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MMElogitsoln

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

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© 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 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

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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
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MMElogitsoln - 2011 Steven Tschantz Logit demand model...

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