Can public arts education replace arts subsidization?

Abstract

The debate about whether the arts should be supported or not is far from new, and most governments support the arts in one way or the other. The literature considers several arguments in favor of such interventions. Public education may seem to be an action which could, in the long run, lead to possible reductions of subsidies. Surveys show that those who have been exposed to the arts when young participate more when adult. However, the “non-market” transmission from parents to children generates an external effect, which has to be taken into account to reach first-best situations. We construct an overlapping generations model in which young consumers are exposed to both public education toward the arts and to non-altruistic transmission of such a taste from their parents. We show that the first-best can be reached only if there is both public cultural education and subsidization of arts consumption. Therefore, education cannot be considered as a substitute for subsidies to arts consumption. However, as is often the case in European countries, government intervention is usually below the first-best level. Using a model calibrated on French data, we show that it is then preferable to subsidize education, while consumption, especially of the older generations, should be taxed rather than subsidized.

Appendix

1.1 First-order conditions of the welfare optimum

An interior optimal solution of the welfare optimum of Sect. 3, should satisfy all the first-order conditions as well as the transversality conditions of the Lagrangian function where Ω^t q _t and Ω^t p _t are the multipliers associated with the constraints of the program. From these first-order conditions, one can derive the following conditions that should be satisfied in the stationary state of the welfare optimum:

$$ \Upomega f^{\prime}(k^{\ast}) = 1 $$

(A1)

$$ u_c^{\prime} = p^{\ast} $$

(A2)

$$ u_d^{\prime} = \Upomega p^{\ast} $$

(A3)

$$ q^{\ast}\phi_a^{\prime} = p^{\ast} $$

(A4)

$$ u_{\mu}^{\prime}\psi_b^{\prime} = \Upomega p^{\ast} $$

(A5)

$$ \Upomega q^{\ast}\phi_e^{\prime} = p^{\ast} $$

(A6)

$$ u_{\lambda}^{\prime} + u_{\mu}^{\prime}\psi_{\lambda}^{\prime} = (1 - \Upomega \phi_{\lambda}^{\prime})q^{\ast} $$

(A7)

$$ \lambda^{\ast} = \phi(e^{\ast}, \lambda^{\ast}, a^{\ast}) $$

(A8)

$$ k^{\ast} + c^{\ast} + a^{\ast} + d^{\ast} + b^{\ast} + e^{\ast} = f(k^{\ast}). $$

(A9)

In all these expressions, the derivatives are taken in the optimum. Combining (A2) and (A3), one sees that

$$ \Upomega u_c^{\prime}=u_d^{\prime} $$

(A10)

Combining (A2), (A4) leeds to $q^{\ast} = u_c^{\prime}/\phi_a^{\prime}.$ Replacing q* by this expression in (A7), one obtains

$$ (u_{\lambda}^{\prime} + u_{\mu}^{\prime}\psi_{\lambda}^{\prime})\phi_a^{\prime} = (1 - \Upomega \phi_{\lambda}^{\prime})u_c^{\prime}. $$

(A11)

Finally, from (A3) and (A5)

$$ u_{\mu}^{\prime}\psi_b^{\prime} = u_d^{\prime}. $$

(A12)

1.2 Steady-state conditions for consumers’ equilibrium

In the steady-state, the following first-order conditions hold (for an interior solution):

$$ u_c^{\prime} = (1+r)u_d^{\prime} $$

(A13)

$$ (1-\theta^a)u_c^{\prime} = (u_{\lambda}^{\prime} + u_{\mu}^{\prime}\psi_{\lambda}^{\prime})\phi_a^{\prime} $$

(A14)

$$ (1-\theta^b)u_d^{\prime} = u_{\mu}^{\prime}\psi_b^{\prime}, $$

(A15)

and the budget constraints

$$ c = w - T^1 - (1-\theta^a)a - s $$

(A16)

$$ d = (1+r)s - T^2 - (1-\theta^b)b $$

(A17)

are satisfied.

1.3 Decentralizing the welfare optimum

We now check whether and how the first-best can be decentralized as a steady-state equilibrium solution. The steady-state resource constraints of the centralized problem are obviously satisfied in every equilibrium. With competitive producers, lump-sum transfers to consumers make it possible to reach the optimal capital stock for which, by (A1), f′(k*) = Ω⁻¹. Therefore, it is sufficient (a) to verify that the first-best satisfies the steady-state first-order conditions (A13)–(A15) of the consumers’ problem, (b) to compute the necessary lump-sum transfers in order to satisfy the consumers’ budget constraints (A16)–(A17) and (c) to verify whether the government’s budget is in equilibrium.

• (a) By (15) and (A1), r* = Ω⁻¹−1, and (A13) coincides with (A10). So do (A14) and (A11) iff $\theta^a = \Upomega \phi_{\lambda}^{\prime}.$ Finally, (A15) coincides with (A12) iff θ^b = 0.

