How many hours of tutoring should be used? An example of production and cost analysis

Production Relationships

The relationship between the input and output is presented in the table below. For those wanting a visual representation of the analysis, you should check out the graphs section at the end of this section.  

Production Relationships
Tammy's Tutoring

How do you read the table?  The first column contains information on the number of units of the input, in this case hours of studying. In other settings you would think of the figures as representing the number of workers, the number of hours, the number of weeks, or any other unit for labor input.  In the second column we have output, the grades that would be achieved in ECN.  If our analysis were of the automobile industry, these would be the number of automobiles produced, while if we were talking about a hospital, output may be the number of patients treated / cured.

In our example, when two hours are spent studying with the tutor, the ECN grade is 22.  If Tammy provides five hours of tutorial help, the grade is 54. This confirms what you would have expected, more inputs produce more output. If you study more, your grade improves. The graph of the relationship between the level of output (grades) and the number of inputs (hours) is the total product curve and it is positively sloped.  You should check out the g= to see what this relationship looks like because it is the graphical version that most of your friends will be seeing in their classes.

In the third column you will find the average product that measures the average grade produced by each hour.  When Tammy provides five hours of tutoring time, the grade would be 54 that means each hour on average produces 10.8 points [54/5].  This figure is computed by dividing the units of output by the units of input.  This figure of average productivity is what the US government measures with its published labor productivity figures.

When the hours spent studying expands from 1 to 2, output increases from 10 points to 22 points.  If Tammy expanded tutorial time from 2 to 3 hours, grades would increase from 22 to 36.  As we will see later, the relationship between the change in inputs and the change in output - the marginal product of labor - is very important in our analysis of profit maximizing behavior.  In this example, the marginal product of the 2nd hour would be 12.  [MP = Doutput/Dinput = (22-10)/(2-1) - 12/1 = 12. 

Before leaving our simple example, you should note one feature of the Marginal Product column. Once we expand beyond the third hour the marginal product of the tutor begins to decline.  It turns out that this is a general feature of production relationships, if you keep adding units of an input to an operation with a fixed facility, eventually the additional output produced by the additional input will begin to decline.  Once this happens we have diminishing marginal product.  If you happen to be looking at the graphs, you will find we are talking about a marginal product graph with a negative slope. Once again this should not surprise you. As you study longer your grade will improve, but at some point each additional hour will improve your grade by a smaller amount. In this instance we would say that you are experiencing diminishing returns to your study. You could work so long through the night that your grade actually falls - a negative marginal product [so don't use the cramming approach!]. 

Tutoring Example: The graphs

Total Product

Total (physical) product: amount of output obtained from given input. The positive slope indicates that output increases as the input increases.

Marginal and Average Product

Average (physical) product: amount of output obtained per unit of input.  This is calculated by dividing total output by number of units of inputs.

Marginal (physical) product: additional output obtained by 1 more unit of input. You will note the marginal product curve has a negative slope after 3 units of input are used that indicates that if you keep adding labor to an operation with a fixed facility, eventually the additional output produced by the additional labor will begin to decline.  When this happens we enter a situation in which we have diminishing marginal product. One of the additional features of the graph is that the MP curve cuts through the maximum point on the AP curve.

Cost Relationships

Let's return to Tammy who hires tutors that cost her $5.00 an hour. RIU also charges Tammy a licensing fee of $3 so she can supply the services to the university's students. The $3.00 would be considered a fixed cost because it will not change as the number of hours changes, while the labor cost would be considered to be the variable cost. Now let's look at the cost of 3 hours of tutoring help that will produce a grade of 36. The fixed cost, the cost of the license from the university, is $3.00 while the variable cost, which is simply the $5 per hour for 3 hours, is $15. Total cost is $18, which is just the addition of fixed and variable costs. The complete relationship between grades and the cost of tutorial time spent with Tammy's Tutors appears in the table below.

Cost Relationships
Tammy's Tutoring

Inputs Hours	Total Product (Grades)	Variable Cost	Fixed Cost	Total Cost	Average Cost	Marginal Cost
1	10	$5	$3	$8	$.80
2	22	$10	$3	$13	$.59	$.42
3	36	$15	$3	$18	$.50	$.36
4	46	$20	$3	$23	$.50	$.5
5	54	$25	$3	$28	$.52	$.63
6	60	$30	$3	$33	$.55	$.83

Tutoring Example: The graphs

Total Cost: all costs of producing points

Fixed Cost: costs of production that do not vary with output

Variable Cost: costs of production that vary with output

Average Cost: costs per unit of output.  Diminishing average product translates into increasing average cost.  As an example, consider the situation where the input labor is paid $8 an hour and that the average productivity of a worker is 4 units of output per hour of labor. Since each unit of output can be produced in .25 hours, the cost per unit of output will be $2 [.25*$8].  If the average productivity of a worker is 2 units of output per hour of labor, then each unit of output can be produced in .5 hours, the cost per unit of output will be $4 [.5*$8].

