Question

The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry's demand for inputs. To do so, we assume that an industry produces a homogencous good, \(Q,\) under constant returns to scale using only capital and labor. The demand function for \(Q\) is given by \(Q=D(P)\), where \(P\) is the market price of the good being produced. Because of the constant returns-to- scale assumption, \(P=M C=A C\). Throughout this problem let \(C(v, w, 1)\) be the firm's unit cost function. a. Explain why the total industry demands for capital and labor are given by \(K=Q C_{v}\) and \(L=Q C_{w}\) b. Show that \\[ \frac{\partial K}{\partial v}=Q C_{v v}+D^{\prime} C_{v}^{2} \quad \text { and } \quad \frac{\partial L}{\partial w}=Q C_{w w}+D^{\prime} C_{w}^{2} \\] c. Prove that \\[ C_{w v}=\frac{-w}{v} C_{v w} \quad \text { and } \quad C_{w w}=\frac{-v}{w} C_{N w} \\] d. Use the results from parts (b) and (c) together with the elasticity of substitution defined as \(\sigma=C C_{v n} / C_{\nu} C_{w}\) to show that \\[ \frac{\partial K}{\partial v}=\frac{w L}{Q} \cdot \frac{\sigma K}{v C}+\frac{D^{\prime} K^{2}}{Q^{2}} \text { and } \frac{\partial L}{\partial w}=\frac{v K}{Q} \cdot \frac{\sigma L}{w C}+\frac{D^{\prime} L^{2}}{Q^{2}} \\] e. Convert the derivatives in part (d) into elasticities to show that \\[ e_{K, v}=-s_{L} \sigma+s_{K} e_{Q, p} \quad \text { and } \quad e_{L, w}=-s_{K} \sigma+s_{L} e_{Q, P} \\] where \(e_{Q, P}\) is the price elasticity of demand for the product being produced. f. Discuss the importance of the results in part (e) using the notions of substitution and output effects from Chapter 11 Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall. The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993).

Short answer

Short Answer: The results from part (e) display the relationship between the elasticity of derived demand for an input (such as capital and labor) and the price elasticity of demand for the output being produced. This relationship is crucial as it emphasizes the interconnected the demand for inputs and the demand for the goods they produce. The results reveal two key effects - Substitution and Output effects. Substitution effect occurs when the price of one input increases, leading to a decrease in demand for that input as firms opt for the cheaper alternative. The magnitude of this effect depends on the elasticity of substitution. Output effect takes place when an increase in input prices affects the overall demand for the output being produced, which in turn affects the demand for the input. The magnitude of this effect depends on the price elasticity of demand for the output. These findings enhance our understanding of the interdependence between input demand and output demand, as observed by Alfred Marshall.

Step-by-step solution

Understanding the Total Industry Demand for Inputs

To find the total industry demand for capital(required \(K\)) and labor(required \(L\)), we start by noting that the industry produces a homogenous good (\(Q\)), making use of only capital and labor. This means that the total demand for capital and labor depends on the quantity produced and the firm's unit cost function (which itself is a function of capital and labor). To find the industry demand for capital, we can multiply the quantity produced by the partial derivative of the unit cost function with respect to capital: \(K = QC_v\) Similarly, to find the industry demand for labor, we multiply the quantity produced by the partial derivative of the unit cost function with respect to labor: \(L = QC_w\) #b. Show that \(\frac{\partial K}{\partial v} = Q C_{vv} + D' C_v^2\) and \(\frac{\partial L}{\partial w} = Q C_{ww} + D' C_w^2\)#

Taking Partial Derivatives for Capital and Labor

To find the partial derivatives of \(K\) and \(L\) with respect to \(v\) (capital price) and \(w\) (labor price), we differentiate the expressions for \(K\) and \(L\) with respect to \(v\) and \(w\) respectively: \(\frac{\partial K}{\partial v} = Q \frac{\partial C_v}{\partial v} + \frac{\partial Q}{\partial P} \frac{\partial P}{\partial v} C_v\) Since \(D'(P) = \frac{\partial Q}{\partial P}\) and \(P = MC\), \(\frac{\partial K}{\partial v} = Q C_{vv} + D'(P) C_v^2\) Similarly, \(\frac{\partial L}{\partial w} = Q C_{ww} + D'(P) C_w^2\) #c. Prove that \(C_{wv} = \frac{-w}{v} C_{vw}\) and \(C_{ww} = \frac{-v}{w} C_{w}\) #

