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# Appendix A: Club reputation dynamics

## The firm’s maximization problem

1The firm faces the problem of maximizing the present value of profits subject to the state equation for reputation and under the constraint that profits de non-negative. Adopting the approach of Winfree and McCluskey (2005), firms follow a stationary Markovian strategy (wherein the firm conditions its actions at a certain period only on the state of affairs at that given perdio), so that qj = φ(R):

2The current-value Hamiltonian (where λi is the current-value shadow price of reputation, a co-state variable) is:

3Apply the Pontryagin recipe:

4Solve for the isoclines:

5Equation (a) represent the first-order conditions with respect to the control variable quality.

6Equation (b) has a standard interpretation as the adjustment equation for the shadow value of reputation along an efficient path. Equation (c) is the reputation constraint.

## Equilibrium

7The steady-state equilibrium is found at the intersection of the isoclines  = 0 and q̇ = 0, with the symmetry assumption:

p'(R*) is the marginal revenue of the representative firm (as output is set at unity); c'(q*) is the output marginal cost, and G'(R*)/N is the club-house marginal cost. The solution involves a price schedule where marginal revenue covers marginal cost at a premium . Under the assumption that p''(R) < 0 (downward-sloped marginal revenue curve), the model implies a lower equilibrium level of quality/reputation than that observed under collective reputation (in the latter case G = 0).

## System stability: saddle-point path equilibrium

8The Jacobian matrix is:

9The determinant of the Jacobian matrix at the steady-state provides information about behaviour near the steady-state. Under the assumptions of the model (c'' > 0, G'' ≥ 0 and c'' > p''), the determinant is unambiguously negative, indicating eigenvalues of opposite signs and a saddle-point path equilibrium. The stability condition < 0 is a curvature condition that asserts that the firm’s marginal revenue either declines (p'' < 0), or, if it rises, it rises no faster than the “relevant” marginal cost increases (the marginal cost includes the premium).

10Although a concave marginal revenue function is usually assumed, the model does not rule out a convex marginal revenue function (p'' > 0), which may occur, for example, in the case of extremely rare wine vintages – such as Champagne – wherein increases in quality are marketed at huge price premia:

11Additionally, the assumption that the function G is concave (G'' ≤ 0) does not rule out a saddle-point path equilibrium either, though it comes at the cost of an additional condition.

12Wherever the determinant is positive, the system is unstable – i.e. there is a direction in a 2-dimensional space in which the system will not tend to return back to the equilibrium point.

## Quality is a function of membership

13The implicit derivative of quality with respect to number of firms at the equilibrium point is:

Otherwise, quality will decrease, particularly at high levels of membership. Rearranging the terms, these conditions reduce to one simple condition: quality will increase with the number of firms as long as the premium over production cost is lower than the per-firm GI-related marginal cost ; if not, quality will decrease. This indicates the potential existence of a scenario in which there is an inflexion point of some lever of N, up to which the equilibrium quality increases and after which it decreases.

# Appendix B: The top-down approach to optimal membership

14In step 2, the social planner maximizes profits at the club lever assuming a steady-stage / symmetric equilibrium:

15Profits are maximized at:

# Appendix C: Welfare perfect competition

## Welfare functions

16The arguments of the integrands are omitted, but G = G[d], V = V[d] and U = U[d], and the other parameters are held constant. Asterisks denoting equilibrium values are also omitted (Y = Y*, Z = Z* and so on).

17Positive GI-variety sales:

18GI-variety sales deterred:

19Under perfect competition, σ = 0 and Foreign firms have zero profits. Total welfaire is therefore given exclusively by Home consumer surplus.

∆W = ∆CS = CS1 – CS0

20The variables of integration (dd) are also omitted from now on (unless the notation is ambiguous).

## GI-variety sales deterred (Y1 = 0), panel C of Figure 4.5.a

21For Home consumer surplus, distribute the integrands into domains determined by consecutive quantities and simplify:

# Appendix D: Welfare oligopolistic competition

## Foreign sales increase

### GI-variety sales deterred (Y1 = 0), panel C of Figure 4.6.a

22Global welfare is the sum of profits and consumer welfare. The difference in profits earned by Foreign firms is cancelled out by the equivalent loss in Home consumer surplus. The switch to GI-original goods (first term) may have a net positive or negative welfare impact, depending on the parameters of the model, because at X0, the first integral is positive, but it might be negative at X1 (in panel C of Figure 4.6.a, parameters were selected so that the welfare gain over compensates for the welfare loss for consumers of GI-original goods):

### Positive GI-variety sales (Y1 > 0), panel D of Figure 4.6.a

23The net welfare impact of the switch to GI-original goods for those consumers is ambiguous:

## Foreign sales decrease

### GI-variety sales deterred (Y1 = 0), panel B of Figure 4.6.b

24Net welfare is unambiguously negative, as both terms are always negative. In particular, (G + V + a) is greater than c at both X1 and X0.

## Positive GI-variety sales (Y1 > 0), not drawn

25Net welfare is unambiguously negative, as all terms are always negative (this is because σ0 > 0 and b' > b, so that σ0 + b' > b.

# Appendix E: Welfare enforcement

26An increase in b affects sales of GI-labelled goods Y[b], sales of GI-original goods X[b] and, under oligopolistic competition, the mark-up σ[X[b]]. In addition:

27Profits:

28Consumer surplus:

29Global welfare:

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