Title: Microeconomics / production duality

Q: Find the conditional input demand from the cost function
A: z*(r,q) = ∂C(r,q) / ∂r_i

Q: Find the unconditional supply from the cost function
A: The unconditional supply function can be derived from the cost function using the concept of profit maximization. A firm chooses its output level to maximize its profit, which is the difference between its revenue and its cost.
max{q} pq - C(r,q) --> π(p, w) = pq* - C(r,q)

Q: Find the unconditional input demand from the cost function
A: 1. derive the firms output supply function q(p,r)
2. plug into the conditional input demand: z*(r,q) = = ∂C(r,q) / ∂r_i
3. unconditional input demand: z*(p,r)= z*(r,q(p,r)

Q: Find the Input distance function from the cost function
A: min{r} rz s.t. C(r,q)

Q: Find the conditional supply given the revenue function
A: q*(p,z) = ∂R(p,z) / ∂p_i

Q: Find the unconditional input demand from the revenue function
A: max{z} π(p, w) = R(p, y) - rz -->  π(p, w) = R(p, y) - rz*

Q: Given the revenue function find the unconditional supply
A: 1. derive the firms input supply function: z(p,r)
2. plug into the conditional input demand and take the derivative: z*(p,r)= z*(r,q(p,r))
 --> q*(p,z) =∂R(p,z) / ∂p_i
3. unconditional input demand: z*(p,r)= z*(r,q(p,r))

Q: Given the revenue function find the output distance function
A: Do(q,z) = max{q} pq s.t. R(p,z)

Q: Given the profit function find the unconditional input demand
A: z*(p,r) = - ∂π(p,r) / ∂r_i

Q: Find the unconditional supply given the profit function
A: q*(p,r) = ∂π(p,r) / ∂p_i

Q: Given profit function find the input distance function
A: min{p,r} pq - π(p,r) >= 1

Q: Given the profit funciton find the output distance function
A: D0(q,z) = max{p,r} rz + π(p,r)

Q: Given the input distance function find the conditional input demand
A: z*(r,q) = min{z} rz s.t. DI >= 1

Q: Given the ouput distance function find the conditional supply
A: q*(p,z) = max{q} pq s.t. D0 <= 1

Q: Given the PPS find the input distance function
A: DI(q,z) = max{δ} { δ >0: (q, (-z/δ) }

Q: Given the PPS find the output distance function
A: Do(q,z) = min{δ} { δ >0: (q/δ, -z}

Q: Find returns to scale
A: Set up θ as (q,-z) = (θq, - θz) and find the θ to satisify the RS

Q: Proof of upward and downward sloping
A: Example:
[latex]$r^a \cdot z^a \le r^a \cdot z^b$[/latex]
[latex]$r^b \cdot z^b \le r^b \cdot z^a$[/latex]
next sum both sides:
[latex]$r^a \cdot z^a + r^b \cdot z^b \le r^a \cdot z^b + r^b \cdot z^a$[/latex]
Simplify:
[latex]$(r^b - r^a)(z^b - z^a) \le 0$[/latex]

Q: Find the revenue cost function
A: 1. Start with the cost funciton
2. [latex]$(R^a = p_aq_a)$[/latex]
[latex]$(R^b = p_bq_b)$[/latex]
3. plug in the [latex]$(q_i)$[/latex] into your cost function
4. This is your revenue cost function

Q: How do you find the Certainty equivalent from the revenue cost function?
A: we know that certainty equivalent (ce) is equal to [latex]$max_{e}(r,p,e) \le C(r,p,R_a,R_b)$[/latex]
1. replace the revenue values with e [latex]$(R_i = e)$[/latex]
2. set the revenue cost function with e to less than or equal to the original revenue cost function
3. find e

Q: How do you find the producers subjective probabilities?
A: 1. take the parital of the profit function with respect to each profit Wa(π) or Wb(π)
2. Next recall that φa=Wa(π)/Wa(π)+Wb(π) and vice versa for state b
3. plug in the values found in step1 into the equation for step 2
4. Simplify each probability

Q: How do you find expected profit in production?
A: E[π]=φg*πg+φb*πb
where φg and φb are subjective probabilities
πg and πb are profits in either states

Find the state contingent output supply from the revenue cost function

Q: How to find the unconditional inputs from the revenue cost funciton
A: Take the partial with respect to input prices (shephard's Lemma)

Q: Find the unconditional demand
for input from the industry profit and quantity
A: Take the aggregate cost and differentiate with respect to r and plug in the industry quantity into Q*

generalized-schur concavity