4. Suppose there is a perfectly competitive market, where consumers maximize utilityv(q, α)+msubject to budget constraintsw= (p+t)q+m, wheretis a tax on consumption ofq, and firms havecostsC(q) =cq. Assume thatvαq(q, α)≥0. (i) Characterize a perfectly competitive equilibrium inthe market, and show how tax revenue —tq∗(t, α) — varies withtandα. (ii) Supposeαchangesand the government adjuststto hold tax revenuetq∗(t, α) constant.Use the implicit functiontheorem to show howt(α) varies withα.(It might help to think about the elasticity ofqwithrespect tot,ε=tq∗∂q∗∂t)We did this in class.5.When an objective functionf(x, y) depends on two controls,xandy, and is subject toa linear constraintc=ax+by, the problem can be simplified to a one-dimensional problem by
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solving the constraint in terms of one of the controls,y=c−axband substituting it into the objective to getmaxxfparenleftbiggx,c−axbparenrightbiggDerive FONCs and SOSCs for this problem. What assumptions guarantee that the SOSC’s hold?

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- Fall '12
- Johnson
- Calculus, lim, Continuous function, Lipschitz