for various levels of output
Exactly the same as the income offer curve (IOC) in consumer theory.
(And, if the optimum is found via a tangency condition, exactly the same as the tangency condition.)
Long-Run Total Cost of \(q\) Units
c^{LR}(q) = wL^c(w,r,q) + rK^c(w,r,q)
Conditional demand for labor
Conditional demand for capital
"The total cost of producing \(q\) units in the long run
is the cost of the cost-minimizing combination of inputs
that can produce \(q\) units of output."
Exactly the same as the expenditure function in consumer theory.
Conditional Demand
in the Short Run
If there's only one variable input,
it's perfectly inelastic -- there's only one choice!
f(L,K) = \sqrt{LK}
Short-run conditional demand for labor
if capital is fixed at \(\overline K\):
Short-Run Total Cost of \(q\) Units
TC(q\ |\ \overline{K}) = wL(q,\overline{K}) + r\overline{K}
Variable cost
"The total cost of producing \(q\) units in the short run is the variable cost of the required amount of the input that can be varied,
plus the fixed cost of the input that is fixed in the short run."
Fixed cost
f(L,K) = \sqrt{LK}
Short-run conditional demand for labor
if capital is fixed at \(\overline K\):
Total cost of producing \(q\) units of output:
L^c(w,r,q, \overline K) = {q^2 \over \overline K}
c(w, r, q,\overline{K}) = wL(w, r, q,\overline{K}) + r\overline{K} =
{wq^2 \over \overline K} + r\overline K
Total, Fixed and Variable Costs
c(q,\overline{K}) =
Fixed Costs \((F)\): All economic costs
that don't vary with output.
Variable Costs \((VC(q))\): All economic costs
that vary with output
explicit costs (\(r \overline K\)) plus
implicit costs like opportunity costs
r\overline{K}
wL(q,\overline{K})
+
TC(q) =
+
F
VC(q)
e.g. cost of labor required to produce
\(q\) units of output given \(\overline K\) units of capital
pollev.com/chrismakler
Generally speaking, if capital is fixed in the short run, then higher levels of capital are associated with _______ fixed costs and _______ variable costs for any particular target output.
Next Time
- Examine the curvature of cost functions
- See the relationship of long run and short run costs
- Examine unit costs: average and marginal
- Analyze economies and diseconomies of scale
Econ 50 | Spring 23 | Lecture 15
By Chris Makler
Econ 50 | Spring 23 | Lecture 15
Production and Cost
