%path = "maths/finance/cost and price theory/cournot" %kind = kinda["problems"] %level = 12 In the following use fractions for numbers.
  1. If there is a linear price-sales function \(p(x)=kx+d\), how does the marginal revenue look like \(E'(x)\) (use \(E(x)=xp(x)\)).
  2. 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))
  3. 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?
  4. Costs consist of fixed costs and variable costs. Are the fixed costs independent from the production quantity and why?
  5. 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))
  6. Write down the expression for the price-sales function p(x).
    \(p(x) =\) %include('chcko/getorshow',idx = 3, show = util.tx(util.tex))
  7. 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))
  8. With what production quantity do we have maximum profit?
    \(x_g =\) %include('chcko/getorshow',idx = 5)
  9. Is the demand at this production quantity elastic and why?
  10. What price does one have to use to maximize the profit?
    \(p(x_g) =\) %include('chcko/getorshow',idx = 6)
  11. What is the name of this point \( (x_g,p(x_g)) \)?