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Combined SOPs

Below is a single, consolidated set of Standard Operating Procedures (SOPs) for the most common microeconomics exercises, combining the major points from both SOP documents (“Different exercises SOPs” and “NLM SOPs”), while removing duplicated content. These SOPs are organized by exercise type, highlighting each topic’s typical problems and the step-by-step approach to solve them.

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COMPREHENSIVE SOPs FOR SOLVING MICROECONOMICS EXERCISES

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CONSUMER THEORY: PREFERENCES, UTILITY, AND CHOICE

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Typical Problems • Understanding Preferences and Utility Functions (e.g., Cobb-Douglas, Perfect Substitutes, Perfect Complements). • Calculating Marginal Utility (MU) and deriving the Marginal Rate of Substitution (MRS). • Utility Maximization under a Budget Constraint. • Identifying corner vs. interior solutions.

Steps to Solve

Identify the Utility Function

Determine the form: Cobb-Douglas (e.g., U = xᵃy^(1−ᵃ)), Perfect Substitutes (U = ax + by), Perfect Complements (U = min(ax, by)), Quasilinear, or other forms.

Set Up the Budget Constraint

Typically of the form: I = pₓx + pᵧy.

For multi-good cases, generalize to Σ pᵢxᵢ = I.

Compute Marginal Utilities and MRS

MUₓ = ∂U/∂x; MUᵧ = ∂U/∂y.

MRS = MUₓ / MUᵧ (or ratio of partial derivatives).

The typical interior-solution condition is MRS = pₓ / pᵧ.

Solve the Utility Maximization Problem

Either use the Lagrangian: L=U(x,y)+λ[I−pxx−pyy] \mathcal{L} = U(x, y) + \lambda [I - p_x x - p_y y] and derive the first-order conditions.

Or check for special forms (e.g., corner solutions for perfect substitutes, or balanced bundles for perfect complements).

Interpret the Result

Determine optimal consumption (x*, y*) given pₓ, pᵧ, and I.

Check for corner vs. interior solution.

For corner solutions, the MRS condition might not hold because the optimum occurs at a boundary.

Perform Comparative Statics

Examine how x* and y* change with income (I) or prices (pₓ, pᵧ).

Relate changes to normal vs. inferior goods, or to income/substitution effects.

Summarize the Findings

Compute and interpret results (optimal bundle, total utility, possible corner solutions, etc.).

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DEMAND THEORY: MARSHALLIAN AND HICKSIAN DEMANDS

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Typical Problems • Deriving demand functions (Marshallian) from the utility maximization problem. • Substitution and income effects (Slutsky decomposition). • Hicksian (compensated) demand functions. • Consumer surplus, Compensating Variation (CV), Equivalent Variation (EV).

Steps to Solve

Obtain Marshallian Demand Functions

From the consumer’s utility maximization under budget constraint.

Express x* and y* in terms of pₓ, pᵧ, and income I.

Distinguish Income vs. Substitution Effects

Use Slutsky decomposition: total effect of a price change = substitution effect + income effect.

For large price changes, depict graphically or use formulas for Slutsky/ Hicks decomposition.

Derive the Hicksian (Compensated) Demands a) Indirect Utility Function: plug the Marshallian demands back into U(x, y) to get V(p, I). b) Expenditure Function: solve the expenditure minimization problem

min? pxx+pyysubject toU(x,y)=Uˉ. \min \, p_x x + p_y y \quad \text{subject to} \quad U(x, y) = \bar{U}.

The minimal expenditure E = E(pₓ, pᵧ, Ū). c) Hicksian Demands: differentiate E with respect to pₓ or pᵧ (Shepard’s Lemma).

Compute Welfare Changes

Consumer Surplus (if approximating changes for small price variations).

For larger changes, use Compensating Variation (CV) or Equivalent Variation (EV):

CV: income adjustment needed before price change to reach new utility.

EV: income adjustment needed after price change to preserve original utility.

Summarize Key Elasticities

Price elasticity of demand, income elasticity, cross-price elasticity.

Interpret normal, inferior, and Giffen goods from sign and magnitude of effects.

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PRODUCTION AND COST FUNCTIONS

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Typical Problems • Identifying the production function type (Cobb-Douglas, CES, Leontief). • Deriving marginal products, returns to scale, and the marginal rate of technical substitution (MRTS). • Cost minimization: short-run vs. long-run costs. • Deriving the cost function and analyzing cost curves (e.g., AC, MC).

Steps to Solve

Identify Production Function

E.g., Cobb-Douglas: q = A Kᵅ Lᵝ, or CES: q = [αK^ρ + βL^ρ]^(1/ρ), etc.

Compute Marginal Products

MPᴷ = ∂q/∂K, MPᴸ = ∂q/∂L.

Check for diminishing marginal returns, returns to scale, or special forms.

Find MRTS

MRTS₍K,L₎ = MPᴷ / MPᴸ.

For cost minimization, set MRTS = w/r (the ratio of input prices, e.g., wages w and rental rate r).

