# Price uncertainty and the labor managed firm: a note.

I. Introduction

A large literature on a labor managed firm (LMF) has been published since Ward's |6~ pioneering work. Further, the analysis on the LMF under uncertainty has recently drawn attention and has been developed in several directions. Among others, Muzondo |4~ and Paroush and Kahana |5~ considered the LMF with output price uncertainty and showed that three perverse results hold.(1) Their conclusions were, however, derived in a short-run model with a single input, labor. So, in a recent paper in this Journal, Choi and Feinerman |2~ extended their short-run model to a long-run model with two variable inputs, capital and labor, and examined whether or not the three perverse effects occur. They demonstrated that those effects do not always carry over to the long-run model. In addition, the other of their purposes is to conduct the comparative static analysis for the risk averse LMF. Although they attempted to examine the effects of changes in mean output price and uncertainty on the risk averse LMF's inputs and output, they could not obtain obvious results. (See their Propositions 2 and 3.)

In this note we show that the comparative static results with respect to mean output price and uncertainty are clarified. For example, if capital and labor are substitutes or independent, then an increase in mean output price induces the risk averse LMF to increase capital and decrease labor. Its increase will decrease (increase) output as long as capital and labor are an inferior (a normal) factor and a normal (an inferior) one, respectively. On the other hand, the effects of uncertainty on the employment and output of the risk averse LMF are opposite to those concerning mean output price: If capital and labor are substitutes or independent, then an increase in uncertainty leads to an increase in labor, and further if capital and labor are an inferior and a normal factors, respectively, then it leads to an increase in output.

II. The Model

An LMF produces a single output Q with two variable inputs, capital K and labor L. The LMF is now assumed to choose the inputs before output price is resolved. Following Choi and Feinerman |2~, the production function is given by Q = F(K, L), where the function is assumed to be monotonically increasing and strictly concave. Dividend per worker is

w = (pQ - rK)/L,

where p denotes uncertain output price and r denotes the rental cost for capital service. Then the objective of the risk averse LMF is to choose K and L to maximize the expected utility of dividend per worker:

|Mathematical Expression Omitted~

where U (|center dot~) is a monotonically increasing and strictly concave von Neumann-Morgenstern utility function, |Mathematical Expression Omitted~ (|center dot~) |is greater than~ 0 and |Mathematical Expression Omitted~ (|center dot~) |is less than~ 0. The first order conditions are

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

We assume that the second order conditions are satisfied. When using (1a) and (1b), we obtain

K|F.sub.K~ + L|F.sub.L~ = Q. (2)

This expression demonstrates that even if the risk averse LMF faces output price uncertainty, its expansion path is determined independent of its attitude towards risk as well as the rental cost for capital service r. Given input price uncertainty, the same result holds.(2)

III. Comparative Statics

Consider how the risk averse LMF responds to changes in parameters, i.e., mean output price and uncertainty. Let the random output price p replace |mu~ + |gamma~z, where |mu~ |is equivalent to~ Ep is the expected (mean) output price and z a random variable with Ez = 0.

We first examine the effects of an increase in mean output price on capital, employment and output. Substituting |mu~ + |gamma~z into p and differentiating the first order conditions with respect to |mu~ yields.

|Mathematical Expression Omitted~

where E|U.sub.KK |is less than~ 0, E|U.sub.LL~ |is less than 0 and |Mathematical Expression Omitted~ |is equivalent to~ E|U.sub.KK~E|U.sub.LL~ - |(E|U.sub.KL~.sup.2~ |is greater than~ 0 from the second order conditions, and further where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Recalling (2), we obtain

|Mathematical Expression Omitted~

By using (5) and arranging (4b), (4c) and (4e), we have

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Hence we can solve (3) by using (4a), |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, (4d) and |Mathematical Expression Omitted~ as follows:

|Mathematical Expression Omitted~ |Mathematical Expression Omitted~

As shown by Choi and Feinerman |2~, when we assume decreasing absolute risk aversion (DARA), E|U.sub.k|mu~~ is positive. The effects of an increase in the expected output price on capital and employment are finally reduced to depend on the functional form of the production function:

|Mathematical Expression Omitted~

These expressions demonstrate that if K and L are substitutes or independent (|F.sub.KL~ |is less than or equal to~ 0), then dK/d|mu~ |is greater than~ O and dL/d|mu~ |is less than~ 0. An increase in mean output price causes the Paroush-Kahana perverse response on employment. However, if K and L are complements (|F.sub.KL~ |is greater than~ 0), then the signs of dK/d|mu~ and dL/d|mu~ are of either sign. A sufficient condition for the Paroush-Kahana perverse effect to occur for the two-input model is that capital and labor are either substitutes or independent.

