The research department in a company that manufactures watches established the following
- p(x) =50-(1.2t)x
- R(X)=X.p^(-1)(X) = X.Q(X),
where x = p^(-1)(X) = Q(X) is the price per watch in thousands of cents when there is demand X = p(x) , and C(X) and R(X) are the cost and revenue respectively in thousands of shillings, and where t is an unknown constant.
1) Find the output that will produce maximum revenue, where the maximum values of revenue is achieved at q = q1.
2) Find the output that will produce maximum profit.
3) Determine the break even points.
The total revenue R(q) made by the company when producing q items is defined as being equal to the price x per item when X = p(x) have been demanded (the inverse demand function x = p^(-1)(X)) multiplied by the number demanded (and produced to according to demand) X. That is
R(X) = X.p^(-1)(X)
p(x) = 50 - (1.2t)x
p^(-1)(X) = 50/(1.2t) - 1/(1.2t)X
where t is an unknown constant of the system, then
R(X) = 50/(1.2t)X - 1/(1.2t)X^2
1) The maximum revenue achievable is the value of this curve (which is an upside-down parabola) at its peak. The peak will be positioned at some particular number of items produced X = X1. We can use calculus to find this, by differentiating the curve with respect to X (giving the gradient curve) and setting the result to zero (the gradient is zero at the curve's peak, where it is flat). The derivative of the revenue curve with respect to X can be found to be
dR(X)/dX = 50/(1.2t) - 2/(1.2t)X (multiply by the current power of X and then reduce the power by 1)
Setting this to zero, the point at which maximum revenue is achieved q = q1 is given by
50/(1.2t) - 2/(1.2t)X1 = 0
X1 = 50/2 = 25
Therefore, to achieve maximum revenue the firm should produce an output of X1 = 25 items.
2) The amount of output X that will produce maximum profit is not necessarily the same as that that produces maximum revenue (calculated in 1) ), as cost of production is now taken into account.
The total profit curve S(X) is defined to be the total revenue R(X) minus the total cost C(X).
S(X) = R(X) - C(X) = 50/(1.2t)X - 1/(1.2t)X^2 - (160 + 10X) = - 1/(1.2t)X^2 + (50/(1.2t) - 10)X - 160
The number of items X2 that the firm need to produce in order to achieve maximum profit is found in a similar way to that needed to achieve maximum revenue (calculated in 1) ). We differentiate the profit curve S(X) with respect to X and set to zero:
dS(X)/dX = -2/(1.2t)X + 50/(1.2t) - 10
This implies that the output X2 required to achieve maximum profit satisfies
-2/(1.2t)X2 + 50/(1.2t) - 10 = 0
that is, that
X2 = 25 - 5(1.2t) = 25 - 6t
If the number of items produced X2 is positive then we must have that t < 25/6.
3) The break even point is the point where revenue and costs are equal. Given the definition of the profit curve, this is the boundary between positive profit (useful profit) and negative profit (technically loss). This is where the profit curve S(X) crosses the X axis. To determine these points we need to factorise the profit curve and find its roots.
It helps to clarify your terminology when endeavoring to work calculations in economics. So we'll start you on your way by defining some of the terms you need to work with.
Price demand is properly called price elasticity of demand (or price sensitivity of demand). It numerically and graphically describes the relationship between price asked and quantity produced. Price demand is represented as:
- Price elasticity of demand = % change in quantity of demand / % change in price offered
Cost is the total cost of production, which includes fixed costs, like capital and facilities, and variable costs, like labor and raw materials.
Revenue is the total income from the total amount of sales of the product made. Revenue does not factor in the costs of production. In this way, revenue is different from profit since profit does exclude total cost of production.
Maximum revenue and maximum profit are different in that since maximum revenue dose not take into account cost of production, owner's profit may be minimized, while maximum profit depends upon decreasing costs, while prices may be increased. maximum profits occur when marginal revenue equals marginal cost as the value of marginal cost increases.
Maximum revenue is found as a derivative function that produces a parabola with the vertical axis R (revenue) and the horizontal axis p (price). Maximum profit is shown on a graph with vertical axis Profits (P) and horizontal axis price (p).
The break even point in break-even analysis shows the relationship between costs (fixed and variable) and revenue (profit that does not exclude costs). It analyzes the point at which revenue and costs are equal: revenue equals costs of production. This is the point that divides breaking even in revenues from costs exceeding revenue. The margin of safety determined by break-even analysis is the margin at which revenue can fall before dropping below costs so that costs exceed revenue and exceed the break-even point.