Cost Minimization

For a given output level q̄, minimize total cost C = wL + rK subject to q(K, L) ≥ q̄.

Use the Lagrangian: L=wL+rK+λ[qˉ−f(K,L)]. \mathcal{L} = wL + rK + \lambda [q̄ - f(K,L)].

Solve for conditional input demands (K*(q̄,w,r), L*(q̄,w,r)).

Derive the Cost Function

Plug K*(q̄,w,r), L*(q̄,w,r) back into C = wL + rK → C(q̄, w, r).

Analyze shapes: AC(q), MC(q), the presence of economies of scale, etc.

Check Short-Run vs. Long-Run

In short run, some factors are fixed.

In long run, all factors are variable and you can re-optimize capital.

Interpret Returns to Scale

Increasing: doubling inputs > doubles output.

Constant: doubling inputs doubles output.

Decreasing: doubling inputs < doubles output.

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PROFIT MAXIMIZATION AND SUPPLY

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Typical Problems • Profit maximization for a price-taking firm. • Short-run vs. long-run production decisions. • Deriving supply curves from marginal cost (MC). • Producer surplus.

Steps to Solve

Set Up the Profit Function

π(q)=p⋅q−C(q). \pi(q) = p \cdot q - C(q).

If multiple inputs, π=p⋅f(K,L)−wL−rK. \pi = p \cdot f(K, L) - wL - rK.

Find First-Order Condition

For a single-output firm, profit max typically satisfies p = MC if interior solution.

For multiple inputs, partial derivatives w.r.t. each input = 0.

Short-Run vs. Long-Run Decisions

Short run: keep fixed inputs, only optimize variable inputs.

Long run: all inputs variable, can exit or enter.

Supply Curve

In a perfectly competitive market, the firm’s supply curve is MC above the average variable cost (AVC) in the short run, and above average total cost (ATC) in the long run.

Sum across identical firms for the industry supply.

Compute Producer Surplus

Producer Surplus (PS) = Revenue − Variable Cost. In the short run, fixed cost does not affect the decision to produce but does affect profit.

Comparative Statics

Examine how optimal q* changes with changes in p, w, r, or technology.

Example: pass-through of taxes or shifts in supply curves.

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MARKET EQUILIBRIUM AND WELFARE ANALYSIS

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Typical Problems • Determining market-clearing price and quantity under perfect competition. • Measuring changes in consumer surplus (CS) and producer surplus (PS) from policy changes. • Effects of price controls, quotas, or taxes. • Understanding deadweight loss (DWL) and efficiency.

Steps to Solve

Set Up the Market Demand and Supply

Demand: Qᴰ = D(p).

Supply: Qˢ = S(p).

Equilibrium: Qᴰ = Qˢ → p* and q*.

Solve for Equilibrium

For linear demand and supply, set D(p) = S(p) to find p*. Then q* = D(p*).

Policy or Shock

Introduce a tax, quota, or price ceiling/floor.

Derive the new equilibrium.

Compare old vs. new price, quantity, and welfare.

Measure Consumer and Producer Surplus

Graphically: area under demand curve above price → CS; area above supply curve below price → PS.

For larger changes, might need exact integrals or CV/EV approaches.

Calculate Deadweight Loss (DWL)

Reduction in total surplus from the policy distortion.

Compare new total surplus vs. original total surplus.

Interpret Efficiency

If no externalities or market failures, competitive equilibrium is (Pareto) efficient.

Government interventions can create DWL unless justified by externalities or other distortions.

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MARKET POWER: MONOPOLY, PRICE DISCRIMINATION, OLIGOPOLY

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Typical Problems • Profit maximization under monopoly; optimal price and quantity. • Welfare loss from monopoly power. • Types of price discrimination (1st, 2nd, 3rd degree). • Basic oligopoly models (Cournot, Bertrand, Stackelberg).

Steps to Solve (A) Monopoly

Set Up the Demand Function

p = D⁻¹(Q), total revenue TR(Q) = p(Q)·Q.

Profit Maximization

π(Q)=p(Q)⋅Q−C(Q). \pi(Q) = p(Q)\cdot Q - C(Q).

Solve MR = MC for Q*. Then p* = p(Q*).

MR = ∂[p(Q)·Q]/∂Q = p(Q) + Q·p′(Q).

Price Discrimination

1st degree: perfect discrimination, each unit sold at highest willingness to pay.

2nd degree: quantity discounts or block pricing.

3rd degree: separate markets or segments, distinct prices for each.

Welfare Effects

Compare competitive vs. monopoly output to identify deadweight loss.

(B) Oligopoly

Identify the Game Type

Cournot: Firms choose quantities Qᵢ.

Bertrand: Firms choose prices pᵢ.

Stackelberg: Sequential move quantity-setting.

Write Down Payoff Functions

Each firm’s profit = (Price - Cost) × Quantity for that model.

Find Nash Equilibrium

For Cournot: each firm sets Qᵢ to maximize its profit given the other firms’ Qⱼ, leading to best response functions Qᵢ = BRᵢ(Qⱼ). Solve simultaneously.

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