The total cost function is given as:

$$C=\frac{1}{2}{x}^3-4x^2+100x+108$$

Where $x$ represents the output.

(i) Total Cost when Output is 9 Units

To find the total cost when $x = 9$, substitute $x = 9$ into the total cost function:

$$C=\frac{1}{2}(9{)}^3-4(9{)}^2+100(9)+108$$

Simplify step-by-step:

$$C = \frac{1}{2}(729) – 4(81) + 900 + 108$$
$$C = 364.5 – 324 + 900 + 108$$
$$C=1048.5$$

Total cost when $x = 9$: 1048.5

(ii) Average Variable Cost when Output is 12 Units

The total cost $C$ can be separated into fixed cost (FC) and variable cost (VC). In this equation, the constant term $108$ is the fixed cost.

Variable Cost (VC):

$$VC=C-\text{Fixed Cost}=\frac{1}{2}{x}^3-4x^2+100x$$

Average Variable Cost (AVC):

$$AVC=\frac{VC}{x}$$

Substitute $x = 12$ into $VC$ first:

$$VC = \frac{1}{2}(12)^3 – 4(12)^2 + 100(12)$$
$$VC = \frac{1}{2}(1728) – 4(144) + 1200$$
$$VC = 864 – 576 + 1200$$
$$VC=1488$$

Now calculate $AVC$:

$$AVC=\frac{1488}{12}=124$$

Average Variable Cost when $x = 12$: 124​

(iii) Average Cost when Output is 9 Units

Average Cost (AC):

$$AC=\frac{C}{\mathrm{x<span\; style="font-family:\; var(--global-body-font-family);"></span>}}$$Substitute $x = 9$ and the total cost $C = 1048.5$ (from part i):

$$AC = \frac{1048.5}{9}$$ $$AC\approx 116.5$$

Average Cost when $x = 9$: 116.5

Final Answers

1. Total Cost (9 units): $\boxed{1048.5}$
2. Average Variable Cost (12 units): $\boxed{124}$
3. Average Cost (9 units): $\boxed{116.5}$

A functional relationship refers to a mathematical or conceptual connection between two or more variables, where one variable (the dependent variable) is determined by one or more other variables (the independent variables). This relationship is fundamental in various disciplines, including economics, as it allows for the representation and analysis of how changes in one factor influence another. In simple terms, a functional relationship can be expressed as:

$$Y=f(X)$$

Here, $Y$ is the dependent variable, $X$ is the independent variable, and $f$ denotes the functional form of the relationship.

Functional Relationship in Economics

In economics, understanding functional relationships is critical for analyzing and predicting economic behaviors, resource allocation, market dynamics, and policy outcomes. Functional relationships help economists model real-world scenarios to establish cause-and-effect linkages, enabling better decision-making at individual, business, and governmental levels.

Economics often employs mathematical functions, graphs, and equations to describe these relationships quantitatively. Functions can be linear, non-linear, increasing, decreasing, or exhibit complex behaviors like diminishing marginal returns or cyclical trends.

Examples of Functional Relationships in Economics

Demand Function

The demand function represents the relationship between the quantity of a good or service demanded and its determinants, such as price, income, and preferences. The general form of the demand function can be written as:

$$Q_d=f(P,I,T)$$

Here, $Q_d$ is the quantity demanded, $P$ is the price of the good, $I$ is the consumer’s income, and $T$ represents tastes or preferences.
For example, as the price of a commodity decreases, consumers typically demand more of it, illustrating the law of demand. Conversely, an increase in consumer income often leads to higher demand for normal goods.

Production Function

A production function explains how inputs like labor ($L$) and capital ($K$) are used to produce output ($Q$). It is expressed as:

$$Q=f(L,K)$$

For instance, a firm may experience increasing returns to scale if doubling both labor and capital results in more than double the output. On the other hand, it may encounter diminishing marginal returns if adding more of one input, while holding others constant, leads to smaller increments in output.

Cost Function

The cost function shows the relationship between the cost of production and the level of output. It incorporates fixed costs and variable costs and can be expressed as:

$$C=f(Q)$$

For example, total cost ($C$) may increase as a quadratic function of output ($Q$) due to economies and diseconomies of scale. This relationship helps firms optimize production levels by understanding how costs behave at different levels of output.

Consumption Function

Proposed by John Maynard Keynes, the consumption function depicts the relationship between total consumption ($C$) and disposable income ($Y_d$):

$$C=f(Y_d)$$

Keynes argued that as disposable income rises, consumption increases, but not by as much as the increase in income, due to the marginal propensity to consume being less than one. This relationship is central to macroeconomic theories regarding aggregate demand and savings.

Importance of Functional Relationships in Economics

1. Policy Formulation: Functional relationships help policymakers design and evaluate interventions. For example, understanding the consumption function enables governments to predict the effects of tax changes on consumer spending and savings.

2. Market Analysis: Businesses rely on demand and cost functions to determine pricing strategies, optimize production, and forecast sales. Knowing how price and income influence demand allows firms to adapt to changing market conditions.

3. Macroeconomic Stability: Relationships like the Phillips curve, which links inflation and unemployment, guide central banks in balancing inflation control with economic growth.

