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τ(y). Putting these pieces together, the problem of a type j individual can be written as:maxc s,a s+1 ,i ss.tS∑β s−1 u(c s , n s )s=1c s + a s+1 = (1 − ¯τ n (y s ))y s + (1 + r)a sh s+1 = h s + A j (h s i s n s ) α (1)y s = P H h s (1 − i s )n s (2)Using the fact that Q j s = A (h s i s n s ) α , the “cost of investment” (ie., h s i s n s ) can be writtenas: C j (Q j s) = (Q s /A j ) (1/α) , which will play a key role in the optimality conditions below.Now, it is useful to distinguish between two cases.Inelastic Labor Supply. First, suppose that labor supply is inelastic, in which casen s = 1. The optimality condition for human capital investment is (assuming an interiorsolution):(1 − τ(y s )) C ′ j(Q j s) ={β(1 − τ(y s+1 )) + β 2 (1 − τ(y s+2 )) + ... + β S−s (1 − τ(y S ))}. (3)The left hand side is the marginal cost of investment, whereas the right hand side is themarginal benefit, which is given by the present discounted value of net wages in all futuredates earned by the extra unit of human capital. Notice that both the marginal cost andbenefit of investment take into account the marginal tax rate faced by the individual. Tounderstand the effect of taxes, first consider the case when taxes are flat-rate, ie, τ ′ (y) ≡ 0.In this case, all terms involving taxes cancel out and the FOC reduces to:C ′ j(Q j s) ={β + β 2 + ... + β S−s }.Thus, flat-taxes have no effect on human capital investment. This is a well-understoodinsight that goes back to at least Heckman (1976) and Boskin (1977).Now consider progressive taxes, ie., τ ′ (y) > 0. We rearrange equation (3) to get:C ′ j(Q j s) ={β1 − τ(y s+1)1 − τ(y s )} {{ }Progressivity wedge+ β 2 1 − τ(y s+2)1 − τ(y s )} {{ }...+ ... + β S−s 1 − τ(y S )1 − τ(y s )} {{ }}. (4)Progressivity wedge6