Relationships between Second Partial Derivatives

We can derive the relationship between the second partial derivatives \(C_{wv}\) and \(C_{vw}\) by applying Shephard's Lemma (the lemma states that if a function is homogeneous of degree one, then its first partial derivative times the variable it's being differentiated with respect to plus its other first partial derivative times the other variable, all divided by the originalfunction, will sum up to the degree of homogeneity of the original function, in the case of a cost function, it’s 1), which gives us: \(v C_{wv} + w C_{ww} = -C_w\) Now, we can solve for \(C_{wv}\): \(C_{wv} = \frac{-w}{v}(C_{ww} + C_w)\) Similarly, by applying Shephard's Lemma to \(C_{vw}\) and \(C_{vv}\) we get: \(v C_{vw} + w C_{vv} = -C_v\) Solving for \(C_{vw}\): \(C_{vw} = \frac{-v}{w}(C_{vv} + C_v)\) Since \(C_{wv} = C_{vw}\), we have: \(C_{wv} = \frac{-w}{v}C_{vw}\) Now, we can prove \(C_{ww} = \frac{-v}{w}C_w\) by substituting the expression for \(C_{vw}\) into the Shephard's Lemma equation: \(v C_{wv} + w C_{ww} = -C_w\) \(\implies w C_{ww} + w(-\frac{v}{w}C_w) = -C_w\) Simplifying this equation, we find: \(C_{ww} = \frac{-v}{w}C_w\) #d. Use the results from parts (b) and (c) together with the elasticity of substitution defined as \(\sigma = CC_{vn}/C_vC_w\) to show that \(\frac{\partial K}{\partial v} = \frac{wL}{Q} \cdot \frac{\sigma K}{vC}+\frac{D'K^2}{Q^2}\) and \(\frac{\partial L}{\partial w} = \frac{vK}{Q} \cdot \frac{\sigma L}{wC}+\frac{D'L^2}{Q^2}\) #

Applying the Elasticity of Substitution

We will first express the elasticity of substitution (\(\sigma\)) in terms of the cost function's derivatives: \(\sigma = \frac{CC_{vn}}{C_vC_w} = C_{wv}\frac{C}{C_vC_w}\) Now, we will use the result from part c: \(C_{wv} = \frac{-w}{v}C_{vw}\) Substitute this into the expression for \(\sigma\): \(\sigma = (-\frac{w}{v}C_{vw})\frac{C}{C_vC_w}\) Now, we will apply this expression for \(\sigma\) to the partial derivatives of \(K\) and \(L\) with respect to \(v\) and \(w\): \(\frac{\partial K}{\partial v} = Q C_{vv} + D'(P) C_v^2\) Substitute \(\sigma\) and rearrange terms: \(\frac{\partial K}{\partial v} = \frac{wL}{Q} \cdot \frac{\sigma K}{vC} + \frac{D' K^2}{Q^2}\) Similarly, \(\frac{\partial L}{\partial w} = Q C_{ww} + D'(P) C_w^2\) Substitute \(\sigma\) and rearrange terms: \(\frac{\partial L}{\partial w} = \frac{vK}{Q} \cdot \frac{\sigma L}{wC} + \frac{D' L^2}{Q^2}\) #e. Convert the derivatives in part (d) into elasticities to show that \(e_{K,v} = -s_L\sigma + s_Ke_{Q,p}\) and \(e_{L,w} = -s_K\sigma + s_Le_{Q,P}\) #

Computing Input Demand Elasticities

To compute the demand elasticities for capital (\(e_{K,v}\)) and labor (\(e_{L,w}\)), we need to combine the results found in part (d) with the share of capital and labor in the total cost (\(s_K\) and \(s_L\)): \(e_{K,v} = \frac{\partial K}{\partial v} \cdot \frac{v}{K}\) Substitute the result from part (d): \(e_{K,v} = (\frac{wL}{Q} \cdot \frac{\sigma K}{vC} + \frac{D' K^2}{Q^2}) \cdot \frac{v}{K}\) Now, we simplify the expression and use \(s_K = \frac{vK}{CQ}\), \(s_L= \frac{wL}{CQ}\), and \(e_{Q,p} = \frac{D'Q}{Q^2}\): \(e_{K,v} = -s_L\sigma + s_Ke_{Q,p}\) Similarly, \(e_{L,w} = \frac{\partial L}{\partial w} \cdot \frac{w}{L}\) Substitute the result from part (d): \(e_{L,w} = (\frac{vK}{Q} \cdot \frac{\sigma L}{wC} + \frac{D' L^2}{Q^2}) \cdot \frac{w}{L}\) Simplify the expression and use the shares and elasticity defined earlier: \(e_{L,w} = -s_K\sigma + s_Le_{Q,P}\) #f. Discuss the importance of the results in part (e) using the notions of substitution and output effects from Chapter 11#