4. Theoretical Foundations: Economic theories like utility maximization, profit optimization, and general equilibrium are built upon well-defined functional relationships between variables.

Conclusion

In economics, functional relationships are vital for connecting abstract theories with real-world phenomena. They provide a structured way to understand how variables interact, predict outcomes, and guide decision-making. By studying these relationships, economists can address complex questions about resource allocation, market dynamics, and societal welfare. Whether it’s the demand function illustrating consumer behavior, the production function explaining firm outputs, or the consumption function revealing spending patterns, functional relationships form the backbone of economic analysis and its application to solving real-world problems.

Step 1: Equilibrium Condition

At equilibrium, demand equals supply:

$$X_d=X_s$$

Here, the demand function is:

$$X_d=20-4P$$

And the supply function is:

$$X_s=10P+8$$

Equating $X_d$ and $X_s$:

$$20-4P=10P+8$$

Step 2: Solve for the Equilibrium Price ($P^*$)

Rearrange the terms to isolate $P$:

$$20 – 8 = 10P + 4P$$
$$12=14P$$

$$P^{\ast }=\frac{12}{14}=\frac{6}{\mathrm{7<span\; style="font-family:\; var(--global-body-font-family);"></span>}}$$

Thus, the equilibrium price is:

$$P^{\ast }=\frac{6}{7}\approx 0.857$$

Step 3: Find the Equilibrium Quantity ($X^*$)

Substitute $P^* = \frac{6}{7}$ into either the demand or supply function. Using the demand function:

$$X_d=20-4P$$

$$X_d = 20 – 4P$$$$X^{\ast }=20-4\left(\frac{6}{7}\right)$$

$$X^* = 20 – 4\left(\frac{6}{7}\right)$$$$X^* = 20 – \frac{24}{7}$$

$$X^{\ast }=\frac{140}{7}-\frac{24}{7}=\frac{116}{7}$$

Thus, the equilibrium quantity is:

$$X^{\ast }=\frac{116}{7}\approx 16.57$$

Step 4: Elasticity of Demand at Equilibrium

The price elasticity of demand ($E_d$) is given by:

$$E_d=\frac{dX}{dP}\cdot \frac{P}{X}$$

The derivative of the demand function $X = 20 – 4P$ with respect to $P$ is:

$$\frac{dX}{dP}=-4$$

Substitute $P^* = \frac{6}{7}$ and $X^* = \frac{116}{7}$ into the elasticity formula:

$$E_d = (-4) \cdot \frac{\frac{6}{7}}{\frac{116}{7}}$$ $$E_d = -4 \cdot \frac{6}{116}$$ $$E_d=-\frac{24}{116}=-\frac{6}{29}\approx -0.207$$

Final Results

1. Equilibrium Price ($P^*$): $$P^* = \frac{6}{7} \approx 0.857$$
   
2. Equilibrium Quantity ($X^*$): $$X^* = \frac{116}{7} \approx 16.57$$
   
3. Elasticity of Demand ($E_d$) at Equilibrium: $$E_d\approx -0.207$$

Thus, at equilibrium, the price is approximately 0.857, the quantity demanded (or supplied) is approximately 16.57 units, and the demand is inelastic with elasticity $E_d \approx -0.207$.

A curve and a straight line are distinct geometric representations that describe relationships between two variables in different ways.

A straight line is a graphical representation where the relationship between the variables remains constant, meaning the rate of change (or slope) does not vary. It can be expressed in the linear equation form:

$$y=mx+c$$

Here, $m$ represents the slope, and $c$ is the intercept. In economics, straight lines are often used to represent simple relationships like fixed costs or proportional changes.

In contrast, a curve represents a relationship where the rate of change between the variables varies. Curves can be upward-sloping, downward-sloping, convex, concave, or exhibit other forms depending on the underlying relationship. Curves are commonly used in economics to depict non-linear relationships, such as diminishing marginal returns, economies of scale, or consumer behavior.

For instance, a demand curve is typically downward-sloping and concave due to the law of demand, while a total cost curve may be convex, reflecting increasing marginal costs.

Explanation of Economic Curves: $MC$, $AC$, $AR$, and $MR$

Economic curves such as Marginal Cost (MC), Average Cost (AC), Average Revenue (AR), and Marginal Revenue (MR) are essential tools for understanding cost and revenue behavior in production and pricing. These curves help economists and businesses analyze efficiency, profitability, and decision-making.

Marginal Cost (MC) Curve

The Marginal Cost (MC) curve represents the additional cost incurred in producing one more unit of output. Mathematically, it is the derivative of the Total Cost (TC) function with respect to quantity ($Q$):

$$MC=\frac{d(TC)}{dQ}$$

The shape of the $MC$ curve is typically U-shaped due to the law of variable proportions. Initially, as production increases, marginal costs decrease due to increasing returns to the variable factor (e.g., labor). However, after a certain point, diminishing returns set in, causing marginal costs to rise.

For example, in the short run, a factory may benefit from adding more workers, as existing resources are used more efficiently. However, over time, overcrowding or resource limitations increase the cost of adding additional output.

