BECC-105 Solved Assignment

INTERMEDIATE MICROECONOMICS – I

Assignment 1

Assignment 2

3. Given the profit function
   $\pi =160x-3x^2-2xy-2y^2+120y-18$$\pi =160x-3x^2-2xy-2y^2+120y-18$pi=160 x-3x^(2)-2xy-2y^(2)+120 y-18\pi=160 x-3 x^2-2 x y-2 y^2+120 y-18$\pi =160x-3x^2-2xy-2y^2+120y-18$ for a firm producing two goods x and y
   Find out (i) the maximizing profits
   (ii) test the second order condition.
4. Distinguish between price elasticity of demand and income elasticity of demand. Given $\mathrm{Q}=700$$\mathrm{Q}=700$Q=700\mathrm{Q}=700$\mathrm{Q}=700$ –

Assignment 3

6. Explain the properties of preferences with example.
7. What is consumer surplus? State the relationship between consumer surplus, compensating variation and equivalent variation.
8. What is risk aversion? How does insurance help in reducing risk? Illustrate.
9. What is CES production function? How does CES production function approaches a Leontief Production function?
10. Make distinction between any three of the following:
    (i) Concave function and convex function.
    (ii) Expected value and Expected utility.
    (iii) General equilibrium and partial equilibrium.
    (iv) Marginal Rate of Substitution and Marginal Rate of Technical Substitution.

Expert Answer:

Assignment 1

Answer:

Budget Constraint

• $P_1$$P_1$P_(1)P_1$P_1$ and $P_2$$P_2$P_(2)P_2$P_2$ are the prices of goods 1 and 2.
• $X_1$$X_1$X_(1)X_1$X_1$ and $X_2$$X_2$X_(2)X_2$X_2$ are the quantities of goods 1 and 2.
• $I$$I$II$I$ is the consumer’s income.

Impact of Income Increase and Price Decrease

Illustration with a Diagram

• The initial budget constraint (BC1) represents the consumer’s original income and prices.
• When the consumer’s income increases and the price of one good decreases, the budget constraint shifts outward (BC2).

• The blue line represents the initial budget constraint, which shows the combinations of goods the consumer can afford with their initial income and prices.
• The green dashed line represents the new budget constraint after the consumer’s income increases and the price of one good decreases.

Answer:

Utility Function

• $X_1,X_2,...,X_n$$X_1,X_2,...,X_n$X_(1),X_(2),…,X_(n)X_1, X_2, …, X_n$X_1,X_2,...,X_n$ represent the quantities of different goods consumed.
• $U$$U$UU$U$ is the utility derived from the consumption bundle $(X_1,X_2,...,X_n)$$(X_1,X_2,...,X_n)$(X_(1),X_(2),…,X_(n))(X_1, X_2, …, X_n)$(X_1,X_2,...,X_n)$.

Direct Utility Function vs. Indirect Utility Function

1. Direct Utility Function:
   • The direct utility function describes the utility derived directly from the consumption of goods and services. It focuses on how much utility the consumer gets from different consumption bundles, assuming they have certain quantities of goods.
   • It can be expressed as:$$U(X_1,X_2,...,X_n)$$$$U(X_1,X_2,...,X_n)$$U(X_(1),X_(2),…,X_(n))U(X_1, X_2, …, X_n)$$U(X_1,X_2,...,X_n)$$
   • The direct utility function does not consider the prices of goods or the consumer’s income but purely reflects their preferences.
2. Indirect Utility Function:
   • The indirect utility function describes the maximum utility that a consumer can achieve given their income and the prices of goods. It takes into account the consumer’s budget constraint.
   • The indirect utility function is derived from the optimization of the consumer’s utility under the budget constraint and is expressed as:$$V(I,P_1,P_2,...,P_n)$$$$V(I,P_1,P_2,...,P_n)$$V(I,P_(1),P_(2),…,P_(n))V(I, P_1, P_2, …, P_n)$$V(I,P_1,P_2,...,P_n)$$
   • Where $I$$I$II$I$ is the consumer’s income and $P_1,P_2,...,P_n$$P_1,P_2,...,P_n$P_(1),P_(2),…,P_(n)P_1, P_2, …, P_n$P_1,P_2,...,P_n$ are the prices of the goods.
   • It represents the highest level of utility that can be attained given income and prices.

