Econ 46 Urban & Regional Economics Lecture : New Economic Geography Instructor: Hiroki Watanabe Summer / 5 Model Assumptions Agricultural Sector Monopolistic Competition Manufacturing Sector Monopolistic Competition Profit Maximization Problem Iceberg Transportation Technology 3 Core & Periphery Cities 3 Cities Objections 4 Summary / 5
Assumptions New Economic Geography model: Attempt to violate the assumption 4 (perfect/complete market) of Starrett s theorem to generate cities. Developed by M. Fujita, became well-known by P. Krugman. 3 / 5 Assumptions sectors: agriculture & manufacture locations (regions) Agricultural workers: immobile Manufacturing workers & firms: mobile Worker attributes are not interchangeable Identical consumers 4 / 5
Assumptions Question to Be Addressed: Generating Cities Given the previous conditions, what would disturb the uniform distribution? How does imperfect competition lead to a city? 5 / 5 Preferences are represented by a Cobb-Douglas utility function u(m, A) = M μ A μ, where A = consumption of agricultural goods. M = of manufactured goods. M is an index of various manufactured goods m,, m n, m N : σ σ M = m + + m σ σ n + + m σ σ N σ σ, where σ denotes the elasticity of substitution. M is called a CES (constant elasticity of substitution) function. 6 / 5
σ σ σ σ preferences perfect substitutes equally weighted Cobb-Douglas perfect complements NEG models assume σ >. 7 / 5 σ, Perfect Substitutes (σ )/σ (σ )/σ (m +m ) σ/(σ ).5.5.5 3.5.5 3 3.5 m.5.5.5.5.5.5 m 8 / 5
5 5.5 σ=, Substitutes 3 3 3 3.5 3.5 (m (σ )/σ +m (σ )/σ ) σ/(σ ) 4.5.5.5.5 m.5.5.5.5.5.5.5 m 9 / 5 5 σ, Cobb-Douglas 5 3 35 (m (σ )/σ +m (σ )/σ ) σ/(σ ).5 3 m 5 5 5.5 5 5 5 5 5.5.5 m / 5
. m.5.5...4.4.6.6 σ=., Complements.8.8 (m (σ )/σ +m (σ )/σ ) σ/(σ )..8.6.6.4.4.4.4.6.4....5.5 m / 5 m.5...4.4 σ, Perfect Complements.8.8.6.6. (m (σ )/σ +m (σ )/σ ) σ/(σ )..4.6.8.4.8.8.5.6.6.4.4.4....5.5 m / 5
Utility maximization problem: max m,,m n,a Mμ A μ subject to GM + p A A Y, where G is a price index for manufactured goods. UMP is actually twofold. The problem above is the second half. See "The Spatial Economy" by Fujita, Krugman & Venables. MRS at (A, M) is μ μ Tangency condition : A M. μ M μ A = G. p A Solution (not a bundle): (A, M ) = ( μ) Y, μ Y p A G. 3 / 5 Fact: manufacturing price index G is negatively correlated with the number of varieties N. 4 / 5
5 4 Price Index G=p M N /( σ) (σ<) G=p M N /( σ) (σ>) Price Index (G) 3 5 5 # of Varieties (N) 5 / 5 Interpret: σ < : Added variety is bad news for consumers. They have to buy each and every variety to maintain the utility level, which costs them more. This scenario will not happen in an NEG model. σ > : Added variety is good news for consumers. They do not have to buy everything. They can buy a similar but cheaper good while maintaining their utility level. 6 / 5
Agricultural Sector One farmer produces one agricultural product: q A (l A ) = l A. Marginal product at l A is MP A (l A ) =. No centripetal nor centrifugal force in agricultural sector. Constant returns to scale technology. Competitive environment. Costless transport. Pick the uniform distribution: μ of farmers in each region. p A is same in both regions. Having immobile workers prevent us from ending up with a degenerate distribution. 7 / 5 Agricultural Sector Agricultural wage w A is found by w A = p A MP A (l A ). Factor price is equated to marginal value product. LHS=additional cost from one more hour worked. RHS=additional revenue from one more hour worked: p A MP A (l A ) = p A $ apples MP A (l A ) additional apple one more hour worked If w A < p A MP A (l A ), additional labor brings in more revenue than the additional cost it incurs. If w A > p A MP A (l A ), additional labor costs more than the additional revenue that he brings in. Due to CRS, MP A (l A ) = everywhere.. 8 / 5
Agricultural Sector Agricultural Factor Supply & Demand Factor Supply Factor Demand Agricultural Wage w A ($) p^a MP^A Agricultural Labor l A (hours) 9 / 5 Agricultural Sector Agricultural factor market is in equilibrium only when w A = p A MP A (l A ). With a production function q A (l A ) = l A, w A = p A. Agricultural profit is π A (l A ) = p A q A (l A ) w A l A =. Question: Marginal Product & Factor Price under CRS Technology p A MP A (l A ) > w A does not only fail to put the factor market in equilibrium but also explodes the profit level. Why? And what is its consequence? / 5
Agricultural Sector Agricultural Profit Total Cost TC(l A )=w A l A Total Revenue TR(l A )=p A l A Total Cost & Revenue ($) p^a MP^A w Agricultural Labor l A (hours) / 5 Model Assumptions Agricultural Sector Monopolistic Competition Manufacturing Sector Monopolistic Competition Profit Maximization Problem Iceberg Transportation Technology 3 Core & Periphery Cities 3 Cities Objections 4 Summary / 5
Manufacturing Sector Same technology for all the manufactured goods. Skilled workers can move between regions: Input requirement: L + L = μ. l n = F + cq n. l n = labor F = fixed input (same across the manufacturing sector) c = additional input requirement (labor/output) q n = output level Production function of a manufacturing good q n : q n (l n ) = c ln F c ( ) Returns to scale? 3 / 5 Manufacturing Sector 4 3.5 Manufacturing Production Function q n (l n )=l n /c F/c=l n 3.5 q n.5.5 3 4 5 l n 4 / 5