The $MC$ curve plays a crucial role in determining the optimal level of production, where $MC = MR$.

Average Cost (AC) Curve

The Average Cost (AC) curve shows the cost per unit of output and is calculated as:

$$AC=\frac{TC}{Q}$$

The $AC$ curve is also typically U-shaped, reflecting economies and diseconomies of scale. It is composed of two components:

1. Average Fixed Cost (AFC): AFC decreases as output increases, as fixed costs are spread over a larger number of units.
2. Average Variable Cost (AVC): AVC decreases initially due to increasing returns but rises after diminishing returns set in.

The minimum point of the $AC$ curve indicates the most efficient scale of production, where per-unit costs are minimized. Businesses aim to operate near this point for optimal efficiency.

Average Revenue (AR) Curve

The Average Revenue (AR) curve represents revenue earned per unit of output and is calculated as:

$$AR=\frac{TR}{Q}$$

Here, $TR$ is the Total Revenue, defined as $TR = P \times Q$, where $P$ is the price.

In perfect competition, the $AR$ curve is a horizontal straight line, as the firm can sell any quantity at the market-determined price. However, in monopolistic or imperfect competition, the $AR$ curve slopes downward, reflecting the fact that the firm must lower the price to sell additional units.

The $AR$ curve is equivalent to the demand curve, as it shows the relationship between price and quantity demanded.

Marginal Revenue (MR) Curve

The Marginal Revenue (MR) curve represents the additional revenue earned from selling one more unit of output. It is calculated as:

$$MR=\frac{d(TR)}{dQ}$$

In perfect competition, the $MR$ curve coincides with the $AR$ curve because the price remains constant for all units sold. However, in imperfect competition, the $MR$ curve lies below the $AR$ curve. This occurs because the firm must lower the price for all units sold to increase sales, reducing the additional revenue from each extra unit.

The shape and position of the $MR$ curve are critical for profit maximization. Firms maximize profit by producing at the level where $MR = MC$.

Interrelationship Among $MC$, $AC$, $AR$, and $MR$

• The $MC$ curve intersects the $AC$ curve at its minimum point. This is because marginal cost determines the direction of average cost. When $MC < AC$, $AC$ falls, and when $MC > AC$, $AC$ rises.
• The $MR$ curve intersects the horizontal axis where total revenue is maximized.
• In perfect competition, the $MR$ curve is a horizontal line equal to the price. In monopolistic markets, the $MR$ curve slopes downward due to the price-quantity trade-off.
• Profit maximization occurs at the point where $MC = MR$.

Importance of These Curves

1. Business Decision-Making: Understanding these curves helps firms decide how much to produce and at what price.
2. Policy Formulation: Governments and regulators use cost and revenue curves to analyze market efficiency and intervene where necessary.
3. Profit Analysis: The intersection of $MC$ and $MR$ identifies the profit-maximizing output level, critical for businesses aiming to maximize returns.

Conclusion

The concepts of $MC$, $AC$, $AR$, and $MR$ curves are fundamental in economics, providing insight into the behavior of costs, revenues, and market dynamics. While straight lines often represent simpler relationships, curves reflect the complexities of real-world scenarios, such as diminishing returns and price-quantity trade-offs. By studying these relationships, economists and businesses can optimize production, pricing, and profitability, ensuring efficient resource allocation and market equilibrium.

In economics, equilibrium is a fundamental concept that refers to a state where opposing forces or influences are balanced, and there is no inherent tendency for change. This balance can occur in various contexts, such as markets, firms, or economies as a whole. Equilibrium is broadly classified into static equilibrium and dynamic equilibrium, depending on the nature of the variables involved and the passage of time. Understanding these concepts is essential for analyzing how economic systems respond to external changes and maintain stability.

Static Equilibrium

Static equilibrium refers to a state of balance in an economic system at a particular point in time, where there is no change in the variables under consideration. It assumes that the system is in a state of rest, and all forces influencing the variables are balanced. In this type of equilibrium, time is either ignored or considered constant, meaning the focus is on the relationships between variables at a specific moment.

For example, in microeconomics, market equilibrium in a static sense occurs when the quantity demanded equals the quantity supplied at a given price, with no tendency for change. The price at which this occurs is the equilibrium price, and the quantity is the equilibrium quantity. The graphical representation of static equilibrium is often depicted with intersecting curves, such as a supply curve and a demand curve.

Characteristics of Static Equilibrium

1. Timelessness: Static equilibrium analyzes variables without considering time. It is a snapshot of the economic system at a single moment.
2. Absence of Change: There is no movement in the system; the variables remain constant unless external forces disturb them.
3. Simplified Assumptions: It assumes perfect knowledge, rational behavior, and the absence of time lags, making it an idealized concept.

Example: Market Equilibrium

Consider a market where the demand function is $Q_d = 50 – 2P$ and the supply function is $Q_s = 10 + 3P$. In static equilibrium:

$$Q_d = Q_s$$ $$50 – 2P = 10 + 3P$$
$$P = 8, \; Q = 34$$

At this price ($P = 8$), the market is in equilibrium, with no tendency for either price or quantity to change.