• (b) Setting s = k*, r = Ω⁻¹−1, w = f(k*) − k*f′(k*) = f(k*) − Ω⁻¹ k*, equilibrium values for the transfers can be computed as:

  $$ T^1=f(k^{\ast}) - \Upomega^{-1} k^{\ast} - (c^{\ast} + (1-\theta^a)a^{\ast} + k^{\ast}) $$

  (A18)
  $$ T^2 = \Upomega^{-1} k^{\ast} - (d^{\ast} + b^{\ast}), $$

  (A19)

  using (A16) and (A17).

• (c) Finally, adding (A18) and (A19) and using (A9), it is straightforward to check that:

  $$ T^{\ast} = T^1 + T^2 = e^{\ast} + \theta^a a^{\ast}, $$

  which shows that the government budget (8) is also in equilibrium.

1.4 Second-best in a Cobb-Douglas economy²⁰

1.4.1 The Marshallian demand functions and calibration

We substitute (18) and (19) in (20), and obtain

$$ u = \hbox{log} c +\beta \hbox{log} d +\alpha_1 \hbox{log} a + \alpha_2 \hbox{log} b + \alpha_3 \hbox{log} e + \alpha_4 \hbox{log} \overline{\lambda}, $$

(B1)

where

$$ \alpha_1 = \alpha \eta_1 (1+ \beta \delta_2), \alpha_2 = \beta \alpha \eta_2, \alpha_3 = \alpha \rho (1+\beta \delta_2), \alpha_4 = \alpha \delta_1 (1+\beta \delta_2). $$

(B2)

To compute the indirect utility function, we first maximize (B1) under the budget constraint (17). This leads to the following Marshallian demand functions:

$$ c = \gamma (w^{\ast} - T), $$

(B3)

$$ d = \beta \gamma (w^{\ast} - T), $$

(B4)

$$ (1-\theta^a)a = \alpha_1 \gamma (w^{\ast} - T), $$

(B5)

$$ (1-\theta^b)b = \alpha_2\gamma (w^{\ast} - T), $$

(B6)

where γ = 1/(1 + β + α₁ + α₂).

Using French national accounts, as well as the results of a survey on cultural expenditures of French households carried out in 1995,²¹ one can compute the (inclusive taxes and subsidies) expenditure shares reproduced in Table A, which also shows the corresponding parameters to which they correspond in the Marshallian demand functions (B3)–(B6).

These are the basic parameters for the Cobb-Douglas economy, from which most of the other parameters will be deduced. See Sect. 4.3.

1.4.2 Computation of the indirect utility function

To derive the indirect utility function, we replace in (B1) c, d, a and b by their expressions in (B3)–(B6), also taking into account that, in the fixed point, $\lambda = \overline{\lambda}$ and that the government budget constraint $e = T - \theta^a a - \theta^b b$ has to be satisfied. This leads to the following expression of the indirect utility function:

$$ \begin{aligned} V(\theta^a, \theta^b, w^{\ast} - T) =& (1/\gamma + \beta_1) \hbox{log} (w^{\ast} - T) + (\alpha_1 + \beta_1) \hbox{log} (1-\theta^a)^{-1} + \alpha_2 \hbox{log} (1-\theta^b)^{-1} \\ &+ \beta_2 \hbox{log} \left[T - (\alpha_1 \theta^a (1-\theta^a)^{-1} +\alpha_2 \theta^b (1-\theta^b)^{-1})\gamma (w^{\ast}-T)\right]+ \hbox{const}. \end{aligned} $$

where

$$ \beta_1=\alpha_4\eta_1/(1-\delta_1), \beta_2=\alpha_3+\alpha_4\rho/(1-\delta_1). $$

(B7)

To check for the concavity properties of this function, in θ^a < 1, θ^b < 1 for any T ≥ 0, w* − T > 0, e > 0, it is convenient to make the following substitution:

$$ x^a = -\theta^a/(1-\theta^a), x^b = -\theta^b/(1-\theta^b), $$

(B8)

which leads to the following expression for the indirect utility function:

$$ \begin{aligned} V(x^a, x^b, w^{\ast} - T) =& (1/\gamma + \beta_1) \hbox{log} (w^{\ast} - T) + (\alpha_1 + \beta_1) \hbox{log} (1-x^a) + \alpha_2 \hbox{log} (1-x^b)\\ +& \beta_2 \hbox{log} \left[T + (\alpha_1 x^a +\alpha_2 x^b)\gamma (w^{\ast}-T)\right]. \end{aligned} $$

It is easy to check that this function is concave in x ^a, x ^b < 1 (which is true when to values θ^a, θ^b < 1). Therefore, its maximum is characterized by the following first-order conditions:

$$ \begin{aligned} \partial V/\partial x^a =& -(\alpha_1 + \beta_1)/(1 - x^a)\\ &+ \alpha_1 \beta_2 \gamma (w^{\ast} - T)/[T + (\alpha_1 x^a + \alpha_2 x^b)\gamma (w^{\ast}-T)] = 0, \end{aligned} $$

