Chapter 6Consumer and Producer Theory
Introduction
Part I treated demand and supply curves as given. We drew them, shifted them, and measured the surplus they generated. But where do these curves come from? This chapter answers that question by deriving demand from the optimization problem of the consumer and supply from the optimization problem of the firm.
The shift in method is significant. Part I used algebra and geometry. This chapter introduces constrained optimization — maximizing an objective function subject to a constraint — using calculus and Lagrangian methods. The payoff is that demand and supply curves stop being assumptions and become consequences of deeper primitives: preferences, technology, and prices.
The chapter is long because it covers two parallel theories — consumer theory and producer theory — that mirror each other in structure. The consumer maximizes utility subject to a budget constraint; the firm minimizes cost subject to an output target (or maximizes profit subject to technology). Both lead to tangency conditions, and both generate the curves we took as given in Part I.
By the end of this chapter, you will be able to:
- Set up and solve a consumer's utility maximization problem using the Lagrangian
- Derive Marshallian demand functions from utility maximization
- Decompose price changes into income and substitution effects (Slutsky equation)
- Set up and solve a firm's cost minimization and profit maximization problems
- Derive short-run and long-run cost curves from a production function
- Classify returns to scale
Prerequisites: Chapters 2 and 3. Mathematical prerequisites: multivariable calculus, constrained optimization (see Appendix A for review).
6.1 Preferences and Utility
The consumer chooses among bundles of goods — combinations like "3 apples and 2 bananas" or "5 hours of leisure and \$100 of consumption." To model this choice, we need a way to represent the consumer's preferences — their ranking of different bundles.
Preferences. A binary relation $\succsim$ on the set of bundles. We write $x \succsim y$ to mean "the consumer weakly prefers bundle $x$ to bundle $y$." Strict preference ($x \succ y$) means $x$ is strictly better. Indifference ($x \sim y$) means both are equally good.
For preferences to be well-behaved enough to model mathematically, we require three axioms:
- Completeness: For any two bundles $x$ and $y$, either $x \succsim y$ or $y \succsim x$ (or both). The consumer can always compare any two options.
- Transitivity: If $x \succsim y$ and $y \succsim z$, then $x \succsim z$. Preferences are logically consistent — no cycles.
- Continuity: Small changes in bundles produce small changes in preference. No "cliff edges."
Completeness. An axiom of rational preferences requiring that for any two bundles $x$ and $y$, the consumer can rank them: either $x \succsim y$, or $y \succsim x$, or both (indifference). The consumer is never "unable to decide."
Transitivity. An axiom of rational preferences requiring that if $x \succsim y$ and $y \succsim z$, then $x \succsim z$. Preferences contain no cycles — logical consistency is maintained.
Continuity. An axiom requiring that small changes in bundles produce small changes in preference ranking. There are no "jumps" — if bundle $x$ is preferred to $y$, bundles sufficiently close to $x$ are also preferred to $y$.
Utility function. A real-valued function $U(x_1, x_2)$ that assigns a number to each bundle such that higher numbers correspond to more preferred bundles. It exists when preferences satisfy completeness, transitivity, and continuity.
Ordinal utility. A utility representation in which only the ranking of bundles matters, not the magnitude of the utility numbers. Any monotonic transformation $V = g(U)$ (where $g$ is strictly increasing) represents the same preferences.
Under these conditions, a fundamental theorem guarantees the existence of a utility function $U(x_1, x_2)$ — a real-valued function that assigns a number to each bundle such that:
$$x \succsim y \iff U(x) \geq U(y)$$
Higher utility means more preferred. But the numbers themselves carry no meaning beyond ranking. Any monotonic transformation $V = g(U)$ (where $g$ is strictly increasing) represents the same preferences. This is what we mean by ordinal utility: only the ordering matters.
Indifference Curves
Indifference curve. The set of all bundles yielding the same utility level: $\{(x_1, x_2) : U(x_1, x_2) = \bar{u}\}$.
Properties of indifference curves (given well-behaved preferences): (1) Downward-sloping: more of one good requires giving up some of the other. (2) Cannot cross: would violate transitivity. (3) Higher curves = higher utility. (4) Convex to the origin (if preferences are convex): mixtures are preferred to extremes.