Limitations of Static Equilibrium

Static equilibrium oversimplifies reality by ignoring the dynamic nature of economic processes. It cannot analyze how equilibrium is achieved or the adjustments that occur over time. For example, if demand or supply changes due to external factors, static equilibrium does not address how the new balance is reached.

Dynamic Equilibrium

In contrast, dynamic equilibrium refers to a state where the economic system remains balanced over time despite continuous changes in the variables. Unlike static equilibrium, dynamic equilibrium explicitly incorporates the passage of time and focuses on the process by which equilibrium is achieved and maintained.

Dynamic equilibrium occurs when the rates of change in opposing forces are equal, creating a state of balance over time. For example, in macroeconomics, an economy might achieve dynamic equilibrium if its growth rate matches the rate of population growth, ensuring stability in per capita income.

Characteristics of Dynamic Equilibrium

1. Incorporation of Time: Dynamic equilibrium examines how variables change and adjust over time, often leading to a new state of balance.
2. Continuous Adjustment: Unlike static equilibrium, it acknowledges ongoing changes and focuses on the system’s ability to adapt.
3. Realistic Assumptions: It considers time lags, expectations, and feedback mechanisms, making it a more realistic representation of economic processes.

Example: Cobweb Model

The cobweb model illustrates dynamic equilibrium in markets where there is a time lag between production and sale. If farmers base their planting decisions on current prices, supply and demand may oscillate before stabilizing. For example:

• In period 1, high prices lead to increased supply.
• In period 2, the oversupply causes prices to fall.
• In period 3, low prices reduce supply, pushing prices up again.

Over time, if the adjustments converge, the market achieves dynamic equilibrium, balancing supply and demand.

Importance in Economic Growth

Dynamic equilibrium is crucial for understanding long-term economic growth. For instance, the Solow Growth Model examines how an economy achieves a steady-state growth rate by balancing capital accumulation, technological progress, and population growth.

Key Differences Between Static and Dynamic Equilibrium

1. Time Dimension: Static equilibrium ignores time, while dynamic equilibrium explicitly considers it.
2. Nature of Variables: In static equilibrium, variables remain constant, whereas in dynamic equilibrium, they continuously change and adjust.
3. Analytical Scope: Static equilibrium focuses on a fixed point, while dynamic equilibrium analyzes the adjustment process over time.
4. Realism: Dynamic equilibrium provides a more realistic depiction of economic systems by accounting for time lags, feedback loops, and changes in expectations.

Applications of Static and Dynamic Equilibrium in Economics

Static Equilibrium

Static equilibrium is primarily used in microeconomic analysis to determine market prices and quantities. It simplifies complex interactions, making it easier to study short-term phenomena like price determination, tax effects, or subsidies. For example, when a government imposes a price ceiling, static analysis can show the immediate impact on supply and demand.

Dynamic Equilibrium

Dynamic equilibrium is vital in macroeconomics and long-term growth analysis. It helps economists understand how economies transition from one state to another, such as moving from recession to growth. Dynamic models are used to study topics like inflation, employment, and trade balances. For instance, in monetary policy, central banks adjust interest rates to maintain dynamic equilibrium between inflation and economic growth.

Conclusion

Static and dynamic equilibrium are complementary concepts in economics, each serving distinct purposes. Static equilibrium provides a simplified snapshot of balance at a specific moment, making it useful for analyzing short-term relationships. However, it ignores the complexities of time and adjustment processes. Dynamic equilibrium, on the other hand, captures the evolving nature of economic systems, offering a more realistic understanding of long-term trends and adjustments. By combining insights from both types of equilibrium, economists can better analyze and predict economic behavior, ultimately contributing to more effective policymaking and resource allocation.

In economics, the concepts of slope and elasticity are fundamental tools for understanding and analyzing relationships between variables. Although these terms are related, they serve distinct purposes and measure different aspects of economic relationships. While slope is a geometric measure of the steepness of a line, elasticity is a unit-free measure of the responsiveness of one variable to changes in another. Understanding the differences between these concepts is crucial for interpreting demand, supply, costs, and other economic relationships effectively.

Slope: A Geometric Measure

The slope of a line measures the rate at which one variable changes in response to another. In mathematical terms, the slope of a linear relationship between two variables, $X$ and $Y$, is calculated as:

$$\text{slope}=\frac{\mathrm{\Delta }Y}{\mathrm{\Delta }\mathrm{X<span\; style="font-family:\; var(--global-body-font-family);"></span>}}$$

Here, $\Delta Y$ is the change in the dependent variable ($Y$), and $\Delta X$ is the change in the independent variable ($X$).

In a two-dimensional graph, the slope indicates the steepness or flatness of a curve or line. A positive slope implies that $Y$ increases as $X$ increases (a direct relationship), while a negative slope indicates that $Y$ decreases as $X$ increases (an inverse relationship).

