# Question: consider the following modified solow growth model the economy wide...

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Consider the following modified Solow Growth model. The economy wide production function is Yt = f(Kt,Lt,At), where Kt is the stock of capital, Lt is labor input, and At is total factor productivity. Here assume L and A are constant over time. The marginal product of capital (fK) is positive but diminishing. Capital accumulates according to ∆K t +1 = I t − δK t , where ∆K t +1 in the change in the capital stock, It is investment expenditures and δ is the depreciation rate. Total savings in the economy is given by St = s(Yt − Tt ) , where s is the marginal propensity to save (or the savings rate) out of current after tax income (Yt − Tt ) . Assume that tax revenue is of the form of an income tax, so that Tt = τ Yt , where τ is the income tax rate. In equilibrium, total investment expenditures, It, equals total savings, St.

a. Using a diagram with capital on the horizontal axis and saving\investment on the vertical axis, show the steady state level of capital. What is the long-run growth rate of output in this case? What happens to the steady state level of capital if the income tax rate, τ , increases.

b. Suppose the marginal product of capital ( fk ) is not diminishing. In fact, assume lim f k = A, where A is a constant. What values of A would imply that the economy will k→∞ growth continuously? Suppose that the tax rate rises, under what conditions would long- run growth stop?

3. Consider the following model of the goods market: Consumption is given by: Ct = C + c(Yt −Tt ) Aggregate demand: ADt = Ct + It + Gt . In equilibrium: Yt = ADt . Assume that Tt , It and Gt are exogenous and equal to constants: Tt = T , It = I , andGt =G. a. Find the (output) multiplier for an increase in government spending. b. Suppose Tt = τYt . Find the new government spending multiplier. Is it larger or smaller than the one derived in part a?

4. Consider a closed economy. Assume that goods prices are fixed in the short-run and that firms produce whatever they need to meet demand. Assume that money demand is negatively related to the nominal interest rate and positively related to the level of output. Consumption depends on current disposable income and the real interest rate. Speculate on what would happen to equilibrium output, price level, consumption, investment, and nominal and real interest rates if:

a. There is a one-time increase in money supply.

b. Government expenditures rise but taxes remain unchanged.

c. Firms become optimistic about the future and increase their investment expenditures.

d. Households decide that they want to hold more dollars (money).

e. Taxes rise (keeping government expenditures constant).