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# ECONOMIC OPTIMIZATION

## Using Marginal Analysis.

Economic Optimization
Effective managerial decision revolves around economic optimization that refers to the best solution to the problem. Among many alternative courses of action, the best decision is consistent with managerial objectives.

Economic Optimization
Optimal Decision the decision that produces a result consistent with the managerial objectives is the optimal decision. The objective is to Maximize the value of the firm n Value = profit/(1 + i)t =TR-TC/(1+i)t t=1

Economic Optimization
As far as value of the firm is concerned it involves the analysis of TR, TC and the discount factor. For optimal decisions integration among marketing, production, and financial aspects is a must.

Marginal Analysis
This is the basic decision making method to identify the optimal location. Marginal Analysis (MA) is the process of considering small changes in a decision and to determine whether the change will improve the ultimate objective.

## Marginal Analysis and Firm

Revenue Relations TR = f (Q) TR = P * Q Hence there is a relationship between TR and Price If the price is \$ 4 regardless of Q sold the relationship can be written as: TR = \$4 * Q

## Marginal Analysis and Firm

With the fall in P Q is expected to rise and the firm can estimate the linear demand function of the form: P = a+bQ If P 1= 20 Q1 = 15 P2 = 18 Q2 = 20

## Total Revenue Relations and Price

Two points on the firms linear demand curve are identified. By solving two unknowns, a and b, it is possible to identify the firms linear demand function: 20 = a + b(15) Minus 18 = a + b(20) ------------------------------2 = - 5b and b = - 2.5

## Total Revenue Relations and Price

By substituting if b = -2.5 20 = a -2.5(15) 20 = a - 37.5 a = 57.5 P = 57.5 2.5Q

## Total Revenue Relations and Price

The functional relationship between TR and P is: TR = P * Q = (57.5 2.5Q) *Q = 57.5 2.5Q2

Revenue Maximization
Revenue maximization occurs at the point of greatest total revenue Set the MR = 0 and solve for Q MR =0 57.5 2.5Q = 0 Q = 11.5 TR = 57.5Q 2.5Q2 = 57.5(11.5) 2.5(11.5)2

Revenue Maximization
Graph

Revenue maximization
= 661.25 - 330.625 TR = \$330.625 If fewer than 11.5 units are sold TR can be increased by expanding output. If more units are sold TR will fall.

## Cost Relations of the Firm

Short-Run Long -RUN In the short-run total costs are comprised of Fixed Costs and variable Costs TC = FC + VC

## Cost Relations of the Firm

If FC = \$ 8 VC = \$4Q + 0.5Q2 TC = 8 + 4Q + 0.5Q2 VC rises with Q.

## Marginal and Average Cost of the Firm

Average Cost: Cost per unit of output TC/Q Marginal Cost: Change in total cost following a one unit change in output TC/ Q

## Marginal and Average Cost of the Firm

Graph of TC page 41

## Marginal and Average Cost of the Firm

1 2 3 AC Falls when MC < AC AC rises when MC > AC AT minimum AC MC = AC

## Average Cost Minimization

Given: TC = 8 + 4Q + 0.5Q2 MC = TC/ Q = 4 + 1Q AC = TC /Q = 8/Q + 4 + 0.5Q For Minimum AC set MC = AC

## Average Cost Minimization

4 + 1Q = 8/Q + 4 + 0.5Q 0.5Q = 8/Q Q2 = 8/0.5 Q = /16 (under root) Q = 4 minimum AC at Q= 4 If Q rises from 4 Ac rises

Profit Relations
Total and Marginal Profit

## Marginal Analysis and Firm

Relationship between Total profit, marginal profit and average profit 1. TP is maximized when MP is zero 2. AP rises when MP is greater than AP. 3. At maximum AP MP is equal to that

## Marginal Analysis and Firm

Inflection Point Point of maximum or minimum slope. Derivates and Marginal Analysis in Decision Making The terms derivates and marginals are interchangeable. Maximization of minimization of a function occurs when its derivative or marginal is zero.

## Derivates and Marginal Analysis in Decision Making

For Maximization or minimization follow the steps: 1. Take the first derivate of the function 2. Equate it to zero 3. Take the second derivate 4 For maximum the second derivate is negative and for minimum it is positive.

## Derivates and Marginal Analysis in Decision Making

For example if the profit function is: = 220 + 20Q 2Q2 Marginal profit M = 20 -4Q (1) First derivate Setting it to zero 20-4Q=0, Q = 5 Take the second derivate of (1) -4 As its is less than 0 profit at Q 5 is maximum

Numerical Examples
Tprofit = -3000 2400Q + 350Q2 - 8.33Q3 MR = 2400 + 700Q 25Q2 Equate to 0 Formula a = 25, b = 700, and c = 2400 X1 = 4 X2 24

Example
Use of Marginal's to maximize the Difference between Two Function TR = 41.5Q 1.1 Q2 TC = 150 + 10Q 0.5Q2 + 0.02Q3 Tprofit = TR-TC TP+ -150 + 31.5Q 0.6Q2 0.02Q3 Marginal profit = = 41.5 1.2 Q 0.06Q2

Example
MC = 10 1.0Q +0.06Q2 MR = 41.5 2.0 MR = MC = 31.5 1.2Q -0.06Q2 = 0 Solving Q1 = -35 and Q2 =15
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