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Problem Set #1 Key Sonoma State University Economics 494- Seminar in Quantitative Marketing II Dr. Cuellar Profit Maximizing Behavior of the Firm Suppose a firm operating in a differentiated product market has the following demand and cost functions: A1/2 Q= TC=4Q2 +10Q +A 1. Calculate the profit maximizing level of output, price and advertising. Solve demand for P and multiply by Q to get total revenue: Π = 100Q – 3Q2 + 4A1/2Q – [4Q2 +10Q +A] The two necessary first order conditions for optimization are: 1. 90 –14Q + 4A1/2 =0. 2. 2QA-1/2 – 1=0 Solve for A1/2 and substitute into (1). A1/2 = 2Q Solving (1) for Q = + A1/2 Q= Q- 14Q-8Q=90 6Q=90 Q=15 A=(2*90)2 = $900 P=$175 MC= 8Q +10 = $130 2. Calculate the price elasticity of demand at the profit maximizing solution. We can calculate the price elasticity of demand from the demand equation: Q= = 3.889 3. Calculate the advertising elasticity of demand at the profit maximizing solution. The advertising elasticity of demand can be derived from the demand equation. = 1.33 4. Verify that the Lerner mark-up index is equal to the negative of the inverse of the price elasticity of demand ( ). = 1/.257 = 3.889 5. Verify that the advertising expenditures-to-sales revenue ratio is equal to the negative of the ratio of the advertising elasticity-to-price elasticity ( ). Estimating Elasticities Suppose that a firm believes that its demand is given by the equation: Q=kPηIβaγAδ, where: P is the price per unit and η is the price elasticity of demand I is income and β is the income elasticity of demand a is current advertising and γ is the short run advertising elasticity of demand A is average advertising in the preceding ten years and (γ+δ) is the long run advertising elasticity of demand 1. The data for problem set #1 contains data for all relevant variables except price. However, you can still estimate all relevant elasticities by estimating the associated total revenue function. Estimate the total revenue function and solve for the price, income, the short-run and long-run advertising elasticities of demand. Are your estimates statistically significant? Show all your work. Solving Q=kPηIβaγAδ for price, multiplying by Q and taking logs on both sides produces: lnTR= ( ) The estimating equation: lnTR=β0 + β1lnQ+ β2lnI+ β3lna+ β4lnA+ui lnqty/lnprice lninc lnadv lnaveadv Constant Question #1 lnrev -0.339 [0.16] 1.147 [0.00]** 0.185 [0.24] 0.119 [0.05]* -8.201 [0.00]** 50 0.67 Question #3 lnqty -0.309 [0.00]** 0.786 [0.00]** 0.363 [0.00]** 0.025 [0.39] -6.065 [0.00]** 50 0.95 Observations Adjusted R2 p-value in brackets * significant at 5% level; ** significant at 1% level Solving the total revenue equation for the respective elasticities: η -0.746826 β 0.856609 γ 0.138163 δ γ+δ 0.088872 0.227035 .2 .19 .18 .17 .16 .15 .14 .13 .12 .11 .1 .09 .08 .07 .06 .05 .04 .03 .02 .01 0 Advertising/Revenue 2. Is the firm advertising optimally? Explain fully and show graphically. The optimal ratio of advertising expenditures to revenue. 1960 1970 1980 1990 YEAR 2000 2010 It appears that the firm is under advertising. 3. Construct the variable price where price=revenue/quantity and re-estimate the elasticity parameters directly. Are they similar to the estimates in (1)? Are they statistically significant? Show all your work. 4. Is the firm advertising optimally? Explain fully and show graphically. The optimal ratio of advertising expenditures to revenue. 1.2 1.1 1 .9 .8 .7 .6 .5 Advertising/Revenue .4 .3 .2 .1 0 1960 1970 1980 It appears that the firm is under advertising. 1990 YEAR 2000 2010