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