Standard Operating Procedures (SOPs) for Solving Microeconomics Assignments
Based on the assignments & solutions 2024/2025 file, the following SOPs provide step-by-step guidelines to solve different types of microeconomics exercises.
1. Consumer Theory: Preferences and Utility Functions
Typical Problems:
Understanding preferences and utility functions Utility maximization subject to budget constraints Steps to Solve:
Identify the Utility Function: Determine whether it is Cobb-Douglas, Perfect Substitutes, Perfect Complements, or a Quasilinear utility function. Compute Marginal Utility (MU): MUx=∂U∂xMU_x = \frac{\partial U}{\partial x} MUy=∂U∂yMU_y = \frac{\partial U}{\partial y} Find the Marginal Rate of Substitution (MRS): MRS=MUxMUyMRS = \frac{MU_x}{MU_y} Set up the Budget Constraint: I=pxx+pyyI = p_x x + p_y y L=U(x,y)+λ(I−pxx−pyy)L = U(x, y) + \lambda (I - p_x x - p_y y) Derive first-order conditions and solve for optimal consumption levels. Interpret Results: Check if the solution makes economic sense (e.g., corner solutions for perfect substitutes). 2. Demand Theory: Marshallian and Hicksian Demand Functions
Typical Problems:
Deriving demand functions Substitution and income effects (Slutsky Equation) Compensating and Equivalent Variation Steps to Solve:
Identify the Demand Function Type: Marshallian Demand: Found by solving the utility maximization problem. Hicksian Demand: Found by solving the expenditure minimization problem. Use the Indirect Utility Function to Find Demand: x1=ap1Ix_1 = \frac{a}{p_1} I, x2=1−ap2Ix_2 = \frac{1-a}{p_2} I for Cobb-Douglas preferences. Apply the Slutsky Equation: dxdp=dxhdp−xdIdp\frac{dx}{dp} = \frac{dx^h}{dp} - x \frac{dI}{dp} Calculate the Compensating and Equivalent Variation: Compensating Variation (CV): Income required to maintain original utility after a price change. Equivalent Variation (EV): Income needed to achieve new utility before price change. 3. Production and Cost Functions
Typical Problems:
Finding the production function (Cobb-Douglas, CES, Leontief) Cost minimization and cost function derivation Returns to scale and marginal rate of technical substitution (MRTS) Steps to Solve:
Identify the Production Function Type: Cobb-Douglas: q=Akαlβq = A k^\alpha l^\beta CES: q=(akρ+blρ)1/ρq = (a k^\rho + b l^\rho)^{1/\rho} Compute Marginal Products: MPK=∂q∂KMP_K = \frac{\partial q}{\partial K} MPL=∂q∂LMP_L = \frac{\partial q}{\partial L} MRTSKL=MPKMPLMRTS_{KL} = \frac{MP_K}{MP_L} Set Up the Cost Minimization Problem: Minimize C=wL+rKC = wL + rK subject to q=f(K,L)q = f(K, L). Use Lagrangian: L=wL+rK+λ(q−f(K,L))L = wL + rK + \lambda (q - f(K, L)). Interpret Returns to Scale: Increasing: f(tK,tL)>tf(K,L)f(tK, tL) > t f(K, L) Constant: f(tK,tL)=tf(K,L)f(tK, tL) = t f(K, L) Decreasing: f(tK,tL)<tf(K,L)f(tK, tL) < t f(K, L) 4. Market Structures: Perfect Competition, Monopoly, Oligopoly
Typical Problems:
Profit maximization under different market structures Price and quantity determination Game theory applications (Nash Equilibrium, Best Response Functions) Steps to Solve:
Identify the firm’s cost function and marginal cost MCMC. Find equilibrium where MC=MR=PMC = MR = P. Check for short-run vs. long-run equilibrium. Use demand function P=f(Q)P = f(Q). Find Total Revenue: TR=P(Q)×QTR = P(Q) \times Q. Derive Marginal Revenue: MR=dTRdQMR = \frac{dTR}{dQ}. Solve MR=MCMR = MC for equilibrium price and quantity. For Oligopoly (Cournot and Bertrand Models): Set up best response functions for firms. Solve for Nash equilibrium using reaction functions. 5. Game Theory and Strategic Behavior
Typical Problems:
Solving normal-form and extensive-form games Finding Nash Equilibria (pure and mixed strategies) Analyzing collusion and cartels Steps to Solve:
Construct the Payoff Matrix: Identify strategies and corresponding payoffs for each player. Highlight dominant strategies. A set of strategies where no player has an incentive to deviate. Check for Mixed Strategies: Solve for probability distributions where expected payoffs are equal. Use the Folk Theorem to determine equilibrium strategies. 6. Externalities and Market Failures