Marginal Rate of Substitution
Marginal rate of substitution (MRS). The rate at which the consumer is willing to trade good 2 for good 1 while maintaining the same utility — the (negative of the) slope of the indifference curve.
Along an indifference curve, $dU = 0$:
$$MRS_{12} = -\frac{dx_2}{dx_1}\bigg|_{U = \bar{u}} = \frac{MU_1}{MU_2}$$
(Eq. 6.1)
The MRS is the ratio of marginal utilities. Diminishing MRS: for convex preferences, the MRS decreases as the consumer moves down the indifference curve (more $x_1$, less $x_2$). Intuitively: the more lemonade you already have, the less you're willing to give up cookies for yet another cup.
Common Utility Functions
| Name | $U(x_1, x_2)$ | MRS | Key feature |
|---|
| Cobb-Douglas | $x_1^a x_2^b$ | $(a/b)(x_2/x_1)$ | Constant budget shares |
| Perfect substitutes | $ax_1 + bx_2$ | $a/b$ (constant) | May buy only one good |
| Perfect complements | $\min(ax_1, bx_2)$ | Undefined at kink | Fixed consumption ratio |
| Quasilinear | $v(x_1) + x_2$ | $v'(x_1)$ | No income effect on $x_1$ |
| CES | $(x_1^\rho + x_2^\rho)^{1/\rho}$ | $(x_2/x_1)^{1-\rho}$ | Nests all of the above |
6.2 The Consumer's Problem
Budget constraint. The set of affordable bundles: $p_1 x_1 + p_2 x_2 \leq m$, where $p_i$ are prices and $m$ is income. The budget line has slope $-p_1/p_2$ and intercepts $m/p_1$ on the $x_1$-axis and $m/p_2$ on the $x_2$-axis.
The slope $-p_1/p_2$ is the market rate of exchange: to buy one more unit of good 1 (costing $p_1$), the consumer must give up $p_1/p_2$ units of good 2.
Interactive: Budget Constraint Explorer
Drag the sliders to change prices and income. Watch the budget line pivot and shift in real time.
Budget line: $x_1$-intercept = 30 | $x_2$-intercept = 60 | Slope = −2.00
The Consumer's Problem
Utility maximization. The consumer's fundamental problem: choose the bundle of goods that yields the highest utility subject to the budget constraint. Formally: $\max U(x_1, x_2)$ subject to $p_1 x_1 + p_2 x_2 \leq m$.
$$\max_{x_1, x_2} \; U(x_1, x_2) \quad \text{subject to} \quad p_1 x_1 + p_2 x_2 = m$$
(Eq. 6.2)
The Lagrangian Method
Lagrangian. A mathematical technique for solving constrained optimization problems. The Lagrangian $\mathcal{L} = U(x_1, x_2) + \lambda(m - p_1 x_1 - p_2 x_2)$ converts a constrained problem into an unconstrained one by introducing a multiplier $\lambda$ that prices the constraint.
$$\mathcal{L} = U(x_1, x_2) + \lambda(m - p_1 x_1 - p_2 x_2)$$
(Eq. 6.3)
The Lagrange multiplier $\lambda$ is the marginal utility of income — the increase in maximum utility from an additional dollar of budget.
First-order conditions:
$$MU_1 = \lambda p_1, \quad MU_2 = \lambda p_2, \quad p_1 x_1 + p_2 x_2 = m$$
(Eq. 6.4)
The consumer allocates spending so that the marginal utility per dollar is the same for both goods: $MU_1/p_1 = MU_2/p_2 = \lambda$. Dividing the first two conditions:
$$MRS = \frac{MU_1}{MU_2} = \frac{p_1}{p_2}$$
(Eq. 6.5)
Tangency condition. At the consumer's optimum, the indifference curve is tangent to the budget line: $MRS = p_1/p_2$. The rate at which the consumer is willing to trade goods equals the rate at which the market allows her to trade.
Marshallian Demand
Marshallian (ordinary) demand. The optimal quantities as functions of prices and income: $x_i^*(p_1, p_2, m)$. These are the demand functions that underlie the demand curves of Chapter 2.
