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Unformatted text preview: Ecn 101 – WQ09 – Answer Key for HW 6 Prof. Emanuel Frenkel Paul Gaggl Seema Sangita Yuan Xu 1 Determinants of Output under Perfect Compe tition Problem (3) To break the production function Y = F ( K, L ) into the four principal components of growth we need to make use of 3 assumptions: (A1) Firms seek to maximize their profits , (A2) firms engage in perfectly competitive markets , and (A3) the production function F ( K, L ) exhibits constant returns to scale . Let’s first define profits as “revenue minus cost”, Π ≡ P · F ( K, L )  {z } revenue ( w · L + r · K )  {z } cost . (1) Assumption (A2) implies that firms have no market power at all and hence have to take all prices as given: Sales prices P , wages w , and the rental rate of capital r . Therefore, what the firms get to choose is how much labor, L , to hire and how much capital, K , to install. In terms of math this means that the firms have to solve the following maximization problem: max K,L { P · F ( K, L ) ( w · L + r · K ) } . (2) This might seem like a simple restatement of equation (1) but in fact it is important to realize a crucial difference between equations (1) and (2). The former simply states the “objective” of the firm. This is what assumption (A1) says. Equation (2) however incorporates a lot more information. It tells us how the firm is maximizing its objective: by choosing the quantities L and K ! As you remember from calculus, the solution to the maximization problem (2) can be found by setting the first (partial) derivatives of Π equal 1 to zero: P · MP K z } { ∂F ( K, L ) ∂K r = (capital) (3) P · ∂F ( K, L ) ∂L  {z } MP L w = (labor), (4) which can also conveniently be written as MP K = r * P * (5) MP L = w * P * . (6) Optimality conditions (5) and (6) have a nice intuitive economic interpre tations: Firms hire their factors of production up to the point where the benefit from an additional unit equals the cost of that additional unit (in real terms). For labor and capital this simply means that firms install cap ital up to the point where the additional output produced from one more unit of capital ( MP K ) equals exactly its cost (the real rental rate of capi tal). Likewise, firms hire new workers up to the point where the additional output produced by that additional worker ( MP L ) equals exactly its cost (the real wage). In a perfectly competitive world (as we assume here) the supplies of labor and capital ( L s , K s ), which we assume are fixed, will be fully employed at the (real) equilibrium prices ( w * P * , r * P * ), which in turn are entirely determined by the marginal products of these factors, MP L and MP K . Consequently, real profits in equilibrium will be Π * P * = F ( K s , L s ) w * P * · L s + w * P * · K s ¶ = F ( K s , L s ) ( MP L · L s + MP K · K s ) . (7) Therefore, at this stage, we can decompose full employment output as fol lows: Y * ≡ F ( K s , L s ) = MP L · L s + MP K · K s + Π...
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This note was uploaded on 10/10/2011 for the course ECN 101 taught by Professor Frenkel during the Winter '10 term at UC Davis.
 Winter '10
 FRENKEL
 Macroeconomics, Perfect Competition

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