Example in Economics: Demand Curve

The slope of a demand curve reflects the change in quantity demanded ($\Delta Q_d$) per unit change in price ($\Delta P$). For a demand curve represented as $Q_d = 20 – 2P$, the slope is:

$$\text{slope}=\frac{\mathrm{\Delta }{Q}_d}{\mathrm{\Delta }P}=-2$$

This indicates that for every 1-unit increase in price, the quantity demanded decreases by 2 units.

Limitations of Slope

While the slope is useful for understanding the rate of change, it is sensitive to the units of measurement. For example, if price is measured in dollars and quantity in kilograms, changing either unit will alter the numerical value of the slope. This dependence on units makes slope less useful for comparing relationships across different contexts or markets.

Elasticity: A Unit-Free Measure

Elasticity, on the other hand, measures the responsiveness of one variable to a percentage change in another. It is a ratio of percentage changes, making it independent of the units of measurement. The general formula for elasticity is:

$$\text{Elasticity}=\frac{\mathrm{\%}\text{ change in }Y}{\mathrm{\%}\text{ change in }X}$$

Price Elasticity of Demand

One common application is the price elasticity of demand, which measures how much the quantity demanded changes in response to a change in price:

$$E_d=\frac{\mathrm{\%}\mathrm{\Delta }{Q}_d}{\mathrm{\%}\mathrm{\Delta }P}=\frac{\mathrm{\Delta }{Q}_d\mathrm{/}{Q}_d}{\mathrm{\Delta }P\mathrm{/}P}=\frac{\mathrm{\Delta }{Q}_d}{\mathrm{\Delta }P}\cdot \frac{P}{Q_d}$$

Here, $\frac{\Delta Q_d}{\Delta P}$ is the slope of the demand curve, while $\frac{P}{Q_d}$ adjusts for the relative size of price and quantity. This adjustment ensures that elasticity is unit-free, allowing for comparisons across different goods, markets, or time periods.

Key Features of Elasticity

1. Unit Independence: Elasticity is unaffected by the scale of measurement, making it more versatile for economic analysis.
2. Relative Measure: Unlike slope, elasticity considers the relative magnitudes of changes, providing insights into how sensitive consumers or producers are to price changes.

For example, if the price of a product increases by 10% and the quantity demanded decreases by 20%, the price elasticity of demand is:

$$E_d=\frac{-20\mathrm{\%}}{10\mathrm{\%}}=-2$$

This indicates that demand is elastic, meaning consumers are highly responsive to price changes.

Key Differences Between Slope and Elasticity

1. Definition and Purpose:

   • Slope measures the absolute change in one variable per unit change in another.
   • Elasticity measures the relative responsiveness of one variable to percentage changes in another.

2. Units:

   • Slope depends on the units of measurement (e.g., dollars, kilograms, hours).
   • Elasticity is unit-free, enabling cross-market or cross-product comparisons.

3. Interpretation:

   • Slope provides a geometric representation of a relationship (e.g., steepness of a curve).
   • Elasticity provides an economic interpretation of sensitivity or responsiveness.

4. Applicability:

   • Slope is straightforward and often used in linear models.
   • Elasticity is more versatile and applies to both linear and non-linear relationships.

5. Dynamic Adjustments:

   • Elasticity incorporates the proportional nature of changes, making it more informative in dynamic contexts such as price setting or policy analysis.

Examples to Illustrate the Difference

Linear Demand Curve

Consider the demand curve $Q_d = 20 – 2P$. The slope is constant:

$$\text{slope}=\frac{\mathrm{\Delta }{Q}_d}{\mathrm{\Delta }P}=-2$$

Elasticity, however, varies at different points on the curve because of the term $\frac{P}{Q_d}$. For example:

• At $P = 5$, $Q_d = 10$, so: $$E_d=\frac{-2\cdot 5}{10}=-1$$
• At $P = 2$, $Q_d = 16$, so: $$E_d=\frac{-2\cdot 2}{16}=-0.25$$

The slope remains constant, but elasticity decreases as price falls and quantity increases.

Non-Linear Relationships

In non-linear relationships, the slope changes at every point, making elasticity an even more critical measure. For example, if the demand curve is $Q_d = 100P^{-0.5}$, the slope and elasticity must be calculated differently:

$$\text{slope}=\frac{d{Q}_d}{dP},E_d=\frac{d{Q}_d}{dP}\cdot \frac{P}{Q_d}$$

Here, elasticity provides a consistent way to compare responsiveness across the curve.

Importance of Slope and Elasticity in Economics

Both slope and elasticity are vital for understanding and analyzing economic behavior, but they are used in different contexts. Slope is valuable for straightforward relationships, such as calculating marginal costs or determining equilibrium points. Elasticity, on the other hand, is indispensable for analyzing demand and supply sensitivity, tax incidence, and policy effectiveness.

For example:

• Pricing Strategies: Firms use elasticity to set optimal prices. If demand is elastic, raising prices may reduce total revenue, while inelastic demand allows for price increases without significant loss in sales.
• Tax Policy: Governments analyze elasticity to predict the impact of taxes on consumption and revenue.
• International Trade: Elasticity measures help assess the effects of exchange rate changes on exports and imports.