Example 6.1 — Cobb-Douglas Demand
$U = x_1^{1/2} x_2^{1/2}$. Tangency: $x_2/x_1 = p_1/p_2$, so $x_2 = (p_1/p_2)x_1$.
Substituting into the budget constraint: \$1p_1 x_1 = m$.
Marshallian demand: $x_1^* = m/(2p_1)$, $x_2^* = m/(2p_2)$.
The consumer spends exactly half her income on each good — the constant budget share property of Cobb-Douglas preferences.
Interactive: Utility Maximization and Demand Derivation
This visualization shows the deep connection: as $p_1$ changes, the optimal bundle traces out the demand curve for good 1. The demand curve IS the set of optimal points at different prices.
Optimal bundle: x₁* = 15.0, x₂* = 30.0 | Utility = 20.1 | MRS = p₁/p₂ = 2.00
Example 6.2 — Quasilinear Utility
$U = \ln(x_1) + x_2$. Tangency: \$1/x_1 = p_1/p_2$, so $x_1^* = p_2/p_1$.
Budget: $x_2^* = m/p_2 - 1$.
Demand for $x_1$ depends only on the price ratio, not on income — the hallmark of quasilinear utility. There are no income effects on good 1.
6.3 Income and Substitution Effects
When the price of a good changes, two things happen simultaneously:
Substitution effect. The change in quantity demanded due solely to the change in relative prices, holding utility constant. The substitution effect is always negative: a price increase always reduces the compensated quantity demanded.
Income effect. The change in quantity demanded due to the change in real purchasing power caused by the price change. For normal goods, a price increase reduces real income and further reduces demand. For inferior goods, the income effect works in the opposite direction.
- Substitution effect: The good becomes relatively cheaper (or more expensive). The consumer substitutes toward the cheaper good. This effect is always negative.
- Income effect: The price change alters real purchasing power. A price decrease is like an income increase. For normal goods, this reinforces the substitution effect. For inferior goods, it works in the opposite direction.
The Slutsky Equation
Slutsky equation. The fundamental decomposition of the total effect of a price change into substitution and income effects: $\partial x_1/\partial p_1 = \partial x_1^h/\partial p_1 - x_1 \cdot \partial x_1/\partial m$. It shows that the demand response to a price change depends on how easily the consumer substitutes and how much the good matters in the budget.
$$\frac{\partial x_1}{\partial p_1} = \underbrace{\frac{\partial x_1^h}{\partial p_1}}_{\text{substitution (−)}} - \underbrace{x_1 \cdot \frac{\partial x_1}{\partial m}}_{\text{income (sign varies)}}$$
(Eq. 6.7)
Normal good (consumer theory). A good for which demand increases when income rises ($\partial x/\partial m > 0$). For normal goods, the income effect reinforces the substitution effect, so the law of demand always holds.
Inferior good. A good for which demand decreases when income rises ($\partial x/\partial m < 0$). For inferior goods, the income effect opposes the substitution effect, but the substitution effect usually dominates.
Giffen good. An extreme inferior good for which the income effect is so large that it dominates the substitution effect, causing demand to increase when the price rises. Giffen goods violate the law of demand and are exceedingly rare in practice.
| Good type | Substitution effect | Income effect | Total effect of price increase |
|---|
| Normal good | − (buy less) | − (poorer → buy less) | Unambiguously − |
| Inferior good | − (buy less) | + (poorer → buy more) | Usually − |
| Giffen good | − (buy less) | + (income effect dominates) | + (demand rises) |
Interactive: Income and Substitution Effects (Hicks Decomposition)
Slide $p_1$ downward to see the price decrease decomposed into a substitution effect (movement along the original indifference curve) and an income effect (movement to a higher indifference curve).
No price change yet. Slide p₁ below \$1.00 to see the decomposition.
Engel Curves
Engel curve. The relationship between income and the quantity demanded of a good, holding prices constant. For normal goods, the Engel curve slopes upward. For inferior goods, it eventually slopes downward.
