Lecture 2A: Firms: demand, markups and sticky prices

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Firms are the second actor. In Lecture 1 prices simply existed; here firms set them. Two ideas carry the chapter. First, market power: because each firm's product is slightly different (think of brands of coffee), a firm can charge more than it costs to make the product without losing all its customers. The gap between price and cost is the markup, and how big it can be depends on how easily customers switch (the elasticity ε). Second, sticky prices: changing a price is a hassle with a cost (new menus, new labels, annoyed customers), so firms do not change prices every day and when they do, they prefer small steps. That reluctance is what gives the central bank its power: if prices adjust slowly, a change in the interest rate changes real spending before prices catch up.

The mathematics looks heavy because of the integral signs and the CES formula. You do not need to reproduce those derivations. What you need: the demand curve (a firm that raises its price above the average loses customers, more so when ε is large), the markup formula M = ε/(ε − 1), and the idea that gradual price changes are cheaper when the adjustment cost is quadratic.

Why two layers of firms?

To set prices, firms need market power: each must sell a good that is not a perfect substitute for the others.
But households buy one final good. Two layers make this tractable:
- Wholesale firms j ∈ [0, 1] each produce a different variety (think brands) and set its price.
- Retail firms buy all varieties and combine them into the final good, in perfect competition.
The retail firms' problem gives the demand curve each wholesale firm faces; the wholesale firms' problem gives pricing, first with flexible prices, then with costs of changing prices.

Where this goes: wholesale firms' pricing decisions, added up, give the Phillips curve.

Retail firms: the CES aggregator

A retail firm combines the varieties with a constant elasticity of substitution (CES, Dixit-Stiglitz) technology:

What the CES formula is doing

A supermarket (the retail firm) buys coffee from many brands (the wholesale firms) and sells "coffee" to households. The CES formula is a recipe for how much "coffee" you get from a mix of brands. It has one knob, ε. If ε is very large, brands are near-perfect substitutes: the supermarket buys whichever is cheapest and brands have no pricing power. If ε is small, customers care about their brand and will keep buying it even at a higher price. Economists like this recipe because it gives a demand curve with a constant elasticity: a 1% higher relative price always costs a firm about ε% of its sales, whatever the starting point.

Y_{t} = (\int_{0}^{1} Y_{j t}^{\frac{ε - 1}{ε}} d j)^{\frac{ε}{ε - 1}}, ε > 1

Y_jt: the quantity of variety j used as an input; Y_t: output of the final good.
ε: how easily the varieties can be substituted. A large ε means close substitutes (little market power); ε → ∞ means perfect substitutes and perfect competition.
The retail firm is perfectly competitive: it takes all prices as given and maximises P_tY_t − ∫₀¹ P_jtY_jt dj.

The demand for each good

The FOC says the value of the marginal product of each variety equals its price: P_t ∂Y_t/∂Y_jt = P_jt, with ∂Y_t/∂Y_jt = Y_t^1/εY_jt^−1/ε. Solving for Y_jt:

Demand for good j

$Y_{j t} = (\frac{P _{j t}}{P _{t}})^{- ε} Y_{t}$ , with the price index $P_{t} = (\int_{0}^{1} P_{j t}^{1 - ε} d j)^{\frac{1}{1 - ε}}$

Demand for good j falls when its price rises relative to the average price P_t; ε is the price elasticity.
Demand rises one-for-one with total output Y_t.
P_t, the aggregate price index, follows from zero profits in retail: P_tY_t = ∫ P_jtY_jt dj.

Worked example

ε = 6. A firm sets its price 2% above the average (P_jt/P_t = 1.02). Its demand is 1.02⁻⁶ = 0.888 times its normal share: about 11% lower. With ε = 2 the loss would be only about 4%: less competition, more market power.

Wholesale firms: technology and marginal cost

Firm j produces with labor only: Y_jt = A_tN_jt, where A_t is productivity, common to all firms.
One more unit needs 1/A_t more hours, each costing the real wage w_t.

Real marginal cost

m c_{t} = \frac{w _{t}}{A _{t}}

The same for all firms: they hire in the same labor market and use the same technology.
Marginal cost rises when wages rise and falls when productivity rises.
Marginal cost is the key link between the real economy and inflation.

Flexible prices: a constant markup

Suppose changing prices is free. Firm j chooses its relative price p_jt ≡ P_jt/P_t to maximise real profits, knowing its demand curve:

p_{j t} max profit per unit (p_{j t} - m c_{t}) units sold p_{j t}^{- ε} Y_{t}

FOC: (1 − ε)p_jt^−ε + ε mc_t p_jt^−ε−1 = 0. Multiply by p_jt^ε+1 and solve:

Optimal pricing: a constant markup over marginal cost

$\frac{P _{j t}}{P _{t}} = markup M \frac{ε}{ε - 1} m c_{t}$ , with the markup M = ε/(ε − 1)

A higher price raises profit per unit but lowers sales; the markup balances the two.
Example: ε = 6 gives M = 6/5 = 1.2, a 20% markup. As ε → ∞, M → 1: price equals marginal cost.
All firms are identical, so P_jt = P_t and mc_t = (ε − 1)/ε = 1/M: constant. With ε = 6, mc = 5/6 = 0.833.

