辅导案例-EC 502

  • May 15, 2020

EC 502 Lecture 2: Production Functions Stephen J. Terry∗ 1 The Source of Output We reviewed some basic facts about the growth and level of output or GDP across coun- tries in the last class. To explain those facts with a model, we must first specify the source of output. Definition: A production function is a function specifying how much output can be pro- duced with a given combination of inputs. Labor L and capital K are the two inputs we will consider most often, but there is no reason in principle to restrict ourselves to this simple list of inputs. In general, microe- conomics uses production functions to describe the output of an individual firm. In this macroeconomics class, we will typically use a production function to instead describe the output or GDP of the entire economy. 2 The Neoclassical Production Function This workhorse of macroeconomics describes the overall output or GDP in an economy Yt at time t as a function F of capital Kt and labor Lt Yt = F (Kt, AtLt). 2.1 Technology In the equation above At is some number which describes the level of technology in the economy. It is also sometimes convenient to think of At as representing the level of knowledge in an economy. For much of this class we will assume that At is exogenous, i.e. we will take the level or path of At over time as given. ∗These notes borrow heavily from notes by Adam Guren, Simon Gilchrist, and Francois Gourio. 1 I have written technology At as labor augmenting above, which means that At enters multiplicatively with respect to Lt. Therefore, increasing the level of technology makes labor Lt more productive, and we refer to AtLt as the efficiency units of labor. Production functions can also be written to assume capital augmenting technology, F (AtKt, Lt), or with Hicks neutral technology AtF (Kt, Lt). The distinction will typically make little difference for our purposes, but for concreteness we will focus on labor aug- menting technology. 2.2 Properties and Definitions We will typically make several natural assumptions about the production function F . First, we will typically assume that F is differentiable with respect to all its inputs and that the higher order derivatives of F always exist. This allows us to lay out some additional terminology and assumptions. • The marginal product of an input is the partial derivative of output with respect to that input. The marginal product of capital MPKt is given by MPKt = ∂ ∂Kt F (Kt, AtLt) = F1(Kt, AtLt). Holding At and Lt fixed, MPKt represents the approximate increase in output Yt that would result from an additional unit of capital. Similarly, the marginal product of labor MPLt is given by MPLt = ∂ ∂Lt F (Kt, AtLt) = F2(Kt, AtLt)At. The first equality is a definition, while the second equality results from application of the Chain Rule. As with capital, MPLt is the approximate increase in output Yt resulting from an additional unit of labor, holding At and Kt fixed. • We typically assume that both inputs are productive or that the production function F is strictly increasing in both inputs: MPKt > 0,MPLt > 0. • We also typically assume diminishing returns to each input by itself: ∂ ∂Kt MPKt = ∂2 ∂K2t F (Kt, AtLt) < 0 ∂ ∂Lt MPLt = ∂2 ∂L2t F (Kt, AtLt) < 0 2 Holding capital fixed, the additional output generated by hiring an additional unit of labor is decreasing in the level of labor already used. Similarly, holding labor fixed, the additional output from an extra unit of capital declines with the level of capital. • Often, we will also assume that capital and labor are complements. This means that the marginal product of capital is increasing in the level of labor, and vice-versa, i.e. ∂ ∂Lt MPKt > 0, ∂ ∂Kt MPLt > 0. In practice, this means that workers with more capital are more productive, and capital with more workers is more productive. It’s worth pausing for a moment to ask whether either of these statements can be true if the other is false? The following theorem is helpful. Young’s Theorem: If the second partial derivatives of f(x, y) are continuous, then ∂2 ∂x∂y f(x, y) = ∂ 2 ∂y∂x f(x, y). • We will also often assume that the production function F satisfies constant returns to scale in capital and labor. This means that for any constant λ > 0 the function F satisfies F (λKt, AtλLt) = λF (Kt, AtLt). This implies, for example, that doubling both capital and labor doubles the output of the economy as can be seen by setting λ = 2. • We also typically assume that the production function satisfies the following techni- cal conditions: – F (Kt, 0) = 0 and F (0, AtLt) = 0 – F (Kt, 1) is concave – limKt→∞ ∂∂KtF (Kt, 1) = 0 – limKt→0 ∂∂KtF (Kt, 1) =∞ The first condition simply states that both inputs