Let us examine the effect of increased mean output price on output. Differentiating the production function with respect to mu and using (6a) and (6b) gives

|Mathematical Expression Omitted~

where ||delta~.sub.K~ |is equivalent to~ |F.sub.K~ - |F.sub.L~|F.sub.KL~/|F.sub.LL~ and ||delta~.sub.L~ |is equivalent to~ |F.sub.L~ - |F.sub.K~|F.sub.KL~/|F.sub.KK~. According to Choi and Feinerman |2~, ||delta~.sub.K~ is positive (negative) if K is a normal (an inferior) factor, and ||delta~.sub.L~ is positive (negative) if L is a normal (an inferior) factor. Now since the sign of dQ/d|mu~ depends on the sign of (||delta~.sub.K~L|F.sub.LL~ - ||delta~.sub.L~K|F.sub.KK~), it follows that dQ/d|mu~ |is less than~ 0 for ||delta~.sub.k~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0. while dQ/d|mu~ |is greater than~ 0 for ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0. Namely, if capital is an inferior factor and labor is a normal factor, then the risk averse LMF decreases output as mean output price rises. In effect, the Ward perverse effect holds for ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0. However, if both K and L are normal factors, then dQ/d|mu~ remains dubious. We establish the following proposition in place of Choi-Feinerman's Proposition 2:

PROPOSITION 2. Assume that the risk averse LMF exhibits DARA. If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|mu~ |is greater than~ 0 and dL/d|mu~ |is less than~ 0. If |F.sub.KL~ |is greater than~ 0, then dK/d|mu~ and dL/d|mu~ are indeterminate. Further, if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|mu~ |is less than~ 0, while if ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than 0, then dQ/d|mu~ |is greater than 0.

We find that the comparative static results for the risk neutral and the risk averse LMFs are the same. Proposition 2 states that the Ward result and the Paroush-Kahana result on employment do not always carry over to the long-run model. Their results are weakened in the multi-factor framework. It should be noted that although the perverse effects on employment and output may simultaneously occur, that possibility appears to be small.

We now examine the effects of a change in uncertainty on capital, labor and output. Differentiating the first order conditions with respect to |gamma~ and then evaluating at |gamma~ = 1 yields.

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

By recalling (3), E|U.sub.L|gamma~~ = (-K/L) E|U.sub.K|gamma~~ is derived. Using this expression and solving (7), we have

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

According to Choi and Feinerman |2~, since E|U.sub.K|gamma~~ |is less than~ 0 under DARA, it follows that

|Mathematical Expression Omitted~

If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|gamma~ |is less than~ 0 and dL/d|gamma~ |is greater than~ 0, while the effects of increased uncertainty remain ambiguous if |F.sub.KL~ |is greater than~ 0. Therefore, it is a sufficient condition for the Paroush-Kahana perverse result on employment to hold that capital and labor are either complements or independent.

The effect of a change in output price uncertainty on output is given by

|Mathematical Expression Omitted~

Since E|U.sub.K|gamma~~ |is less than~ 0 under DARA, the sign of dQ/d|gamma~ depends on the sign of (-||delta~.sub.K~L|F.sub.LL~ + ||delta~.sub.L~K|F.sub.KK~): If ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0, then dQ/d|gamma~ is less than 0, while if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|gamma~ |is greater than~ 0. Namely, when capital is an inferior factor and labor is a normal factor, increased uncertainty induces the risk averse LMF to increase output. This result is the same as the Paroush-Kahana perverse result but is a sharp contrast to that of a risk averse capitalist twin.(3) We establish the following proposition in place of Choi-Feinerman's Proposition 3:

PROPOSITION 3. Assume that the risk averse LMF exhibits DARA. If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|gamma~ |is less than~ 0 and dL/d|gamma~ |is greater than~ 0, while if |F.sub.KL~ |is greater than~ 0, then dK/d|gamma~ and dL/d|gamma~ remain dubious. Moreover, if ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0, then dQ/d|gamma~ |is less than~ 0, while if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|gamma~ |is greater than~ 0.