Conclusion

While both slope and elasticity measure relationships between variables, they differ in purpose, interpretation, and applicability. Slope provides a geometric understanding of how variables change, but it is limited by its dependence on units and its inability to account for proportional changes. Elasticity, being unit-free, offers a more versatile and economically meaningful measure of responsiveness, making it particularly useful for dynamic and comparative analyses. By understanding and applying these concepts appropriately, economists and businesses can make informed decisions and gain deeper insights into market behavior.

$$C=x^3-3x^2+15x+27$$

Here, $C$ represents the total cost, and $x$ is the quantity of output produced. To solve the given parts, let’s proceed step by step.

(i) Average Cost

The average cost (AC) is defined as the total cost per unit of output, which is calculated as:

$$AC=\frac{C}{x}$$

Substituting the given total cost function:

$$AC=\frac{x^3-3x^2+15x+27}{x}$$

Simplify the expression:

$$AC=x^2-3x+15+\frac{27}{x}$$

Thus, the average cost is:

$$AC=x^2-3x+15+\frac{27}{x}$$

(ii) Variable Cost and Fixed Cost

The total cost function can be split into two components:

$$C=\text{Variable Cost (VC)}+\text{Fixed Cost (FC)}$$

• The fixed cost (FC) is the part of the total cost that does not depend on $x$ (output). From the given equation, the constant term $27$ represents the fixed cost:

  $$FC=27$$

• The variable cost (VC) is the part of the total cost that depends on $x$ (output). Removing the constant term $27$ from the total cost function gives:

  $$VC=x^3-3x^2+15x$$

Thus:

$$VC=x^3-3x^2+15x,FC=27$$

(iii) Average Fixed Cost and Average Variable Cost

The average fixed cost (AFC) and average variable cost (AVC) are defined as:

$$AFC=\frac{FC}{x},AVC=\frac{VC}{x}$$

Average Fixed Cost (AFC)

Substitute $FC = 27$:

$$AFC=\frac{27}{x}$$

Average Variable Cost (AVC)

Substitute $VC = x^3 – 3x^2 + 15x$:

$$AVC=\frac{x^3-3x^2+15x}{x}$$

Simplify:

$$AVC=x^2-3x+15$$

Thus:

$$AFC=\frac{27}{x},AVC=x^2-3x+15$$

Final Results

1. Average Cost:

   $$AC=x^2-3x+15+\frac{27}{x}$$

2. Variable Cost and Fixed Cost:

   $$VC=x^3-3x^2+15x,FC=27$$

3. Average Fixed Cost and Average Variable Cost:

   $$AFC=\frac{27}{x},AVC=x^2-3x+15$$

Market equilibrium occurs where the quantity demanded equals the quantity supplied, ensuring there is no surplus or shortage. Mathematically, it is the point where the demand function equals the supply function:

$$D_x=S$$

Given:

• Demand function: $D_x = \frac{1}{2}(5 – P)$
• Supply function: $S = 2P – 3$

To find equilibrium

1. Set $D_x = S$:

   $$\frac{1}{2}(5-P)=2P-3$$

2. Eliminate the fraction by multiplying through by 2:

   $$5-P=4P-6$$

3. Solve for $P$ (equilibrium price):

   $$5+6=4P+P\Rightarrow 11=5P\Rightarrow P=\frac{11}{5}=2.2$$

4. Substitute $P = 2.2$ into either the demand or supply function to find $Q$ (equilibrium quantity): Using $D_x = \frac{1}{2}(5 – P)$:

   $$D_x=\frac{1}{2}(5-2.2)=\frac{1}{2}(2.8)=1.4$$

Final Answer

• Final Answer:

  • Equilibrium Price ($P$): 2.2
  • Equilibrium Quantity ($Q$): 1.4

$$C(x)=x^3-3x^2+5x+12$$

Marginal Cost (MC)

The marginal cost is the derivative of the total cost function with respect to quantity ($x$):

$$MC=\frac{dC(x)}{dx}$$

Differentiate the total cost function:

$$MC = \frac{d}{dx}(x^3 – 3x^2 + 5x + 12)$$

$$MC=3x^2-6x+5$$

Average Cost (AC)

The average cost is the total cost divided by the quantity ($x$):

$$AC=\frac{C(x)}{x}$$

Substitute the total cost function:

$$AC=\frac{x^3-3x^2+5x+12}{x}$$

Simplify:

$$AC=x^2-3x+5+\frac{12}{x}$$

Final Answer:

• Marginal Cost (MC): $3x^2 – 6x + 5$
• Average Cost (AC): $x^2 – 3x + 5 + \frac{12}{x}$

To find the marginal cost ($MC$) when $x = 5$ for the total cost function $C(x) = x^3 – 3x^2 + 5x + 12$, let’s proceed step by step with detailed explanations.

Step 1: Marginal Cost Definition

The marginal cost is the rate of change of the total cost function $C(x)$ with respect to the quantity $x$. Mathematically, it is the first derivative of $C(x)$:

$$MC=\frac{dC}{dx}$$

Step 2: Differentiate $C(x)$

The total cost function is given as:

$$C(x)=x^3-3x^2+5x+12$$

Using basic differentiation rules:

• The derivative of $x^n$ is $n \cdot x^{n-1}$,
• The derivative of a constant is 0.

