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Lecture 2: Applied general equilibrium (Chapter 3) • • • Introduction of formats Comparing the formats Model implementation – – – – Classification The Social Accounting Matrix (SAM) Missing commodities and markets Closure rules Aim of lecture 2 • Highlighting the characteristics of different representations of general equilibrium models (formats) • Illustrating the steps from a general equilibrium to an applied general equilibrium model Importance of existence proofs • Non-existence of a solution implies model is inconsistent and therefore useless • Existence proofs highlight crucial necessary assumptions that have to be made • For applications: – Necessary assumptions indicate scope for variation of parameters – Construction of fixed point mapping to prove existence may suggest algorithm for numerical implementation Excess demand format • For every producer j, y *j solves max y p* y j y j Yj j • For every consumer i, xi* solves max x 0 ui ( xi ) p* xi hi* , where i hi* p*i j ij p* y*j • All markets are in equilibrium: * * x y i i j j i i 0 • Define excess demand function z( p) i xi ( p) j y j ( p) i i • The price p* 0, p* 0 , and the excess demand z( p* ) define an excess demand equilibrium if z ( p* ) 0 . Excess demand format (continued) • Debreu (1959) assumes convex production sets and quasiconcave utility functions. Therefore, excess demand correspondence is not single-valued. • This is problematic since compatibility of allocations is not guaranteed • Therefore, Debreu assumes consumer demand functions to be single-valued, and aggregates production by different producers into one single set Negishi format (a) Welfare program maxxi 0,all i,y j ,all j i i ui ( xi ) subject to i xi - j y j i i (p) y j Y j (b) Adjustment of welfare weights i S m such that budget constraints pxi pi j ij j ( p ) hold for every i, where j ( p ) max y py j y j Y j is the profit function j of producer j Full format Constrained welfare optimum: maxxi 0,all i,y j ,all j i i ui ( xi ) subject to i xi j y j i i (p) y j Yj pxi j ij py j pi ( i ) for given S m and p S r such that i 0 is a shadow price and p= p for some scalar >0 . Open economy format (a) Optimization with given consumptions x̂i : maxm,e0,y j ,all j p ee p m m subject to j y j m e i ˆxi i i ( p) y j Yj (b) A feedback relation that sets x̂i by solving the consumer problem: x̂i arg max ui ( xi ) pxi pi j ij j ( p ),xi 0 ,all i where j ( p ) max py j , y j Y j CGE format • CGE format is a system of simultaneous equations: – balance equation for factors (f) and goods (g) g g f i xi ( p , p f g f i xi ( p , p g ,hi ) A ( p , p )q q ,hi ) A f ( p g , p f )q g i if g g f g – complemented with individual budgets: hi p f if – and price relations: p g p g Ag ( p g , p f ) p f A f ( p g , p f ) CGE format (continued): assumptions • Constant returns to scale in production – no profits in equilibrium • Factors are not produced and used in production – boundedness of production • Goods are not available as endowments • Utility functions are continuous, strictly quasi-concave and non-satiated. For at least one consumer, utility increases in all factors CGE format (continued): extensions • Decreasing returns – Firm-specific inputs in CRTS technology imply DRTS in remaining inputs – Profits have to be included in budgets • Markups – Compensation for inputs not included in the model – Caused by imperfect competition • Closure rules Comparing formats: merits • Excess demand – Microfoundations • Negishi and Full format – Direct link to welfare analysis – Weaker assumptions with respect to production technology – Solvable even if decentralization is problematic • Open economy format – Econometric estimation of model parameters – Possibility to include non-optimizing behavioral rules • CGE format – Easiest for application – Possibility to include non-optimizing behavioral rules Comparing formats: mathematical requirements Production sets Utility functions Endowments Excess demand format Possibility of inaction, Continuous, strictly Strictly positive for compact, strictly concave, non-satiated, consumers convex increasing in all commodities* i 0, i 0 for all i Negishi Possibility of inaction, Continuous, strictly format compact, convex concave, nonsatiated, 0 for i=1 i increasing for I=1 i 0, i 0 for all i Full format Possibility of inaction, Continuous, strictly compact, convex concave, nonsatiated, i 0 for i=1 increasing for I=1 i 0, i 0 for all i Open Possibility of inaction, Continuous, strictly economy compact, convex concave, nonsatiated, i 0 for i=1 format increasing for I=1 i 0, i 0 for all i CGE format CRTS or DRTS Continuous, strictly production technology quasiconcave, i 0 for i=1 for goods. Factors are nonsatiated, not produced increasing for I=1 for factors only *These are the assumptions made in G/K needed for single-valued excess demand. all Mapping from theorems to “work horses” Concavity of objective 1.4 Excess demand 3.2 Negishi 3.3 Full format Assumptions on u ( x) Assumptions on u ( x) Assumptions on u ( x) Constraint set of welfare program Constraint set of welfare program including budgets Slater Homogeneity Prices can be constrained to lie on a simplex. Welfare weights can be constrained to lie on a simplex Welfare weights and prices can be constrained to lie on a simplex Maximum theorem Continuity of consumer demand (strict concavity of utility function) ; Continuity of supply functions (strict convexity of production sets) Continuity of consumer demand in welfare weights and endowments, upper semicontinuity of price correspondence in welfare weights, compactness and convexity of set of prices Continuity of consumer demand in welfare weights and endowments, upper semicontinuity of Lagrangian multiplier correspondence in welfare weights and budget prices, compactness and convexity of set of Lagrangian multipliers Kakutani Fixed point in p Fixed point in and p Fixed point in ,p, p, Driven by excess demand (Brouwer) Driven by budget surplus Driven by p, 3.4 Open economy Constraint set of optimization program Upper semicontinuity , compactness and convex-valued ness of price correspondence in consumption of individuals. Fixed point in p Driven by p Classifications • Commodities – Focus on relevant characteristic that allows aggregation of different products in some common unit – Maintain link to data availability • Agents – Consumers • Dissimilarities between groups with respect to reactions on policy changes (employers vs employees) • Similarities between agents within the same group (income sources, consumption patterns) – Producers • Different production technologies for single good? Organizing data: the SAM Goods Factors Firms Consumers Goods Input Consumption Factors Input Consumption Firms Production (Production) Consumers (Endowments) Endowments (Profits) *Terms within parentheses are zero in the basic CGE model *This SAM represents a closed economy without government The SAM (Continued) • SAM structure and classification follow model architecture • SAM entries are in value terms • Rows and associated columns have equal value since they represent balances – – – – – Commodity balances in value terms Budgets Firm profits (Government budget) (Balance of payments) • Prices follow from confronting SAM with balances in quantity terms Missing commodities and markets • Rough markets – Only one market for different qualities • Markets for characteristics – Consumer preferences in terms of characteristics – Markets usually in terms of products – Introduce mapping from characteristics to products • Commodities traded but not fully represented – Missing markets for goods implies these go unpriced • Markups – Reflect reward for input not accounted for in the model Closure rules • Including macro-economic mechanisms in general equilibrium model – define a new variable – define balancing constraint for this variable – Example: Taxes (closure variable) and government budget (associated constraint) – Beware: Closure rule can become dominant mechanism in model! Closure rules (continued) • CGE model with parameters: • Closure rule: • Relation: z ( p; q) 0 g ( p, q ) 0 z ( p; q) 0 g ( p, q ) 0