are essential for positive produc- tion. Later on, the other conditions will help to guarantee a positive and finite level of output per capita in the long run. 2.3 The Intensive Form of the Production Function With a constant returns to scale production function, we will often work with the intensive form of the production function. This form expresses output per efficiency units of labor 3 as a function of capital per efficiency units of labor. We will use undercase variables to refer to these quantities: yt = YtAtLt and kt = Kt AtLt . With this notation in place, we can write yt = Yt AtLt = 1 AtLt F (Kt, AtLt) = F ( Kt AtLt , AtLt AtLt ) = F ( Kt AtLt , 1 ) = F (kt, 1). We can now define a new function, the intensive form of the production function, f(kt) = F (kt, 1). With constant returns to scale we therefore have yt = f(kt). Because we assumed that capital is productive and has diminishing returns in the original function F , we can immediately conclude that the intensive form f satisfies the same properties ∂ ∂kt f(kt) = ∂ ∂kt F (kt, 1) > 0 ∂2 ∂k2t f(kt) = ∂2 ∂k2t F (kt, 1) < 0 The intensive form is convenient because it allows us to summarize a two-input constant returns to scale production function using a single input. If we are only interested in the output per efficiency unit of labor yt, then we only need to track the ratio of capital to the efficiency units of labor kt. 2.4 Example: Cobb-Douglas Production Function The Cobb-Douglas production function, first proposed by Charles W. Cobb and Paul Dou- glas in the 1920s, is given by Yt = K α t (AtLt) 1−α for some α ∈ (0, 1). The constant α is often referred to as the “capital elasticity” or “capital share” for reasons that will become clear later. We can easily compute each of the quantities above and verify each of the assumptions above for this functional form: • The marginal products of capital and labor are given by MPKt = ∂ ∂Kt Kαt (AtLt) 1−α = αKα−1t (AtLt) 1−α = α Yt Kt MPLt = ∂ ∂Lt Kαt (AtLt) 1−α = (1− α)Kαt (AtLt)−αAt = (1− α) Yt Lt • The expressions above imply that MPLt > 0 and MPKt > 0 for any Kt, Lt > 0, so both capital and labor are productive inputs. 4 • Both capital and labor have diminishing returns. Note that ∂ ∂Kt MPKt = ∂ ∂Kt αKα−1t (AtLt) 1−α = α(α− 1)Kα−2t (AtLt)1−α = −α(1− α) Yt K2t ∂ ∂Lt MPLt = ∂ ∂Lt (1−α)Kαt (AtLt)−αAt = (1−α)(−α)Kαt (AtLt)−α−1A2t = −α(1−α) Yt L2t . Since 0 < α < 1, ∂ ∂Kt MPKt < 0 and ∂∂LtMPLt < 0. • Labor and capital are complements. Note that ∂ ∂Lt MPKt = ∂ ∂Lt αKα−1t (AtLt) 1−α = α(1− α)Kα−1t (AtLt)−αAt. Since 0 < α < 1, ∂ ∂Lt MPKt > 0. By Young’s theorem MPLt is also increasing in Kt. • The Cobb-Douglas production function exhibits constant returns to scale. For λ > 0 F (λKt, AtλLt) = (λKt) α(AtλLt) 1−α = λαλ1−αKαt (AtLt) 1−α = λKαt (AtLt) 1−α = λF (Kt, AtLt). • The Cobb-Douglas production function satisfies our technical conditions, since – F (0, AtLt) = 0α(AtLt)1−α = 0 and F (Kt, 0) = Kαt 01−α = 0 – F (Kt, 1) = Kαt is concave with ∂2 ∂K2t Kαt = −α(1− α)Kα−2t < 0 – limKt→∞ ∂∂KtF (Kt, 1) = limKt→∞ αK α−1 t = 0 – limKt→0 ∂∂KtF (Kt, 1) = limKt→0 αK α−1 t =∞ At this point, it is useful to point out a few facts about the Cobb-Douglas production func- tion which will come in handy later: • The marginal product of each input is proportional to the average product: MPKt = α Yt Kt ∝ Yt Kt = Average Product of Capital MPLt = (1− α)Yt Lt ∝ Yt Lt = Average Product of Labor • Because the Cobb-Douglas production function satisfies constant returns to scale, the intensive form will often be useful. The intensive form is given by f(kt) = k α t . 5 • The elasticities of output with respect to capital and labor are given by α and 1− α, respectively. To see this, note that ∂Yt ∂Kt Kt Yt = ∂ log Yt ∂ logKt = α ∂Yt ∂Lt Lt Yt = ∂ log Yt ∂ logLt = 1− α. For each equation, the first expression on the left hand side above is the definition of the elasticity. The first equality then follows from the argument below. The sec- ond equality follows from taking the logarithm of the production function to get the equation log Yt = α logKt + (1− α) logAt + (1− α) logLt. Log Differentials & Elasticities: Let Y = f(X) for differentiable X. Then define the transformations x = logX and y = log Y . We have Y = f(X) (assumed) ey = f(ex) (substitute) ∂ ∂x ey = ∂ ∂x f(ex) (differentiate) ey ∂y ∂x = f ′(ex)ex (Chain Rule) ∂ log Y ∂ logX = f ′(X) X Y (re-arrange and substitute for elasticity formula) 3 An Argument for an Aggregate Production Function The assumption that we can use a single production function F to describe the output of an entire economy is clearly quite strong. A more palatable starting point might be to use the same production function to describe the output of a single firm. However, it