Typical Problems:
Finding socially optimal quantity Pigouvian taxes and subsidies Coase theorem applications Steps to Solve:
Determine the Marginal Social Cost (MSC) and Marginal Social Benefit (MSB). Find the Socially Optimal Output: Tax = Difference between MSC and Private Cost. Check if property rights and bargaining can resolve the externality. 7. Public Goods and Asymmetric Information
Typical Problems:
Solving for Nash equilibrium in public goods provision Calculating optimal provision level Analyzing moral hazard and adverse selection Steps to Solve:
Determine the Public Good’s Characteristics: Non-rivalry and non-excludability. Set Up the Samuelson Condition: ∑MRSi=MRT\sum MRS_i = MRT (Sum of Marginal Benefits = Marginal Cost). Analyze Free Rider Problem: Check if individual contributions match optimal provision. For Asymmetric Information: Use Screening and Signaling models. Final Notes
Use Graphs: Many microeconomic problems are best understood using graphs. Check Second-Order Conditions: Ensure the solution satisfies concavity/convexity conditions. Interpret Economic Meaning: Always relate mathematical results to real-world implications. These SOPs will help in solving various types of microeconomic exercises efficiently.
Below is a “cook‐book” style set of Standard Operating Procedures (SOPs) that cover many common microeconomic exercises. These SOPs have been written in great detail so that even someone completely new to microeconomics can follow the steps. The instructions below are based on the types of problems found in the “assignments & solutions 20242025.pdf” file and are organized by topic. In each SOP, we include variations (for example, when to use a Lagrangian versus direct substitution) and explain every subtask step by step.
SOP 1. Refreshing Your Differentiation Skills
Many microeconomic problems start with taking derivatives. Use these detailed steps whenever you see a function you need to differentiate (for example, to find marginal utility or the slope of a production function).
Identify the Function & Rewrite if Needed: Look at the function. For example, if you have a radical like √x, rewrite it as x0.5x^{0.5}. If the function is given as a fraction, decide whether to use the quotient rule directly or rewrite it as a product (e.g., 1/x1/x as x−1x^{-1}). Power Rule: ddxxn=n⋅xn−1\frac{d}{dx} x^n = n \cdot x^{n-1} Constant Multiple Rule: ddx[a⋅f(x)]=a⋅f′(x)\frac{d}{dx}[a \cdot f(x)] = a \cdot f'(x) Sum/Difference Rule: ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) Apply Product, Quotient, or Chain Rules (as needed): Product Rule: If f(x)=u(x)⋅v(x)f(x) = u(x) \cdot v(x), then f′(x)=u′(x)v(x)+u(x)v′(x)f'(x) = u'(x)v(x) + u(x)v'(x) Quotient Rule: If f(x)=u(x)v(x)f(x) = \frac{u(x)}{v(x)}, then f′(x)=u′(x)v(x)−u(x)v′(x)[v(x)]2f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} Chain Rule: If f(x)=h(g(x))f(x) = h(g(x)), then f′(x)=h′(g(x))⋅g′(x)f'(x) = h'(g(x)) \cdot g'(x) Differentiate Step by Step: Write down each term separately. Apply the rule that fits best. For example, for f(x)=ln?(2x2+3x)f(x) = \ln(2x^2+3x): First, note that the derivative of ln?(u)\ln(u) is 1u⋅u′\frac{1}{u} \cdot u'. Then compute u′u' for u=2x2+3xu = 2x^2+3x using the power rule. Combine your results carefully. Rewrite expressions in simplest form to check for cancellation or combining like terms. Double-Check Using an Alternate Approach: If possible, solve the same derivative by rewriting the function (for example, convert a quotient into a product with a negative exponent) and verify the answer is the same. SOP 2. Solving a Utility Maximization Problem (Consumer Theory)
This SOP explains how to solve a standard consumer choice problem using a Lagrangian approach. (Many assignments use utility functions like Stone‐Geary, Cobb–Douglas, or Quasilinear.)
