PATSON DAKA-D21040

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THE ZAMBIA CATHOLIC UNIVERSITY FACULTY OF BUSINESS, BANKING AND FINANCE DEPARTMENT OF ECONOMICS EC350: MANAGERIAL ECONIMICS AND OPTIMIZATIONTECHNIQUES ASSIGNMENT REPORT ONE NAMES : PATSON DAKA STUDENT NUMBER: D21040 LECTURER : Mr. BEN CHANSA PROGRAM : BACHELOR OF BUSINESS ADMINISTRATION DUE DATE : 1 ST SEPTEMBER, 2023

QUESTION 1- THE SCOPE OF MANAGERIAL ECONOMICS a) State the theory of the firm: The theory of the firm refers to the microeconomic approach devised in neoclassical economics that every firm operates in order to make profits. Companies ascertain the price and demand of the product in the market, and make optimum allocation of resources for increasing their net profits [ CITATION CHR20 \l 2057 ].. b) How does the theory of firm differ from short term maximization? Is the former superior to the latter? The theory of the firm and short-term profit maximization are related concepts, but they differ in their focus and objectives. Here's how they differ, and whether one is superior to the other depends on various factors and perspectives: 1. Objective: The theory of the firm is a broader economic concept that seeks to explain the fundamental reasons for a firm's existence and how it operates in the long run. It considers various aspects such as profit maximization, cost minimization, market structure, and the firm's role in the economy were us Short-term profit maximization focuses specifically on the goal of maximizing profits in the immediate future, often within a single accounting period (e.g., a quarter or a year). 2. Timeframe: The theory of the firm can apply to both the short term and the long term, as it encompasses a wide range of economic and strategic decisions that firms make over time whereas It is a short-term orientation and tends to prioritize actions that can boost profits quickly, sometimes at the expense of long-term considerations. 3. It takes into account not only profit maximization but also other factors like market competition, cost efficiency, strategic positioning, and the relationship between a firm's internal organization and external environment. Whereas in Short-term profit maximization may lead to decisions that involve cost-cutting, reducing investments in research and development, or neglecting sustainability and social responsibility in pursuit of immediate financial gains. Whether the theory of the firm is superior to short-term profit maximization depends on the perspective and goals of various stakeholders, including business owners, managers, investors, employees, and society as a whole: 1. Long-Term Sustainability: From a broader perspective, the theory of the firm is often considered superior because it recognizes that firms operate within dynamic and complex environments. Focusing solely on short-term profit maximization can lead to decisions that harm a company's long-term sustainability and reputation. 2. Stakeholder Interests: Firms that consider the theory of the firm typically take into account the interests of various stakeholders, such as employees, customers, suppliers, and the community. This approach can lead to more balanced and responsible decision- making. 3. Shareholder Perspective: However, some argue that short-term profit maximization aligns with the interests of shareholders who are primarily concerned with immediate financial returns. Shareholders may view short-term profit maximization as the superior goal if they can reinvest those profits in other opportunities to create long-term value.