Average Variable Cost: variable costs per unit of output. Tends to look very similar to the Average Cost curve except that it lies above it.

Average Fixed Cost: fixed costs per unit of output. This curve declines with output since you are dividing ever larger outputs into the same level of costs.  Average fixed costs equals the difference between average cost and average variable.

 

Marginal Cost: additional costs of producing more output.  Diminishing marginal product translates into increasing marginal cost. As an example, consider the situation where the input labor is paid $8 an hour and that the marginal productivity of a worker is 4 units of output per additional hour of labor.  Since each unit of additional output can be produced in .25 hours, the cost per unit of additional output will be $2 [.25*$8]. If the marginal productivity of a worker is 2 units of output, then each additional unit of output can be produced in .5 hours, the cost per unit of additional output will be $4 [.5*$8].

You will find there is a relationship between average and marginal cost curves.  The marginal cost curve and average cost curves tend to be U-shaped and the marginal cost curve cuts through the average cost curves at their minimum points.

Profit Maximization

The first two columns in the table below represent the demand for grades (points) - how much students are willing to pay for the grades achieved with the tutoring help.  These two columns would be the demand schedule that would correspond to the demand curve for the grade points.  If a student receives an improvement in one's grade of 46 points, then the student will pay $7 per point while a 60 point grade improvement would generate a price of $5 a point.  Not surprisingly, the price they pay per point declines with the number of points of improvement in the grade.  With this demand information, the rest of the table is easy to fill in. The total revenue associated with a grade of 36 would be $288 [$8*36], while the average revenue per point would be $8 [$288/36].  Marginal revenue is the change in revenue generated by a one unit change in output. When the decision is made to increase output (grades) from 36 to 46, revenue increases from $288 to $322. In this case we would compute the marginal revenue of expanding from 36 to 46 units of output as (MR = ($322-$288)/(46-36) = $34/10 = $3.40).

Demand and Revenue
Tammy's Tutoring

Grade (Q)	Price (P)	Total Revenue (TR)	Average Revenue (AR)	Marginal Revenue (MR)
10	$10	$100	$10
22	$9	$198	$9	$8.17
36	$8	$288	$8	$6 .43
46	$7	$322	$7	$3.40
54	$6	$324	$6	$0.25
60	$5	$300	$5	$-4.00

What do we see in the table?  There are a few important features that are even more obvious if you happen to review the corresponding graphs in the next section. 

1. Average Revenue and Marginal Revenue both decline as output increases.  This is the result of the downward sloping demand curve.
2. Average Revenue and Price are the same. This will always be true.
3. Marginal Revenue is less than Average Revenue.  This is the result of the downward sloping demand curve.  To get students to pay for more hours to improve their grades Tammy must lower price so the additional revenue from the extra customer will be less that the price Tammy charges.  For example, for a grade of 36, Tammy received a price of $8 a point.  For a grade of 46 points, the price of grades fell to $7.  Tammy gets an additional $70 by supplying the additional 10 points, but now loses $36 on the first 36 points since the price per point dropped by $1.  The net increase in revenue of $34.00 for the 10 points gives us a marginal revenue of $3.40, which is substantially lower than the price.

To demonstrate the logic of profit maximization and the 'rules,' we need to add the revenue and cost data together, which has been done in the table below.  Below is the table containing Cost and Revenue data for Tammy, data you have already seen. 