For Cobb-Douglas, the Engel curve is a straight line through the origin: $x_1 = am/p_1$, linear in $m$. The budget share is always $a$, regardless of income.
6.4 Production Functions
Production function. A mathematical relationship describing the maximum output obtainable from given inputs: $Y = f(K, L)$, where $K$ is capital and $L$ is labor.
Cobb-Douglas Production
$$Y = AK^\alpha L^{1-\alpha}$$
(Eq. 6.8)
where $A > 0$ is total factor productivity and $\alpha \in (0,1)$ is the output elasticity of capital.
Marginal products: $MP_K = \alpha Y/K$, $MP_L = (1-\alpha)Y/L$. Both are positive and diminishing.
Isoquants and MRTS
Isoquant. The set of input combinations producing the same output: $\{(K, L) : f(K,L) = \bar{Y}\}$. Isoquants are the production analog of indifference curves.
Marginal rate of technical substitution (MRTS). The rate at which a firm can substitute one input for another while keeping output constant — the (negative of the) slope of the isoquant. $MRTS_{LK} = MP_L/MP_K$.
$$MRTS_{LK} = \frac{MP_L}{MP_K} = \frac{(1-\alpha)K}{\alpha L}$$
(Eq. 6.9)
Returns to Scale
Returns to scale. How output changes when all inputs are scaled by the same factor. Constant returns to scale (CRS): output scales proportionally. Increasing returns to scale (IRS): output more than scales proportionally (economies of scale). Decreasing returns to scale (DRS): output scales less than proportionally (diseconomies of scale).
| Type | Condition | Meaning |
|---|
| CRS | $f(tK,tL) = tY$ | Doubling inputs doubles output |
| IRS | $f(tK,tL) > tY$ | Doubling inputs more than doubles output |
| DRS | $f(tK,tL) < tY$ | Doubling inputs less than doubles output |
Example 6.3 — Returns to Scale
$Y = K^{0.3}L^{0.8}$: $f(tK,tL) = t^{1.1}Y$. Since \$1.1 > 1$: increasing returns to scale.
6.5 Cost Minimization
Cost minimization. The firm's problem of choosing the combination of inputs that produces a given output level at the lowest total cost: $\min wL + rK$ subject to $f(K,L) = \bar{Y}$.
$$\min_{K, L} \; wL + rK \quad \text{subject to} \quad f(K,L) = \bar{Y}$$
(Eq. 6.10)
Isocost line. All combinations of $K$ and $L$ that cost the same: $C = wL + rK$. Slope: $-w/r$.
The cost-minimizing condition (from the FOCs of the Lagrangian):
$$MRTS = \frac{MP_L}{MP_K} = \frac{w}{r}$$
(Eq. 6.11)
This perfectly parallels the consumer's $MRS = p_1/p_2$.
Interactive: Isoquant/Isocost Cost Minimization
The firm chooses inputs to minimize cost. Adjust factor prices and watch the isocost line pivot and the optimal $K/L$ ratio change.
Cost minimum: L* = 141.4, K* = 70.7 | K/L = 0.50 | TC = \$1,828
Example 6.4 — Cost Minimization
$Y = K^{0.5}L^{0.5}$, $w = 10$, $r = 20$. Produce $\bar{Y} = 100$.
$MRTS = K/L = w/r = 0.5$, so $K = 0.5L$.
$(0.5L)^{0.5} \cdot L^{0.5} = 100 \Rightarrow L^* = 141.4$, $K^* = 70.7$.
$TC = 10(141.4) + 20(70.7) = \\$1{,}828$. Since labor is cheaper, the firm uses more labor than capital.
6.6 Cost Curves
Short Run vs. Long Run
In the short run, at least one input is fixed (typically capital: $K = \bar{K}$). In the long run, all inputs are variable.
Short-Run Cost Functions
Fixed cost (FC). The cost of inputs that cannot be adjusted in the short run (e.g., rent, equipment leases). Fixed costs do not change with the level of output.
Variable cost (VC). The cost of inputs that vary with the level of output (e.g., labor, raw materials). Variable cost rises as the firm produces more.
Marginal cost (MC). The additional cost of producing one more unit of output: $MC = dTC/dQ$. Marginal cost typically falls initially (increasing returns to the variable input), then rises (diminishing returns).