Utility Functions for Perfect Substitutes and Perfect Complements

1. Perfect Substitutes:

• Goods are considered perfect substitutes when a consumer is willing to substitute one good for another at a constant rate.
• The utility function for perfect substitutes is usually of the form:$$U(X_1,X_2)=a{X}_1+b{X}_2$$$$U(X_1,X_2)=a{X}_1+b{X}_2$$U(X_(1),X_(2))=aX_(1)+bX_(2)U(X_1, X_2) = aX_1 + bX_2$$U(X_1,X_2)=a{X}_1+b{X}_2$$
• Where $a$$a$aa$a$ and $b$$b$bb$b$ are constants representing the rate at which one good can be substituted for the other without changing the overall utility.

2. Perfect Complements:

• Goods are considered perfect complements when they are always consumed together in fixed proportions, such as left and right shoes.
• The utility function for perfect complements is typically of the form:$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$U(X_(1),X_(2))=min(aX_(1),bX_(2))U(X_1, X_2) = \min(aX_1, bX_2)$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$
• Here, $a$$a$aa$a$ and $b$$b$bb$b$ are constants that indicate the fixed proportion in which the goods must be consumed to generate utility.

Summary of Mathematical Expressions

1. Perfect Substitutes Utility Function:$$U(X_1,X_2)=a{X}_1+b{X}_2$$$$U(X_1,X_2)=a{X}_1+b{X}_2$$U(X_(1),X_(2))=aX_(1)+bX_(2)U(X_1, X_2) = aX_1 + bX_2$$U(X_1,X_2)=a{X}_1+b{X}_2$$
2. Perfect Complements Utility Function:$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$U(X_(1),X_(2))=min(aX_(1),bX_(2))U(X_1, X_2) = \min(aX_1, bX_2)$$U(X_1,X_2)=min(a{X}_1,b{X}_2)$$

Answer:

Cost Minimization

Approaches to Cost Minimization

1. Isoquant-Isocost Approach
2. Marginal Product Approach

1. Isoquant-Isocost Approach

• Isoquant: An isoquant represents all combinations of two inputs (say labor and capital) that can produce a given level of output. It is similar to an indifference curve in utility analysis but applies to production.
• Isocost Line: An isocost line represents all combinations of two inputs that have the same total cost. The equation for an isocost line is:
  $$C=wL+rK$$$$C=wL+rK$$C=wL+rKC = wL + rK$$C=wL+rK$$
  Where:
  • $C$$C$CC$C$ is the total cost,
  • $w$$w$ww$w$ is the wage rate (cost of labor),
  • $r$$r$rr$r$ is the rental rate of capital (cost of capital),
  • $L$$L$LL$L$ is the quantity of labor,
  • $K$$K$KK$K$ is the quantity of capital.

• $M{P}_L$$M{P}_L$MP_(L)MP_L$M{P}_L$ is the marginal product of labor,
• $M{P}_K$$M{P}_K$MP_(K)MP_K$M{P}_K$ is the marginal product of capital.

2. Marginal Product Approach

• If the marginal product per dollar of labor is higher than that of capital, the firm should hire more labor and use less capital.
• If the marginal product per dollar of capital is higher than that of labor, the firm should use more capital and less labor.

Illustration of Cost Minimization

• Labor costs: $10 per unit.
• Capital costs: $20 per unit.
• The firm needs to decide how much labor and capital to use to produce 100 units of output.

• The isoquant curve shows different combinations of labor and capital that can produce 100 units of output.
• The isocost line shows different combinations of labor and capital that can be hired for a given total cost.