In practice, the optimal approach for a firm often lies somewhere in between. Striking a balance between short-term profitability and long-term sustainability is a key challenge for businesses. The choice depends on a firm's strategic goals, competitive environment, industry, and the values of its stakeholders. It's important for firms to consider both short- term and long-term objectives to achieve sustainable success. c) How does the theory of the firm provide an integrated framework for the analysis of managerial decision making across the functional areas of business? We show that Firm theory is a microeconomic concept that firms the existence of a company and makes decisions to maximize profits. The company's profits are maximized by creating a distinction between revenue and expenditure. Strong theory influences many decisions, including resource àllocation, production processes, price volatility, and production volume. Long- term goals, such as sustainability, and short-term motivations, such as increased profitability, are sometimes separated from modern methods of solid theory. A company's revenue under its stated cost is called a business profit The expense represents the actual cost of the company out of pocket. Economic profits are calculated by deducting the firm's income from its open and hidden costs. The amount of inputs owned and used by a company in its production processes is called the expense. If the company's goal is to increase short- term profits, it may look for ways to increase revenue while reducing costs. Companies that rely on fixed assets, such as machinery, will need, however, large investments to make a profit over time. Short-term profits may be disrupted if the money is spent to invest in assets, but it can help the company to work longer. Decision- making for strong managers can be influenced by competition (not just profit). If the dispute is heated, the company will need to restructure and change its offers so that it can not only increase wages but also stay one step ahead of its competitors. As a result, long-term gains can only be increased if the balance is obtained between short-term gains and future investments [ CITATION CHR02 \l 2057 ]. QUESTION 2 – OPTIMISATION TECHNIQUES a) Define average and marginal (i) Revenue  Average Revenue (AR): Average revenue is the total revenue generated by a firm divided by the quantity of goods or services sold. It is essentially the revenue per unit sold. Mathematically, it's calculated as AR = TR / Q, where TR is the total revenue and Q is the quantity sold.  Marginal Revenue (MR): Marginal revenue is the additional revenue a firm earns by selling one more unit of a good or service. In other words, it's the change in total revenue resulting from selling one additional unit. Mathematically, it's calculated as MR = ΔTR / ΔQ, where ΔTR is the change in total revenue and ΔQ is the change in quantity sold. (ii) Product  Average Product (AP): Average product is the total output or production divided by the quantity of input used (typically labor or capital). It represents

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the average productivity of each unit of input. Mathematically, it's calculated as AP = TP / L, where TP is the total product (output) and L is the quantity of the input (e.g., labor or capital).  Marginal Product (MP): Marginal product is the additional output or production resulting from using one more unit of input while keeping other inputs constant. In other words, it's the change in total product resulting from using one additional unit of input. Mathematically, it's calculated as MP = ΔTP / ΔL, where ΔTP is the change in total product and ΔL is the change in the quantity of input. (iii)Cost  Average Cost (AC): Average cost is the total cost incurred by a firm divided by the quantity of goods or services produced. It represents the cost per unit of output. Mathematically, it's calculated as AC = TC / Q, where TC is the total cost and Q is the quantity produced.  Marginal Cost (MC): Marginal cost is the additional cost a firm incurs by producing one more unit of a good or service. In other words, it's the change in total cost resulting from producing one additional unit. Mathematically, it's calculated as MC = ΔTC / ΔQ, where ΔTC is the change in total cost and ΔQ is the change in quantity produced (iv)Profit  Average Profit (AP): Average profit is the total profit earned by a firm divided by the quantity of goods or services sold. It represents the profit per unit sold. Mathematically, it's calculated as AP = π / Q, where π is the total profit and Q is the quantity sold.  Marginal Profit (MP): Marginal profit is the additional profit a firm earns by selling one more unit of a good or service. In other words, it's the change in total profit resulting from selling one additional unit. Mathematically, it's calculated as MP = Δπ / ΔQ, where Δπ is the change in total profit and ΔQ is the change in quantity sold. b) Examine the relationship between average product and marginal product. Figure 1:The marginal product (MP) curve crosses the average product (AP) curve at the point where the average product curve is at a maximum.

