MMEchoicesoln

MMEchoicesoln - 2011 - Steven Tschantz Choice demand model...

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© 2011 - Steven Tschantz Choice demand model Steven Tschantz 2/22/11 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 ± 1. [Calibration] Suppose consumers are choosing whether or not to buy a single product. Suppose their values for the product are normally distributed, that 20% of consumers buy the product at a price of $20, and they have an elasticity of demand of -3 at this price. This first exercise is a calibration problem. Assuming a model and some basic information, determine coefficients of the model fitting the given information. Use the method of undetermined coefficients assuming the distribution of consumer values is In[1]:= Clear @ dist, mu, sigma, choiceprob D In[2]:= dist = NormalDistribution @ mu, sigma D Out[2]= NormalDistribution @ mu, sigma D and the choice probability at a price p implied by these values is given by In[3]:= choiceprob @ p_ D = 1 - CDF @ dist, p D Out[3]= 1 + 1 2 - 1 - Erf B - mu + p 2 sigma F Write down a definition of the elasticity of this demand (probability) . .. In[4]:= elast @ p_ D = D @ choiceprob @ p D , p D * p ± choiceprob @ p D Out[4]= - ª - H - mu + p L 2 2 sigma 2 p 2 Π sigma 1 + 1 2 - 1 - Erf B - mu + p 2 sigma F then set up the system of equations defining the given conditions.
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In[5]:= conditions = 8 choiceprob @ 20. D ± 0.20, elast @ 20. D ± - 3. < Out[5]= : 1 + 1 2 - 1 - Erf B 20. - mu 2 sigma F ± 0.2, - 7.97885 ª - H 20. - mu L 2 2 sigma 2 sigma 1 + 1 2 - 1 - Erf B 20. - mu 2 sigma F ± - 3. > Now you may have to experiment some to find the parameters solving these conditions. You won't be able to solve for the parameters using Solve; this isn't an algebraic system of equations. So you have to use FindRoot. If you assigned the system of equations to a variable conditions for example, you would use the command FindRoot[conditions,{{mu,startmu},{sigma,start- sigma}}] where startmu and startsigma are an initial guess at a solution. You may have to try several combinations of values
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MMEchoicesoln - 2011 - Steven Tschantz Choice demand model...

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