Microeconomics: Taxes, Subsidies, and Utility Maximization

Posted on Feb 11, 2026 in Economics

1. Taxes, Subsidies, and Deadweight Loss (DWL)

This topic analyzes how government intervention (taxes or subsidies) affects market equilibrium, consumer surplus (CS), producer surplus (PS), and total efficiency (DWL).

Key Formulas & Concepts

  • Tax Wedge:


    A tax ($\tau$) drives a wedge between the price buyers pay ($P_d$) and the price sellers receive ($P_s$).

    • $P_d = P_s + \tau$

    • The new equilibrium quantity ($Q_t$) is found where $Q_d(P_d) = Q_s(P_s)$.

  • Subsidy Wedge:


    A subsidy ($s$) also creates a wedge, but in the opposite direction.

    • $P_s = P_d + s$ (Sellers receive more than buyers pay)

    • The new equilibrium quantity ($Q_s$) is found where $Q_d(P_d) = Q_s(P_s)$.

  • Deadweight Loss (DWL):


    The value of lost trades. It’s the triangle pointing to the efficient, no-tax equilibrium.

    • DWL (Tax):


      $0.5 \times \tau \times (Q^* – Q_t)$

    • DWL (Subsidy):


      $0.5 \times s \times (Q_s – Q^*)$

Hard Example: Subsidy Analysis (like PS #3, Part E)

A common “hard” problem is solving for the DWL of a subsidy, as it involves over-production rather than under-production.

Let’s use your problem set’s functions:

  • Demand: $Q_d = 30 – 2P$

  • Supply: $Q_s = P$

  • Subsidy: $s = \$6$ given to consumers.

Step 1: Find the No-Intervention Equilibrium ($Q^*$)


  • Set $Q_d = Q_s$: $30 – 2P = P$

  • $30 = 3P$

  • $P^ = 10$* and $Q^ = 10$*

Step 2: Set up the Subsidy “Wedge”

  • The subsidy is given to consumers, so the price they pay ($P_d$) is $6 less than the price sellers receive ($P\_s$).

  • $P_d = P_s – 6$

  • Alternatively (and often easier), $P_s = P_d + 6$.

Step 3: Find the New Equilibrium Quantity ($Q_s$)


  • Plug the wedge equations into the S/D equations.

    • $Q_d = 30 – 2P_d$

    • $Q_s = P_s \implies Q_s = P_d + 6$

  • Set $Q_d = Q_s$:

    • $30 – 2P_d = P_d + 6$

    • $24 = 3P_d$

    • $P_d = 8$ (This is what buyers pay)

  • Find the quantity and seller price:

    • $Q_s = 30 – 2(8) = 14$ (This is the new quantity)

    • $P_s = P_d + 6 = 8 + 6 = 14$ (This is what sellers receive)

    • Check: $Q_s = P_s \implies 14 = 14$. It works.

Step 4: Calculate DWL

  • The subsidy induced 4 extra units of production (from $Q^*=10$ to $Q_s=14$).

  • The DWL is the cost of producing these units minus their value.

  • Formula: $DWL = 0.5 \times s \times (Q_s – Q^*)$

  • $DWL = 0.5 \times 6 \times (14 – 10)$

  • $DWL = 3 \times 4 = \mathbf{12}$

2. Preferences, ICs, and the MRS (from PS #4)

This is about the shape of preferences. The MRS (Marginal Rate of Substitution)
is the slope of the indifference curve (IC). It measures your willingness to trade $q_2$ for one more unit of $q_1$.

Key Formulas & Concepts

  • MRS:
    $MRS = – \frac{MU_1}{MU_2}$ where $MU_1 = \frac{\partial U}{\partial q_1}$

  • Diminishing MRS:


    The MRS decreases (in absolute value) as $q_1$ increases. This means ICs are convex (bowed in).
    This is the standard assumption.

  • Constant MRS:


    ICs are straight lines (Perfect Substitutes)
    .

  • Undefined MRS:


    ICs are L-shaped (Perfect Complements)
    .

Walk-through: Identifying Preference Types & MRS

This is a key skill from PS #4. Given a utility function, you must know what it is and how to find its MRS.

Utility Function (U)


Preference Type


MU1​MU2​MRS (−MU2​MU1​​)

Diminishing?


$U = q_1^a q_2^b$ (e.G., $q_1 q_2$ or $q_1^{0.5} q_2^{0.5}$)
Cobb-Douglas(Standard Convex)		$a \cdot q_1^{a-1} \cdot q_2^b$$b \cdot q_1^a \cdot q_2^{b-1}$$-\frac{a \cdot q_2}{b \cdot q_1}$Yes
$U = a q_1 + b q_2$ (e.G., $2q_1 + 3q_2$)
Perfect Substitutes		$a$$b$$-\frac{a}{b}$(Constant)
No
$U = \min(a q_1, b q_2)$ (e.G., $\min(q_1, q_2)$)

Perfect Complements

UndefinedUndefinedUndefined (Kink)
No
$U = q_1^{0.5} + q_2$
Quasi-Linear$0.5 q_1^{-0.5}$$1$$-\frac{1}{2\sqrt{q_1}}$Yes
$U = q_1^2 + q_2^2$
Concave$2q_1$$2q_2$$-\frac{q_1}{q_2}$No(Increasing!)

