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Assignment 1
Answer Key
Correct answers are in bold.
Assignment 1 Due Date: January 30, 2012
Name:
1. In Figure 1 (Part II) of this handout, we mapped the model in words and symbols using a
diagram to represent model relationships. Describe the addition we need to make to the diagram
in Figure 1 when we change the quantity of pies demanded and sold (Q) to a variable that
depends on the price charged?
To alter the influence diagram to reflect the demand equation, we need to trace the quantity
(Q) variables in the revenue, processing cost, and ingredients cost equations to the
formulated demand equation (Q = intercept –slope*price). This shows that quantity of pies
demanded and sold is not exogenously chosen but endogenously determined by price.
2. In Figure 3 above, three schedules are plotted for Revenue, Costs, and Profits. The plot for
profits can be used to find the correct price to charge by finding the highest point it reaches.
Describe how would you find the highest profit using this graph if it only had revenue and costs
plotted?
We know that profit is revenues less costs. Keep in mind that revenues must exceed cost if
there is to be positive profit and the profits will be highest at the largest vertical difference
between revenue and cost. Therefore, by looking at the graph, we can look at the
relationship between revenue and cost at each price and determine at which price this
vertical difference is the largest. In this graph, the difference appears to be largest at $9.00.
Note that this is a simple model with nice curves. With more complicated models it is not
always clear where profits are highest just by looking and further calculation will be
required.
3. Assume that the manager of Simon Pies Co. chooses $9.00 as the price to set. Using the
worksheet Sens.Analysis.9dollars in the Laboratory 2 Spreadsheet discuss how sensitive profits
are to this choice of price. (In other words, how do changes in the price around the $9.00 level
affect profits the company earns and do profits increase or decrease when the price changes?).
The decision on price to charge is not all that sensitive to small price changes. If we
charged five percent more or less than nine dollars we would not see much change in the
optimal profit levels. Changing price by a lot will result in much larger changes in profits so
our conclusion that nine dollars is the best price is sensitive to large price changes.
4. Using the spreadsheet model (and the graphs in the spreadsheet) what would the new profit
maximizing price be if the slope of the demand curve was increased to -3,600? Explain this
change. (Note: Use the worksheet Add_Demand_Schedule for this question and change the
slope parameter for all columns).
Assignment 1
Answer Key
The new profit-maximizing price is $10.00 and the new profit level is 38,040.00, assuming we
only looked at prices plotted in the spreadsheet. If we use the graph to estimate prices that
are not in whole dollar amounts $9.50 might be a good choice. Remember that we want to
look at the curve and evaluate the highest point and not necessarily the highest point from
the prices we used in the plot.
We can solve this problem using calculus by combining the information into a single
equation for profits.
Profits = Revenue – Costs
Profits = P*(48000-3600*P) – 5.83*(48000-3600*P)
(5.83 is the unit cost)
Differentiating:
dProfits/dP = 48000-7200P+20988 = 0
Which gives an optimal price to set of $9.58 when you solve for P.
5. Assume the demand curve slope is again -4,000. What is the new profit maximizing price when
fixed costs are increased by 100 percent to $ 24,000? What is the new profit level? Explain this
change.
The profit-maximizing price is again $9.00. The new profit level is $14,040.00. The fact that
the profit-maximizing price did not change is not surprising. Doubling the fixed cost will
not change the decision. The increase in the fixed cost decreases the profit level uniformly
at any given price, unlike a change in the unit cost which would penalize the profit level for
every additional pie produced and sold. Changes in the fixed cost simply shift the profit
curve up or down with no change in the shape, so the highest point on the curve is still the
same. Notice, the new profit level is $12,000 less than the original profit-level, which is the
amount of the fixed cost increase!
The only time fixed costs would affect an output decision would be if profits for the best
price became negative (the entire profit curve fell below zero). At this point, if shutting
down production will save the fixed costs then shutting down is appropriate.