Write Down the Given Information: Utility Function: For instance, the Stone–Geary function U(x1,x2)=(x1−c1)b(x2−c2)1−bU(x_1, x_2) = (x_1 - c_1)^b (x_2 - c_2)^{1-b} Budget Constraint: I=p1x1+p2x2I = p_1 x_1 + p_2 x_2 List all known parameters (prices, income, constants c1,c2,bc_1, c_2, b).e Write the Lagrangian function L\mathcal{L}: L=U(x1,x2)+λ(I−p1x1−p2x2)\mathcal{L} = U(x_1, x_2) + \lambda \left( I - p_1 x_1 - p_2 x_2 \right) Here, λ\lambda is the Lagrange multiplier that represents the marginal utility of income. Find the First-Order Conditions (FOCs): Differentiate L\mathcal{L} with respect to each choice variable and λ\lambda: ∂L∂x1=∂U∂x1−λp1=0\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial U}{\partial x_1} - \lambda p_1 = 0 ∂L∂x2=∂U∂x2−λp2=0\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial U}{\partial x_2} - \lambda p_2 = 0 ∂L∂λ=I−p1x1−p2x2=0\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_1 x_1 - p_2 x_2 = 0 For the Stone–Geary utility, compute the partial derivatives carefully using the chain rule. Solve the System of Equations: Step 4a: Divide the FOC for x1x_1 by the FOC for x2x_2 to eliminate λ\lambda. This usually yields an equation equating the marginal rate of substitution (MRS) to the price ratio: MUx1MUx2=p1p2\frac{MU_{x_1}}{MU_{x_2}} = \frac{p_1}{p_2} Step 4b: Solve this equation for one variable in terms of the other. Step 4c: Substitute the expression into the budget constraint to solve for the numerical values of x1x_1 and x2x_2. Check Second-Order Conditions (Optional but Recommended): Verify that the second-order conditions (concavity of the utility function or negative definiteness of the bordered Hessian) are satisfied to ensure a maximum. Express the optimal consumption bundle (x1∗,x2∗)(x_1^*, x_2^*) in terms of income II, prices p1,p2p_1, p_2, and constants. Note any conditions (for example, if a corner solution might occur when the minimum consumption c1c_1 or c2c_2 is binding). Alternate Method – Direct Substitution: In simpler cases (like Cobb–Douglas), you may solve by expressing one variable in terms of the other from the budget line and substituting directly into the utility function. Explain that this method is an alternative when the utility function is easily separable. SOP 3. Deriving Marshallian (Uncompensated) Demand Functions
Once the utility maximization problem is solved, you need to derive the Marshallian demand functions (how much of each good is purchased).
Start from the FOCs or the Optimized Expressions: Use the results from your utility maximization problem. For instance, if you obtained an equation like: x1−c1x2−c2=b1−bp1p2\frac{x_1 - c_1}{x_2 - c_2} = \frac{b}{1-b}\frac{p_1}{p_2} solve for x1x_1 in terms of x2x_2 (or vice versa). Substitute into the Budget Constraint: Replace the variable (e.g., x1x_1) in the budget constraint I=p1x1+p2x2I = p_1 x_1 + p_2 x_2 with the expression derived in step 1. Solve the resulting equation for the remaining variable. Solve for Each Demand Function: Once one variable (say x2x_2) is determined in terms of II, p1p_1, p2p_2, and constants, substitute it back to get x1x_1. Your final answers should be functions: x1∗=f1(I,p1,p2,c1,c2,b)x_1^* = f_1(I, p_1, p_2, c_1, c_2, b) x2∗=f2(I,p1,p2,c1,c2,b)x_2^* = f_2(I, p_1, p_2, c_1, c_2, b) If the consumer’s income is just high enough to cover the subsistence levels c1c_1 and c2c_2, note that the demands may “kink” (corner solutions). For other utility types (e.g., Cobb–Douglas), the procedure is similar, and the demands often have the form: x1∗=aIp1,x2∗=(1−a)Ip2x_1^* = \frac{aI}{p_1},\quad x_2^* = \frac{(1-a)I}{p_2} Check Economic Intuition: Verify that as income II increases, both x1∗x_1^* and x2∗x_2^* increase. Confirm that if the price of a good rises, its demand falls (all else equal). SOP 4. Calculating Elasticities of Demand
Elasticity measures the responsiveness of demand to changes in prices or income. Use these steps once you have your Marshallian demand functions.
