MMElogitsoln

MMElogitsoln - 2011 - Steven Tschantz Logit demand model...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
© 2011 - Steven Tschantz Logit demand model Steven Tschantz 3/1/11 Exercises Complete the following exercises. You can save a copy of this notebook, delete the material above, insert and evaluate your commands computing answers below, save your results, and email your final version to tschantz@math.vanderbilt.edu Your task in this assignment is to implement the logit demand model for three products. Calibrate the demand model to initial prices and quantities and specified aggregate elasticity and an own price elasticity for one product. Then determine the marginal costs that would make the initial prices and quantities of the products in Nash equilibrium for separate firms owning the three products. Compute the Nash equilibrium if two of the firms merge and price their two products jointly so as to maximize their total profit in competition with the third firm. Compare the initial prices and quantities to the prices and quantities predicted for the post merger scenario. ± 1. Let eta1, eta2, eta3, lambda, and M be parameters for a logit demand model with three products. Write definitions for the demand functions q1[p1,p2,p3], q2[p1,p2,p3], and q3[p1,p2,p3] of the three products and for q0[p1,p2,p3] the demand for the outside good. Clear the parameters eta1, eta2, eta3, lambda, and M, and the variables p1, p2, p3, q1, q2, q3, and q0. Define q1 @ p1_, p2_, p3_ D = .. writing out the logit demand formula for product 1. Do the same for q2, q3, and q0. As a check, verify that q1[p1,p2,p3]+q2[p1,p2,p3]+q3[p1,p2,p3]+q0[p1,p2,p3] simplifies to M. In[1]:= Clear @ eta1, eta2, eta3, lambda, M, p1, p2, p3, q1, q2, q3, q0 D ; In[2]:= q1 @ p1_, p2_, p3_ D = M * Exp @H eta1 - p1 L ± lambda D ± H 1 + Exp @H eta1 - p1 L ± lambda D + Exp @H eta2 - p2 L ± lambda D + Exp @H eta3 - p3 L ± lambda DL Out[2]= ª eta1 - p1 lambda M 1 + ª eta1 - p1 lambda + ª eta2 - p2 lambda + ª eta3 - p3 lambda In[3]:= q2 @ p1_, p2_, p3_ D = M * Exp @H eta2 - p2 L ± lambda D ± H 1 + Exp @H eta1 - p1 L ± lambda D + Exp @H eta2 - p2 L ± lambda D + Exp @H eta3 - p3 L ± lambda DL Out[3]= ª eta2 - p2 lambda M 1 + ª eta1 - p1 lambda + ª eta2 - p2 lambda + ª eta3 - p3 lambda In[4]:= q3 @ p1_, p2_, p3_ D = M * Exp @H eta3 - p3 L ± lambda D ± H 1 + Exp @H eta1 - p1 L ± lambda D + Exp @H eta2 - p2 L ± lambda D + Exp @H eta3 - p3 L ± lambda DL Out[4]= ª eta3 - p3 lambda M 1 + ª eta1 - p1 lambda + ª eta2 - p2 lambda + ª eta3 - p3 lambda
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
In[5]:= q0 @ p1_, p2_, p3_ D = M ± H 1 + Exp @H eta1 - p1 L ± lambda D + Exp @H eta2 - p2 L ± lambda D + Exp @H eta3 - p3 L ± lambda DL Out[5]= M 1 + ª eta1 - p1 lambda + ª eta2 - p2 lambda + ª eta3 - p3 lambda In[6]:= Simplify @ q1 @ p1, p2, p3 D + q2 @ p1, p2, p3 D + q3 @ p1, p2, p3 D + q0 @ p1, p2, p3 DD Out[6]= M ± 2. Write definitions for the aggregate elasticity ae[p1,p2,p3] and elasticity matrix elasticitymatrix[p1,p2,p3]. Define ae
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 10/28/2011 for the course MATH 256 at Vanderbilt.

Page1 / 9

MMElogitsoln - 2011 - Steven Tschantz Logit demand model...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online