Ch 2 #6  Computer Sales

(a)  The main model.

We consider computer sales per month, but we only need to consider one month (we do not have any changes over time.)

The base number of computers sold (per month) is 10,000 units.
The cost to manufacture is 700 $/unit.
The base wholesale selling price is 950 $/unit.

For every $100 decrease in selling price, we increase the number sold by 50%, to
 10,000 + (50%) (10,000) ((950 - p)/100)
 where p is the new selling price.
 
 The advertising base cost is 50,000 $.  
 
 For evey $10,000 increase in advertising budget, we increase the  number sold by 200 uunits, to 
 10,000 + 50(950-p) + 200(a-50,000)/10,000 
 where a is the new advertising budget.
 
 a <= 100,000.
 
 p-700 is the profit per unit.

Clear[Profit,AdBudget,SellPrice,L,ConstraintFcn]

Profit[AdBudget_, SellPrice_] = (SellPrice-700) *(10000 + 50(950-SellPrice) + 1/50 (AdBudget - 50000)) - AdBudget

D[Profit[AdBudget,SellPrice], AdBudget]

D[Profit[AdBudget,SellPrice], SellPrice]

ConstraintFcn[AdBudget_, SellPrice_] =AdBudget-100000

D[ConstraintFcn[AdBudget,SellPrice], AdBudget]

D[ConstraintFcn[AdBudget,SellPrice], SellPrice]

Solve[{-1+(1/50) (-700+SellPrice) == L * 1,10000+(1/50) (-50000+AdBudget)+50 (950-SellPrice)-50 (-700+SellPrice) == L*0, ConstraintFcn[AdBudget,SellPrice] == 0 }, {SellPrice, AdBudget, L}]

Profit[100000,935]

p1=Plot3D[Profit[AdBudget,SellPrice], {AdBudget,50000,150000},{SellPrice,700,1000}]

p2 = ListPointPlot3D[{{100000,935,2661250}},PlotStyle->PointSize[0.03]];

Show[p1,p2]

We are constrained by the  $100, 000 maximum value of the advertising budget.  For AdBudget <= 100 000, we can see from the graph that we have a max for the profit.

Under this model, the company's profit would increase to $2,661,250 by increasing their advertising budget to $100,000 and decreasing their wholesale selling price to $935.

Since the Lagrange multiplier has value 3.7, if we allowed the advertising budget to increase past the constraint of $100,000 by $1, we would expect to see a $3.70 increase in profit.  That's a very good return on our investment!

(b) Sensitivity to price elasticity, the 50% number.

In part (a), we had the price elasticity of 50%, meaning that every dollar decrease in price was assumed to generate 50 more units sold per month.

Originally, we had...
		For every $100 decrease in selling price, we increase the number sold by 50%, to
		 10,000 + (50%) (10,000) ((950 - p)/100)
Now we have ...
		For every $100 decrease in selling price, we increase the number sold by PriceElasticity, to
		10,000 + (PriceElasticity) (10,000) ((950 - p)/100)

Clear[Profit,AdBudget,SellPrice,L,ConstraintFcn,PriceElasticity,OptSellPrice]

Profit[AdBudget_, SellPrice_] = (SellPrice-700) *(10000 + PriceElasticity*1000*(950-SellPrice) + 1/50 (AdBudget - 50000)) - AdBudget

D[Profit[AdBudget,SellPrice], AdBudget]

D[Profit[AdBudget,SellPrice], SellPrice]

ConstraintFcn[AdBudget_, SellPrice_] =AdBudget-100000

D[ConstraintFcn[AdBudget,SellPrice], AdBudget]

D[ConstraintFcn[AdBudget,SellPrice], SellPrice]

Solve[{-1+(1/50) (-700+SellPrice) == L * 1,10000+(1/50) (-50000+AdBudget)+1000 PriceElasticity (950-SellPrice)-1000 PriceElasticity (-700+SellPrice) == L*0, AdBudget == 100000 }, {SellPrice, AdBudget, L}]

Simplify[%]

The advertising budget does not change with the price elasticity.  In other words, advertising budget  is completely insensitive to the price elasticity.  The sensitivity function for advertising budget based on  price elasticity is zero.

OptSellPrice[PriceElasticity_] = 825+(11/(2 PriceElasticity))

D[   OptSellPrice[PriceElasticity], PriceElasticity   ]

Recall that, in general, the sensitivity of variable x on parameter r is S(x,r) = (dx/dr) (r/x).

If we hide the dependence of OptSellPrice on PriceElasticity, the sensitivity function is

-(11/(2 (PriceElasticity^2))) * (( PriceElasticity)/OptSellPrice)

When the price elasticity is 50% or 0.50,

OptSellPrice[.5]

(-(11/(2 (PriceElasticity^2))) * (( PriceElasticity)/OptSellPrice))/.{PriceElasticity->.5, OptSellPrice->836}

Another way to say this... we know that the optimum selling price is actually a function of the price elasticity.

-(11/(2 (PriceElasticity^2)))*(( PriceElasticity)/(OptSellPrice[PriceElasticity]))

Simplify[%]

When the price elasticity is 50% or 0.50, the sensitivity function value is

(1/(-1-150 PriceElasticity))/. {PriceElasticity->.5}

so that with each 100% increase in the value of the price elasticity (say from 50% to 50% + (10%*50) = 55%),  we expect the optimum selling price to decrease by about 13%.  That is, if the price elasticity is 13% greater than we expected,, then the optimal seling price will be about 100% lower than expected, and the optimal advertising budget stays the same.