Introduction to Quantitative Economics by kmb15358

VIEWS: 0 PAGES: 5

• pg 1
```									                                               Topic 10

Integration Solutions

1. A profit maximising firm has MR  34  3Q and MC  Q  10Q  26 .
2

i) How much will the profit maximising firm produce (i.e. where MR = MC)?

Step 1: set MR=MC and find output that maximises profit, q*
Q 2  10Q  26  34  3Q
Q 2  7Q  8  0

 b  b 2  4a c 
Q
2a 
Solve the quadratic for value of Q using formula                                          :

a=1, b=-7, c=-8

7  49  41 8 7  9 so
Q                     
21           2
Q  1 (inadmissible) or Q  8

Thus 8 units produced by profit max firm

ii)     Find expressions for total revenue TR and total Cost TC, and hence profits.

Integrate MR and MC to find TR & TC, and thus profits

  TR  TC
TR   MR.dQ   34  3Q dQ  34Q  Q 2  c
3
2
In this case, the constant of integration c  0 , since the firm makes no revenue when Q=0
3 2
So   TR  34Q          Q
2
        1

TC   MC.dQ   Q 2  10Q  26 dQ  Q 3  5Q 2  26Q  F
3
F = the constant of integration = Fixed Costs

1
 = TR - TC
3         1
  34Q  Q 2  Q 3  5Q 2  26Q  F
2         3

1         7
   Q 3  Q 2  8Q  F
3         2

iii) What level of profits will the firm make when producing the profit maximising level
of output (found in part (i))

substitute in q* to TR and TC to get profit max values when producing q*

Substituting in Q  8 for profit max.

          8  8  88  170 2  224  64  117 1  F
1 3 7 2
3      2                  3                  3

iv) Find the level of Fixed Costs at which the firm will make zero profits.
Set profit =0 (thus TR – TC = 0), & solve for F

Setting     0 , gives               1
0  117  F
3
1
Thus, value of F at  = 0 is   F  117
3

2. A firm which has NO fixed costs has MC and MR given as follows:

MC=2Q2 – 6Q + 6;

MR = 22 – 2Q;

Find total profit for profit maximising firm when MR=MC?

Solution:
1) Find profit max output Q where MR = MC

2
22 – 2Q = 2Q2 – 6Q + 6
gives Q2 – 2Q – 8 = 0

Solve quadratic for Q, by using formula, or

(Q - 4)(Q + 8) = 0 so Q = +4 or Q =-2

so Q = +4     (since Q=-2 inadmissable)

2) Find TR and TC
TR( Q )   22  2Q dQ
TR( Q )  22 dQ  2 QdQ

TR( Q )  22Q  Q 2  c
TR = c when Q=0; but TR = 0 when Q = 0; so therefore c = 0

so TR = 22Q – Q2

MC = f (Q) = 2Q2 – 6Q + 6

TC ( Q )   2Q 2  6Q  6 dQ 
TC ( Q )  2 Q 2dQ  6 QdQ 6  dQ
2 3
TC ( Q )      Q  3Q 2  6Q  F
3
2 3
F = Fixed Cost = 0        (from question) so…. TC ( Q )      Q  3Q 2  6Q
3

3. Find profit = TR-TC, by substituting in value of q* when MR = MC

Profit = TR – TC
TR if q*=4:            22(4) - 42 = 88-16 = 72
2
TC if q* =4:             /3 (4)3 – 3(4)2 + 6(4)
= 2/3(64) – 48 + 24
= 182/3
so total profit when producing at MR=MC at q*=4 is

TR – TC = 72 - 182/3 = 53 1/3

3.    The inverse demand and supply functions for a good are, respectively:

pD  100  1 q            and   pS  20  q
2

3
(a)    Find the market equilibrium values of p and q.

For market equilibrium, pD  100  2 q  pS  20  q
1

So 120 = 3/2 q

Hence 3q = 240 and so q = 80

Since q = 80 and p = 100 – ½ (80) = 60

Equilibrium q = 80 and p = 60

(b) Sketch the diagram and highlight Consumer and Producer Surplus at equilibrium

Demand curve: pD  100  1 q so intercepts: at q = 0, p = 100 so (0,100) and at p =
2
0 , q = 50 so (50, 0)

Supply curve: pS  20  q , so intercept: at q = 0, p = - 20

P                          CS S
100              14

p*=60

PS
D
0
q* = 80       50
-20  Consumer Surplus
Q

(c) Find the consumers’ surplus when the market is in equilibrium.

Difference between value to consumers and to the market…. Area above price line and under
Demand curve

4
CS   Dq dq  p * q *
q*

0

80
Consumer surplus (CS)                  (100  2 q )dq  pq
1

0

 100q  4 q 2 
1            100q  4 q 2 
1             80  60  8000  1600  4800  1600
             q 80                q 0

d) Find the producers’ surplus when the market is in equilibrium

Producer Surplus
Difference between market value and total cost to producers… area below price line and
above Supply curve

PS  p * q *   S q .dq
q*

0

PS  60 80            20  q dQ
80

0


PS  4800   20 q  1 q 2
2                80
0

PS  4800   20 (80 )  1 (6400 )   20 0  1 0
2                       2

PS  4800  (1600  3200)  4800  1600  3200  3200

(e) What is the measure of Total Surplus?

Total Surplus = CS + PS = 1600 + 3200 = 4800

5

```
To top