 When MP > AP: This indicates that the marginal product of the last unit of input is greater than the average product. In this scenario, the average product tends to rise as additional units of input are used, which means that the production process is becoming more efficient. When MP is greater than AP, firms may want to increase the use of that input to take advantage of increasing returns to scale and improve overall efficiency.  When MP = AP: When the marginal product equals the average product, the average product is at its maximum point. This implies that the production process is operating at its peak efficiency, and additional units of input do not affect the average product. When MP equals AP, the firm is operating optimally, and there is no need to change the level of input.  When MP < AP: If the marginal product is less than the average product, it suggests that the last unit of input added contributes less than the current average. In this case, the average product tends to fall as more units of input are added, indicating diminishing returns to that input. When MP is less than AP, the firm may need to consider reducing the use of that input or exploring alternative production methods, as adding more of the input is resulting in diminishing returns. Firms often use the relationship between AP and MP to make decisions about resource allocation and input levels. They seek to maximize production efficiency while minimizing costs. c) How does a firm determine the profit maximizing output? Profit maximization is the process of finding the level of production that generates the maximum amount of profit for a business. A firm determines its profit-maximizing output level by analyzing the relationship between its total revenue (TR) and total cost (TC). The primary goal is to find the output level at which the difference between total revenue and total cost, known as profit (π), is maximized. This can be done using marginal analysis, which involves examining marginal revenue (MR) and marginal cost (MC). Here are the steps a firm typically follows to find the profit-maximizing output: 1. Calculate Marginal Revenue (MR): Marginal revenue is the additional revenue a firm earns by selling one more unit of its product. It can be calculated by finding the change in total revenue resulting from selling one additional unit: MR = ΔTR / ΔQ, where ΔTR is the change in total revenue, and ΔQ is the change in quantity sold. 2. Calculate Marginal Cost (MC): Marginal cost is the additional cost incurred by producing one more unit of the product. It can be calculated by finding the change in total cost resulting from producing one additional unit: MC = ΔTC / ΔQ, where ΔTC is the change in total cost, and ΔQ is the change in quantity produced. 3. Analyze the Marginal Relationships: Compare MR and MC. The profit-maximizing output level occurs where MR equals MC (MR = MC). This is because, at this point, the additional revenue earned from producing one more unit is exactly equal to the additional cost incurred in producing that unit.

4. Calculate Profit: Determine the level of output (Q*) at which MR equals MC. This is the profit-maximizing quantity. 5. Verify Profit Maximization: To ensure that the output level Q* indeed maximizes profit, calculate the total profit (π) at this level. Profit is calculated as π = TR - TC, where TR is total revenue, and TC is total cost. 6. Check for Profit Maximization: If the total profit at the Q* level is greater than at any other level of output, then Q* is the profit-maximizing output. If not, adjust the level of output until the maximum profit is achieved. d) (i) What is meant by the “concept of the derivative”? the "concept of the derivative" refers to the application of calculus, specifically the concept of the derivative, to analyze and understand various economic phenomena. The derivative is a fundamental concept in calculus that measures the rate of change of a function with respect to one of its independent variables. In economics, this concept is used to study how economic variables change in response to changes in other variables [ CITATION JAM201 \l 2057 ]. (ii) Why are the concept of the derivative and the use of differential calculus so important to marginal analysis? One of the most common applications of the derivative in economics is in marginal analysis. The marginal concept refers to the incremental change in a variable resulting from a one-unit change in another variable. For example, marginal cost (MC) is the derivative of the total cost (TC) function with respect to the quantity produced (Q), which measures the additional cost incurred when producing one more unit. Similarly, marginal revenue (MR) is the derivative of the total revenue (TR) function with respect to quantity, indicating the additional revenue from selling one more unit. Marginal analysis uses the derivative (or rate of change) to determine the rate at which a particular quantity is increasing or decreasing. In this section, the marginal functions that we will cover are those for the cost, average cost, revenue, and profit functions. The last topic that will be covered is the elasticity of demand. No matter which function we are dealing with, the word “marginal” indicates to us that we need to find the derivative of the function. For example, if we are asked to find the marginal cost function then we need to find the derivative of the cost function. e) (i) What is meant by the “second derivative”? The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to is increasing or decreasing. In calculus, the "second derivative" of a function represents the rate of change of the first derivative. In other words, it measures how the slope (or gradient) of a function changes as you move along its graph. Mathematically, if you have a function denoted as "f(x),"

