Technical issues studied

Here the impact of provision of public goods is considered. It is assumed that revenue raised from taxes are used by the elite to provide public goods 𝐺_(𝑡) in addition to transfers at any time 𝑡.

The utility is still given as equation 1 but the government budget constraint and the production technology are now given as:

𝑇_(𝑡)^𝑐 + 𝑇 _𝑡 ^𝑚 + 𝑇_(𝑡)^𝑒 + 𝐺_(𝑡) ≤ 𝜏_(𝑡) 𝑌_𝑖(𝑡)𝑑𝑖 (18)

𝑌_𝑖(𝑡)= ¹

𝛼(𝐾_𝑖(𝑡))^𝛼 (𝐴 _𝑡 𝐿_{𝑖 𝑡} )^{1− 𝛼} (19)⁴

Equation 19 is an augmented Cobb–Douglas production technology where 𝐴_(𝑡) measures the productivity of labour (by labour here we refer to both citizens and entrepreneurs contributions to production). 𝐴_(𝑡) is specified as time varying because it is assumed that productivity changes over time depending on the level of public goods investment at each time. In other words, the more access to public goods such as education, healthcare, security, sanitation, and basic infrastructure, the higher the productivity of labour. Assume that:

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𝛼 is a convenience normaliser

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57 𝐴_(𝑡) = ^𝛼∅

1−𝛼𝐺_(𝑡) ^1/Ø (20)

Where: 𝐺_(𝑡) denotes government spending on public goods, Ø > 1 ensures that the technology for public goods investment exhibits diminishing returns. One important point here is that certain amount of government investment in public goods is necessary for private citizens to function productively. The political elite sets the tax rate 𝜏_(𝑡) ∈ 0, 𝜏 on total output where 𝜏 is maximum tax rate. Then tax revenues are:

𝑇𝑎𝑥 𝑅𝑒𝑣𝑒𝑛𝑢𝑒_(𝑡) = 𝜏_(𝑡) 𝑌𝑖(𝑡)𝑑𝑖 = 𝜏_(𝑡)𝑌_(𝑡) (21)

Before examining what determine elite‟s decision to invest in public goods, let us first examine the impact of public goods on the state variable (capital – labour ratio). The utility of an entrepreneur with capital stock 𝐾_𝑖(𝑡) at time 𝑡 is:

𝑈_𝑖( 𝐾_{𝑖 𝑡} , 𝐴_(𝑡), 𝐿_{𝑖 𝑡} /𝑃^𝑡,𝑤^∗) = ^∞_𝑡=0𝛽^𝑡 1 − 𝜏 _𝑡 𝑌_(𝑡) − 𝐾_{𝑖 𝑡+1} − 1 − 𝛿 𝐾_{𝑖 𝑡} − 𝑤_(𝑡)𝐿_𝑖(𝑡)+ 𝑇 _𝑡 ^𝑚 (22)

Where: 𝑌_(𝑡) is given as in equation 19, thus maximising (22) with respect to the sequence of capital stock yields:

1 − 𝜏 _𝑡 (𝐾 _𝑡 )^𝛼−1(𝐴 _𝑡 𝐿_{𝑖 𝑡} )^1−𝛼 + 1 − 𝛿 = 𝛽⁻¹ (23) Equation 23 is expressed in term of capital–labour as:

1 − 𝜏 _𝑡 (𝑘 _𝑡 )^𝛼−1(𝐴 _𝑡 )^1−𝛼 + 1 − 𝛿 = 𝛽⁻¹ Therefore, the equilibrium capital–labour ratio is :

𝑘 _(𝑡) = 𝛽⁻¹+ 𝛿 − 1 1 − 𝜏 _𝑡

1𝛼−1

𝐴_(𝑡) (24)

Given equation 20, (24) can be written as:

𝑘 _(𝑡)= ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

1𝛼−1 𝛼∅

1−𝛼𝐺_(𝑡) ^1/Ø , thus the future capital–labour ratio is:

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58 𝑘 _(𝑡+1) = ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡+1

1𝛼−1 𝛼∅

1−𝛼𝐺_(𝑡+1) ^1/Ø (25) The equilibrium wage rate is also given as:

𝑤_(𝜏,𝐺)= 1 − 𝜏 _𝑡 𝑓 𝑘 _𝜏,𝐺 − 𝑘 _𝜏,𝐺 𝑓^/ 𝑘 _𝜏,𝐺 (26)

Equations 25 and 26 are important results which show that investment in public goods do not only impact the choice of future capital–labour ratio but equally has impact on the equilibrium wage rate indirectly. Since, investments in public goods is important in the equilibrium, let us now examine what determines the decision to investment in public goods.