Revenue, Cost, and Profit
Tammy's Tutoring

Grade (Q)	Price (P)	Total Revenue (TR)	Average Revenue (AR)	Marginal Revenue (MR)	Total Cost (TC)	Average Cost (VC)	Marginal Cost (MC)	Total profit	Marginal Profit
10	$10	$100	$10		$8	$0.80		$92	$0
22	$9	$198	$9	$8.17	$13	$0.59	$0.42	$185	$7.75
36	$8	$288	$8	$6.43	$18	$0.50	$0.36	$270	$6.07
46	$7	$322	$7	$3.40	$23	$0.50	$0.50	$299	$2.9
54	$6	$324	$6	$0.25	$28	$0.52	$0.63	$296	-$0.38
60	$5	$300	$5	($4.00)	$33	$0.55	$0.83	$267	-$4.83

What is new are the last two columns.  Total Profit is simply Total Revenue - Total Cost.  The profit of a grade of 36 is $270 [$288 - $18].  Marginal Profit (MP), using the now familiar concept of marginal, is defined as the change in profit associated with increased output.  The formula would be the change in profit divided by the change in output (MP = DP/DQ).  At an output of 46 points, marginal profit is 2.9.  This means that for every point improvement in the grade from 36 to 46 profit increases by $2.9 [ (299-270)/(46-36) = $29/10 = $2.90].  If you look at marginal revenue and marginal cost @ 46 points you find that MR = $3.40 and MC = $.50.  If you subtract MR - MC you have $3.40 - $.50 = $2.90.  What you have 'proved' is that Marginal Profit = Marginal Revenue - Marginal Cost. 

So how many points should be produced to maximize profit?  You can look at the profit column and see that the maximum profit occurs when output is 46.  The profit is $299, more than at any other level of output.  You could also look at the Marginal profit column and see that marginal profit is negative if you expand output to 54 points so this decision would reduce your profit.  You can use the marginal profit column to find the maximum profit output by expanding output as long as marginal profit is positive.  Finally, you can look at the MR and MC columns and simply compare them.  If MR >MC then you would expand output (keep moving down in the table), while if MR<MC then you would reduce output. 

Now let's look at the graphs. 

Tutoring Example: The graphs

The demand curve gives the firm information concerning the relationship between how much the firm can sell and the price it charges.  This is the same information you get from the first two columns in the Demand and Revenue table.  

Total Revenue

These demand data could also be used to draw the relationship between output and total revenue.  As you would expect, as output increases revenue will increase.  At some point, however, the curve slopes downward.  At this point the price drop needed to increase sales will be large enough so total sales revenue will fall.  You should not be surprised this has something to do with elasticity and you are encouraged to explore the link.  If you have a good mastery of elasticity, you should be able to convince yourself the curve slopes upward as long as demand is elastic - as long at demand is responsive to price changes.

Average Revenue

These data could also be used to draw the relationship between output and average revenue.  As you saw in the table, this curve is identical to the demand curve.  

Marginal Revenue

Finally, these data could also be used to draw the relationship between output and marginal revenue.  Here we find that as output increases revenue increases, but marginal revenue decreases.  As output is expanding, the increments to revenue will be less than the price so we find the MR curve lies below the AR curve.  [If you are not bad in math, you may have already realized MR is also equal to the slope of the TR curve.  Slope is equal to the rate of change and MR = DR/DQ, the formula for rate of change in total revenue.  This is why MR can be negative.  MR is negative when the TR curve's slope is negative.]

Profit maximization

We can now look at the profit maximization problem graphically, the method you will find in most texts. Part of the confusion students tend to have here is we can use a variety of graphs to demonstrate the profit maximizing (optimal) choice for output / price. We can look at the decision in terms of marginal concepts (cost and marginal revenue, marginal profit) or total concepts (total cost and total revenue, or total profit).  For this reason you will find 4 graphs that point out the optimal output choice. [Note: the result using marginal analysis is 3.5 while using total analysis we get 3. The reason is we have used only integer values for output and thus our results are approximations. What we know from the total is that 3 is too low (MR>MC) and 4 is too high (MR<MC)] The actual optimal value would be somewhere in between the two.

Approach #1: Total Revenue and Cost    

Maximum profit occurs when the vertical distance between the curves is greatest

Approach #2: Marginal Revenue and Marginal Cost

Maximum profit occurs when the two curves intersect.   At that level of output MR = MC.  When the MR curve is above the MC curve, you have MR > MC so the firm should expand output.

Approach #3: Total Profit

Maximum profit occurs when the Profit curve reaches its maximum.  This is the same point where the gap between revenue and cost is largest in the first diagram.

Approach #4: Marginal Profit

Maximum profit occurs when the curve crosses the axes (when it equals 0).  Since marginal profit is defined as MR - MC, then it is just the difference between the MR and MC graphs.  If marginal profit is positive, then MR > MC and the firm should continue to expand.  It will do so as long as marginal profit is > 0.