Policy Uncertainty and Technology Adoption

Mohammad H. Dehghani

doi:10.1515/bejeap-2013-0167

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Policy Uncertainty and Technology Adoption

Mohammad H. Dehghani

Veröffentlicht/Copyright: 7. Juni 2014

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Aus der Zeitschrift The B.E. Journal of Economic Analysis & Policy Band 14 Heft 4

Abstract

This paper considers technology adoption under both technological and subsidy uncertainties. Uncertainty in subsidies for green technologies is considered as an example. Technological progress is exogenous and modeled as a jump process with a drift. The analytical solution is presented for cases when there is no subsidy uncertainty and when the subsidy changes once. The case when the subsidy follows a time invariant Markov process is analyzed numerically. The results show that improving the innovation process raises the investment thresholds. When technological jumps are small or rare, this improvement reduces the expected time before technology adoption. However, when technological jumps are large or abundant, this improvement may raise this expected time.

Keywords: real options; technology adoption; policy uncertainty; clean technology; regime switching

JEL Classification: C63; D81; O33; Q58

Acknowledgments

I would like to thank Svetlana Boyarchenko for her continued advice and encouragement. I also thank Thomas Wiseman, Maxwell Stinchcombe, Andres Almazan, Stathis Tompaidis, Charles F. Mason, Till Requate, Maurizio Iacopetta, Eugenio Miravete, Randal Watson Maedeh Faraji, Marzieh Mirghasemi, Ali Akbar Merrikh and anonymous referees. I am thankful to seminar participants at University of Texas at Austin in Fall 2009 for insightful comments. Also I benefited from comments of participants of the Midwest Economic Theory and Trade Meetings (Madison, Fall 2010) and CU Environmental and Resource Economics Workshop (Vail, 2010).

Appendix

Derivation of adoption thresholds and value function

Let Eq be the normalized expected present value operator or EPV-operator. This operator gives the present value of a stream of payoffs. The value depends on the present observed x and it is normalized such that Eq of a stream of payoff that gives one unit at any time is equal to one. Note that Eq=q(q−L)−1. See Boyarchenko and Levendorskiĭ (2007) for details. I also define Eq+ to be the normalized EPV-operator under the supremum process X¯t=sup0 ≤ s<tXs and Eq− to be the normalized EPV-operator under the infimum process X_t=inf0≤s<tXs. Let a1=q(λ−β1)μ(β2−β1) and a2=q(λ−β2)μ(β1−β2), I derive the action of operators for supremum and infimum processes to be

[30]{Eq−u(x)=u(x)Eq+u(x)=∑i=12ai∫0+∞e−βiyu(x+y)dy.

Refer to Boyarchenko and Levendorskiĭ (2007) for more details on how to derive the EPV-operators. For exponential functions they reduce to

[31]{Eq−ezx=ezxEq+ezx=∑i=12ai∫0+∞e−βiyez(x+y)dy=[a1β1−z+a2β2−z]ezx.

I define κq+z and κq−z such that Eq±ezx=κq±(z)ezx. It also holds that

[32]{κq−(z)=1κq+(z)=qq−Ψ(z)=a1β1−z+a2β2−z

According to Wiener-Hopf factorization formula, the normalized EPV operator can be expressed as Eq=Eq+Eq−. Hence, κq(z)=qq−Ψ(z)=κq+(z)κq−(z). See Boyarchenko and Levendorskiĭ (2007) for more details on Wiener-Hopf factorization, EPV-operators and how they are derived.

To make the problem [4] look like the standard exit problem, I define V(x;h)=V(x;h)−A0Bqeσx(1−qθ). Then, problem [4] is equivalent to the following in which the firm chooses the optimal time to exit:

[33]{(q−L)V(x;h)=A0B−A0Bκq(σ)eσx(1−qθ)x<hV(x;h)=0x≥h

The adoption threshold in problem [4] is the same as the exit threshold in problem [33]. From the paper by Boyarchenko and Levendorskiĭ (2006), it follows that the solution for the value function for x<h is

[34]V(x;h)=A0BqEq+1(−∞,h)(x)Eq−1−1−qθκq(σ)eσx.