The elite at time 𝑡 decides how much of the revenue to spend on public goods for the next date 𝐺_(𝑡+1). The elite is assumed to take this decision such as to maximise the consumption of representative elite given as:

𝐶_(𝑡)^𝑒 = 𝑇𝑎𝑥 𝑅𝑒𝑣𝑒𝑛𝑢𝑒_(𝑡)− 𝐺 _𝑡 + 𝑇 _𝑡 ^𝑖 + 𝑇 _𝑡 ^𝑒 ; 𝑖 = 𝑚, 𝑐 (27) Given the tax rate 𝜏_(𝑡) ∈ 0, 𝜏 , the capital–labour ratio given by equation 24 and output per capita written from equation 19 as: 𝑦 ≡ 𝑓 𝑘 = ¹

𝛼 𝑘_𝑖(𝑡) ^𝛼 𝐴_(𝑡) ^1−𝛼 (19^*) Substituting equation 24 into 19^* yields : 𝑦 ≡ 𝑓 𝑘 = ¹

𝛼 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

1𝛼−1

𝐴_(𝑡)

𝛼

𝐴_(𝑡) ^1−𝛼

𝑦 ≡ 𝑓 𝑘 = ¹

𝛼

𝛽⁻¹+ 𝛿−1 1− 𝜏 _𝑡

𝛼 𝛼−1

𝐴_(𝑡) (28) Using equation 28, equilibrium tax revenue from equation 21 can be written as:

𝑇_{(𝐴 𝑡 )}= ¹

𝛼 𝜏 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

1𝛼−1

𝐴_(𝑡)

𝛼

𝐴_(𝑡) ^1−𝛼 , which gives:

𝑇_{(𝐴 𝑡 )} = ¹

𝛼 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

𝛼 𝛼−1

𝜏 𝐴_(𝑡) (29)⁵

Equation 29 implies that equilibrium tax revenue is a function of investment in public goods.

5 We substituted equation 24 into 𝑦 ≡ 𝑓(𝑘) and substituted the result in equation 21

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The elite then choose public investment 𝐺_(𝑡) to maximise their consumption 𝐶_(𝑡)^𝑒 . The problem of the elite can be written recursively as:

𝑉^𝑒 𝐴_(𝑡) = max

𝐴_(𝑡+1). 𝑇 𝐴_(𝑡) − 1 − 𝛼

𝛼∅ 𝐴_(𝑡+1)^∅− 𝑇 _𝑡 ^𝑖 + 𝑇 _𝑡 ^𝑒 + 𝛽𝑉^𝑒(𝐴_(𝑡+1)) (30)

Where: ^1−𝛼

𝛼∅ 𝐴_(𝑡+1)^∅ = 𝐺_(𝑡) from equation 20

The first order condition (FOC) with respect to how much the elite invest in future public goods 𝐴 _𝑡+1 gives :

1 − 𝛼

𝛼 𝐴_(𝑡+1)^∅−1= 𝛽(𝑉^𝑒)^/ 𝐴_(𝑡+1) (31)

Equation 31 links the marginal cost of greater investment in public goods to the greater value that follows from this investment.

