3.3 Marginal functions

(Section 4.3 of Jacques)


        Calculate marginal revenue and marginal cost

        Calculate marginal product of labour

        Study the law of diminishing returns

        Partial derivatives



3.3.1 Revenue and cost



Total revenue is

TR = P Q

where P is the price of a good and Q is quantity demanded

Given a demand function, e.g.

P = 100 - 2Q

we can write TR in terms of Q:

TR = PQ  = 100Q - 2Q2.


In order to study how TR reacts to a small change in Q, we look at the derivative of TR with respect to Q.


Marginal revenue MR is the derivative of total revenue with respect to demand. That is,





In the example above, we had

TR = 100Q - 2Q2.

Differentiating gives

MR = 100 - 4Q.

This formula allows us to calculate MR at any value of Q.



In some books, MR is defined as


 the change in the value of TR brought about by a one unit increase in the value of demand Q.


This definition approximates the slope of the graph of TR (i.e. the tangent slope) by the slope of a nearby line which cuts the curve at two points. However this approximation is not always accurate.



The definition of differentiation tells us that



So if DQ is small, then

.      (*)


In particular, if DQ = 1, this gives the non-calculus version of MR:



If we read equation (*) from right to left, we see that we can use MR to predict the change in TR brought about by a small change in demand Q:



change in total revenue ~ marginal revenue X change in demand




Given the demand function P = 60 - Q, calculate TR and then

(1)                find MR in terms of Q, and evaluate MR at Q = 50.

(2)                Calculate TR at Q = 50 and at Q = 51 and hence confirm that the one-unit-increase approach gives a reasonable approximation to the exact value of MR.


Solution: We have

TR = PQ = (60 - Q)Q = 60Q - Q2.


(1)                Differentiating this expression gives


When Q = 50, this gives

MR = 60 - 2(50) = -40.


(2)                When Q = 50, TR = 60(50) - (50)2 = 500.

When Q = 51, TR = 60(51) - (51)2 = 459.


Then using the approximation formula


we get

MR ~ 459 - 500 = -41,


which is a good approximation to the actual value of -40.



The total revenue of a good is

TR = 100Q - Q2.

Write down an expression for marginal revenue.

If the current demand is Q=60, estimate the change in TR brought about by a 2 unit increase in Q.



Differentiating gives

MR = 100 - 2Q.

At Q = 60, this gives

MR = -20



Now we use the formula


This predicts that the change in TR will be


D(TR) ~ -20 2 = -40.


A two-unit increase in demand leads to a 40 unit decrease in total revenue.






The total revenue function for a good is given by

TR = 1000Q - 4Q2.

Write down an expression for the marginal revenue.

If demand is currently 30, estimate the change in TR due to a

(1)                three-unit increase in Q

(2)                two-unit decrease in Q





What we have done here for total revenue can be done for any economic function. For example, given the total cost function TC, we define the marginal cost MC by


marginal cost is the derivative of total cost

with respect to output Q


Just as we did for revenue, we can derive the approximation

change in total cost ~ marginal cost X change in output



Find the marginal cost given the average cost function


If the current output is 15, estimate the effect on TC of a three-unit decrease in Q.


(Remember that .)




3.3.2 Production


We studied production functions of the form

Q = f(K,L),

i.e. the production function f determines output Q in terms of labour L and capital K.

Here, we will consider the case where K does not change, so that Q depends only on L:

Q = f(L).





Then the derivative of Q with respect to L tells us how Q reacts to changes in L. This derivative is called marginal product of labour:



marginal product of labour is the derivative of output

with respect to labour.

As usual,

change in output ~ marginal product of labour X change in labour




If the production function is

Q = 300L - 4L,

calculate the value of MPL when

(a)               L = 1

(b)              L = 9

(c)               L = 100

(d)              L = 2500.


Comment on the results.





We have

Q = 300 L1/2 - 4L



(a)                When L = 1,

(b)               When L = 9,

(c)               When L = 100,

(d)               When L = 2500,


Thus we see that MPL is decreasing, and eventually becomes negative.


Thus the size of the increase in output brought about by a one-unit increase in labour L is decreasing.