Note on PS #4, Q4:


The MRS for $U=q_1 q_2$ is $-q_2/q_1$. The MRS for $U=q_1^{0.5} q_2^{0.5}$ is also $-q_2/q_1$. This is because the second function is just the square root of the first. They represent the exact same preferences (one is a “monotonic transformation” of the other).

3. Utility Maximization (from PS #5 & #6)

This is the main event: finding the one affordable bundle $(q_1^*, q_2^*)$ that gives the highest utility. The method depends entirely on the preference type.

Case 1: Standard (Cobb-Douglas) / Convex (PS #5, Q3)

This is the most common problem. You find the point where the budget line is tangent to the highest possible indifference curve.

  • Problem:


    Max $U = q_1^{2/3} q_2^{1/3}$ given $Y=12, P_1=1, P_2=2$.

  • The Rules (2 Conditions):


    1. Tangency:


      $|MRS| = \frac{P_1}{P_2}$

    2. Budget:


      $P_1 q_1 + P_2 q_2 = Y$

  • Step 1: Find $|MRS|$

    • $MU_1 = \frac{2}{3} q_1^{-1/3} q_2^{1/3}$

    • $MU_2 = \frac{1}{3} q_1^{2/3} q_2^{-2/3}$

    • $|MRS| = \frac{MU_1}{MU_2} = \frac{\frac{2}{3} q_1^{-1/3} q_2^{1/3}}{\frac{1}{3} q_1^{2/3} q_2^{-2/3}} = \frac{2 q_2}{q_1}$

  • Step 2: Set Tangency Condition

    • $|MRS| = P_1 / P_2$

    • $\frac{2q_2}{q_1} = \frac{1}{2} \implies 4q_2 = q_1$

  • Step 3: Substitute into Budget Constraint

    • $1q_1 + 2q_2 = 12$

    • $1(4q_2) + 2q_2 = 12$

    • $6q_2 = 12 \implies \mathbf{q_2^* = 2}$

  • Step 4: Solve for $q_1$

    • $q_1^* = 4q_2^* = 4(2) \implies \mathbf{q_1^* = 8}$

  • Solution:


    $(q_1=8, q_2=2)$.

Case 2: Special Cases (Perfect Substitutes & Complements) (PS #6)

This is a classic “hard” question because you cannot use the tangency rule.
Students who try to take derivatives of a $\min()$ function will get zero points.

Walk-through: Perfect Substitutes (PS #6, 1b)

  • Problem:


    Max $U = 2q_1 + 3q_2$ given $Y=12, P_1=1, P_2=2$.

  • Rule:


    Compare the “Bang-for-your-Buck” ($\frac{MU}{P}$) for each good. You will spend all your money on the good that gives more utility per dollar.

  • Step 1: Calculate $\frac{MU}{P}$ for each good


    • Good 1:


      $\frac{MU_1}{P_1} = \frac{2}{1} = 2$ “utils per dollar”

    • Good 2:


      $\frac{MU_2}{P_2} = \frac{3}{2} = 1.5$ “utils per dollar”

  • Step 2: Compare


    • $2 > 1.5 \implies \frac{MU_1}{P_1} > \frac{MU_2}{P_2}$

  • Step 3: Conclude


    • Good 1 gives a better bang-for-your-buck. You should spend all your income on Good 1.

  • Solution:


    • $q_1^* = Y / P_1 = 12 / 1 = \mathbf{12}$

    • $q_2^* = \mathbf{0}$

    • This is a corner solution.

Walk-through: Perfect Complements (PS #6, 1d)

  • Problem:


    Max $U = \min(q_1, 2q_2)$ given $Y=12, P_1=1, P_2=2$.

  • Rule:


    You only get utility from consuming at the “kink” of the L-shaped IC. The kink is where the two terms inside the $\min()$ function are equal.

  • Step 1: Find the Kink/Ratio


    • $q_1 = 2q_2$

  • Step 2: Substitute this ratio into the Budget Constraint


    • $P_1 q_1 + P_2 q_2 = Y$

    • $1(q_1) + 2(q_2) = 12$

    • $1(2q_2) + 2(q_2) = 12$

  • Step 3: Solve


    • $2q_2 + 2q_2 = 12$

    • $4q_2 = 12 \implies \mathbf{q_2^* = 3}$

  • Step 4: Find $q_1$ using the ratio


    • $q_1^* = 2q_2^* = 2(3) \implies \mathbf{q_1^* = 6}$

  • Solution:


    $(q_1=6, q_2=3)$.