(B9)

and

$$ \begin{aligned} \partial V/\partial x^b =& - \alpha_2/(1 - x^b)\\ & + \alpha_2 \beta_2\gamma (w^{\ast} - T) /[T + (\alpha_1 x^a + \alpha_2 x^b)\gamma (w^{\ast}-T)] = 0. \end{aligned} $$

(B10)

In the solution, the government budget constraint e =  T − θ^a a − θ^b b must also hold. Replacing a and b by (B5) and (B6), and using (B8), this constraint can be rewritten as:

$$ e = T +(\alpha_1 x^a +\alpha_2 x^b)\gamma (w^{\ast}-T). $$

(B11)

1.4.3 Solving for the second-best optimum

Solving the system of three equation (B9), (B10) and (B11) in three unknowns x ^a, x ^b and e leads to:

$$ e = \beta_2 [T +\gamma (\alpha_1 +\alpha_2)(w^{\ast}-T)]/(\alpha_1 + \beta_1 + \alpha_2 + \beta_2), $$

(B12)

$$ (1-x^a)^{-1} = (1-\theta^a)= \frac{\alpha_1(\alpha_1+\beta_1+\alpha_2+\beta_2)} {(\alpha_1+\beta_1)[(\alpha_1+\alpha_2)+T/\gamma(w^{\ast}-T)]}, $$

(B13)

$$ (1-x^b)^{-1} = (1-\theta^b)= \frac{(\alpha_1+\beta_1+\alpha_2+\beta_2)} {(\alpha_1+\alpha_2)+T/\gamma(w^{\ast}-T)}. $$

(B14)

Using (B12)–(B14), it is easy to check that:

• (a) e is increasing in T since γ(α₁ + α₂) < 1,

• (b) θ^a and θ^b are increasing in T,

• (c) θ^a >  θ^b.

Since θ^b is increasing in T and equal to zero in the first-best, we necessarily have θ^b < 0 in the second-best. The condition θ^a > 0 is equivalent to (α₁ β₂ − β₁ α₂) γ (w* − T) < (α₁ + β₁)T. There are thus two possibilities. Either α₁β₂ − β₁α₂ ≤ 0 so that θ^a > 0 for all T > 0. Or α₁β₂ − β₁α₂ > 0 and then θ^a < 0 for small T, i.e., for T such that T < γ(α₁β₂ − β₁α₂)/[α₁ + β₁ + γ (α₁β₂ − β₁α₂)]w*.

1.4.4 Some first-best results

To compare first-best and second-best results, we need T*, the first-best value of the government budget. This is obtained as follows. The first-best is the maximal value of the utility, obtained by substituting λ and μ by (18) and (19) in (20) and, in the fixed-point, setting $\overline{\lambda} = \lambda,$ so that

$$ u^{\ast} = \hbox{log} c + \beta \hbox{log} d + (\alpha_1 + \beta_1) \hbox{log} a + \alpha_2 \hbox{log} b + \beta_2 \hbox{log} e. $$

The maximum of u* subject to the golden rule resources constraint f(k*) − k* = c + a + d + b + e, leads to

$$ e^{\ast} = \beta_2w^{\ast}/(1 + \beta + \alpha_1 + \beta_1 + \alpha_2 + \beta_2), $$

(B15)

$$ a^{\ast} = (\alpha_1 + \beta_1)w^{\ast}/(1 + \beta + \alpha_1 + \beta_1 + \alpha_2 + \beta_2), $$

since f(k*)−k* = w*. Using the fact that the values of the optimal subsidy rates are θ^a = δ₁,   θ^b = 0, we obtain T* = e* +  δ₁ a*, and

$$ T^{\ast} = [\beta_2 + \delta_1 (\alpha_1 + \beta_1)]w^{\ast}/(1 + \beta + \alpha_1 + \beta_1 + \alpha_2 + \beta_2). $$

(B16)

1.4.5 Parameterization of the Cobb-Douglas economy

From the values in Table A, it is easy to compute β ≃ 0.5, α₁ = 0.018, and α₂ = 0.006. We assume that the effect on the cultural stock of consuming arts is the same when young or old, which implies η₁ = η₂. Using (B2), one can check that α₁/α₂ = (1 + βδ₂)/β = 3 and δ₂ ≃ 1, which can be interpreted as “nothing is lost” in the transmission of cultural capital to oneself (when “switching” from young to old). We also assume that the effect of education (ρ) on the stock of culture of the young is equal to (1 − δ₁), to represent in a parametric way, the relative effects of education and family transmission.

Table A Values of the main parameters

Since it is hard to find values for α and δ₁, we shall, in the simulations, parameterize α for values ranging from 0.05 to 0.2 and δ₁ for values between 0.125 and 0.50.

Now the values of all the other parameters (α₃, α₄, β₁, β₂, η₁, η₂, ρ) can be computed as functions of α and δ₁, using (B2) and (B7).