Transportation Economics Lecture 12: Production and Costs in Transportation Economics, Study notes of Economics

Download Transportation Economics Lecture 12: Production and Costs in Transportation Economics and more Study notes Economics in PDF only on Docsity! ECN 145 Lecture 12 Transportation Economics: Production and Costs I Demand and Supply • Chapters from Essays textbook: Q0 P P0 Q Supply (Chapter 3) Pricing (Chapter 4) Demand (Chap. 2) Production Function • Illustrate with “iso-quants”: f(x)=y0λx0+(1-λ)x1 x0 x1 X2 X1 f(x)=y1>y0 Returns to Scale • Suppose that there is one input x, and, • α=1 ⇒ constant returns to scale • doubling all inputs will just double output • α>1 ⇒ Increasing returns to scale • doubling all inputs more than doubles output • α<1 ⇒ decreasing returns to scale. • doubling all inputs will less than double output α== x)x(fy Input prices • Prices for inputs xi are wi • E.g. labor – wage • Capital – rental price • Fuel – cost of oil; • Total costs are, ∑ = = n 1i ii xwC Change in Price: • Suppose price falls from w1 to w1’: f(x)=yB A x2 x1 Slope = -w1/w2 Slope = -w1′/w2 Change in Price (cont’d): • Fall in w1 will increase demand for x1, and reduce demand for x2, moving from A to B. • - pure “substitution” effect0 w x i i < ∂ ∂ Demand Curve • This gives us downward sloping demand: D wi Xi Average costs: • Hold the (single) input price w fixed: • Define average costs, • E.g. Total costs =$100, y=5, so AC=$20 ,wy y wy y )w,y(CAC 1/1 α α−α === Marginal costs • Hold the (single) input price w fixed: • Define marginal costs, • E.g. Total costs=$100 when y=5, $115 when y=6 • So marginal costs are $15. α α− α = ∂ ∂ = 1 yw y )w,y(CMC Returns to Scale: ,wyAC 1 α α− = α α− α = 1 wy1MC • so, • which is a measure of returns to scale! , MCy CostsTotal MC AC ⋅ =α= Decreasing Returns to Scale: α<1 (Say, α=1/2) $ MC AC Decreasing returns y Eg: Cobb-Douglas Production Function • Notice that doubling both x1 and x2 : • So, • constant returns to scale • increasing returns to scale • decreasing returns to scale 0 , ,xx)x(fy 21 >βα== βα )x(f2)x2()x2()x2(f 21 β+αβα == 1⇒=β+α 1⇒>β+α 1⇒<β+α Cobb-Douglas Cost Function • Use a Lagrangian, • Find first-order condition w.r.t x1, x2: yxx subject to xwxwmin 212211 ≥+ βα )xx -(yxwxwL 212211 βαλ++= )x(fxw0xxw x L 112 1 11 1 λα=⇒=λα−= ∂ ∂ β−α )x(fxw0xxw x L 22 1 212 2 λβ=⇒=λβ−= ∂ ∂ −βα Returns to Scale: • First write the log of costs as: • Differentiating this w.r.t. y, we see that, • measures the returns to scale! 21 wlnwlnyln 1BCln       β+α β +      β+α α +      β+α += )( yln Cln )y/C(y Costs MC AC 1 β+α=      ∂ ∂ = ∂∂ = − Cobb-Douglas Input Demands: • Demands for inputs are obtained by differentiating costs: β+α β β+α −α β+αα= ∂ ∂ = 2 1 1 1 1 211 wwAyw C)w,w,y(x β+α −β β+α α β+αβ= ∂ ∂ = 1 21 1 2 212 wwAyw C)w,w,y(x Costs Shares: • Comparing x1 and x2 with total costs: • we see that, • which are constant! β+α α = C xw 11 β+α β = C xw 22 )ww(Ay)(C 21 1 β+α β β+α α β+αβ+α= Translog Cost Function (cont’d) • Note that the first line is just Cobb-Douglas, • (in logs, with multiple inputs and outputs): • The “extra” terms on the second and third lines allow for more general substitution between inputs and outputs. ∑∑ == ++= n 1i ii m 1i ii0 wlnbylnaa)w,y(Cln Translog Cost Shares: • Differentiating the cost function, • so that, • this allows for a wide pattern of substitution between inputs. j m 1i ij n 1j jiji ii i ylng wlnbb C wx wln Cln ∑∑ == ++== ∂ ∂ 0b wlnwln Cln ij ji 2 ≠= ∂∂ ∂ Returns to Scale: • so if aij=0, then, • is a measure of returns to scale! 1 i iyln/Cln)y/C(y Costs MC AC −       ∂∂= ∂∂⋅ = ∑ 1 i iaMC AC −       = ∑