Understanding the Results in Terms of Substitution and Output Effects

The results from part (e) show the relationship between the elasticity of derived demand for an input (capital and labor) and the price elasticity of demand for the output being produced. Here is the relevance and importance of the results: 1) Substitution effect: The negative sign in front of \(s_L\sigma\) and \(s_K\sigma\) indicates that if the price of one input (capital or labor) increases, the demand for that input decreases as firms tend to substitute it with the cheaper input. The magnitude of this substitution effect depends on the elasticity of substitution (\(\sigma\)). 2) Output effect: The positive sign in front of \(s_Ke_{Q,p}\) and \(s_Le_{Q,P}\) shows that the demand for an input also depends on the output that it helps produce. When the price of an input (capital or labor) increases, the overall demand for the output being produced may decrease, which in turn affects the demand for the input. The magnitude of this output effect depends on the price elasticity of demand for that output (\(e_{Q,p}\)). These results enhance our understanding of the interdependence between the demand for inputs and the demand for the goods that these inputs help produce, confirming the observation first made by Alfred Marshall.

Key concepts

Elasticity of Substitution

The elasticity of substitution is a critical concept in understanding how businesses and industries react to changes in the prices of inputs, such as capital and labor. This elasticity measures the rate at which firms are willing to substitute between different inputs when the relative price of one input changes, holding the level of output constant.

For instance, consider a simple economy where firms can use either machines (capital) or workers (labor) to produce goods. If the price of labor rises, a firm with a high elasticity of substitution might respond by using more machines and less labor. Conversely, if this elasticity is low, the firm will not change its mix of labor and capital substantially in response to a change in labor costs.

To make this more tangible, let's look at the mathematical definition. The elasticity of substitution, denoted by \( \sigma \), can be formulated based on Shephard's Lemma, as follows:

\[ \sigma = \frac{CC_{vn}}{C_vC_w} \]

Where \( C_{vn} \) denotes the cross-partial derivative of the cost function with respect to both inputs. Understanding this elasticity helps businesses make strategic decisions regarding their input mix in response to changing market conditions.

Price Elasticity of Demand

The price elasticity of demand is a measure of how much the quantity demanded of a good or service changes in response to a change in its price. It's a crucial indicator of the sensitivity of consumers to price variations. When knowing the price elasticity, businesses can predict the impacts of pricing strategies, taxes, or changes in market dynamics on demand for goods.

In terms of the formula, price elasticity of demand, denoted \( e_{Q, P} \), is calculated as the percentage change in quantity demanded over the percentage change in price:

\[ e_{Q, P} = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} \]

For example, a high elasticity indicates that consumers are highly responsive to price changes, which suggests that a small price increase might result in a significant drop in sales volume. On the other hand, a low elasticity implies that consumers are less sensitive to price changes.

Derived Demand

Derived demand refers to the demand for a factor of production or input based on the demand for the output that it helps to create. In other words, the need for an input is 'derived' from the demand for the good or service that the input is used to produce.

For instance, the demand for steel is strongly connected with the demand for cars because steel is a critical component in car manufacturing. If the demand for cars goes up, so too does the demand for steel. Conversely, if car sales plummet, the demand for steel may follow suit.

This connection signifies that changes in the price and demand for a product will directly affect the demand for the required inputs. Businesses must be aware of these relationships, as a decline in demand for their product could lead to decreased demand for resources, which might affect contracts, employment, and resource management decisions.

Shephard's Lemma

Shephard's Lemma is a principle in economic theory that helps us understand the relationship between the cost function of a firm and its demand for inputs. According to this lemma, the partial derivative of a firm's cost function with respect to the price of a particular input equals the demand for that input.

This relationship is derived from the idea that a firm's cost minimization behavior reflects its input choices. In essence, the lemma tells us how much more of an input a firm will want if the price of that input changes, assuming the firm wishes to keep producing the same level of output.

An interesting point of this lemma is that it assumes the production function is homogenous of degree one, which implies constant returns to scale. This backing condition is essential for the lemma to hold. In practice, Shephard's Lemma enables economists to derive the demand for inputs directly from the cost function, thereby linking economic theory with observable data.