Markup in euros

With ε = 6 the markup is 1.2: a product that costs €10 to make sells for €12. With ε = 11 (more competition) the markup is 1.1 and the price is €11. With ε = 2 (strong brand loyalty) the markup is 2 and the price is €20. The formula is the firm's answer to one question: how much can I charge above cost before the customers I lose cost me more than the extra margin earns?

Costs of changing prices

Menu costs: any cost of the act of changing a price: printing menus, managerial time, upsetting customers.
Fixed menu costs are independent of the size of the change; convex menu costs make larger changes disproportionately more expensive.

Rotemberg (1982): quadratic adjustment cost

Firm j pays a real cost, in units of goods: (ψ/2)(P_jt/P_j,t−1 − 1)² Y_t

ψ > 0 measures the importance of the cost; ψ = 0 gives back flexible prices.
No change in price, no cost. Increases and decreases cost the same.
Convexity gives an incentive to adjust gradually: two changes of 5% are cheaper than one of 10%. Prices become sticky.

The alternative, Calvo (1983) pricing, gives the same linear Phillips curve.

Worked example: why gradual is cheaper

ψ = 50 and Y_t = 100. One 10% increase costs (50/2) × 0.10² × 100 = 25. Two increases of 5% cost 2 × (50/2) × 0.05² × 100 = 12.5: half as much. So firms spread price changes over time.

The firm's problem becomes dynamic

A price chosen today affects tomorrow's adjustment cost, because tomorrow's change is measured from today's price. So the firm looks ahead and maximises the present value of real profits:

{P_{j t}} max E_{0} t = 0 \sum \infty Q_{0, t} profits from sales (\frac{P _{j t}}{P _{t}} - m c_{t}) (\frac{P _{j t}}{P _{t}})^{- ε} Y_{t} - cost of changing price \frac{ψ}{2} (\frac{P _{j t}}{P _{j, t - 1}} - 1)^{2} Y_{t}

Q_0,t is the stochastic discount factor. Firms are owned by households, so future profits are valued as households value them: Q_t,t+1 = β(C_t+1/C_t)^−σ, from the Euler equation.
P_jt appears in two periods: in t (sales and today's adjustment cost) and in t + 1 (tomorrow's adjustment cost, through P_j,t+1/P_jt).

The first-order condition, term by term

(a) Static markup trade-off: the same terms as with flexible prices; on their own they give the markup rule.
(b) Cost today: raising the price today increases today's adjustment cost.
(c) Saving on cost tomorrow: raising the price today reduces the increase needed tomorrow, and so tomorrow's adjustment cost.

In plain words

The firm balances three things: being close to its desired markup today, the cost of changing its price today, and the cost of changing it tomorrow. If it expects to raise prices tomorrow, it already starts today.

Symmetric equilibrium: the nonlinear Phillips curve

All firms face the same problem and choose the same price, so impose P_jt = P_t and define net inflation by 1 + π_t = P_t/P_t−1:

The New Keynesian Phillips curve (nonlinear)

ψ π_{t} (1 + π_{t}) = (1 - ε) + ε m c_{t} + E_{t} [Q_{t, t + 1} \frac{Y _{t + 1}}{Y _{t}} ψ π_{t + 1} (1 + π_{t + 1})]

If ψ = 0: mc_t = (ε − 1)/ε, the flexible-price markup rule.
Higher marginal cost → higher inflation: firms raise prices to restore their markups.
Higher expected inflation → higher inflation today: if costs will rise later, firms start raising prices now to spread the adjustment.
Steady state with zero inflation: mc = (ε − 1)/ε.

Linearising the Phillips curve, step by step

π_t is small, so π_t(1 + π_t) ≈ π_t (π_t² is negligible).
In steady state Q_t,t+1Y_t+1/Y_t = β; deviations multiply π_t+1, which is already small, so use β.
Write marginal cost as a percentage deviation: mc_t − mc ≈ mc · m̂c_t = ((ε − 1)/ε) m̂c_t. Then (1 − ε) + ε mc_t = (ε − 1) m̂c_t, and ψπ_t = (ε − 1) m̂c_t + βψE_tπ_t+1.

Linear Phillips curve in marginal cost (divide by ψ)

π_{t} = β E_{t} π_{t + 1} + \frac{ε - 1}{ψ} m c_{t}

Stickier prices (larger ψ): inflation reacts less to marginal cost. More competition (larger ε): inflation reacts more.
With Calvo pricing (a fraction θ of firms cannot reset their price each period) the coefficient is (1 − θ)(1 − βθ)/θ: same equation, different interpretation. Example: θ = 0.75 (prices last about a year on average) and β = 0.99 give (0.25 × 0.2575)/0.75 ≈ 0.086.

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