It should be noted from this proposition that the possibility is small that the Paroush-Kahana perverse effects on employment and output simultaneously occur. Thus, their perverse effects are not robust in the two-input model. Comparing with Proposition 2, we find that the effects of uncertainty on capital, employment and output are just the opposite of those concerning mean output price.

1. See Choi and Feinerman |2~ since the three perverse results are well summarized.

2. See Haruna |3~. On the other hand, Batra and Ullah |1~ showed that the expansion path of a risk averse capitalist firm with output price uncertainty depends on input prices, but not its attitude towards risk.

3. Batra and Ullah |1~ demonstrated that an increase in output price uncertainty leads to a decline in the output of the risk averse capitalist firm.

References

1. Batra, Raveendra N. and Aman Ullah, "Competitive Firm and the Theory of Input Demand under Price Uncertainty." Journal of Political Economy, May/June 1974, 537-48.

2. Choi, E. Kwan and Eli Feinerman, "Price Uncertainty and the Labor Managed Firm." Southern Economic Journal, June 1991, 43-53.

3. Haruna, Shoji, "Random Input Price and the Theory of the Competitive Cooperative Firm." Journal of Comparative Economics, March 1987, 81-95.

4. Muzondo, Timothy R., "On the Theory of the Competitive Labor-Managed Firm under Price Uncertainty." Journal of Comparative Economics, June 1979, 127-44.

5. Paroush, Jacob and Nava Kahana, "Price Uncertainty and the Cooperative Firm." American Economic Review, March 1980, 212-16.

6. Ward, Benjamin, "The Firm in Illyria: Market Syndicalism." American Economic Review, September 1958, 566-89.

A large literature on a labor managed firm (LMF) has been published since Ward's |6~ pioneering work. Further, the analysis on the LMF under uncertainty has recently drawn attention and has been developed in several directions. Among others, Muzondo |4~ and Paroush and Kahana |5~ considered the LMF with output price uncertainty and showed that three perverse results hold.(1) Their conclusions were, however, derived in a short-run model with a single input, labor. So, in a recent paper in this Journal, Choi and Feinerman |2~ extended their short-run model to a long-run model with two variable inputs, capital and labor, and examined whether or not the three perverse effects occur. They demonstrated that those effects do not always carry over to the long-run model. In addition, the other of their purposes is to conduct the comparative static analysis for the risk averse LMF. Although they attempted to examine the effects of changes in mean output price and uncertainty on the risk averse LMF's inputs and output, they could not obtain obvious results. (See their Propositions 2 and 3.)

In this note we show that the comparative static results with respect to mean output price and uncertainty are clarified. For example, if capital and labor are substitutes or independent, then an increase in mean output price induces the risk averse LMF to increase capital and decrease labor. Its increase will decrease (increase) output as long as capital and labor are an inferior (a normal) factor and a normal (an inferior) one, respectively. On the other hand, the effects of uncertainty on the employment and output of the risk averse LMF are opposite to those concerning mean output price: If capital and labor are substitutes or independent, then an increase in uncertainty leads to an increase in labor, and further if capital and labor are an inferior and a normal factors, respectively, then it leads to an increase in output.

II. The Model

An LMF produces a single output Q with two variable inputs, capital K and labor L. The LMF is now assumed to choose the inputs before output price is resolved. Following Choi and Feinerman |2~, the production function is given by Q = F(K, L), where the function is assumed to be monotonically increasing and strictly concave. Dividend per worker is

w = (pQ - rK)/L,

where p denotes uncertain output price and r denotes the rental cost for capital service. Then the objective of the risk averse LMF is to choose K and L to maximize the expected utility of dividend per worker:

|Mathematical Expression Omitted~

where U (|center dot~) is a monotonically increasing and strictly concave von Neumann-Morgenstern utility function, |Mathematical Expression Omitted~ (|center dot~) |is greater than~ 0 and |Mathematical Expression Omitted~ (|center dot~) |is less than~ 0. The first order conditions are