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(ii) How is the second derivative used in distinguishing between a maximum and a minimum point? The second derivative test is useful when trying to find a relative maximum or minimum if a function has a first derivative that is zero at a certain point. Since the first derivative test fails at this point, the point is an inflection point. The second derivative test relies on the sign of the second derivative at that point. If it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum. f) (i) What is meant by the “partial derivative”? a "partial derivative" is a derivative that measures the rate of change of a multivariable function with respect to one of its independent variables, while keeping the other independent variables constant. Partial derivatives are particularly useful when dealing with functions of several variables, where changes in one variable may affect the function while others are held constant. Mathematically, suppose you have a function f ( x , y ) that depends on two independent variables, x and y. The partial derivative of f with respect to x, denoted as ∂ y ∂ x represents how f changes concerning x, with y held constant. Similarly, the partial derivative of f with respect to y, denoted as δy δx , measures how f changes concerning y, with x held constant (ii) How is it determined? For single-variable functions, determining the derivative is straightforward, and it's essentially the same as finding the derivative in ordinary calculus. For example, if you have a function f(x) and you want to find ∂f/∂x, you use the rules of differentiation. Multivariable Functions: In the case of multivariable functions, partial derivatives are calculated by treating one variable as the variable of interest (e.g., x) and treating all other variables (e.g., y, z, etc.) as constants. To find ∂f ∂x , you differentiate the function f ( x, y ) with respect to x while treating y (and any other variables) as constants. To find ∂f ∂ y , you differentiate the function f ( x, y ) with respect to y while treating x (and any other variables) as constants [ CITATION Rob16 \l 2057 ] (iii) Why is the concept of the partial derivative important in managerial economics?  The concept of the partial derivative is important in managerial economics because it enables decision-makers to analyze and understand how changes in multiple variables affect the behavior of economic functions, systems, and models. In managerial economics, where complex decisions are often made considering various factors, partial derivatives provide valuable insights and tools for optimization, sensitivity analysis, and understanding the relationships between variables.

 Managerial economics often deals with models and situations that involve multiple variables. For instance, in production analysis, firms consider how factors like labor, capital, and technology affect output. In marketing, variables such as price, advertising, and market size interact. The partial derivative allows economists and managers to analyze how one specific variable impacts the outcome while holding other factors constant.  Marginal analysis is fundamental in managerial decision-making. It involves studying how a small change in one variable affects another variable, such as how a change in price affects demand or how an increase in labor affects production. Partial derivatives provide a precise way to measure these marginal effects. For example, ∂Q/∂P represents how quantity (Q) changes with a change in price (P), holding other factors constant.  In production and cost analysis, firms use partial derivatives to determine the optimal level of input factors (like labor and capital) to minimize costs or maximize production. The marginal product of labor (∂Q/∂L) and the marginal product of capital (∂Q/∂K) are key partial derivatives used in production analysis. Similarly, marginal cost (∂C/∂Q) is a crucial partial derivative in cost analysis.  Firms aim to maximize revenue or profit. Partial derivatives help determine the optimal pricing and production levels to achieve these objectives. For example, firms use the marginal revenue (∂R/∂Q) and marginal cost (∂C/∂Q) to find the profit-maximizing quantity (Q) and price (P).  the concept of the partial derivative is a fundamental tool in managerial economics that empowers decision-makers to analyze complex interactions between variables and make well-informed choices. By providing insights into the rate of change of functions concerning individual variables, partial derivatives contribute to efficient resource allocation, optimization, risk assessment, and overall effective decision-making in a business context. (iv) How can we use partial derivatives to optimize a multivariate function? Using partial derivatives to optimize a multivariate function involves finding the critical points of the function and determining whether these points correspond to maxima, minima, or saddle points. Partial derivatives can be used to optimize an objective function which is a function of several variables subject to a constraint or a set of constraints, given that the functions are differentiable. Mathematically, the constrained optimization problem requires to optimize a continuously differentiable function f ( x 1 ,x 2 ,...,x n ) subject to a set of constraints. General form of the constrained optimization problem where the problem is to maximize the objective function can be written as

