Aggregate Supply Econ 208 Lecture 16 April 3, 2007 Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 1 / 12
Introduction rices might be xed for a brief period, but we need to look beyond this The di culties of price-setting theory (there is no Walrasian auctioneer) Why imperfect competition is useful First pass - the model from Romer, ch. 6.4 Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 2 / 12
Imperfect Competition and rice-setting Basic Assumptions: 1 n individuals and n goods 2 Each individual is a monopoly producer of one of the goods, but likes to consume all goods. 3 The only input to production is labor, used one-for-one: Q i = L d i output of good i = demand for labor by individual i 4 There is an economy-wide competitive labor market 5 At rst, we assume no uncertainty Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 3 / 12
The order of events 1 Each individual i sets the price i 2 Then the labor market meets, in which the wage rate W is determined along with each individual s and each individual s supply of labor L i and each individual s supply Q i. 3 Each individual i chooses her aggregate demand C i and her demand C ij for each individual product j Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 4 / 12
Individual demands Utility function for individual i U i = C i (1/γ) L γ i, γ > 1, where " # η 1 C i = n η 1 η 1 nc η ij, η > 1 j (η is the elasticity of substitution between goods) Maximizing U i subj to j p j C ij = M i (i s nominal income: wages + pro t) yields: C ij = 1 M η i j, j = 1, 2,..., n n where is the aggregate price level, de ned as = 1 n j 1 j η! 1 1 η Note: C i = M i Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 5 / 12
Market demands for goods, and the supply and demand for labor The market demand for each good j is Q d j = C ij = Y i η j, where Y = 1 n M i (average real income) i Each individual i will face the budget constraint: C i = ( i W ) Q i + WL i So she will choose ( i, Q i, L i ) so as to max U i = ( i W ) Q i + WL i η i subj to Q i = Y (1/γ) L γ i Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 6 / 12
Optimal rice-setting Individual i s decision problem: i W max First-order condition for i : 1 η Y i multiply by Y 1 i 1+η: i Y i η η i + W L i i η (1/γ) L γ i W Y 1 η 1 i = 0 W = 0 So the optimal relative price is a xed markup over the real marginal cost: where the markup η η 1 i = η W η 1 is a decreasing function of the elasticity of demand. Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 7 / 12
The supply of labor Recall the decision problem: i max W η i Y + W L i (1/γ) L γ i First-order condition for L i : W L γ 1 i = 0 So the supply of labor by individual i is: L i = 1 W γ 1 Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 8 / 12
Equilibrium In a symmetric equilibrium, we have: L i = Q i = Y for all i (Labor-market clearing) In logs: or: W = Y γ 1 (Marginal cost increasing in Y ) i = η η 1 Y γ 1 (Relative price = markup over MC) p i = p + ln η η 1 p i = p + c + φy + (γ 1) y where c > 0 is the log of the markup and φ > 0 is the elasticity of marginal cost wrt output. Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 9 / 12
Capacity output The economy cannot go forever with people trying to set a relative price greater than unity. Thus capacity output y is the value of y such that the average price-setter is content to set p i = p So the price-setting equation can be written as: 0 = c + φy p i = p + c + φy p i = p + φ (y y ) Note the resemblance to an expectations-augmented hillips Curve (in price-output space) where the p i on the LHS is the price that each price-setter will set, the p on the RHS is the price-level that each price-setter expects to prevail in the market, and the coe cient φ is the elasticity of marginal cost (an inverse measure of real rigidity ) Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 10 / 12
Dynamics and uncertainty (The Fischer model) Suppose people set relative prices at date t the same way but based on expectations of p, y and y : p t = E t 1 p t + φe t 1 (y t y t ) where E t 1 is the (rational) expectation conditional on information up to and including period t 1 We have: E t 1 p t = E t 1 p t + φe t 1 (y t y t ), so E t 1 (y t y t ) = 0 p t = E t 1 p t Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 11 / 12
olicy Ine ectiveness Suppose that nominal aggregate demand is controlled by the central bank, subject to error y t = m t p t + ε t where ε t is an iid random variable with E ε t = 0 Then: E t 1 y t = E t 1 m t E t 1 p t + E t 1 ε t E t 1 yt = E t 1 m t p t So: y t = E t 1 yt + (m t E t 1 m t ) + ε t For example, if m t = a 1 ε t 1 + u t, where u t is iid with Eu t = 0, so E t 1 m t = a 1 ε t 1, then: y t = E t 1 yt + u t + ε t, independent of the feedback parameter a 1 Econ 208 (Lecture 16) Aggregate Supply April 3, 2007 12 / 12