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

We assume that the second order conditions are satisfied. When using (1a) and (1b), we obtain

K|F.sub.K~ + L|F.sub.L~ = Q. (2)

This expression demonstrates that even if the risk averse LMF faces output price uncertainty, its expansion path is determined independent of its attitude towards risk as well as the rental cost for capital service r. Given input price uncertainty, the same result holds.(2)

III. Comparative Statics

Consider how the risk averse LMF responds to changes in parameters, i.e., mean output price and uncertainty. Let the random output price p replace |mu~ + |gamma~z, where |mu~ |is equivalent to~ Ep is the expected (mean) output price and z a random variable with Ez = 0.

We first examine the effects of an increase in mean output price on capital, employment and output. Substituting |mu~ + |gamma~z into p and differentiating the first order conditions with respect to |mu~ yields.

|Mathematical Expression Omitted~

where E|U.sub.KK |is less than~ 0, E|U.sub.LL~ |is less than 0 and |Mathematical Expression Omitted~ |is equivalent to~ E|U.sub.KK~E|U.sub.LL~ - |(E|U.sub.KL~.sup.2~ |is greater than~ 0 from the second order conditions, and further where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Recalling (2), we obtain

|Mathematical Expression Omitted~

By using (5) and arranging (4b), (4c) and (4e), we have

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Hence we can solve (3) by using (4a), |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, (4d) and |Mathematical Expression Omitted~ as follows:

|Mathematical Expression Omitted~ |Mathematical Expression Omitted~

As shown by Choi and Feinerman |2~, when we assume decreasing absolute risk aversion (DARA), E|U.sub.k|mu~~ is positive. The effects of an increase in the expected output price on capital and employment are finally reduced to depend on the functional form of the production function:

|Mathematical Expression Omitted~

These expressions demonstrate that if K and L are substitutes or independent (|F.sub.KL~ |is less than or equal to~ 0), then dK/d|mu~ |is greater than~ O and dL/d|mu~ |is less than~ 0. An increase in mean output price causes the Paroush-Kahana perverse response on employment. However, if K and L are complements (|F.sub.KL~ |is greater than~ 0), then the signs of dK/d|mu~ and dL/d|mu~ are of either sign. A sufficient condition for the Paroush-Kahana perverse effect to occur for the two-input model is that capital and labor are either substitutes or independent.

Let us examine the effect of increased mean output price on output. Differentiating the production function with respect to mu and using (6a) and (6b) gives

|Mathematical Expression Omitted~

where ||delta~.sub.K~ |is equivalent to~ |F.sub.K~ - |F.sub.L~|F.sub.KL~/|F.sub.LL~ and ||delta~.sub.L~ |is equivalent to~ |F.sub.L~ - |F.sub.K~|F.sub.KL~/|F.sub.KK~. According to Choi and Feinerman |2~, ||delta~.sub.K~ is positive (negative) if K is a normal (an inferior) factor, and ||delta~.sub.L~ is positive (negative) if L is a normal (an inferior) factor. Now since the sign of dQ/d|mu~ depends on the sign of (||delta~.sub.K~L|F.sub.LL~ - ||delta~.sub.L~K|F.sub.KK~), it follows that dQ/d|mu~ |is less than~ 0 for ||delta~.sub.k~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0. while dQ/d|mu~ |is greater than~ 0 for ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0. Namely, if capital is an inferior factor and labor is a normal factor, then the risk averse LMF decreases output as mean output price rises. In effect, the Ward perverse effect holds for ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0. However, if both K and L are normal factors, then dQ/d|mu~ remains dubious. We establish the following proposition in place of Choi-Feinerman's Proposition 2:

PROPOSITION 2. Assume that the risk averse LMF exhibits DARA. If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|mu~ |is greater than~ 0 and dL/d|mu~ |is less than~ 0. If |F.sub.KL~ |is greater than~ 0, then dK/d|mu~ and dL/d|mu~ are indeterminate. Further, if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|mu~ |is less than~ 0, while if ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than 0, then dQ/d|mu~ |is greater than 0.

We find that the comparative static results for the risk neutral and the risk averse LMFs are the same. Proposition 2 states that the Ward result and the Paroush-Kahana result on employment do not always carry over to the long-run model. Their results are weakened in the multi-factor framework. It should be noted that although the perverse effects on employment and output may simultaneously occur, that possibility appears to be small.