Worse still, at the stage when MPL < 0, an increase in labour L would cause a decrease in output Q.


This is what we would expect in the real world; problems such as overcrowding on the workshop floor and administrative overheads decrease the efficacy of increasing the size of the workforce.


This example illustrates the law of diminishing marginal productivity (also known as the law of diminishing returns).


"The increase in output due to a one-unit increase in labour will eventually decline"

Translating this into maths, we obtain


the marginal product of labour MPL will eventually be a decreasing function.


That is, for sufficiently large values of L, we should have


Since we know that MPL is the derivative of output Q, we can write this as 


To show that a production function satisfies the law of diminishing returns, we must show that

holds for all sufficiently large values of L.



Show that the Cobb-Douglas production function

Q = 5L1/2K1/2,

with K = 100, satisfies the law of diminishing returns.



3.3.3 Partial Derivatives

(cf. Jacques, Section 5.2)


Consider the Cobb-Douglas function above

Q = 5L1/2K1/2.

This depends on both L and K.


If we take K to have a fixed value and allow L to vary, we can analyse changes in Q by looking at the derivative dQ/dL.


Similarly, If we take L to have a fixed value and allow K to vary, we can analyse changes in Q by looking at the derivative dQ/dK.

To emphasise the fact that Q depends on both variables, we call these derivatives partial derivatives and use a different notation:


= derivative of Q with respect to L, holding K constant.


= derivative of Q with respect to K, holding L constant.






(i) Calculate  and  for the Cobb-Douglas function

Q = 5L2K3/2.


(ii) If the current values are L=10, K = 12, which would bring about a bigger increase in Q: increasing L by one unit, or increasing K by one unit?





3.3.4Consumption and Savings


Y = national income, C = consumption, S = savings  connected by   Y= C + S


To analyse changes in C and S due to changes in Y use marginal propensity to consume MPC and marginal propensity to save MPS.



These agree with the special case of linear consumption and savings curves given earlier.


We can find MPC given MPS (or vice versa): we know that


Y = C + S


Ddifferentiate both sides with respect to Y:


1 = MPC + MPS

Example:  Given

C = 0.01Y2+0.2Y + 50


Find MPC and MPS when Y=30.


When Y=30,

MPC = 0.02(30) + 0.2 = 0.8


To find MPS: MPC + MPS = 1 =>

MPS + 0.8 = 1 => MPS = 0.2
















3.4 Elasticity


Elasticity deals with the sensitivity of demand (supply) to changes in price. Remember that TR = PQ and in our models, P decreases => Q increases.


Change in TR depends on percentage changes in P and Q.


Demand is said to be  elastic, if the percentage change in Q is higher than the percentage change in P.

Then we can increase revenue by lowering price.

Demand is said to be inelastic, if the percentage change in Q is lower than the percentage change in P.

Then we can increase revenue by raising price.

Price Elasticity of Demand


E = -   (% change in demand)       

         (% change in price)


Minus sign is a convention to ensure E > 0.


Demand is

        inelastic if E < 1

        unit elastic if E = 1

        elastic if E > 1     


Change in P = DP, change in Q = DQ

% change in P  = 100 (DP/P)

% change in Q  = 100 (DQ/Q)






Mathematical definition of elasticity


Elasticity at P:

Let the arc PQ shrink down to P by moving Q closer to P

Thus DP and DQ shrink to zero and             


Price elasticity of demand at a point is given by               




Demand function is usually P = f (Q);

To find , we use





Example: Given the demand function

P = - Q2 - 4Q + 96,

(i) Find price elasticity of demand when      P = 51.

(ii) If this price rises by 2 %, find the corresponding  percentage rise in demand.


First, find Q corresponding to P = 51

- Q2 - 4Q + 96 = 51

roots: Q = - 9 (reject), Q = 5.


Next, we calculate .

 =  -2Q - 4 =>


Q = 5 gives  = -1/14




Answer (i): When P = 51,  E = 0.73


(ii) recall that:


We found E = 0.73, we are told  (% change in P) = 2. Thus



=> % change in Q = - 1.46 %

Q decreases by 1.46 %



Price elasticity of supply