Q1. Two firms compete with each other in the output market but have geographically separate, competitive labor markets, and labor is immobile. The profit functions for the two firms are therefore R1 (q1,q2) - w1L1 and R2 (q1,q2) - w2L2, where q stands for output, L stands for labor input, w stands for the wage rate, and superscripts identify firms The revenue function is left general, but each firm has some market power so, using subscripts to signify derivatives, Rii > 0, Rij < 0; revenue for firm i will increase when its own output increases, but its revenue will decrease when its competitor increases output. Each firm produces output using labor as the only variable input: ql = F1(L1) and q2 = P (L2, α) where α is a parameter measuring technological change for firm 2's production function: an increase in α increases output for a given amount of labor and it also increases the marginal product of a given amount of labor. Each firm chooses its labor input in order to maximize profits, treating the other firm's quantity as fixed. Find the first-and second-order conditions for both firms' profit maximization problems and use them to find and sign (if possible) the comparative static derivative ∂L1*/∂w1. Explain the economics of your answer.
Q2. For the model, confirm by substituting in the explicit formula for L that the formula for dL/dw obtained by implicit differentiation of the first-order condition is identical to that obtained by differentiating the explicit choice function L* (w).
Q3. Add to the IS-LM model. An equation relating the health of the natural environment, N, to the pollution caused by the production of output: N = e(Y), where e' < 0. This results in a system of three equations in the three endogenous variables γ, r and N. Find the total differentials of the three equations and use them to find and sign, if possible, ∂Y*/∂G. Explain why your answer is either the same or different than the usual IS-LM result.
Q4. For the model of the problem above, find and sign, if possible ∂N*/∂G and ∂N*/∂M. Explain the economics.
Q5. For the model of problem above, suppose the government commits to environmental sustainability and assume this means that the health of the natural environment must stay at its current level: dN = 0. What monetary policy must accompany an expansionary fiscal policy in order to ensure sustainability? (That is, find a relationship between dM and dG when dN = 0)
Q6. For the Mundell-Fleming model, find and sign (if possible) the effect of monetary policy on the exchange rate (when exchange rates are flexible). If the sensitivity of investment to the interest rate increases, does that make your answer large or smaller? Explain the Economics.
Q7. The government wants to increase government purchases but have equilibrium output stay constant. Using the model, by how much and in which direction should taxes change, relative to the increase in government purchases?
Q8. Show that dr > 0 in the IS-LM model when there is a balanced budget increase in government purchases and taxes.
Q9. Suppose taxes are a function of income, T = t(Y) Y, instead of being exogenous. If the tax system is progressive, then dt/dY > 0. Let G increase and let t increase such that, in equilibrium, dT = dG. What is the value of the balanced budget multiplier in the IS-LM model?
Q10. In an IS-LM model, suppose that government purchases of goods and services increase while taxes remain constant. The Fed changes the money supply in such a way that interest rates stay constant. (Does the Fed have to increase or decrease the money supply?) Find and sign (if possible) a formula for dY. How is dY affected by various parameters? Explain the economics of your results.
Q11. For the aggregate demand-aggregate supply model, suppose that there is an increase in government spending and the Fed changes the money supply in such a way that interest rates do not change. Find and sign (if possible) expressions for dY and dP and explain the economics of your results.
Q12. Using the method of two competitive markets derive expressions that can be used to show the effects on P1* and Q2* of an increase in a unit subsidy granted to producers of good 1. Provide economic interpretations for the cases in which goods 1 and 2 are complements, substitutes or neither.
Q13. Assume that b1b2 > 1/4. Then specify conditions for the remaining parameters so that the equilibrium levels of outputs will be positive. Provide economic interpretations for your conditions.
Q14. Using the model, determine the unit tax or subsidy for firm 1 that will ensure that both firms have the same equilibrium price. To simplify the calculations, assume b1=b2. Express the required unit tax or subsidy in terms of the cost and demand parameters of the problem.
Q15. For the model of duopoly, assume that the conjectural variations satisfy dq2/dq1 = 3 (dq1/dq), in which dq1/dq2 = h, a positive constant. Further assume that c1 = 2c2, and that c2 is a positive constant. Derive the expressions for P*, q1* and q2* that are analogous to equation. Provide an economic interpretation of why your results differ from equation.