Average cost (AC). Total cost per unit of output: $AC = TC/Q = AFC + AVC$. The AC curve is U-shaped, reaching its minimum where $MC = AC$.
Average variable cost (AVC). Variable cost per unit of output: $AVC = VC/Q$. The AVC curve is also U-shaped. Its minimum is the shutdown point — the lowest price at which the firm is willing to produce in the short run.
Shutdown point. The output level (and corresponding price) at which price equals the minimum of average variable cost ($P = AVC_{min}$). Below this price, the firm loses more by producing than by shutting down entirely, because revenue fails to cover even variable costs.
Minimum efficient scale. The smallest output level at which long-run average cost reaches its minimum. Firms operating below this scale have higher unit costs and are at a competitive disadvantage.
| Cost concept | Symbol | Definition |
|---|
| Fixed cost | $FC$ | Cost of fixed inputs ($r\bar{K}$) |
| Variable cost | $VC$ | Cost of variable inputs ($wL(Q)$) |
| Total cost | $TC$ | $FC + VC$ |
| Marginal cost | $MC$ | $dTC/dQ$ |
| Average total cost | $AC$ | $TC/Q$ |
| Average variable cost | $AVC$ | $VC/Q$ |
| Average fixed cost | $AFC$ | $FC/Q$ (always declining) |
Key relationships:
- $AC = AVC + AFC$. Since $AFC$ always declines, $AC$ and $AVC$ converge at high output.
- MC intersects AC at AC's minimum. When $MC < AC$, producing one more unit pulls the average down. When $MC > AC$, it pulls the average up.
- The shutdown point is where $P = AVC_{min}$. Below this, the firm shuts down.
Interactive: Cost Curves and Profit
The firm has $TC = 50 + 2Q + 0.05Q^2$. Adjust the market price to see the firm's profit-maximizing output and whether it earns profit or loss.
At P = \$1.00: Q* = 60 | TR = \$180 | TC = \$150 | Profit = \$130
Long-Run Average Cost
In the long run, the firm can choose any level of capital. The long-run average cost (LRAC) curve is the envelope of all short-run AC curves — each corresponding to a different level of fixed capital.
Why LRAC is typically U-shaped:
- At low output: economies of scale (LRAC falling) — spreading fixed costs, specialization, bulk purchasing.
- At medium output: constant returns — the flat bottom of the U.
- At high output: diseconomies of scale (LRAC rising) — coordination costs, monitoring problems, inflexibility.
The output level at the bottom of the LRAC is the minimum efficient scale (MES) — the smallest output at which LRAC is minimized.
Interactive: Short-Run vs. Long-Run Average Cost
Each short-run AC curve corresponds to a different capital level. Drag the slider to highlight a specific SRAC curve and see how it relates to the LRAC envelope.
Capital K\u0304 = 3: SRAC minimum at Q = 47, AC = \$1.32 | MES at Q ≈ 60
6.7 Profit Maximization
Profit maximization. The firm's objective: choose output to maximize profit $\Pi = P \cdot Q - TC(Q)$. For a competitive firm (price-taker), the first-order condition yields $P = MC$ — produce where price equals marginal cost.
$$\max_Q \; \Pi = P \cdot Q - TC(Q)$$
(Eq. 6.12)
First-order condition:
$$P = MC(Q)$$
(Eq. 6.13)
The profit-maximizing rule: produce where price equals marginal cost. The firm should keep producing as long as the revenue from one more unit ($P$) exceeds the cost ($MC$). The firm's supply curve is the portion of its MC curve above $AVC_{min}$.
Why $P = MC$ is the supply curve — the deep connection. In Chapter 2, we drew the supply curve as upward-sloping. Now we see where it comes from: it is the firm's marginal cost curve. The supply curve slopes upward because marginal cost is increasing — not because we assumed it, but because it follows from diminishing marginal returns.
Example 6.5 — Profit Maximization
$TC = 50 + 2Q + 0.5Q^2$. At $P = 12$: $P = MC$ gives \$12 = 2 + Q$, so $Q^* = 10$.