Identify the Demand Function: For example, suppose x(p,I)x(p,I) is your demand for good xx. Own-Price Elasticity Calculation: Use the formula: εx,p=∂x∂p⋅px\varepsilon_{x,p} = \frac{\partial x}{\partial p} \cdot \frac{p}{x} Differentiate the demand function xx with respect to its own price pp. Multiply the derivative by the ratio px\frac{p}{x}. Interpretation: A 1% increase in price will decrease demand by ∣εx,p∣|\varepsilon_{x,p}|% if the elasticity is negative. Income Elasticity Calculation: Use the formula: εx,I=∂x∂I⋅Ix\varepsilon_{x,I} = \frac{\partial x}{\partial I} \cdot \frac{I}{x} Differentiate xx with respect to II and multiply by Ix\frac{I}{x}. Cross-Price Elasticity (if applicable): For two goods xx and yy: εx,py=∂x∂py⋅pyx\varepsilon_{x, p_y} = \frac{\partial x}{\partial p_y} \cdot \frac{p_y}{x} Substitute typical values (or leave in algebraic form) and explain what the elasticity implies about consumer behavior. SOP 5. Deriving the Indirect Utility Function
The indirect utility function expresses the maximum utility achieved given prices and income.
Start with the Optimal (Marshallian) Demands: You should have functions x1∗(p1,p2,I)x_1^*(p_1, p_2, I) and x2∗(p1,p2,I)x_2^*(p_1, p_2, I) from the previous SOP. Substitute into the Utility Function: Replace x1x_1 and x2x_2 in the original utility function U(x1,x2)U(x_1, x_2) with x1∗x_1^* and x2∗x_2^*. For example: V(p1,p2,I)=U(x1∗(p1,p2,I), x2∗(p1,p2,I))V(p_1, p_2, I) = U\Bigl(x_1^*(p_1, p_2, I),\, x_2^*(p_1, p_2, I)\Bigr) Simplify algebraically so that VV is written purely as a function of p1p_1, p2p_2, and II. Explain that the indirect utility function shows the highest utility achievable at given market conditions. It is “indirect” because it does not show the direct choice but rather the resulting level of satisfaction. SOP 6. Deriving the Expenditure Function
The expenditure function tells you the minimum expenditure needed to achieve a given utility level at given prices.
Start with the Indirect Utility Function: Write V(p1,p2,I)=UˉV(p_1, p_2, I) = \bar{U} where Uˉ\bar{U} is the target utility level. Rearrange the equation to express II as a function of Uˉ\bar{U} and prices p1p_1 and p2p_2. I=e(p1,p2,Uˉ)I = e(p_1, p_2, \bar{U}) This e(p1,p2,Uˉ)e(p_1, p_2, \bar{U}) is the expenditure function. Alternate Approach – Duality: In some cases, you can set up and solve the expenditure minimization problem directly: Minimize: E=p1x1+p2x2E = p_1 x_1 + p_2 x_2 Subject to: U(x1,x2)=UˉU(x_1, x_2) = \bar{U} Use the Lagrangian method as in the utility maximization problem but with the constraint reversed. Interpret the Expenditure Function: Explain that this function tells the least amount of money the consumer must spend to reach the desired utility level given current prices. It is useful for welfare analysis, for example, in calculating compensating variation. 