• The cost-minimizing point occurs where the isoquant for 100 units of output is tangent to the isocost line. At this point, the marginal rate of technical substitution (MRTS) of labor for capital is equal to the ratio of the prices of labor and capital:$$\frac{M{P}_L}{M{P}_K}=\frac{w}{r}$$$$\frac{M{P}_L}{M{P}_K}=\frac{w}{r}$$(MP_(L))/(MP_(K))=(w)/(r)\frac{MP_L}{MP_K} = \frac{w}{r}$$\frac{M{P}_L}{M{P}_K}=\frac{w}{r}$$

• Suppose at the tangency point, the firm uses 5 units of labor and 3 units of capital to produce the 100 units of output. The total cost at this point is:$$\text{Total Cost}=wL+rK=(10\times 5)+(20\times 3)=50+60=110$$$$\text{Total Cost}=wL+rK=(10\times 5)+(20\times 3)=50+60=110$$“Total Cost”=wL+rK=(10 xx5)+(20 xx3)=50+60=110\text{Total Cost} = wL + rK = (10 \times 5) + (20 \times 3) = 50 + 60 = 110$$\text{Total Cost}=wL+rK=(10\times 5)+(20\times 3)=50+60=110$$
• This combination minimizes the total cost of production for 100 units of output.

• The blue curve is the isoquant, representing all the combinations of labor and capital that produce a given level of output (100 units in this case).
• The green dashed line is the isocost line, which shows all combinations of labor and capital that result in a total cost of $110.

Answer:

Given Total Cost Function:

(i) The Average Cost Function

(ii) The Critical Value at Which AC is Minimized

1. Differentiate the AC function:

2. Set the derivative equal to zero to find the critical point:

(iii) The Marginal Cost

Summary of Results:

1. Average Cost function (AC):$$AC=Q^2-5Q+60$$$$AC=Q^2-5Q+60$$AC=Q^(2)-5Q+60AC = Q^2 – 5Q + 60$$AC=Q^2-5Q+60$$
2. Critical value at which AC is minimized:$$Q=2.5$$$$Q=2.5$$Q=2.5Q = 2.5$$Q=2.5$$
3. Marginal Cost function (MC):$$MC=3Q^2-10Q+60$$$$MC=3Q^2-10Q+60$$MC=3Q^(2)-10 Q+60MC = 3Q^2 – 10Q + 60$$MC=3Q^2-10Q+60$$

Assignment 2

Answer:

(i) Finding the Maximum Profit

1. Partial derivative with respect to $x$$x$xx$x$:

2. Partial derivative with respect to $y$$y$yy$y$:

(ii) Testing the Second-Order Condition

1. Second-order partial derivatives:

2. Determinant of the Hessian:

3. Principal minors:

• The first principal minor is $H_{11}=-6$$H_{11}=-6$H_(11)=-6H_{11} = -6$H_{11}=-6$, which is negative.
• The determinant of the Hessian $\text{Det}(H)=20$$\text{Det}(H)=20$“Det”(H)=20\text{Det}(H) = 20$\text{Det}(H)=20$, which is positive.

Conclusion

1. The maximizing profit is $\pi =4782$$\pi =4782$pi=4782\pi = 4782$\pi =4782$ at $x=20$$x=20$x=20x = 20$x=20$ and $y=20$$y=20$y=20y = 20$y=20$.
2. The second-order condition is satisfied, confirming that this is a maximum point.

Answer:

Price Elasticity of Demand vs. Income Elasticity of Demand

Given Information:

• $P=25$$P=25$P=25P = 25$P=25$
• $y=500$$y=500$y=500y = 500$y=500$

Step 1: Calculate the Quantity Demanded (Q)

(i) Price Elasticity of Demand (PED)

1. First, calculate the partial derivative of $Q$$Q$QQ$Q$ with respect to $P$$P$PP$P$:

2. Now, use the formula for Price Elasticity of Demand:

(ii) Income Elasticity of Demand (YED)

1. First, calculate the partial derivative of $Q$$Q$QQ$Q$ with respect to $y$$y$yy$y$:

2. Now, use the formula for Income Elasticity of Demand:

Summary:

1. Price Elasticity of Demand (PED): $-0.076$$-0.076$-0.076-0.076$-0.076$
2. Income Elasticity of Demand (YED): $0.015$$0.015$0.0150.015$0.015$

Answer:

Walrasian Equilibrium and Pareto Optimality

Definition of Walrasian Equilibrium

1. Consumers maximize utility subject to their budget constraints.
2. Firms maximize profit subject to production constraints.
3. Market clearing conditions are satisfied (i.e., total demand equals total supply for each good).

Definition of Pareto Optimality

The First Fundamental Theorem of Welfare Economics

• Every Walrasian (competitive) equilibrium is Pareto optimal, provided that certain conditions hold, including:
  1. Perfect competition: No individual agent has the power to influence prices.
  2. Complete markets: Every possible good or service is traded.
  3. No externalities: Consumption or production of goods does not directly affect other agents.
  4. Convex preferences and production sets: Consumers have convex preferences (indifference curves are convex to the origin), and production sets are convex.