$\Pi = 12(10) - [50 + 20 + 50] = 0$. Zero economic profit — the long-run competitive equilibrium.
Example 6.6 — Profit Maximization from Production Function
A competitive firm has production function $Y = 10L^{0.5}$, faces wage $w = 20$ and output price $P = 8$.
Step 1 — Find the profit function. Revenue: $R = PY = 8 \times 10L^{0.5} = 80L^{0.5}$. Cost: $C = wL = 20L$. Profit: $\Pi = 80L^{0.5} - 20L$.
Step 2 — FOC. $d\Pi/dL = 40L^{-0.5} - 20 = 0 \implies L^{-0.5} = 0.5 \implies L^* = 4$.
Step 3 — Compute output and profit. $Y^* = 10(4)^{0.5} = 20$. Revenue = \$1 \times 20 = 160$. Cost = \$10 \times 4 = 80$. Profit = \$10.
Verify: $P \times MP_L = w$ at the optimum: \$1 \times 10 \times 0.5 \times 4^{-0.5} = 8 \times 2.5 = 20 = w$. ✓
6.8 The Firm's Supply Curve
- Short-run supply: MC curve for $P \geq AVC_{min}$. Below that, $Q = 0$ (shutdown).
- Long-run supply: Long-run MC curve for $P \geq LRAC_{min}$. Below that, exit.
- Market supply: Horizontal sum of all firms' supply curves. Free entry/exit drives $P \to LRAC_{min}$.
Thread Example: Maya's Enterprise
Maya's Lemonade Stand — The Full Cost Analysis
Cost structure: $FC = \\$10$/day (stand rental). Materials: $\\$1.50$/cup. Maya's labor: 10 cups/hour at opportunity cost $\\$15$/hr, so $\\$1.50$/cup.
$TC = 20 + 3Q$, $MC = 3$, $AVC = 3$, $AC = 20/Q + 3$.
From Chapter 2: $P^* = \\$1.75$. But $MC = \\$1.00 > P^*$. Maya should not operate. Every cup loses $\\$1.25$.
However, if we exclude her opportunity cost (accounting profit only), $AVC_{materials} = \\$1.50$, and $P = 2.75 > 1.50$. She earns $\\$16.25$/day in accounting profit but $-\\$13.75$/day in economic profit. The economist says: Maya, your time is worth $\\$120$/day at the bookstore.
Summary
- Consumer theory: The consumer maximizes $U(x_1, x_2)$ subject to $p_1 x_1 + p_2 x_2 = m$. The tangency condition $MRS = p_1/p_2$ says the consumer's willingness to trade equals the market rate.
- Marshallian demand gives optimal quantities as functions of prices and income. For Cobb-Douglas, budget shares are constant. For quasilinear, there are no income effects on good 1.
- The Slutsky equation decomposes price effects into a substitution effect (always negative) and an income effect (sign depends on normal vs. inferior). Giffen goods require a dominant income effect.
- Production functions map inputs to output. Returns to scale: CRS, IRS, or DRS.
- Cost minimization requires $MRTS = w/r$, paralleling the consumer's $MRS = p_1/p_2$.
- Short-run cost curves: MC crosses AC and AVC at their minima. Shutdown point: $P = AVC_{min}$.
- Profit maximization: $P = MC$. The firm's supply curve is the MC curve above the shutdown point.
- Economic profit (including opportunity cost) vs. accounting profit (excluding it) determines whether a firm should operate.
