cost minimizing problem

Question 1

If a firm is maximising its profits how it lower the costs?

Answer 1

When a company chooses a production plan that maximizes its profits, it means there’s no cheaper way to produce the same amount of goods.
- if a firm maximizes profits by producing q* using the input bundle x* then x* is also the input bundle that produces q* at minimum cost
- if there was another bundle x’ that allowed to produce q* at a lower cost, then the firm should have used x’

Cost minimization is a necessary condition for profit maximization.

Question 2

What can we find with the cost minimization problem?

Answer 2

For any given level of output q, we can determine the set of inputs that will produce that amount of output at the lowest possible cost, based on the current prices of those inputs.

Question 3

How can the CMP be stated?

Answer 3

min wx f(x)=q
x≥0

The firm chooses its input bundle so as to set the absolute value of the MRTS between any two inputs equal to the ratio of their prices.

Question 4

what is the interior solution of the CMP?

Answer 4

With two inputs, an interior solution corresponds to a point of tangency between:
- the q-level isoquant (The term q-level isoquant simply refers to an isoquant that corresponds to a specific output level q)
- An isocost line of the form w₁x₁+w₂x₂=c

Question 5

what is an isoquant?

Answer 5

Shows all combinations of two inputs that produce the same level of output q

Question 6

what is an isocost line?

Answer 6

Shows all combinations of two inputs that cost the same total amount c
isolate x₂ from w₁x₁+w₂x₂=c and you will get the equation of an isocost

An increase in c leads to an increase in the vertical (and horizontal) intercepts

Question 7

how do we find the optimal point?

Answer 7

When q is fixed, the problem of the firm is to find a cost-minimizing point
on a given isoquant.

The firm chooses a point where the isoquant is tangent to the isocost line.

That point is characterized by the tangency condition that the slope of the isoquant must be equal to the slope of the isocost line
|MP₁/MP₂|=w₁/w₂

Question 8

when we have a unique solution?

Answer 8

if f(*) is strictly quasi-concave, the solution is
unique.
The unique solution is x^∼(w,q) and it is called “CONDITIONAL FACTOR DEMAND FUNCTION”

Question 9

COST FUNCTION

Answer 9

If we plug x^∼ into the objective function we get the cost function
c=wx
it measures the minimum cost of producing output q when input prices are w.

Answer 10

For f (·) continuous and strictly increasing, the cost function c(w,q) is:
1. homogeneous of degree one in w;
2. non-decreasing in w; for all w ≫0, strictly increasing in q;
3. concave in w;
4. continuous in w, for w ≫0;
5. if f (·) is homogeneous of degree one (i.e., exhibits constant returns
to scale), c(w,q) is homogeneous of degree one in q;
6. if f (·) is concave (i.e., exhibits decreasing returns to scale), c(w,q)
is convex in q.

Question 11

Shepard Lemma - production version

Answer 11

If f (·) is continuous, strictly increasing, and strictly quasi-concave,
then the conditional demand for factor ℓ can be obtained by
differentiating the cost function with respect to wℓ:
x^∼ ℓ(w,q) = ∂c/∂w