g) (i) What is meant by “constrained optimization”? Constrained optimization" refers to the process of finding the maximum or minimum of a function while subject to certain constraints or limitations. In other words, it involves optimizing a function while adhering to specific conditions or restrictions on the values that the variables can take. (ii) How important is this to managerial economics? Constrained optimization is crucial to managerial economics for several reasons:  constrained optimization is integral to managerial economics because it mirrors the complexity of real-world decision-making. It allows decision- makers to find practical, feasible solutions that maximize or minimize objectives while adhering to the constraints that are inherent in business and economic environments.  Realistic Decision-Making: In many real-world scenarios, decision-makers face constraints on resources, budgets, time, and other factors. Constrained optimization models reflect the practical limitations that businesses and organizations encounter.  Resource Allocation: Managerial economics often involves allocating limited resources efficiently to maximize some objective. Constraints represent the scarcity of resources, and optimization helps determine how to allocate them optimally.  Production and Cost Management: In production and cost analysis, firms aim to optimize production levels, minimize costs, or achieve production targets while dealing with constraints like labor availability, machine capacity, and budget limitations.  Market Behavior: In pricing and market analysis, firms optimize pricing strategies while considering factors such as demand, competition, and production capacity. Constraints may include pricing floors, market share targets, or production limits.  Risk Management: Constrained optimization helps firms manage risk by finding optimal strategies that balance risk and return within predefined constraints. For example, portfolio optimization in finance aims to maximize return within risk tolerance limits. (iii) How can a constrained optimization problem be solved? The commonly used mathematical technique of constrained optimizations involves the use of Lagrange multiplier and Lagrange function to solve these problems followed by checking the second order conditions using the Bordered Hessian. When the objective function is a function of two variables, and there is only one equality constraint, the constrained optimization problem can also be solved using the geometric approach discussed earlier given that the optimum point is an interior optimum. h) (i) What is meant by the “Lagrangian multiplier method”?

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The Lagrange method of multipliers is named after Joseph-Louis Lagrange, the Italian mathematician. The primary idea behind this is to transform a constrained problem into a form so that the derivative test of an unconstrained problem can even be applied. Also, this method is generally used in mathematical optimization. The method of Lagrange’s multipliers is an important technique applied to determine the local maxima and minima of a function of the form f(x, y, z) subject to equality constraints of the form g(x, y, z) = k or g(x, y, z) = 0. That means it is subject to the condition that one or more equations are satisfied exactly by the desired variable’s values. To determine the minimum or maximum value of a function f(x) subject to the equality constraint g(x) = 0 will form the Lagrangian function as: L ( x, λ )= f ( x ) – λg ( x ) Here, ℒ = Lagrange function of the variable x λ = Lagrange multiplier (ii) How is the Lagrangian function formed? Imagine youwant ¿ minimize somefunctionof two variables,say f ( x, y )= x 2 + y 2 Clearly,thishas aminimumat ( x, y )=( 0,0 ) . But whatif werequired x + y = 1 ?Whereis minimumthat obeysthis condition? Thebruteforce wayis ¿ isolate y = 1 − x ,so f ( x, y )= f ( x, y ( x )) ¿ x 2 +( 1 − x ) 2 = 2 x 2 + 1 − 2 x ¿ thusthe minimum ,by differentiating : f ' ( x )= 4 x − 2 = 0 isat x = 1 / 2 ,ie.at ( x , y )=( 1 / 2,1 / 2 ) . Wecanalso writethe conditionlike so x + y − 1 = 0 = λ ( x + y − 1 )