Differentiating equation 30 with respect to current state of public goods, 𝐴_(𝑡) gives the envelope condition (EC) as:

(𝑉^𝑒)^/𝐴_(𝑡) = 𝑇^/𝐴_(𝑡) = ¹

𝛼𝜏 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

𝛼 𝛼−1

(32)⁶

Equation 32 implies that elite value greater public goods because of the additional tax revenue that could be generated from it. Combining the FOC and the EC, the Markov Perfect Equilibrium (MPE) features the choice of the elite as:

𝐴_(𝑡+1) = 𝐴 _𝜏 ≡ (1 − 𝛼)⁻¹ 𝜏 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

𝛼 𝛼−1 1∅−1

(33)

Notice that from equation 31 𝐴_(𝑡+1) = 𝐴 _𝜏 = ^𝛼

1−𝛼𝛽(𝑉^𝑒)^/ 𝐴_(𝑡+1)

1∅−1

, substitution from equation 32 yields 33. It is clear from equation 33 that investment in public goods is a fraction 1 ∅ of tax revenue.

6 We used equation 29 to obtained 32

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Substituting equation 32 and 33 into the elite‟s value function, equation 30 yields:

𝑉^𝑒 𝐴_(𝑡) = ^(𝛽

−1+𝛿−1) ^{𝛼(1−𝛼 )}(1− 𝜏 )^{𝛼 (1−𝛼 )}𝜏 𝐴(𝑡)

𝛼 + ^{(𝛽−1+𝛿−1)}

𝛼 (1−𝛼 ) ∅−1 (1− 𝜏 )^{𝛼(1−𝛼 )}𝜏

(1− 𝛽)∅𝛼 𝐴 _𝜏 (34)⁷

From equation 34 it follows that the value function of the elite depends on current state of public goods, 𝐴_(𝑡) inherited from previous period and tax revenue that could be generated from such goods. It implies that if past political leaders did not spend on public goods, the current elite in power is not equally likely to invest on them.

Proposition 3: In the economy described above, there exist a unique MPE where for all 𝑡, 𝜏 𝜏 = 𝜏 for all 𝐴, 𝐴_(𝑡) is given by 𝐴 𝜏 as in equation 33 for all 𝑡 > 0, and the capital – labour ratio of each entrepreneur 𝑖 at each 𝑡 is given by equation 24. For all 𝑡 > 0, the equilibrium level of aggregate output is:

𝑌_(𝑡)= 𝑌 _𝜏 ≡ 1

𝛼 𝛽⁻¹ + 𝛿 − 1 1 − 𝜏 _𝑡

𝛼 1−𝛼

𝐴_(𝑡) (35)

The level of 𝜏 that maximises output is derived by solving :

max 𝑌 _𝜏 = 1

𝛼 𝛽⁻¹ + 𝛿 − 1 1 − 𝜏 _𝑡

𝛼 1−𝛼

𝐴_(𝑡)

Substituting for 𝐴_(𝑡) gives: max 𝑌 _𝜏 = ¹

𝛼 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

𝛼 1−𝛼

(1 − 𝛼)⁻¹ 𝜏 ^{𝛽−1+ 𝛿−1}

1− 𝜏 _𝑡

𝛼𝛼 −1 1 ∅−1

The output maximising level of tax 𝜏 is thus:

𝜕𝑌_(𝑡)

𝜕𝜏 ≡ 𝜏 ^∗ = 1 − 𝛼

1 − 𝛼 + 𝛼∅ (36)

Let 𝜏 be interpreted as state power or elite power to raise future taxes. If 𝜏 > 𝜏 ∗ , then the elite is powerful enough to raise future taxes above 𝜏 . On the other hand, if 𝜏 < 𝜏 ∗, the elite is not powerful enough to raise future taxes above 𝜏 ∗. Thus, if the power of the elite to raise future

7 The second term follows from (33) that public goods investment is a fraction of 1

∅ of tax revenue

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taxes is limited, there is no incentive for the elite to increase the future productive capacity of the economy by investing more in public goods.

Remarks:

 Public goods investment matter for productivity of labour, aggregate output and tax revenue.

 Elite decision to invest in public goods is determined by level of public goods inherited from past regime and the revenue that could be generated from such investment.

 An excessively powerful elite will impose taxes above the output maximising level of taxes 𝜏 ∗ while excessively weak elite will not invest in public goods because they can not raise future taxes.

In document Strategic customer relationship marketing and re intermediation models in the insurance industry (Page 34-38)