Alex_Of_Darkness
New member
- Joined
- Nov 16, 2017
- Messages
- 29
Need help w/: 1. E(p,m) = Ap−amb, 2. strict concavity/convexity, 3. maximize profit
Hi everyone, in preparation to my exam, I need help solving 3 exercices. These are the only 3 that I can't solve so if you could help I would be really grateful!
Exercice #1: E(p,m) = Ap−amb
P is the price and m is the income, A, a and b are all constants and > 0.
a) Suppose also that p and m are both differentiable functions of time t. E is a function of t only. Find an expression of E '/ E in terms of p' / p and m '/ m.
b) Put p = p0 (1,06) t, m = m0 (1,08) t, where p0 is the price and m0 the income at time t = 0. Show that, in this case, E '/ E = ln Q , where Q = (1, 08) b / (1, 06) a.
Exercice #2: f(x,y) = ax2 +2bxy+cy2 + px+qy+r
a) Show that the general quadratic function is strictly concave if ac - b2> 0 and a <0, while it is strictly convex if ac - b2> 0 and a> 0.
b) Find the necessary and sufficient condition for f (x, y) to be convex / concave.
Exercice #3 :
A company holds the monopoly of the production of a good on two different markets A and B. The company imposes the different prices PA and PB on these two markets. Consumers in these two markets then decide QA and QB quantities of the good they want to consume as follows:
QA = 20-PA and QB = 10-PB.
The total cost of manufacturing this product is given, for its part, by
CT (QA, QB) = 1+ (QA + QB) 2.
a) Write the profit π (PA, PB) as a function of the prices PA and PB.
b) Determine the price-quantity combination that maximizes profits (you must prove that the point in question corresponds to a maximum).
Thanks a lot to everyone in advance
Hi everyone, in preparation to my exam, I need help solving 3 exercices. These are the only 3 that I can't solve so if you could help I would be really grateful!
Exercice #1: E(p,m) = Ap−amb
P is the price and m is the income, A, a and b are all constants and > 0.
a) Suppose also that p and m are both differentiable functions of time t. E is a function of t only. Find an expression of E '/ E in terms of p' / p and m '/ m.
b) Put p = p0 (1,06) t, m = m0 (1,08) t, where p0 is the price and m0 the income at time t = 0. Show that, in this case, E '/ E = ln Q , where Q = (1, 08) b / (1, 06) a.
Exercice #2: f(x,y) = ax2 +2bxy+cy2 + px+qy+r
a) Show that the general quadratic function is strictly concave if ac - b2> 0 and a <0, while it is strictly convex if ac - b2> 0 and a> 0.
b) Find the necessary and sufficient condition for f (x, y) to be convex / concave.
Exercice #3 :
A company holds the monopoly of the production of a good on two different markets A and B. The company imposes the different prices PA and PB on these two markets. Consumers in these two markets then decide QA and QB quantities of the good they want to consume as follows:
QA = 20-PA and QB = 10-PB.
The total cost of manufacturing this product is given, for its part, by
CT (QA, QB) = 1+ (QA + QB) 2.
a) Write the profit π (PA, PB) as a function of the prices PA and PB.
b) Determine the price-quantity combination that maximizes profits (you must prove that the point in question corresponds to a maximum).
Thanks a lot to everyone in advance