Production Functions
*Assumption: we have two inputs labor l and capital k. ( l and k called factors of productions)
*Marginal product of inputs or marginal physical product shows that if i use one more unit of input e.g. labor or capital by how much more unit i will produce. (second derivative of production function for l or k or derivative of marginal product with respect to input.) (ceteris paribus, holding other inputs constant)
*Why Diminishing marginal productivity: Without labor it makes no sense to increase capital e.g. machine numbers. Same logic with diminishing marginal utility. In marginal utility we said consumers prefer mixed bundles rather than extremes. Here we say producers prefer mixed bundle of inputs l and k rather than extremes.
*Thomas Maltus: He proposed diminishing marginal productivity. He interpreted that population is increasing but land is fixed so (by dimishing marginal productivity of labor ) labor productivity will decrease so he said there will be mass-starvation. But he was wrong because of total factor productivity was also increasing (tech improvements and innovations)
MRTS= how one input can be traded for another, holding output constant at what rate we can substitute labor for capital. (-dk/dl)|q=q0 also MRTS is ratio of inputs' marginal productivities.
Isoquants show the alternative combinations of inputs that can be used to produce a given level of output.
Cross-partial derivatives are Fkl Flk and we expect hem to be greater than zero (if labors have more machine their marginal productivity is higher) but we cannot generalize it to all production functions.
Returns to scale is for showing how output q changes if we increase all inputs by same amount. IRS can occur since when we, for example, double both labor and capital we might be more flexible with the division of labor. DRS may occur when we have, for example, more labor and capital it might become harder to effectively manage all. Production function determines which will occur.
Effect on Output Returns to Scale
f (tk, tl ) = tf (k, l ) = tq Constant (CRS)
f (tk, tl ) < tf (k, l ) =tq Decreasing (DRS)
f (tk, tl ) > tf (k, l ) =tq Increasing (IRS)
*If CRS then Production function is homogenous of degree 1 ->Marginal productivity functions derived from a constant returns-to-scale production function are homogeneous of degree 0. (general rule is if a function is homogeneous of degree k, its derivatives are homogeneous of degree k - 1)-> MRTS for CRS production functions only depends on the ratio of inputs (k/l)->Such functions are homothetic->their isoquants are radial expansions of one another. -> the isoquant labels increase proportionately with the inputs.
Elasticity of substitution shows how easy it is to substitute k for l ."If the MRTS does not change at all for changes in k/l, we might say that substitution is easy because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the MRTS changes rapidly for small changes in k/l, we would say that substitution is difficult because minor variations in the input mix will have a substantial effect on the inputs’ relative productivities." [1]
= percent change (k/l) / percent change in MRTS = dln(k/l)/dln(MRTS) when changes are small
k/l = capital intensity
If elasticity of subst. is high, then the RTS will not change much relative to k/l and the isoquant will be close to linear. On the other hand, a low value of s implies a rather sharply curved isoquant; the RTS will change by a substantial amount as k/l changes[1]
*IF CRS then elasticity of substution is 1 (k/l=MRTS)
Prod Func. Types
Linear isoquants-> perfectly substitutable-> elasticity of sub is infinite
Fixed prpoportions->no substitution-> min(xk,yl)
Cobb douglas x^ay^b; depends if a+b>1 IRS ıf a+b=1 CRS if a+b<1 DRS
1.Microeconomic Theory,Basic Principles and Extensions
WALTER NICHOLSON,CHRISTOPHER SNYDER