Key Equations
| Label | Equation | Description |
|---|
| Eq. 6.1 | $MRS = MU_1/MU_2$ | Marginal rate of substitution |
| Eq. 6.2 | $\max U(x_1,x_2)$ s.t. $p_1 x_1 + p_2 x_2 = m$ | Consumer's problem |
| Eq. 6.3 | $\mathcal{L} = U + \lambda(m - p_1 x_1 - p_2 x_2)$ | Lagrangian |
| Eq. 6.4 | FOCs: $MU_i = \lambda p_i$; budget binds | First-order conditions |
| Eq. 6.5 | $MRS = p_1/p_2$ | Tangency condition |
| Eq. 6.6 | $x_i^* = a_i m / p_i$ | Cobb-Douglas Marshallian demand |
| Eq. 6.7 | $\partial x_1/\partial p_1 = \partial x_1^h/\partial p_1 - x_1 \partial x_1/\partial m$ | Slutsky equation |
| Eq. 6.8 | $Y = AK^\alpha L^{1-\alpha}$ | Cobb-Douglas production function |
| Eq. 6.9 | $MRTS = MP_L/MP_K$ | Marginal rate of technical substitution |
| Eq. 6.10 | $\min wL + rK$ s.t. $f(K,L) = \bar{Y}$ | Cost minimization problem |
| Eq. 6.11 | $MRTS = w/r$ | Cost-minimizing input ratio |
| Eq. 6.12 | $\max \Pi = PQ - TC(Q)$ | Profit maximization |
| Eq. 6.13 | $P = MC$ | Profit-maximizing output rule |
Exercises
Practice
- A consumer has utility $U = x_1^{1/3} x_2^{2/3}$, prices $p_1 = 4$, $p_2 = 2$, income $m = 120$. (a) Write the Lagrangian. (b) Derive the tangency condition. (c) Solve for the Marshallian demand for both goods. (d) Compute the optimal bundle and verify it satisfies the budget constraint.
- A consumer has quasilinear utility $U = 2\sqrt{x_1} + x_2$, $p_1 = 1$, $p_2 = 1$, $m = 10$. (a) Solve for optimal consumption. (b) What is the income elasticity of demand for $x_1$? (c) What happens to $x_1^*$ if income doubles?
- A firm has production function $Y = 4K^{0.5}L^{0.5}$, $w = 8$, $r = 2$. (a) Find the cost-minimizing input combination to produce $Y = 40$. (b) What is the total cost? (c) If $w$ doubled, how would the optimal $K/L$ ratio change?
- A competitive firm has $TC = 100 + 5Q + Q^2$. (a) Derive MC, AC, and AVC. (b) Find the shutdown point. (c) At $P = 25$, find profit-maximizing output and profit. (d) At $P = 5$, should the firm produce or shut down?
- Classify returns to scale: (a) $Y = 3K + 2L$, (b) $Y = K^{0.4}L^{0.4}$, (c) $Y = (KL)^{0.6}$, (d) $Y = \min(2K, 3L)$.
Apply
- For Cobb-Douglas utility $U = x_1^a x_2^{1-a}$, derive the Marshallian demands and show that the consumer always spends fraction $a$ on good 1. Then use $V = \ln U$ and show the same demands emerge. What does this confirm about ordinality?
- A price decrease for good 1 leads a consumer to buy less of good 1. (a) Is this irrational? (b) What type of good must this be? (c) What conditions are necessary? (d) Why are Giffen goods so rare?
- A firm can produce with Technology A ($TC_A = 100 + 2Q$) or Technology B ($TC_B = 10 + 5Q$). (a) For what output levels is each cheaper? (b) What does this imply about firm size and technology choice?
- Derive the short-run supply curve for a firm with $TC = 50 + Q^2/2$. Graph it, label the shutdown price, and shade profit at $P = 10$.
- Using $Y = K^{0.3}L^{0.7}$ with $w = 14$, $r = 6$: (a) Find the cost-minimizing $K/L$ ratio. (b) Derive $TC(Y)$. (c) What are the returns to scale?
Challenge
- Prove that for Cobb-Douglas utility $U = x_1^a x_2^{1-a}$, the indirect utility function is $V(p_1, p_2, m) = m \cdot (a/p_1)^a \cdot ((1-a)/p_2)^{1-a}$. Then verify Roy's identity: $x_1^* = -(\partial V/\partial p_1)/(\partial V/\partial m)$.
- Show that a profit-maximizing firm with Cobb-Douglas CRS production earns zero economic profit in long-run equilibrium. (Hint: Euler's theorem.) Why does IRS pose a problem for competitive markets?
- A consumer's demand for good 1 is $x_1 = m/p_1 - p_2$. (a) Is it homogeneous of degree zero? (b) Does it satisfy Slutsky symmetry? (c) Can it be generated by utility maximization?