Wemultiplied by ,whichisallowed since weequate ¿ zeroanyway . ≤ t ' sadd zero , thatis this specialkind of zero, ¿ f ( x, y ) : f ( x, y )= x 2 + y 2 + λ ( x + y − 1 ) . Nowlet ' stry ∧ minimize again,but now consider x ¿ y independent : ∂f / ∂x = 2 x + λ = 0 ∂f / ∂ y = 2 y + λ = 0 Eliminating λ yields x = y Knowing x = y immediately givesus ( x , y )=( 1 / 2,1 / 2 ) ¿ x + y = 1 .Soadding this λ allowed us ¿ skipthe substitutionstep .Wecanalso solve for λ if wewanted ¿ : λ =− 1 . λ isknownas a Lagrangemultiplier .I t ' s simply asmart trick .I t ' salso not very hard ¿ prove that works,but let me skipthat here . ∈ conclusion : A lagrangemultiplier isavariablethat weintroduce ∈ order ¿ find anextrema. (iii) How can constrained optimization problem be solved by the Lagrangian method? The locations of the maximum and minimum of a function f ( x, y ) subject to the constraint g ( x, y ) = k can be found using Lagrange multipliers. In this method, the critical points are the solutions to the system of equations of the form ∇ f ( x , y ) = λ ∇ g ( x, y ) and g ( x, y ) = k where λ is the Lagrange multiplier. Once we have the solutions (critical points), we classify them by comparing their objective function values. The largest function value is the maximum and the smallest function value is the minimum.

PART B: PROBLEMS QUESTION 1 Mr. Chanda started earning $ 2,000 a month in a multinational company in Lusaka in 2016. As per his terms of appointment; he gets a salary hike of 10 per cent every year. Suppose that in Zambia the rate of annual inflation has been 5 per cent for the last 5 years. How much will his salary be 2021? YEAR Calculation for increment and inflation Yearly salary ($) 2016 12000 × 12 24 000 2017 24000 × ( 1 + 0.1 1 + 0.05 ) 1 25 142.85714 2018 24000 × ( 1 + 0.1 1 + 0.05 ) 2 26 340.13605 2019 24000 × ( 1 + 0.1 1 + 0.05 ) 3 27 594.42825 2020 24000 × ( 1 + 0.1 1 + 0.05 ) 4 28 908.44864 2021 24000 × ( 1 + 0.1 1 + 0.05 ) 5 30 285.04143 Therefore, ∈ 2021 the salary per month = 30285.04143 12 = $ 2523.753453 QUESTION 2 A middle aged man managing photocopy business for K25,000 per year decides to open his own duplicating place. His revenue during the first year of operation is K120,000 and his expenses are as follows: Salaries to hired help K 45,000 Supplies K 15,000 Rent K 10,000 Utilities K 1,000 Interest on bank loan K 10,000

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Required: Distinguish between business profits and economic profits and calculate: a) The explicit costs The explicit costs are the actual out-of-pocket expenses incurred by the man in running his business. In this case, the explicit costs are: Salaries to hired help: $45,000 Supplies: $15,000 Rent: $10,000 Utilities: $1,000 Interest on bank loan: $10,000 Totalexplicit costs = 45000 + 15000 + 1000 + 10000 = $ 89,000 b) The implicit costs The implicit costs are the opportunity costs of the man's resources that he could have earned if he had not started his own business. In this case, the implicit costs are: The man's salary as a manager of the photocopying establishment: $15,000 c) The business profit The business profit is the revenue earned by the man minus the explicit costs. In this case, the business profit is: Revenue: $120,000 Explicit costs: $81,000 Business profit = 120000 − 81000 = $ 39,000 d) The economic profit The economic profit is the revenue earned by the man minus both the explicit and implicit costs. In this case, the economic profit is: Revenue: $120,000, Explicit costs: $81,000 Implicit costs: $25,000 Economic profit = 120000 − 81000 − 25000 = $ 14,000 The positive economic profit indicates that the man's resources is better utilized here. e) The normal return on investment in this business The normal return on investment is the minimum return required to keep the man in the business. It is the opportunity cost of the man's resources, including his time and money. In this case, the normal return on investment is: Totalcosts ( explicit + implicit ) :81000 + 25000 = $ 106,000 Normal return on investment = $106,000 x 10% = $10,600 The man's business profit of $39,000 is greater than the normal return on investment of $10,600, indicating that his business is profitable enough to justify his decision to start it.