Proof of Pareto Optimality in Walrasian Equilibrium

Step 1: Utility Maximization

• $U_i(x_i)$$U_i(x_i)$U_(i)(x_(i))U_i(x_i)$U_i(x_i)$ is the utility function of consumer $i$$i$ii$i$,
• $p$$p$pp$p$ is the price vector,
• $e_i$$e_i$e_(i)e_i$e_i$ is the endowment vector for consumer $i$$i$ii$i$.

Step 2: Profit Maximization

• $y_j$$y_j$y_(j)y_j$y_j$ is the output level of firm $j$$j$jj$j$,
• $C_j(y_j)$$C_j(y_j)$C_(j)(y_(j))C_j(y_j)$C_j(y_j)$ is the cost of producing output $y_j$$y_j$y_(j)y_j$y_j$.

Step 3: Market Clearing

Step 4: Proving Pareto Optimality

1. Some consumer is strictly better off (i.e., their utility is higher with the new allocation).
2. No other consumer is worse off.

Conclusion

Assignment 3

Answer:

1. Completeness

• Definition: Preferences are complete if, for any two bundles of goods $A$$A$AA$A$ and $B$$B$BB$B$, the consumer can say whether they prefer $A$$A$AA$A$ to $B$$B$BB$B$, prefer $B$$B$BB$B$ to $A$$A$AA$A$, or are indifferent between them. In other words, the consumer has a clear preference ranking for all possible bundles.
• Example: Suppose a consumer is choosing between two bundles: Bundle $A$$A$AA$A$ (3 apples and 2 bananas) and Bundle $B$$B$BB$B$ (2 apples and 3 bananas). If preferences are complete, the consumer can either:
  • Prefer Bundle $A$$A$AA$A$ to Bundle $B$$B$BB$B$,
  • Prefer Bundle $B$$B$BB$B$ to Bundle $A$$A$AA$A$, or
  • Be indifferent between them.

2. Transitivity

• Definition: Preferences are transitive if, for any three bundles $A$$A$AA$A$, $B$$B$BB$B$, and $C$$C$CC$C$, if the consumer prefers $A$$A$AA$A$ to $B$$B$BB$B$ and $B$$B$BB$B$ to $C$$C$CC$C$, then they must also prefer $A$$A$AA$A$ to $C$$C$CC$C$.
• Example: Suppose a consumer prefers:
  • Bundle $A$$A$AA$A$ (3 apples, 2 bananas) over Bundle $B$$B$BB$B$ (2 apples, 3 bananas),
  • Bundle $B$$B$BB$B$ over Bundle $C$$C$CC$C$ (1 apple, 4 bananas).

3. Reflexivity

• Definition: Preferences are reflexive if, for any bundle $A$$A$AA$A$, the consumer considers $A$$A$AA$A$ at least as good as itself. In other words, any bundle is at least as good as itself.
• Example: Bundle $A$$A$AA$A$ (2 apples, 3 bananas) must be considered at least as good as Bundle $A$$A$AA$A$. This property is straightforward but ensures that preferences are self-consistent.

4. Monotonicity (or Non-Satiation)

• Definition: Preferences are monotonic if more of a good is always preferred to less, assuming that all other goods remain constant. This means that "more is better" and consumers prefer larger quantities of goods.
• Example: A consumer has a choice between two bundles:
  • Bundle $A$$A$AA$A$: 3 apples, 2 bananas,
  • Bundle $B$$B$BB$B$: 4 apples, 2 bananas.

5. Convexity

• Definition: Preferences are convex if consumers prefer averages or combinations of goods to extremes. This implies that consumers prefer diversified bundles of goods rather than putting all resources into one good.
• Example: Suppose a consumer has the following choice between two bundles:
  • Bundle $A$$A$AA$A$: 6 apples, 0 bananas,
  • Bundle $B$$B$BB$B$: 0 apples, 6 bananas.

• Bundle $C$$C$CC$C$: 3 apples, 3 bananas,
  because it offers a mix of both goods, which is more satisfying than an extreme bundle of only one good.

