I have a Marginal analysis assignment I needed help with in my Business Calculus
ID: 1201586 • Letter: I
Question
I have a Marginal analysis assignment I needed help with in my Business Calculus course. We are asked to pick a a product to sell as if we started our own business, and I chose to sell sunglasses at the price of $10 per unit. Total fixed costs are $5,750 per month
The first requirement is that we construct a linear cost function, C(x), for the product. We can afford $200,000 in total costs per month, and I have to figure out how many units I am able to produce without going over that budget.
A.) I have to let x represent the quantity of units of the product demanded each month and let "p" represent the price per unit which I sell the product in dollars. The price-demand equation will be x=f(p) = 12500-50pp. Then I have to construct the revenue function, r(x), for the product. Without consideration for the budget, I need to determine the feasible range of units demanded per month. (smallest number of products I could feasibly sell and the largest number of products I could feasibly sell in a month with no limit on the budget.) Then, I need to find the profit function, p(x). Next I need to determine the break-even points and interpret the result.
B.) If my company is currently producing 4,000 units per month, I need to determine the marginal cost and marginal revenue at a production level of 4,000 units and interpret the results. I need to use the price-demand equation to find the price at which I would sell each unit when 4,000 units are being produced each month. Then I need to write a function for the elasticity of demand, e(p) and determine whether the demand is elastic, inelastic, or has unit elasticity at this price, and discuss how increasing or decreasing the price would affect the revenue.
C.)I then need to determine if I should increase or decrease production based on the analysis done so far, and determine the optimal production level that will maximize profits and explain how. What is the maximum profit? At what price should I sell each unit so that I can maximize profit? How did this price change affect the revenue, and draw conclusions about my business's optimal operations.
Explanation / Answer
(A)
Total cost, TC = Fixed cost + Variable cost
TC = 5,750 + 10X (TC <= 200,000)
Demand: X = 12,500 - 50P
50P = 12,500 - X
P = 250 - 0.02X
Total revenue, TR = P. X = 250X - 0.02X2
Profit, Z = TR - TC = 250X - 0.02X2 - 5,750 - 10X
Z = 240X - 0.02X2 - 5,750
In break-even, Z = 0
240X - 0.02X2 - 5,750 = 0
This is a quadratic equation, solving which (using online solver) we get:
X = 24, or X = 11,976
This means that profits will be exactly zero if either 24 or 11,976 units are produced.
(B)
MC = dTC / dX = 10
MR = dTR / dX = 250 - 0.04X
When X = 4,000, MR = 250 - (0.04 x 4,000) = 250 - 160 = 90
This means that revenue earned by one additional unit of output is higher than the cost incurred in producing that additional unit, by $80 (= $90 - $10).
When X = 4,000, P = 250 - (0.02 x 4,000) = 250 - 80 = 170
Elasticity of demand, eP = (dX/ dP) x (P / X) = - 50 x (170 / 4,000) = - 2.125
Since absolute value of eP is higher than 1, demand is elastic. With elastic demand, a(n) decrease (increase) in price will increase (decrease) total revenue.
(C)
MR > MC, signifying marginal profit is positive. So I should gain by increasing production beyond 4,000 units.
Profit is maximized by equating MR with MC:
250 - 0.04X = 10
0.04X = 240
X = 6,000
When X = 6,000, P = 250 - (0.02 x 6,000) = 250 - 120 = 130
Maximum Profit = TR - TC = (P. X) - 5,750 - 10X = (130 x 6,000) - 5,750 - (10 x 6,000) = 780,000 - 5,750 - 60,000
= 714,250
Revenue at profit-maximizing output = 780,000
This price (= 130) is lower than price at Q = 4,000 (= 170). So, lower price has increased revenue.
Concluded:
To maximize revenue, price should be kept low as demand is elastic. To maximize profits, the MR = MC rule should be followed. At current production of 4,000 units the firm is operating at a sub-optimal level.
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