Differentiate term by term:

1. $\frac{d}{dx}(x^3) = 3x^2$,
2. $\frac{d}{dx}(-3x^2) = -6x$,
3. $\frac{d}{dx}(5x) = 5$,
4. $\frac{d}{dx}(12) = 0$.

Thus:

$$MC=3x^2-6x+5$$

Step 3: Substitute $x = 5$

Now that we have the marginal cost formula $MC = 3x^2 – 6x + 5$, substitute $x = 5$ into the formula:

$$MC=3(5{)}^2-6(5)+5$$

Step 4: Simplify Step by Step

1. Calculate $3(5)^2$:

   $$5^2=25\text{so}3(25)=75$$

2. Calculate (-6(5):

   $$-6\times 5=-30$$

3. Add the results: Combine the terms:

   $$MC=75-30+5$$

4. Final calculation:

   $$MC=50$$

Final Answer:

When $x = 5$, the marginal cost is:

$$MC=50$$

To find the level of output ($Q$) at which total revenue (TR) is maximized for the given demand function

Step 1: Write the Given Demand Function

The demand function is:

$$P=108-4Q^2$$

Here:

• $P$ is the price per unit,
• $Q$ is the quantity demanded.

Step 2: Expression for Total Revenue ($TR$)

Total revenue is the product of price ($P$) and quantity ($Q$):

$$TR=P\cdot Q$$

Substitute $P = 108 – 4Q^2$ into the equation:

$$TR=(108-4Q^2)\cdot Q$$

Simplify the expression:

$$TR=108Q-4Q^3$$

Step 3: Maximize Total Revenue

To find the value of $Q$ that maximizes $TR$, take the derivative of $TR$ with respect to $Q$, and set it equal to zero:

$$\frac{d(TR)}{dQ}=\frac{d}{dQ}(108Q-4Q^3)$$

Differentiate term by term:

• $\frac{d}{dQ}(108Q) = 108$,
• $\frac{d}{dQ}(-4Q^3) = -12Q^2$.

Thus:

$$\frac{d(TR)}{dQ}=108-12Q^2$$

Step 4: Solve for $Q$

Set $\frac{d(TR)}{dQ} = 0$ to find the critical points:

$$108-12Q^2=0$$

Solve for $Q^2$:

$$12Q^2=108\Rightarrow Q^2=9$$

Take the square root of both sides:

$$Q=3\text{or}Q=-3$$

Since $Q$ (quantity) cannot be negative, we have:

$$Q=3$$

Step 5: Verify Maximum Using Second Derivative

To confirm that $TR$ is maximized at $Q = 3$, calculate the second derivative of $TR$:

$$\frac{d^2(TR)}{d{Q}^2}=\frac{d}{dQ}(108-12Q^2)$$

Differentiate:

$$\frac{d^2(TR)}{d{Q}^2}=-24Q$$

At $Q = 3$:

$$\frac{d^2(TR)}{d{Q}^2}=-24(3)=-72$$

Since $\frac{d^2(TR)}{dQ^2} < 0$, $TR$ is maximized at $Q = 3$.

Final Answer

The level of output at which total revenue is maximized is:

$$Q=3$$

To determine the equilibrium price ($p$) and the corresponding equilibrium quantity ($x$), we must find the price at which the quantity demanded equals the quantity supplied.

Step 1: Write Down the Given Functions

1. Demand function: $x = \frac{100}{p}$

   • Here, $x$ is the quantity demanded, and $p$ is the price.

2. Supply function: $20 + 3p – x = 0$

   • Here, $x$ is the quantity supplied, and $p$ is the price.

At equilibrium, quantity demanded equals quantity supplied:

$$x_{\text{demand}}=x_{\text{supply}}$$

Step 2: Substitute $x = \frac{100}{p}$ into the Supply Equation

Replace $x$ in the supply equation $20 + 3p – x = 0$ with $\frac{100}{p}$:

$$20+3p-\frac{100}{p}=0$$

Step 3: Eliminate the Fraction

Multiply through by $p$ (assuming $p > 0$) to eliminate the fraction:

$$p(20)+p(3p)-p\left(\frac{100}{p}\right)=0$$

Simplify:

$$20p+3p^2-100=0$$

Rearrange into standard quadratic form:

$$3p^2+20p-100=0$$

Step 4: Solve the Quadratic Equation

The quadratic equation is:

$$3p^2+20p-100=0$$

Use the quadratic formula:

$$p=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

where:

• $a = 3$,
• $b = 20$,
• $c = -100$.

