Profit maximizing firms
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math 3660 – Spring 2019
Mathematical Models in Economics
assignment 2
Instructions
Save a copy of this notebook, complete the exercises, save and submit on Brightspace your final
version by start of class Tues. Jan. 22.
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Exercises
Clear all of the symbols we will be using.
In[1]:=Clear [ p1, p2, q1, q2, a11, a12, a21, a22, b1, b2, mc1, mc2, profit1, profit2 ] ;
As above, suppose the demands for products are 1 and 2 are linear functions given by
In[2]:=q1 [ p1 _ , p2 _] : = a11 * p1 + a12 * p2 + b1;
q2 [ p1 _ , p2 _] : = a21 * p1 + a22 * p2 + b2;
1. At times x, y, and z, we observe the prices and demand for products 1 and 2,
obtaining the following (hypothetical) data. Determine the coefficients a11,
a12, a21, a22, b1, and b2 of the linear demand functions that match this data.
(Set up a system of equations, given as a list, and solve for the coefficients.)
In[4]:=p1x = 50.; p2x = 40.; q1x = 100.; q2x = 120.;
p1y = 55.; p2y = 40.; q1y = 80.; q2y = 130.;
p1z = 50.; p2z = 45.; q1z = 105.; q2z = 100.;
TableForm [{{ p1x, p2x, q1x, q2x } , { p1y, p2y, q1y, q2y } , { p1z, p2z, q1z, q2z }} , TableHeadings
{{ "time x", "time y", "time z" } , { "price 1", "price 2", "demand 1", "demand 2" }}]
Out[7]//TableForm=
price 1 price 2 demand 1 demand 2
time x 50. 40. 100. 120.
time y 55. 40. 80. 130.
time z 50. 45. 105. 100.
2. Determine the own elasticities and cross elasticities for these products at
time x. (The Mathematica function D[formula,variable] gives the partial
derivative of the formula with respect to the variable.)
3. Suppose constant marginal costs for the products are as given below and
take fixed costs to be 0. Define functions profit1 and profit2 for the profits on
products 1 and 2 (profit1[p1_,p2_]:=… and profit2[p1_,p2_]:=…).
In[8]:=mc1 = 25; mc2 = 15;
4. If firm 1 sets it price at 50. what is the optimal price for firm 2? If firm 2 sets its
price to this amount what is the optimal price for firm 1? And then if firm 1
takes this price what would firm 2 do, and then what would firm 1 respond?
5. Find the Nash equilibrium prices, i.e., the prices p1e and p2e such that
profit1[p1,p2e] is a maximum at p1=p1e and profit2[p1e,p2] is a maximum at
p2=p2e, so that firm 1 is doing the best it can given what firm 2 is doing, and
firm 2 is doing the best it can given what firm 1 is doing.
6. If the firms merge, they will only be concerned about the total profit on the
two products. Find the prices the merged firm will charge to maximize total
profit, i.e., the prices p1m and p2m such that profit1[p1,p2]+profit2[p1,p2] is a
maximum at p1=p1m and p2=p2m.
2 02.2-Assignment02.nb