We now examine the effects of a change in uncertainty on capital, labor and output. Differentiating the first order conditions with respect to |gamma~ and then evaluating at |gamma~ = 1 yields.

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

By recalling (3), E|U.sub.L|gamma~~ = (-K/L) E|U.sub.K|gamma~~ is derived. Using this expression and solving (7), we have

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

According to Choi and Feinerman |2~, since E|U.sub.K|gamma~~ |is less than~ 0 under DARA, it follows that

|Mathematical Expression Omitted~

If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|gamma~ |is less than~ 0 and dL/d|gamma~ |is greater than~ 0, while the effects of increased uncertainty remain ambiguous if |F.sub.KL~ |is greater than~ 0. Therefore, it is a sufficient condition for the Paroush-Kahana perverse result on employment to hold that capital and labor are either complements or independent.

The effect of a change in output price uncertainty on output is given by

|Mathematical Expression Omitted~

Since E|U.sub.K|gamma~~ |is less than~ 0 under DARA, the sign of dQ/d|gamma~ depends on the sign of (-||delta~.sub.K~L|F.sub.LL~ + ||delta~.sub.L~K|F.sub.KK~): If ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0, then dQ/d|gamma~ is less than 0, while if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|gamma~ |is greater than~ 0. Namely, when capital is an inferior factor and labor is a normal factor, increased uncertainty induces the risk averse LMF to increase output. This result is the same as the Paroush-Kahana perverse result but is a sharp contrast to that of a risk averse capitalist twin.(3) We establish the following proposition in place of Choi-Feinerman's Proposition 3:

PROPOSITION 3. Assume that the risk averse LMF exhibits DARA. If |F.sub.KL~ |is less than or equal to~ 0, then dK/d|gamma~ |is less than~ 0 and dL/d|gamma~ |is greater than~ 0, while if |F.sub.KL~ |is greater than~ 0, then dK/d|gamma~ and dL/d|gamma~ remain dubious. Moreover, if ||delta~.sub.K~ |is greater than~ 0 and ||delta~.sub.L~ |is less than~ 0, then dQ/d|gamma~ |is less than~ 0, while if ||delta~.sub.K~ |is less than~ 0 and ||delta~.sub.L~ |is greater than~ 0, then dQ/d|gamma~ |is greater than~ 0.

It should be noted from this proposition that the possibility is small that the Paroush-Kahana perverse effects on employment and output simultaneously occur. Thus, their perverse effects are not robust in the two-input model. Comparing with Proposition 2, we find that the effects of uncertainty on capital, employment and output are just the opposite of those concerning mean output price.

1. See Choi and Feinerman |2~ since the three perverse results are well summarized.

2. See Haruna |3~. On the other hand, Batra and Ullah |1~ showed that the expansion path of a risk averse capitalist firm with output price uncertainty depends on input prices, but not its attitude towards risk.

3. Batra and Ullah |1~ demonstrated that an increase in output price uncertainty leads to a decline in the output of the risk averse capitalist firm.

References

1. Batra, Raveendra N. and Aman Ullah, "Competitive Firm and the Theory of Input Demand under Price Uncertainty." Journal of Political Economy, May/June 1974, 537-48.

2. Choi, E. Kwan and Eli Feinerman, "Price Uncertainty and the Labor Managed Firm." Southern Economic Journal, June 1991, 43-53.

3. Haruna, Shoji, "Random Input Price and the Theory of the Competitive Cooperative Firm." Journal of Comparative Economics, March 1987, 81-95.

4. Muzondo, Timothy R., "On the Theory of the Competitive Labor-Managed Firm under Price Uncertainty." Journal of Comparative Economics, June 1979, 127-44.

5. Paroush, Jacob and Nava Kahana, "Price Uncertainty and the Cooperative Firm." American Economic Review, March 1980, 212-16.

6. Ward, Benjamin, "The Firm in Illyria: Market Syndicalism." American Economic Review, September 1958, 566-89.

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Author: | Haruna, Shoji |
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Publication: | Southern Economic Journal |

Date: | Jan 1, 1993 |

Words: | 2027 |

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