6. Continuity

• Definition: Preferences are continuous if small changes in the quantities of goods do not cause sudden jumps in the ranking of bundles. That is, if Bundle $A$$A$AA$A$ is preferred to Bundle $B$$B$BB$B$, then bundles very close to $A$$A$AA$A$ should also be preferred to $B$$B$BB$B$.
• Example: If a consumer prefers a bundle of 4 apples and 3 bananas over a bundle of 3 apples and 3 bananas, they should also prefer a bundle of 3.9 apples and 3 bananas over 3 apples and 3 bananas. Small changes in quantities should not lead to a sudden switch in preferences.

Example of Preferences with All Properties

• Bundle $A$$A$AA$A$: 3 apples, 2 bananas
• Bundle $B$$B$BB$B$: 2 apples, 3 bananas
• Bundle $C$$C$CC$C$: 1 apple, 4 bananas

• $A$$A$AA$A$ is preferred to $B$$B$BB$B$,
• $B$$B$BB$B$ is preferred to $C$$C$CC$C$.

Answer:

Consumer Surplus

• Example: If a consumer is willing to pay $100 for a product but buys it for $80, the consumer surplus is $20.

Compensating Variation and Equivalent Variation

1. Compensating Variation (CV):
   • Definition: Compensating variation is the amount of money that would need to be given to a consumer after a price increase (or taken away after a price decrease) to compensate them for the price change, so they reach their original utility level.
   • Interpretation: It measures how much compensation is needed to offset the harm from a price increase or how much benefit is gained from a price decrease.
   • Example: If the price of a good rises, compensating variation is the amount of money that must be given to the consumer to make them as well off as they were before the price increase.
2. Equivalent Variation (EV):
   • Definition: Equivalent variation is the amount of money that would need to be taken away from (or given to) the consumer before a price change occurs so that they would be as well off as after the price change.
   • Interpretation: It measures how much money would have to be taken from the consumer to make them as bad off as they would be after a price increase or how much money would need to be given to them to match the welfare gain from a price decrease.
   • Example: If the price of a good rises, equivalent variation is the amount of money that would need to be taken from the consumer (before the price rise) to make them as bad off as they would be after the price rise.

Relationship Between Consumer Surplus, Compensating Variation, and Equivalent Variation

• Consumer Surplus (CS): This is a simpler, more commonly used measure of welfare changes and is based on the consumer’s demand curve. It measures the area between the demand curve and the price line but assumes no changes in income or utility.
• Compensating Variation (CV): This measures how much compensation is needed to restore the consumer to their original utility level after a price change. CV focuses on the consumer’s post-change utility level and ensures that the consumer is made as well off as they were before the price change.
• Equivalent Variation (EV): This measures how much money would have to be taken away from (or given to) the consumer to bring them to the same utility level as after the price change. EV focuses on the consumer’s pre-change utility level and asks how much income would equate to the change in welfare.

Key Points of Relationship:

1. Direction of Measurement:
   • CS measures the direct difference between what consumers are willing to pay and what they actually pay, without considering income effects or utility changes.
   • CV measures how much compensation would restore consumers to their original welfare (after the price change).
   • EV measures how much income would equate to the change in utility before the price change occurs.
2. Ordering:
   • For a price increase: $EV>CS>CV$$EV>CS>CV$EV > CS > CVEV > CS > CV$EV>CS>CV$
   • For a price decrease: $CV>CS>EV$$CV>CS>EV$CV > CS > EVCV > CS > EV$CV>CS>EV$
   This ordering reflects the fact that CV and EV take into account the full welfare impact of price changes, including income effects, whereas consumer surplus is a simpler approximation.

Graphical Illustration:

• Consumer Surplus (CS) is the area between the demand curve and the price line.
• Compensating Variation (CV) measures the change in income that would compensate for the change in utility after the price change.
• Equivalent Variation (EV) measures the change in income that would be needed before the price change to bring the consumer to the same utility level they would experience after the price change.

Summary:

• Consumer Surplus: Measures the net benefit consumers receive from purchasing a good at a lower price than they are willing to pay.
• Compensating Variation (CV): Measures how much income is needed to restore the consumer to their original utility level after a price change.
• Equivalent Variation (EV): Measures how much income would equate to the change in utility before the price change occurs.