turns out that under the assumption of constant returns to scale there is a natural link between the firm and aggregate production functions. First, let’s review the standard argument in support of constant returns to scale which is known as the replication argument. It relies on the following thought experiment. First, take an entire economy, say Mexico, which uses some given levels of capital, labor, and technology to produce output. Then, imagine that we snap our fingers and magically duplicate Mexico to create a second, identical economy alongside the first one. The output of the two Mexicos together should simply be twice the output of the original Mexican economy, as long as they are using the same production practices and technology to generate output. This result is exactly the constant returns to scale definition with λ = 2. 6 If we accept the replication argument and hence the constant returns to scale assumption, we can now given an example of conditions under which there is a tight link between the firm-level and aggregate production functions. 3.1 Example Problem: The Irrelevance of Microeconomics There are N firms in an economy indexed by i = 1, ..., N . Each firm produces output Yi using capital Ki and labor Li with access to the same production function Yi = F (Ki, ALi) featuring fixed technology A. The total amount of capital and labor in the economy are given by K = ∑N i=1Ki and L = ∑N i=1 Li, respectively. Assume that F exhibits constant returns to scale. Furthermore, assume that each firm uses capital and labor in the same proportion satisfying Ki ALi = κ for some constant κ. Prove that “microeconomics is ir- relevant,” i.e. show that the total level of output Y = ∑N i=1 Yi must always satisfy the aggregate production function Y = F (K,AL) regardless of the allocation of inputs across firms {(Ki, Li)}Ni=1. Solution: Under these assumptions of identical factor proportions Ki = κALi for all firms. Write K = N∑ i=1 Ki = N∑ i=1 κALi = κA N∑ i=1 Li = κAL. Therefore, the aggregate capital and labor inputs satisfy the same proportionality as the firm-by-firm inputs with κ = K AL . Then, we can write Y = N∑ i=1 Yi (definition) = N∑ i=1 F (Ki, ALi) (definition) = N∑ i=1 ALiF ( Ki ALi , 1) (constant returns to scale) = N∑ i=1 ALiF (κ, 1) (assumption) = AF (κ, 1) N∑ i=1 Li (constants) = ALF (κ, 1) (definition) = ALF ( K AL , 1) (result above) = F (K,AL) (constant returns to scale) This relationship, Y = F (K,AL) is exactly the desired aggregate production function. 7 3.2 Profit Maximization with Capital and Labor The aggregation example 3.1 above relies upon the assumption of identical input ratios at firms. So far we have only discussed possibilities for production, so we do not yet have a framework for predicting the choice of inputs. In this section, we’ll offer one simple set of predictions for firm behavior relying upon the assumption of profit maximization. Consider a firm i with a constant returns to scale production function F (Ki, ALi) facing a fixed technology level A. The firm can hire labor at the wage rate W per unit and may rent capital at the rental rate R. The profits of the firm are given by output minus input costs, i.e. F (Ki, ALi)−WLi − RKi. This expression embeds the assumptions of perfect competition or price taking in both the output and input markets. Under these conditions, a firm which seeks to maximize its profits will choose optimal levels of capital and labor by solving max Ki,Li F (Ki, ALi)−WLi −RKi. It is straightforward to show that the first-order conditions for optimality in this problem are MPKi = R, MPLi = W. The firm must set the marginal product of each input equal to its marginal cost, which is the rental rate R for capital and the wage rate W for labor. If we are willing to assume a Cobb-Douglas form for the production function, we can proceed further. In that case, Yi = F (Ki, ALi) = Kαi (ALi)1−α. Above we showed that with a Cobb-Douglas production function the marginal product of an input is proportional to its average product: MPKi = α Yi Ki , MPLi = (1− α)Yi Li . The first-order conditions for profit maximization can then be written α Yi Ki = R, (1− α)Yi Li = W. Taking the ratio of the two first-order conditions, rearranging, and multiplying by 1/A yields Ki ALi = Wα AR(1− α) . If we consider N firms i = 1, ..., N which all face the same wage W and rental rate R, then each firm’s optimal choice of capital per efficiency unit of labor is equal to the constant Wα AR(1−α) . Setting κ = Wα AR(1−α) yields profit-maximizing inputs at each firm exactly satisfying the assumptions of Example 3.1 and implying the existence of an aggregate production function. 8

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