The right hand side of the Bellman equation in problem [33] is non-increasing in x. Based on what Boyarchenko and Levendorskiĭ (2006) have shown, to maximize the value function, h needs to solve Eq−A0B1−1−qθκq(σ)eσx=0.

[35]⇒eσh=κq+(σ)1−qθ=κq(σ)1−qθ

Knowing the adoption threshold, h, I can find the value function for x<h from eq. [34].

[36]V(x)={A0Bq[1+∑i=12aiσβi(βi−σ)e−βi(h−x)]x<hA0Bq(1−qθ)eσxx≥h.

Proof of Proposition 2.1

Proof. From eq. [8] it follows that

[37]h=1σlnκq(σ)1−qθ=1σlnq−lnq−Ψ(σ)−ln(1−qθ).

θ: By direct differentiation with respect to θ I have: ∂h∂θ=qσ(1−qθ)>0. Hence h increases as θ goes up.
σ: ∂h∂σ=1σκq′(σ)κq(σ)−1σ2lnκq(σ)1−qθ=1σa1(β1−σ)2+a2(β2−σ)2a1β1−σ+a2β2−σ−1σlna1β1−σ+a2β2−σ1−qθ which goes to −∞ as σ gets close to zero from the right and goes to +∞ when σ gets close to β1.

For the rest of the proof remember that q−Ψ(σ)=q−μσ−cσλ−σ

μ: ∂h∂μ=1q−Ψ(σ)>0 because the valid values of σ are between zero and β1 which makes q−Ψ(σ) positive.
c: ∂h∂c=1(λ−σ)(q−Ψ(σ))>0.
λ:∂h∂λ=−c(λ−σ)2(q−Ψ(σ))<0.
q: ∂h∂q=1σ1q−1q−Ψ(σ)+θ(1−qθ). As q increases from Ψ(σ) to 1θ, this derivative change from −∞ to +∞.

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1
To derive the profit flow it is assumed that the monopolist uses Lt units of labor to produce quantity Qt of a product. Let w be the wage for one unit of labor. Eq. [1] can be derived for two different scenarios. In the first one technology improves the productivity. For this case assume the production function is Qt=DtLtρ, where ρ∈(0,1] and the firm faces a market demand Qt=Pt−ϵ for its output, where Pt is the price of the product and −ϵ is the price elasticity of demand and ϵ>1. Using these assumptions, the solution for the profit of the firm is given by eq. [1] where σ=(ϵ−1)ϵ−ρ(ϵ−1) and B=wρ(ϵ−1)ρ(ϵ−1)−ϵϵρ(ϵ−1)ϵρ(ϵ−1)−ϵϵρ(ϵ−1)−1.
In the second scenario technology raises the demand. For this case assume the production function is Qt=Ltρ, where ρ∈(0,1] and the firm faces a market demand Qt=DtPt−ϵ for its output, where Pt is the price of the product and −ϵ is the elasticity of demand and ϵ>1. Using these assumptions, the solution for the profit of the firm is given by eq. [1] where σ=1ϵ−ρ(ϵ−1) and B=wρ(ϵ−1)ρ(ϵ−1)−ϵϵρ(ϵ−1)ϵρ(ϵ−1)−ϵϵρ(ϵ−1)−1.
If innovation improves the productivity, then σ can take any positive value depending on the elasticity of demand and the parameter of the production function. If innovation raises the demand for the product, σ cannot take values less than or equal to one.
2
The work by Boyarchenko and Levendorskiĭ (2006) is an example of models with multiple adoptions of technology. Another example is the work by Farzin, Huisman, and Kort (1998) which was improved by Doraszelski (2001).
3
Having the cost of adoption proportional to the profit flow after adoption is a better assumption than a constant value for the cost of adoption. The cost of adoption of a more advanced technology is larger due to higher adjustment costs. So it is expected that the cost of adoption increases as the frontier technology advances. This assumption is also used by Boyarchenko and Levendorskiĭ (2006).

Published Online: 2014-6-7

Published in Print: 2014-10-1

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https://doi.org/10.1515/bejeap-2013-0167

Schlagwörter für diesen Artikel

real options; technology adoption; policy uncertainty; clean technology; regime switching