Answer:

Risk Aversion

Example:

1. A certain gain of $50.
2. A gamble with a 50% chance of winning $100 and a 50% chance of winning nothing.

How Insurance Reduces Risk

1. Risk Pooling: Insurers pool the risk of many individuals. Not all policyholders will experience adverse events at the same time, so the insurance company can pay claims from the pool of premiums collected from all policyholders.
2. Predictable Premium: For a fixed cost (the insurance premium), the individual avoids the possibility of a large financial loss. This reduction in uncertainty is particularly appealing to risk-averse individuals, as they prefer known outcomes to uncertain, potentially damaging outcomes.
3. Compensation for Losses: In the event of a loss, the insurance company compensates the insured party according to the terms of the policy, thereby reducing the financial impact of the loss.

Illustration of Risk Reduction through Insurance

• Suppose an individual faces a 10% probability of incurring a loss of $10,000 (e.g., due to an accident or damage) in a given year, and a 90% probability of incurring no loss. The expected loss would be:
  $$\text{Expected Loss}=(0.10\times 10,000)+(0.90\times 0)=1,000$$$$\text{Expected Loss}=(0.10\times 10,000)+(0.90\times 0)=1,000$$“Expected Loss”=(0.10 xx10,000)+(0.90 xx0)=1,000\text{Expected Loss} = (0.10 \times 10,000) + (0.90 \times 0) = 1,000$$\text{Expected Loss}=(0.10\times 10,000)+(0.90\times 0)=1,000$$
• Without insurance, the individual faces two possibilities:
  • A 10% chance of losing $10,000.
  • A 90% chance of losing nothing.

• With insurance, the individual can pay a premium of, say, $1,200 annually. In exchange, the insurance company will cover the $10,000 loss if it occurs. The individual now faces a certain outcome: paying $1,200 every year, regardless of whether the loss occurs or not.

Graphical Illustration of Risk Aversion and Insurance

• The blue curve represents the utility of wealth for a risk-averse individual, which is concave due to diminishing marginal utility (more wealth increases utility, but at a decreasing rate).
• The green point shows the utility at the individual’s wealth if no loss occurs.
• The red point shows the utility at the lower wealth level if a loss occurs.
• The orange point represents the expected utility without insurance, which lies between the green and red points due to the uncertainty of loss.
• The purple point represents the utility with insurance, where the individual pays a fixed premium (shown as a certain reduction in wealth).

Conclusion:

Answer:

CES Production Function

• $Q$$Q$QQ$Q$ is the quantity of output,
• $A$$A$AA$A$ is a productivity parameter (scale factor),
• $\alpha$$\alpha$alpha\alpha$\alpha$ and $1-\alpha$$1-\alpha$1-alpha1 – \alpha$1-\alpha$ are distribution parameters representing the input shares,
• $K$$K$KK$K$ is the quantity of capital,
• $L$$L$LL$L$ is the quantity of labor,
• $\rho$$\rho$rho\rho$\rho$ is the substitution parameter, which determines the elasticity of substitution between inputs.

Properties of the CES Production Function

• Elasticity of Substitution: The CES function allows for different degrees of substitutability between inputs (labor and capital). The elasticity of substitution determines how easily labor and capital can be substituted for one another in the production process.
• Special Cases: Depending on the value of $\rho$$\rho$rho\rho$\rho$, the CES function can represent different types of production functions:
  • If $\rho =1$$\rho =1$rho=1\rho = 1$\rho =1$, the function becomes a linear production function (perfect substitutes).
  • If $\rho =0$$\rho =0$rho=0\rho = 0$\rho =0$, the function reduces to a Cobb-Douglas production function, where the elasticity of substitution is 1.
  • If $\rho \to -\mathrm{\infty }$$\rho \to -\mathrm{\infty }$rho rarr-oo\rho \to -\infty$\rho \to -\mathrm{\infty }$, the function approaches a Leontief production function (perfect complements).

Leontief Production Function

• $K$$K$KK$K$ is the quantity of capital,
• $L$$L$LL$L$ is the quantity of labor,
• $a$$a$aa$a$ and $b$$b$bb$b$ are constants that represent the fixed input coefficients.

How CES Production Function Approaches the Leontief Production Function