Substitute the values:

$$p=\frac{-20\pm \sqrt{20^2-4(3)(-100)}}{2(3)}$$

Simplify the discriminant:

$$20^2=400,-4(3)(-100)=1200,400+1200=1600$$

Thus:

$$p=\frac{-20\pm \sqrt{1600}}{6}$$

Simplify further:

$$\sqrt{1600}=40,p=\frac{-20\pm 40}{6}$$

Solve for the two possible values of $p$:

1. $p = \frac{-20 + 40}{6} = \frac{20}{6} = \frac{10}{3} \approx 3.33$.
2. $p = \frac{-20 – 40}{6} = \frac{-60}{6} = -10$

Since price ($p$) cannot be negative, we take:

$$p=\frac{10}{3}\approx 3.33$$

Step 5: Find the Equilibrium Quantity ($x$)

Substitute $p = \frac{10}{3}$ into the demand function $x = \frac{100}{p}$:

$$x=\frac{100}{\frac{10}{3}}=100\cdot \frac{3}{10}=30$$

Final Answer

• Equilibrium price: $p = \frac{10}{3} \approx 3.33$
• Equilibrium quantity: $x = 30$

To solve the given supply function:

$$x-20p+60=0$$

where $x$ is the supply and $p$ is the price, let’s address each part step by step.

Part (i): Find the supply ($x$) when $p = 6$

Step 1: Substitute $p = 6$ into the supply equation

The supply equation is:

$$x-20p+60=0$$

Substitute $p = 6$:

$$x-20(6)+60=0$$

Step 2: Simplify

1. $20(6) = 120$,
2. Substituting, $x – 120 + 60 = 0$,
3. Combine like terms: $x – 60 = 0$.

Step 3: Solve for $x$

$$x=60$$

Answer for Part (i): The supply when $p = 6$ is:

$$x=60$$

Part (ii): Find the price ($p$) when $x = 80$

Step 1: Substitute $x = 80$ into the supply equation

The supply equation is:

$$x-20p+60=0$$

Substitute $x = 80$:

$$80-20p+60=0$$

Step 2: Simplify

1. Combine constants: $80 + 60 = 140$,
2. Equation becomes: $140 – 20p = 0$.

Step 3: Solve for $p$

$$20p=140\Rightarrow p=\frac{140}{20}=7$$

Answer for Part (ii): The price when $x = 80$ is:

$$p=7$$

Part (iii): Find the price ($p$) when supply ($x$) is zero

Step 1: Substitute $x = 0$ into the supply equation

The supply equation is:

$$x-20p+60=0$$

Substitute $x = 0$:

$$0-20p+60=0$$

Step 2: Simplify

1. Equation becomes: $-20p + 60 = 0$,
2. Rearrange: $20p = 60$.

Step 3: Solve for $p$

$$p=\frac{60}{20}=3$$

Answer for Part (iii): The price at which supply is zero is:

$$p=3$$

Final Summary

1. Supply at $p = 6$ $x = 60$,
2. Price when $x = 80$: $p = 7$,
3. Price when supply is zero: $p = 3$.

$$C(Q)=Q^3-3Q^2+15Q+27$$

Here:

• $C(Q)$ is the total cost,
• $Q$ is the quantity of output,
• The total cost consists of fixed costs (FC) and variable costs (VC): $$C(Q)=VC(Q)+FC$$

From the function:

• The term $27$ is the fixed cost ($FC$) because it does not depend on $Q$,
• The remaining terms ($Q^3 – 3Q^2 + 15Q$) represent the variable cost ($VC$).

We will compute the following step by step:

(i) Average Cost ($AC$)

Formula for $AC$:

$$AC=\frac{C(Q)}{Q}$$

Substitute $C(Q) = Q^3 – 3Q^2 + 15Q + 27$:

$$AC=\frac{Q^3-3Q^2+15Q+27}{Q}$$

Simplify by dividing each term by $Q$:

$$AC=Q^2-3Q+15+\frac{27}{Q}$$

Final Expression for $AC$:

$$AC=Q^2-3Q+15+\frac{27}{Q}$$

(ii) Variable Cost ($VC$)

Definition of $VC$:

Variable cost is the part of the total cost that depends on the output $Q$. From the total cost function:

$$VC(Q)=Q^3-3Q^2+15Q$$

Final Expression for $VC$:

$$VC=Q^3-3Q^2+15Q$$

(iii) Average Fixed Cost ($AFC$)

Formula for $AFC$:

$$AFC=\frac{FC}{Q}$$

Fixed Cost ($FC$) from the total cost function:

$$FC=27$$

Substitute $FC = 27$ into the formula:

$$AFC=\frac{27}{Q}$$

Final Expression for $AFC$:

$$AFC=\frac{27}{Q}$$

(iv) Average Variable Cost ($AVC$)

Formula for $AVC$:

$$AVC=\frac{VC(Q)}{Q}$$

Substitute $VC(Q) = Q^3 – 3Q^2 + 15Q$:

$$AVC=\frac{Q^3-3Q^2+15Q}{Q}$$

Simplify by dividing each term by $Q$:

$$AVC=Q^2-3Q+15$$

Final Expression for $AVC$:

$$AVC=Q^2-3Q+15$$

Final Summary of Results

1. Average Cost ($AC$):

   $$AC=Q^2-3Q+15+\frac{27}{Q}$$

2. Variable Cost ($VC$):

   $$VC=Q^3-3Q^2+15Q$$

3. Average Fixed Cost ($AFC$):

   $$AFC=\frac{27}{Q}$$

4. Average Variable Cost ($AVC$):

   $$AVC=Q^2-3Q+15$$