QUESTION 3 Given the following total-cost schedule: Q 0 1 2 3 4 TC 1 12 14 15 20 Required: a) Derive the average, and marginal cost schedules Average revenue ( AR )= total revenue ( TR )/ Q Marginalrevenue ( MR )= change ∈ total revenue / change ∈ Q Q TC AR MC 0 1 1 12 12 11 2 14 7 2 3 15 5 1 4 20 5 5 b) On the same set of axes, plot the total- cost, average cost and marginal cost schedules 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 5 10 15 20 25 TC AC MC Axis Title Axis Title c) Explain the relationship among the total cost, average cost and marginal profit curves in part (b) Note that we cannot calculate the marginal cost for Q=0, since there is no change in total cost. Also, we can see that the marginal cost is increasing as the quantity increases. This is due to the law of diminishing returns (each additional unit produced requires more resources, and therefore increases the cost). Note that we cannot calculate the average cost for Q=0, since it would involve dividing by zero. Also, we can see that the average cost decreases as the quantity increases up to Q=2, and then it starts to increase again. This is due to the presence of fixed costs (the cost of

producing one unit is high when the quantity is low, but it becomes lower as more units are produced). QUESTION 4 Find the best profit point of a firm whose total revenue and total cost functions are as follows: R = 260Q -- 3Q 2 C = 500 – 20Q (Hint : ie. How many units of Q will the firm need to produce to maximize profit) To find the level of output at which the firm maximizes total profit, we need to find the level of output where marginal revenue (MR) equals marginal cost (MC) MR = d ( R ) dQ = 260 − 6 Q MC = d ( C ) dQ =− 20 MR = MC 260 − 6 Q =− 20 Q = 140 3 units QUESTION 5 a) Calculate the approximate change in y on the function y = x 2 + x – 2 as x increases from 2 to 2.1 dy dx = 2 X + 1 x + Δ x = 2.1 Δ x = 2.1 − 2 = 0.1 dy dx ¿ ¿ 2 ∗ Δ x = ( 2 ∗ 2 + 1 ) ∗ 0.1 = 0.5 Δ y = ¿ y + Δ y = final y final y = 2 2 + 2 − 2 + 0.5 = 4.5 b) Find turning points and points of inflection(if any) for the following curves (i) Y = x 2 + 12x – 20 dy dx = 2 x + 12 = 0 x = − 12 2 =− 6

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y = ( − 6 ) 2 + 12 ( − 6 ) − 20 =− 56 Turning points (− 6 , − 56 ) a point of inflection corresponds ¿ d 2 y d x 2 = 2 ≠ 0. Henceit hasno pointof inflation (ii) Y = 40 + 3x – 2x 2 + x 2 dy dx = 3 − 10 x 3 = 0 x = − 3 − 10 3 = 9 / 10 y = 40 + 3 ( 9 10 ) − 2 ( 9 10 ) 2 + ( 9 10 ) 2 3 = 41.35 Turning points ( 0.9,41.35 ) a point of inflection corresponds ¿ d 2 y d x 2 = − 10 3 ≠ 0. Henceit hasno point of inflation QUESTION 6: Constrained optimization by substitution A firm seeks to maximize its total – profit function given by: = 80X – 2X 2 – XY – 3y 2 + 100y But faces the constraint that the output of commodity X plus the output of commodity Y must be 12. That is, X + Y =12 Required : Determine the production mix of X and Y units that will maximize total profits UsetheLagrangian method ¿ maximizethe following profit function : π = 80 X – 2 X 2 – XY – 3 Y 2 + 100 Y Subject ¿ the following constraint : X + Y = 12 ( output capacity constraint ) Set the constraint functionequal ¿ zero ∧ obtain 0 = 12 – X – Y

