5. A planner makes resource allocation decisions for a closed economy, maximizing the utility function of a representative household: where c is consumption and l is the labor input. Labor is the only factor of production, and the economy’s production function is t t t Y = A l , t t where A is the exogenous level of productivity. At time t , a fraction g of the economy’s output must be allocated to government purchases. So, the planner’s optimization is subject to t t t c # (1 -g ) Y A. (7 points) Find the solution for Y as a function of A and g . Answer : The optimality conditions for consumption and labor imply In equilibrium, this gives and B. (7 points) Suppose g is constant most of the time but unusually high occasionally. How does a high value of g affect the optimal solutions for consumption and employment? Explain the effects using the relevant derivations and give some intuition for the results.
Answer : The optimal solution for output implies and When government purchases rise, the planner cuts consumption and raises employment. Since the government requires more goods, households should either work more to make more goods or cut back on consumption. The planner (optimal) solution involves adjusting on both margins (work more and consume less). C. (5 points) What is the basic message of this exercise for fiscal multipliers in an economy with flexible prices and perfectly competitive firms (operating the same production function as above)? Make sure you state clearly why this exercise is informative about fiscal multipliers in such an economy. Answer : Since the competitive equilibrium is efficient, it replicates the planner’s solution. This means the fiscal multiplier is less than one. A fiscal expansion increases output, but by less than the increase in government purchases (because consumption falls). D. (4 points) Comparing the setup of this exercise to standard Keynesian models (like the short-run IS-LM model), what are the key differences in assumptions that matter for the models’ implications about fiscal multipliers? Answer : Short-run Keynesian models are different because they assume some rigidity in wages and / or prices. So, the resulting equilibria are not efficient. Many simple models (like the IS-LM) also incorporate an arbitrary consumption function that links consumption to current disposable income. In these models, a fiscal expansion increases consumption, and this makes the fiscal multiplier larger than one.
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