Problem 649449 · hard · Economics · Level 06 Dynamics and General Equilibrium

A Wage Tax on Coconut Island

production economy · Robinson Crusoe economy · labour market · distortionary tax · deadweight loss

On a small island one household owns the only firm, a coconut plantation, and has T hours a week to divide between work and leisure. Coconuts are the numeraire (price 1); the wage is w coconuts per hour. Both sides take prices as given.

The firm hires L hours and harvests Q = A L^{\beta} coconuts (0 < \beta < 1). It chooses L to maximise its profit \pi = A L^{\beta} - wL, which it pays to the household.

The government taxes wages at the rate t (0 \le t < 1) and hands the whole revenue back to the household as a lump sum R.

The household chooses consumption c and leisure \ell (hours, any real number in [0, T]) to maximise u = c^{a} \ell^{1-a} subject to

c = (1 - t)\, w\, (T - \ell) + \pi + R

taking w, \pi and R as given (it does not see that its own work changes the transfer).

Equilibrium is a wage at which the hours the firm wants equal the hours the household offers, with R = t\,w\,L. (All coconuts harvested are then eaten: c = Q.)

Write island_economy(A, beta, a, T, t) that returns a dict with the equilibrium values under the tax rate t:

  • "wage": w; "hours": L; "output": Q; "profit": \pi; "revenue": R; "utility": u;
  • "loss": the cost of the tax in coconuts. Let u_0 be the household's utility in the equilibrium with no tax (t = 0, same other parameters). The loss is how many extra coconuts the household would need, on top of its consumption and keeping its leisure in the taxed equilibrium, to be exactly as well off as with u_0. It is 0 when t = 0.

Examples

Input:  A = 10, beta = 0.5, a = 0.5, T = 12, t = 0
Output: {'wage': 2.5, 'hours': 4.0, 'output': 20.0, 'profit': 10.0, 'revenue': 0.0,
         'utility': 12.64911064, 'loss': 0.0}

Input:  A = 10, beta = 0.5, a = 0.5, T = 12, t = 0.5
Output: {'wage': 3.22748612, 'hours': 2.4, 'output': 15.49193338, 'profit': 7.74596669,
         'revenue': 3.87298335, 'utility': 12.19518596, 'loss': 1.17473328}
Explanation: the tax halves the reward for an hour of work, so the household works
2.4 hours instead of 4. All revenue comes back, yet it is worse off: it would need
1.17 more coconuts with its 9.6 hours of leisure to match the untaxed utility.

Constraints

  • answers are compared with an absolute and relative tolerance of 1e-6; do not round

Goals

  • Combine a profit-maximising firm and a utility-maximising household into one general equilibrium
  • Close the model with a lump-sum transfer that must equal the tax revenue
  • Measure the cost of a distorting tax in coconuts, against the untaxed equilibrium
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