Marginal analysis The pricedemand equation for a GPS device

Marginal analysis. The price-demand equation for a GPS device is p(x) = 1,000e-0.02x where x is the monthly demand and p is the price in dollars. Find the production level and price per unit that produce the maximum revenue. What is the maximum revenue?

Solution

p (x) = 1000 exp(-.02*x) Hence total revenue = x*p(x) = 1000x(exp -.02X) for maximum revenue, differentiation of revenue expression at maximum gives value 0. Hence 1000/exp(x/50) - (20*x)/exp(x/50) = 0. Hence x = 50. hence, p = 1000 exp(-.02*x) = $367.88 revenue = 1000x(exp -.02X) = $18393.97 Answer: production level = 50 units price = $367.88 Max revenue = $18393.97