Question

1. The demand function for the Baye Firm is: P = 100 – 0.5 Q

The firm’s total cost function is: 1500 – 10 Q + 0.5Q2

(a) Is this a perfectly competitive firm? (5 Points)

(b) Find the output level and price at which the firm’s total revenue is maximized. (10 Points)

(c) Find the output level and price at which the firm’s total profit is maximized. (10 Points)

(d) Is demand elastic, unitary elastic, or inelastic at the output level where total revenue
is maximized? (10 Points)

(e) Is demand elastic, unitary elastic, or inelastic at the output level where total profit
is maximized? (10 Points)

(f) What is the value of the firm’s total fixed cost at the output level where total revenue
is maximized? What about at the output level where total profit is maximized? (10 Points)

(g) What is the value of the firm’s average variable cost at the output level where total
revenue is maximized? What about at the output level where total profit is
maximized? (10 Points)

Answer 1

Part a) We are given the demand curve. Under perfect competition, the demand curve or the average revenue curve is constant at a particular price for all values of output. But the given curve shows the relation that as Q goes on increasing, P should also fall, which means the the AR curve is downwards sloping, which is the case under imperfect competitions such as monopoly or monopolistic competitions.

For example, let us assume a few values for the Q to find out P and graph the curve,:

Q	P
0	100-0.5x0=	100
10	100-0.5x10=	95
20	100-0.5x20=	90
30	100-0.5x30=	85
40	100-0.5x40=	80
50	100-0.5x50=	75

Part B) Demand curve is also the AR curve. we need to calculate the TR and MR curves from the same:

Now TR is maximized at the point where MR or the slope of TR = 0 and the slope of MR is decreasing or

So

Now checking if this is a maximum point, we need to differentiate the MR function

So at Q = 100 TR is maximum, at this the price will be

Part C) We are given the TC curve and we need to find the MC curve from it

Now point of Profit maximization is where MR=MC

At this level the Average revenue or price is as follows:

Part D & E) Using the Point elasticity method, we will find out the elasticity of demand at the point of revenue maximization:

Let's assume various values of Q to find out the Prices and respective price elasticities:

Q	P		Point elasticity	Absolute value of Elasticity
10	100-0.5x10=	95
20	100-0.5x20=	90	-19	19
30	100-0.5x30=	85	-9	9
40	100-0.5x40=	80	-5.666666667	5.6666667
50	100-0.5x50=	75	-4	4
60	100-0.5x60=	70	-3	3
70	100-0.5x70=	65	-2.333333333	2.3333333
80	100-0.5x80=	60	-1.857142857	1.8571429
90	100-0.5x90=	55	-1.5	1.5
100	100-0.5x100=	50	-1.222222222	1.2222222
110	100-0.5x110=	45	-1	1
120	100-0.5x120=	40	-0.818181818	0.8181818
130	100-0.5x130=	35	-0.666666667	0.6666667
140	100-0.5x140=	30	-0.538461538	0.5384615
150	100-0.5x150=	25	-0.428571429	0.4285714
160	100-0.5x160=	20	-0.333333333	0.3333333
170	100-0.5x170=	15	-0.25	0.25
180	100-0.5x180=	10	-0.176470588	0.1764706
190	100-0.5x190=	5	-0.111111111	0.1111111
200	100-0.5x200=	0	-0.052631579	0.0526316

The graph of the above table is as follows:

The demand is said to be inelastic when the absolute value of demand is <1 and elastic when it is > 1 when it is = 1 then the demand is unitary elastic

We can see from the above table, that the absolute value = 1 at Q =110, for all values of Q<110, the demand is elastic.

The TR is maximum when Q = 100 as per the above table at Q=100, the absolute value of elasticity > 1 so demand is elastic

Profit is maximum at Q = 55 and as the absolute value of elasticity > 1 so demand is elastic

Part F) For this lets look at the TC function give to us:

Fixed cost is the constant value in the function as the fixed cost remains constant for all levels of production. So at Q = or Q = 100, the Fixed cost = 1500

Part G) For this lets look at the TC function give to us:

TC = Fixed cost +Variable cost

The part of the Function related to the variable cost is

AVC when TR is maximum:

AVC when Total profit is maximum: