%path = "maths/finance/cost and price theory/cournot"
%kind = kinda["problems"]
%level = 12
In the following use fractions for numbers.
-
If there is a linear price-sales function \(p(x)=kx+d\), how does
the marginal revenue look like \(E'(x)\) (use \(E(x)=xp(x)\)).
-
Through market observations an upper price limit (intersection
between \(p(x)\) and p axis) is fixed at €{{ g.pmax }}.
The saturation quantity (intersection between \(p(x)\) and the x axis)
is estimated with {{ 2*g.xp0 }}.
One shall make a linear model for the marginal revenue
\(E'(x)=\)
%include('chcko/getorshow',idx = 0, show = util.tx(util.tex))
-
Through integration this marginal revenue one gets the total revenue.
\(E(x) =\)
%include('chcko/getorshow',idx = 1, show = util.tx(util.tex))
How do we determine the integration constant?
-
Costs consist of fixed costs and variable costs.
Are the fixed costs independent from the production quantity and why?
-
The marginal costs are assumed to be quadratic
\(K'(x)= {{ util.tex(g.Kp) }} \).
Which model describes the total costs (integration)?
To get the integration constant consider that even with no production
there is €{{ g.Ko }} cost.
\(K(x)=\)
%include('chcko/getorshow',idx = 2, show = util.tx(util.tex))
-
Write down the expression for the price-sales function p(x).
\(p(x) =\)
%include('chcko/getorshow',idx = 3, show = util.tx(util.tex))
-
Write down the price elasticity \(\epsilon(x)\)
(use the Amoroso-Robinson-Relation or the definition of \(\epsilon\))
\(\epsilon(x) =\)
%include('chcko/getorshow',idx = 4, show = util.tx(util.tex))
-
With what production quantity do we have maximum profit?
\(x_g =\)
%include('chcko/getorshow',idx = 5)
-
Is the demand at this production quantity elastic and why?
-
What price does one have to use to maximize the profit?
\(p(x_g) =\)
%include('chcko/getorshow',idx = 6)
-
What is the name of this point \( (x_g,p(x_g)) \)?