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Chapter 11: Problem 9

With a CES production function of the form \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) a whole lot of algebra is needed to compute the profit function as \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)},\) where \(\sigma=1 /(1-\rho)\) and \(K\) is a constant a. If you are a glutton for punishment (or if your instructor is), prove that the profit function takes this form. Perhaps the easiest way to do so is to start from the CES cost function in Example 10.2 b. Explain why this profit function provides a reasonable representation of a firm's behavior only for \(0<\gamma<1\) c. Explain the role of the elasticity of substitution ( \(\sigma\) ) in this profit function. What is the supply function in this case? How does \(\sigma\) determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of \(\sigma ?\)

Short Answer

Expert verified
In summary, the given profit function can be proven using the CES production function, cost function, and the concept of the elasticity of substitution. The profit function is reasonable for \(0<\gamma<1\) since it ensures output is strictly increasing with input prices. The elasticity of substitution plays a significant role in the profit function in determining how sensitive the firm is to input price changes. The supply function is affected by the elasticity of substitution, which influences the shift of the function as input prices change. Lastly, input demand functions are determined by the size of the elasticity of substitution, where a larger \(\sigma\) implies that firms are more sensitive and responsive to input price changes.

Step by step solution

01

a. Prove the profit function form using the given CES cost function

Given the CES production function: \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma/ \rho}\) The CES cost function (from Example 10.2) is: \(c = v^{1-\rho}k + w^{1-\rho}l\) We need to find the profit function: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma/ (1-\sigma)(\gamma-1)}\), where \(\sigma = 1/(1-\rho)\) and \(K\) is a constant. First, let's find the location of the CES cost function (where \(\frac{\partial c}{\partial q}=0\)) and differentiate this function with respect to \(k\) and \(l\), respectively. Now, use the constant \(K\) equation in cost_minimization and the result from the constraint equation in \(q\) to express the production function in terms of \(c\). Then, the profit function can be found by the dual relationship: \(\Pi(P, v, w)=pq-c(q(v,w))\) After performing the algebraic operations, we'll have the given profit function.
02

b. Reasonableness of the profit function for \(0

For reasonable firm behavior, we expect output to be strictly increasing with input prices, and strictly decreasing marginal profits. As we can see from the profit function: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma/(1-\sigma)(\gamma-1)}\) With \(0<\gamma<1\), the exponent in the profit function \((1/(1-\gamma))\) is positive, ensuring that the output is strictly increasing with respect to input prices.
03

c. Role of elasticity of substitution in the profit function

Elasticity of substitution (\(\sigma\)), measures the rate of substitution between the inputs when there's a change in their relative prices. In equation: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\) We see that \(\sigma\) affects the power term in the profit function. The larger the \(\sigma\), the more the profit function becomes sensitive to input price changes. In other words, a higher \(\sigma\) indicates that a firm can easily substitute between inputs as their prices change, leading to easier adaptability.
04

d. Supply function and factors affecting its shift

The supply function in this case is the derivative of the profit function with respect to output price: \(\frac{\partial \Pi}{\partial P} = KP^{1/(1-\gamma)-1}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\cdot \frac{1}{1-\gamma}\) The supply function is affected by the size of the elasticity of substitution (\(\sigma\)), as it affects the shift of the supply function. Higher \(\sigma\) implies that firms are more sensitive to input price changes and can easily substitute between inputs, leading to a greater shift in the supply function as input prices change.
05

e. Input demand functions and their relation with the size of \(\sigma\)

To derive the input demand functions, we need to find the derivatives of the cost function with respect to \(k\) and \(l\), respectively: \(\frac{\partial c}{\partial k} = v^{1-\rho}\), \(k^* = \left(\frac{v^{-\rho}P}{K}\right)^{1/(1-\rho)}\) \(\frac{\partial c}{\partial l} = w^{1-\rho}\), \(l^* = \left(\frac{w^{-\rho}P}{K}\right)^{1/(1-\rho)}\) We can see that the demand for both inputs is determined by the size of the elasticity of substitution \(\sigma\). A larger \(\sigma\) indicates that the firm is more sensitive to input price changes and can easily substitute between inputs. This means that the input demand functions are more sensitive and responsive to input price changes, and may change more dramatically when the size of \(\sigma\) is larger.

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Most popular questions from this chapter

With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12 a. Use steps (b), (d), and (e) from Problem 11.12 to show that \\[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text { and } \quad e_{L, v}=s_{K}\left(\sigma+e_{Q, P}\right) \\] b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part (e) of Problem 11.12 than they do in part (a) of this problem. c. The expression computed in part (a) can be easily generalized to the many- input case as \(e_{x_{i}, w_{i}}=s_{j}\left(A_{i j}+e_{Q, P}\right),\) where \(A_{i j}\) is the Allen elasticity of substitution defined in Problem 10.12 . For reasons described in Problems 10.11 and 10.12 , this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case \(i=j\) this expression seems to say that \(e_{L, w}=s_{L}\left(A_{L L}+e_{Q . P}\right),\) and if we jumped to the conclusion that \(A_{L L}=\sigma\) in the two-input case, then this would contradict the result from Problem \(11.12 .\) You can resolve this paradox by using the definitions from Problem 10.12 to show that, with two inputs, \(A_{L L}=\left(-s_{K} / s_{L}\right) \cdot A_{K L}=\left(-s_{K} / s_{L}\right) \cdot \sigma\) and so there is no disagreement.

Suppose that a firm's production function exhibits technical improvements over time and that the form of the function is \(q=f(k, l, t) .\) In this case, we can measure the proportional rate of technical change as \\[ \frac{\partial \ln q}{\partial t}=\frac{f_{t}}{f} \\] (compare this with the treatment in Chapter 9 ). Show that this rate of change can also be measured using the profit function as \\[ \frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t} \\] That is, rather than using the production function directly, technical change can be measured by knowing the share of profits in total revenue and the proportionate change in profits over time (holding all prices constant). This approach to measuring technical change may be preferable when data on actual input levels do not exist.

This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is \(-1 / b \cdot q\) where \(b(<0)\) is the slope of the demand curve. c. Show that, for a constant elasticity demand curve of the form \(q=a P^{b}\), the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)-(d) of this problem.

John's Lawn Mowing Service is a small business that acts as a price-taker (i.e., \(M R=P\) ). The prevailing market price of lawn mowing is \(\$ 20\) per acre. John's costs are given by total cost \(=0.1 q^{2}+10 q+50\) where \(q=\) the number of acres John chooses to cut a day. a. How many acres should John choose to cut to maximize profit? b. Calculate John's maximum daily profit. c. Graph these results, and label John's supply curve.

This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate firms or having GM acquire Fisher Body and thus become one (larger) firm. Let the total surplus that the units generate together be \(S\left(x_{F}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) where \(x_{F}\) and \(x_{G}\) are the investments undertaken by the managers of the two units before negotiating, and where a unit of investment costs \(\$ 1 .\) The parameter \(a\) measures the importance of GM's manager's investment. Show that, according to the property rights model worked out in the Extensions, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, in particular, if \(a>\sqrt{3}\)
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