Partial Equilibrium

Prices you face in your daily life In this section, we will understand how these prices are determined. Figure: Prices YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 3 / 31

Competitive Markets Many firms. Each firm will have to take into account not only the behavior of the consumers but also the behavior of the other producers. Definition (Competitive Market) A competitive market is one in which a large number of producers compete with each other to satisfy the demand of a large number of consumers. All producers and consumers are price-takers. That is, no one has the power to control prices. Key assumptions 1 Large number of buyers and sellers. 2 Homogeneous product: many substitutes. 3 No Barrier: free entry and exit, free information flow. Key result: price takers. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 6 / 31

Supply Function and FOC Definition (Supply function) The supply function, y ( p ), gives the profit-maximizing output at each price. Therefore, it must satisfies the FOC (and SOC). That is, the supply function must satisfies the two conditions: p = c ′ [ y ( p )] (1) The inverse supply function is given by “ p = MC ( y )”, but subject to 1) SOC: c ′′ [ y ( p )] ≥ 0, and 2) revenue can cover total variable cost. These conditions can help us find the supply functions graphically, and compute it mathematically. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 9 / 31

Profit and Prices: positive profit In the short term, a competitive firm may have positive, negative profits, or zero profits. Depends on the firm’s technology relative to market conditions (price). Figure: Positive profit: p > AC min The profit: π ( p ) = py ∗ ( p ) − C ( y ∗ ( p )) = [ p − AC ( y ∗ ( p ))] y ∗ ( p ) Positive profit ⇐⇒ p > AC min YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 12 / 31

Why does not the supply function include the MC(y) under AVC min ? Exercise: The cost function of a firm is C ( y ) = 1 3 y 3 − 2 y 2 + 4 y + 1. Suppose the firm is a price taker. Find its short-term supply function. Solution + Graph YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 15 / 31

Short-run (Partial) Competitive Equilibrium YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 18 / 31

Graphic Illustration Aggregate demand (from consumer optimization): X ( p ) = ∑ i x i ( p ) Aggregate supply (from producer optimization): Y ( p ) = ∑ j y j ( p ) Market Clearing: X ( p ) = Y ( p ) Figure: Market Clearing and Market equilibrium price Question: factors that can change the equilibrium? (Umax + Πmax) YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 21 / 31

Case study: Curbing the Housing Price in Hong Kong Bitter fact: a small two bedroom apartment (40 m 2 ) costs 25 years of median household income (28000HKD in 2022) in Hong Kong (no eating, no other expenses). Can you propose some policies to curb (or even reduce) the price? YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 22 / 31

Example: short-run Competitive Equilibrium Suppose the industry demand is: X ( p ) = a − bp . The industry has m identical firms, each with cost function c ( y ) = y 2 + 1. Find the competitive equilibrium of this industry. Solution: 1. Check consumer optimization: it is satisfied because the industry demand is derived from consumer optimization. 2. Check producer optimization: solve individual firm profit maximization gives the firm supply function p = mc ( y j ) ⇒ y j ( p ) = p / 2 The industry aggregate supply function is: Y ( p ) = ∑ j y j ( p ) = mp / 2 3. Market clearing condition: X ( p ) = Y ( p ) ⇒ a − bp = mp / 2 — So the competitive equilibrium price is: p ∗ = a b + m / 2 — The competitive equilibrium industry output: X ∗ = Y ∗ = am 2 b + m — Output of each firm: y ∗ j ( p ) = a 2 b + m — Profit of each firm: π ∗ j ( p ) = a 2 b + m 2 − 1 YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 23 / 31

Long-Run Competitive Equilibrium YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 24 / 31

Firm number and profit The above example shows: firm profit DECREASES in the number of firms in the industry (m), in the short-run competitive equilibrium. π ∗ j ( p ) = a 2 b + m 2 − 1 A natural question is: In the long run, — if the profit is positive, more firms will enter the industry until profit is zero. — if the profit is negative, some firms will exit. So the profit increases until zero. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 25 / 31

Entry and Exit: zero profit condition in the long run competitive equilibrium Figure: p ∗ is the price at which firm’s maximized profit is zero. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 26 / 31

Long-term Competitive Equilibrium: additional condition In the short-run CE, number of firms is fixed. In the long-run, number of firms is endogenous. Long-term competitive equilibrium condition. In long-term competitive equilibrium: profit of all firms is “nearly” zero. Zero-profit condition implies two equivalent conditions: 1. MC = AC. 2. Price equals the minimum average cost. This condition determines the equilibrium prices and number of firms. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 27 / 31

Long-run supply curve Figure: p 0 is the price at which this firm’s maximized profit is zero. In the long run, if the market price p ∗ < p 0 , this firm will exit. The long-run supply curve is: p = MC ( y ) , if p > p 0 y = 0 , if p < p 0 YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 28 / 31

Example: Solve Long-term competitive equilibrium The market demand function is: X ( p ) = 20 − 5 p . There are many identical firms with firm cost function C ( y ) = y 2 + 1. Solve for the long-term competitive equilibrium price and number of firms in the market. Solution: Step 1: suppose there are m firms in the market, solve the short-term equilibrium given m. ▶ Firm profit maximization ⇒ p = MC ( y ) ⇒ y j ( p ) = p / 2. ▶ Industry supply given m: Y ( p , m ) = my j ( p ) = mp / 2 ▶ Market clearing: Y ( p , m ) = X ( p ) ⇒ p ∗ ( m ) = 20 5+ m / 2 Step 2: find the number of firms in the industry m ∗ , using the “zero profit” condition. ▶ Solve for p 0 : MC = AC ⇒ 2 y = y + 1 / y ⇒ y = 1 , p 0 = 2. ▶ The number of firms (m) satisfies the condition: p ∗ ( m ) ≥ p 0 p ∗ ( m + 1) < p 0 This implies: 20 5+ m / 2 ≥ 2 , 20 5+( m +1) / 2 < 2 ⇒ m ∗ = 10 . p ∗ = 2 YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 29 / 31

Thinking In the above example, we assumed that all firms have the same cost function to simplify the analysis. What if firms are heterogeneous such that some have lower costs? Industry dynamics: 5-10% entry and exit simultaneously each year. 1 Why? 2 Impact? Industry upgrading: good firms enter; bad firms exit. YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 30 / 31

Reading Varian : Chapter 13.1-13.5 YF Fong (HKUST) Competitive Market: Partial Equilibrium October 6, 2023 31 / 31