Monopolistic Competition No firm will produce the same product: Unlike Hotelling s model, consumers substitute this ice cream and that ice cream only imperfectly. Since technology is IRS, introducing a new, differentiated product to create demand works. Market is assumed monopolistically competitive (monopolistic competition). 5 / 5 Monopolistic Competition Perfectly competitive market: Product homogeneity Many firms 3 Free entry / exits environment ➊ ➋ ➌ perfect competition monopoly monopolistic competition 6 / 5
Monopolistic Competition Monopolistic competition: An entrant will produce a slightly different product than the existing varieties. This will rip off some of the incumbents profit. 3 The entry keeps happening till all the firms profits go down to zero. Differs from perfect competition: inherits monopolistic pricing (Compare to the agricultural sector). Differs from monopoy: profit is pushed back to zero. Differs from oligopoly: firms are in a different market. A firm s action affects others only via price index. 7 / 5 Monopolistic Competition Discussion: Location Choice Each manufactured good n is produced at most one city in equilibrium. Why? 8 / 5
Monopolistic Competition 9 / 5 Profit Maximization Problem Profit maximization problem: max π n (q n ) = p n q n w n (F + cq n ). If the firm were in a perfectly competitive market, optimal production plan satisfies p n = w n c, i.e., additional revenue from producing one more unit is cancelled out by additional cost of producing one more unit. (Consider what happens if they do not equate). 3 / 5
Profit Maximization Problem Fact: as a firm in a monopolistically competitive market, the firm can score a slight margin: p n = w n c σ σ σ σ is called a markup ratio. 3 / 5 Profit Maximization Problem 9 Markup Ratio Perfect Competition Monopolistic Competition σ/(σ ) 7 Markup Ratio 5 3 3 4 5 σ (Elasticity of Substitution, Price Elasticity) 3 / 5
Profit Maximization Problem Note that markup will be competed away in equilibrium due to free entrance. Due to the lack of first nature advantage (i.e., c and F are same everywhere), optimal output level q n is same for all n =,, N. So is l n (l n = l for all n). # of varieties produced in city is N = L l. Firms in the same city share the factor price and product price (become indistinguishable except by products they produce). 33 / 5 Iceberg Transportation Technology Instead of introducing an independent transportation industry, assume iceberg transportation technology. origin (city of production) Notation: x destination (city of consumption). Of T ( ) units dispatched from city, one unit will be received in city. p = p T $ = $. 34 / 5
Model Assumptions Agricultural Sector Monopolistic Competition Manufacturing Sector Monopolistic Competition Profit Maximization Problem Iceberg Transportation Technology 3 Core & Periphery Cities 3 Cities Objections 4 Summary 35 / 5 Cities Suppose that initially L = λμ and L = ( λ)μ. UMP, PMP, and zero profit condition lead to equilibrium conditions (μ, λ, T are exogenous): Y = μλw + μ Y = μ( λ)w + μ G = λw σ + ( λ)(w T) σ σ G = λ(w T) σ + ( λ)w σ σ w = Y G σ + Y G σ T σ σ w = Y G σ T σ + Y G σ σ ω = w G μ ω = w G μ 36 / 5
Cities Skilled workers change its location depending on the equilibrium real wage (purchasing power) ω, ω. Suppose the following migration dynamics: If ω > ω, λ increases. If ω < ω, λ decreases. 3 If ω = ω, λ remains the same. Recall N = L l G. and N is negatively correlated with 37 / 5 Cities.593 T=.5 Differential Y Y G G w w ω ω.593.5.5.75 λ 38 / 5
Cities.6 T=.7 Differential Y Y G G w w ω ω.7.5.5.75 λ 39 / 5 Cities.75 T=. Y Y G G w w Differential ω ω.75.5.5.75 λ 4 / 5
Cities If the differential ω ω is downward-sloping when it goes across ω = ω line, the equilibrium is stable. Otherwise, the equilibrium is unstable. 4 / 5 Cities Comparative static on T. Pitchfork bifurcation: When transportation cost is negligible, there will be a core (manufacturing workers and half the agricultural worker) and a periphery (half the agricultural worker). When transportation cost is in the intermediate range, there are three stable equilibria, one of which is trivial. When transportation cost is high, the distribution is trivial. 4 / 5
Cities Equilibria Fraction of Manufacturing Workers in City, λ.9.8.7.6.5.4.3...5.7. Iceberg Transportation Cost T 43 / 5 Cities Equilibria (T =.T ) Fraction of Manufacturing Workers in City, λ.9.8.7.6.5.4.3....4.6.8. T 44 / 5
3 Cities Similar patterns for 3 cities. 45 / 5 3 Cities 46 / 5
3 Cities 47 / 5 3 Cities 48 / 5
Objections New Economic Geography model explains 9th century US well, when raw materials were costly to ship. Some objections to NEG models: Sensitivity issues: Not known if the limiting cases hold (σ or σ ). Recall that prediction of monocentric city models are robust against preference specifications. Indeterministic multiple equilibria. 49 / 5 Model Assumptions Agricultural Sector Monopolistic Competition Manufacturing Sector Monopolistic Competition Profit Maximization Problem Iceberg Transportation Technology 3 Core & Periphery Cities 3 Cities Objections 4 Summary 5 / 5
Monopolistic competition Markup ratio Zero profit condition Migration dynamics Ptichfork Bifurcation Drawbacks of NEG models 5 / 5