Competitive producer theory. A competitive firm produces a commodity in amount y using two inputs in quantities x1 and x2

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Competitive producer theory. A competitive firm
produces a commodity in amount y using two inputs in
quantities x1 and x2. Let p be the
output price and w = (w1, w2) the input price
vector.
Production function f (x1, x2)
can take either of the following forms:
a. f (x1, x2) =
x1 + ln x2
b. f (x1, x2) =
[a1x1p +
a2x2p]1/p , with ρ ∈
(−∞, 1), a1 > 0, a2 > 0 (CES)
c. f (x1, x2) =
ln(x1 − b) + ln(x2 − b)
For each of the above technologies, compute:
1. Conditional factor demand x(w, q).
2. Cost function c(w, q).
3. Supply function q(p).
4. Profit function π(p).
and verify their properties.
Process innovation. A firm has already minimized
its costs, and its cost function is c(q) = q2,
where q its the amount of output. It can further reduce
costs by investing r in process innovation. Process innovation
yields a cost reduction rαq, with 0 < α < 1. Model
the profit maximization problem, find the optimal values
q∗ and r∗, and comment on the relationship
between the optimal decisions and the model parameters.
The silo problem. A firm wishes to build a silo.
The silo is an open cylinder of height h and its base has
radius r. It has to buy a piece of land whose size is
equal to the area of a circle with radius r. The firm pays
pα per square meter of the base and pl per
square meter of the lateral area of the silo. The silo can store a
quantity of grain equal to the volume of the silo, and the firm
wishes to store a given quantity q. Model the cost
minimization problem and find the optimal radius and height of the
silo.
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