Formthe Lagrangian function L = 80 X – 2 X 2 – XY – 3 Y 2 + 100 Y + λ ( 12 − X − Y ) Findthe ∂ derivatives ∧ solvesimultaneously ∂L ∂ X = 80 – 4 X –Y − λ = 0 ∂ L ∂Y = – X – 6 Y + 100 − λ = 0 ∂L ∂ λ = 12 − X – Y = 0 Subtract the second equation ¿ the first equation,weget ( 80 − 4 X − Y − λ ) − ( − X − 6 Y + 100 − λ ) = 0 − 3 X + 5 Y − 20 = 0 Solve the system of equations − 3 X + 5 Y − 20 = 0 12 − X – Y = 0 Solution : X = 5, Y = 7, ∧ λ = 53 Find : ∂ 2 L ∂ X 2 = – 4, ∂ 2 L ∂Y 2 = – 6, ∂ 2 L ∂ X ∂Y = – 1. ¿ ( ∂ 2 L ∂ X 2 ∂ 2 L ∂Y ∂ X ∂ 2 L ∂ X ∂Y ∂ 2 L ∂Y 2 ) = ( − 4 − 1 − 1 − 6 ) is negativedefinitebecause ∆ 1 =− 4 < 0, ∆ 2 = | − 4 − 1 − 1 − 6 | = ( − 4 ) ∗ ( − 6 ) − ( − 1 ) ∗ ( − 1 ) = 24 − 1 = 23 > 0. So ,indeed ,wehave found profit maximizingoutput levelof commodities X ∧ Y . So ,the profit willbe

π = 80 ∗ 5 – 2 ∗ 5 2 – 5 ∗ 7 – 3 ∗ 7 2 + 100 ∗ 7 = 400 − 50 − 35 − 147 + 700 = $ 868 QUESTION 7. CONSTRAINED OPTIMIZATION (Mansfield ) The Kloster Company produces two products, and that its total cost equals TC = 4Q 1 2 + 5Q 2 2 – Q 1 Q 2 Where Q 1 equals its output per hour of the first product, and equals its output per hour of the second product. Because of commitments to customers, the amount produced of both products combined cannot be less than 30 per hour. Required: Determine the levels of output of the two products which will minimize the firm’s costs using : a) The substitution method b) The Lagrangian Multiplier, Lambda, λ I. Calculate the value of Lambda,λ Q 1 + Q 2 = 30 L ( Q 1 ,Q 2 ,λ ) = 4 Q 1 2 + 5 Q 2 2 − Q 1 Q 2 − λ ( Q 1 + Q 2 − 30 ) ∂ L ∂Q 1 = 8 Q 1 − Q 2 − λ = 0 ∂ L ∂Q 2 = 10 Q 2 − Q 1 − λ = 0 ∂L ∂ λ = Q 1 + Q 2 − 30 = 0 Solving these equations simultaneously, 8 Q 1 − Q 2 = λ …… 1 10 Q 2 − Q 1 − λ = 0 …… .2 10 Q 2 − Q 1 −( 8 Q 1 − Q 2 )= 0 − 9 Q 1 + 11 Q 2 = 0 …… .3 Q 1 =− Q 2 + 30 …… .4 − 9 (− Q 2 + 30 )+ 11 Q 2 = 0 Q 2 = 27 2 Q 1 = − 27 2 + 30 = 33 2

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λ = 8 ∗ 33 2 − 27 2 = 237 2 II. Explain the significance of Lambda, λ, in economic analysis The Lagrange multiplier λ represents the marginal cost of production. It is the amount by which the cost increases when one more unit of output is produced. In this case, λ = 237 2 , which means that the cost increases by $118.5 for each additional unit of output. References CHEN, J. (2020, April 7). Economic Derivative: What it Means, How it Works . Retrieved from investopedia.: https://www.investopedia.com/terms/e/economic_derivatives.asp Graham, R. J. (2016, 03 26). How to Use Partial Derivatives in Managerial Economics . Retrieved from Dummies: https://www.dummies.com/article/business-careers- money/business/economics/how-to-use-partial-derivatives-in-managerial-economics- 167044/ MURPHY, C. B. (202, December 23). Theory of the Firm: What It Is and How It Works in Economics . Retrieved from investopedia: https://www.investopedia.com/terms/t/theory-firm.asp

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